Heat equation with an exponential nonlinear boundary condition in the half space

Abstract

We consider the initial-boundary value problem for the heat equation in the half space with an exponential nonlinear boundary condition. We prove the existence of global-in-time solutions under the smallness condition on the initial data in the Orlicz space $\mathrm {exp}L^2({\mathbb {R}}^N_+)$. Furthermore, we derive decay estimates and the asymptotic behavior for small global-in-time solutions.

1 Introduction

We consider the initial-boundary value problem for the heat equation in the half space ${{\mathbb {R}}}^N_+=\{x=(x',x_N)\in {{\mathbb {R}}}^N : x_N>0\}$ with a nonlinear boundary condition

$$\begin{aligned} \left\{ \begin{array}{ll} \partial _t u= \Delta u, \quad &{} x\in {{\mathbb {R}}}^N_+,\quad t>0, \\ u(x,0)=\varphi (x) , \quad &{} x\in {{\mathbb {R}}}^N_+, \\ \partial _\nu u=f(u),\quad &{} x\in \partial {{\mathbb {R}}}^N_+, \quad t>0, \end{array} \right. \end{aligned}$$

(1.1)

where $N\ge 1$, $\partial _t=\partial /\partial t$, $\partial _\nu =-\partial /\partial x_N$, and $\varphi$ is the given initial data. Here f(u) is the nonlinearity which has an exponential growth at infinity with $f(0)=0$. More precisely, the condition for the nonlinearity (see (1.9)) covers certain limiting cases which are critical with respect to the growth of the nonlinearity and the regularity of the initial data. In this paper, under a smallness condition on the initial data, we prove the existence of global-in-time solutions to problem (1.1). Furthermore, we derive some decay estimates and the asymptotic behavior of small global-in-time solutions.

The nonlinear boundary value problem such as (1.1) can be physically interpreted as a nonlinear radiation law. The case of power nonlinearities $f(u)=|u|^{p-1}u$ with $p>1$, that is,

$$\begin{aligned} \left\{ \begin{array}{ll} \partial _t u= \Delta u, \quad &{} x\in {{\mathbb {R}}}^N_+,\quad t>0, \\ u(x,0)=\varphi (x) , \quad &{} x\in {{\mathbb {R}}}^N_+, \\ \partial _\nu u=|u|^{p-1}u,\quad &{} x\in \partial {{\mathbb {R}}}^N_+, \quad t>0, \end{array} \right. \end{aligned}$$

(1.2)

has been extensively studied in many papers (see e.g. [5, 6, 11, 13, 17,18,19,20,21,22, 25, 26] and the references therein). It is well-known that problem (1.2) satisfies a scale invariance property, namely, for $\lambda \in {\mathbb {R}}_+$, if u is a solution to problem (1.2), then

$$\begin{aligned} u_\lambda (x,t):=\lambda ^{\frac{1}{p-1}}u(\lambda x,\lambda ^2 t) \end{aligned}$$

(1.3)

is also a solution to problem (1.2) with initial data $\varphi _\lambda (x):=\lambda ^{1/(p-1)}\varphi (\lambda x)$. In the study of the nonlinear boundary value problem (1.2), it seems that all function spaces invariant with respect to the scaling transformation (1.3) play an important role. In fact, for Lebesgue spaces, we can easily show that the norm of the space $L^q({{\mathbb {R}}}^N_+)$ is invariant with respect to (1.3) if and only if $q=q_c:=N(p-1)$, and, for the given nonlinearity $|u|^{p-1}u$, the Lebesgue space $L^{q_c}({{\mathbb {R}}}^N_+)$ plays the role of critical space for the local well-posedness and the existence of global-in-time solutions to problem (1.2) (see e.g. [13, 18, 20]).

On the other hand, the case of the Cauchy problem with the power nonlinearity, that is,

$$\begin{aligned} \partial _t u= \Delta u +|u|^{p-1}u, \quad x\in {\mathbb {R}}^N,\quad t>0, \qquad u(x,0)=\varphi (x) , \quad x\in {\mathbb {R}}^N, \end{aligned}$$

(1.4)

also satisfies a scale invariance property, namely, for $\lambda \in {\mathbb {R}}_+$, if u is a solution to problem (1.4), then

$$\begin{aligned} u_\lambda (x,t):=\lambda ^{\frac{2}{p-1}}u(\lambda x,\lambda ^2 t) \end{aligned}$$

(1.5)

is also a solution to problem (1.4) with the initial data $\varphi _\lambda (x):=\lambda ^{2/(p-1)}\varphi (\lambda x)$. So we can easily show that the norm of the space $L^q({\mathbb {R}}^N)$ is invariant with respect to (1.5) if and only if $q={\tilde{q}}_c:=N(p-1)/2$, and it is well-known that the Lebesgue space $L^{{\tilde{q}}_c}({\mathbb {R}}^N)$ plays the role of critical space for the well-posedness of problem (1.4) (see e.g. [3, 12, 27, 30, 31] and references therein). Furthermore, the scaling property (1.5) also holds for the nonlinear Schrödinger equation

$$\begin{aligned} i\partial _t u+ \Delta u =|u|^{p-1}u, \quad x\in {\mathbb {R}}^N,\quad t>0, \qquad u(x,0)=\varphi (x) , \quad x\in {\mathbb {R}}^N, \end{aligned}$$

(1.6)

and it is well known that the Sobolev space $H^{s_c}({\mathbb {R}}^N)$ with $s_c:=N/2-2/(p-1)$ plays the role of critical space for the well-posedness of problem (1.6) (see e.g. [4]). From these results, we have two critical growth rates of the nonlinearity, that is, $p_h:=1+(2q)/N$ and $p_s:=1+4/(N-2s)$, and these two critical exponents are connected by the Sobolev embedding, $\dot{H}^s({\mathbb {R}}^N)\hookrightarrow L^q({\mathbb {R}}^N)$, where s and q satisfy $0\le s<N/2$ and $1/q=1/2-s/N$. The case $s_c=N/2$ is a limiting case from the following points of view:

• (i) for $s>N/2$, $H^s({\mathbb {R}}^N)$ embeds into $L^\infty ({\mathbb {R}}^N)$;

• (ii) any power nonlinearity is subcritical, since $H^{N/2}({\mathbb {R}}^N)$ embeds into any $L^q({\mathbb {R}}^N)$ space (for $q\ge 2$);

• (iii) $H^{N/2}({\mathbb {R}}^N)$ does not embed into $L^\infty ({\mathbb {R}}^N)$, and thanks to Trudinger’s inequality [29] one knows that $H^{N/2}({\mathbb {R}}^N)$ embeds into the Orlicz space $\mathrm {exp}L^2({\mathbb {R}}^N)$.

For this limiting case, Nakamura and Ozawa [24] consider the nonlinear Schrödinger equation with an exponential nonlinearity of asymptotic growth $f(u)\sim e^{u^2}$ and with a vanishing behavior at the origin, and they show the existence of global-in-time solutions under a smallness assumption of the initial data in $H^{N/2}({\mathbb {R}}^N)$.

As a natural analogy to the results of [24], the third author of this paper and Ruf [28] and Ioku [14] consider the Cauchy problem of the semilinear heat equation with exponential nonlinearity of the form

$$\begin{aligned} f(u)=|u|^{\frac{4}{N}}ue^{u^2} \end{aligned}$$

(1.7)

and the initial data $\varphi$ belonging to the Orlicz space $\mathrm {exp}L^2({\mathbb {R}}^N)$ defined as

$$\begin{aligned} \mathrm {exp}{L^2}({\mathbb {R}}^N) :=\left\{ u\in L^1_{\mathrm{loc}}({\mathbb {R}}^N); \int _{{\mathbb {R}}^N}\left( {\mathrm {exp}}\left( \frac{|u(x)|}{\lambda }\right) ^2-1\right) dx<\infty \quad \hbox { for some}\ \lambda >0\right\} \end{aligned}$$

(see also Definition 2.1). They consider the corresponding integral equation

$$\begin{aligned} u(t)=e^{t\Delta }\varphi +\int _0^te^{(t-s)\Delta }f(u(s))\,ds, \end{aligned}$$

(1.8)

and prove the existence of local/global-in-time (mild) solutions to this equation (1.8) under the smallness assumption of initial data in $\mathrm {exp}L^2({\mathbb {R}}^N)$. Furthermore, the authors of this paper and Ruf [10] show the equivalence between mild solutions (solution to the integral equation (1.8)) and weak solutions to the heat equation with the nonlinearity f(u) as in (1.7), and derive some decay estimates and the asymptotic behavior for small global-in-time solutions. The growth rate of (1.7) at infinity seems to be optimal in the framework of the Orlicz space $\mathrm {exp}L^2({\mathbb {R}}^N)$. In fact, if $f(u)\sim e^{|u|^r}$ with $r>2$, there exist some positive initial data $\varphi \in \mathrm {exp}L^2({\mathbb {R}}^N)$ such that problem (1.8) does not possess any classical local-in-time solutions (see [15]). For the fractional diffusion case and general power-exponential nonlinearities, see e.g. [8, 10, 23]. Furthermore, for $\varphi \in \mathrm {exp}L^2({\mathbb {R}}^N)$, which implies $\varphi \in L^p({\mathbb {R}}^N)$ for $p\in [2,\infty )$, the decay rate of (1.7) near origin, that is, $f(u)\sim |u|^{4/N}u$, is optimal in the framework of $L^2({\mathbb {R}}^N)$. See e.g. [3, 30].

The above limiting case in ${\mathbb {R}}^N$ appears from the relationship between $p_h$ and $p_s$ by the Sobolev embedding. For problem (1.2), we can easily show that the norm of the space $H^s({{\mathbb {R}}}^N_+)$ is invariant with respect to (1.3) if and only if $s={\tilde{s}}_c:=N/2-1/(p-1)$, and we have two critical growth rate of the nonlinearity, that is, ${\tilde{p}}_h=1+q/N$ and ${\tilde{p}}_s=1+2/(N-2s)$. These two exponents are also connected by the Sobolev embedding, $\dot{H}^s({{\mathbb {R}}}^N_+)\hookrightarrow L^q({{\mathbb {R}}}^N_+)$, where s and q satisfy the same conditions as in the case of ${\mathbb {R}}^N$. This means that the same limiting case appears for problem (1.2). On the other hand, as far as we know, there are no results which treat the exponential nonlinearity for the nonlinear boundary problem (1.1).

Based on the above, in this paper, we assume that the nonlinearity f satisfies the following: there exist $C_f>0$ and $\lambda >0$ such that

$$\begin{aligned}&|f(u)-f(v)|\le C_f|u-v|(|u|^{\frac{2}{N}}e^{\lambda u^2}+|v|^{\frac{2}{N}}e^{\lambda v^2})\nonumber \\&\quad \text{ for } \text{ every }\quad u,v\in {\mathbb {R}},\qquad f(0)=0. \end{aligned}$$

(1.9)

This assumption covers the case

$$\begin{aligned} f(u)=\pm |u|^{\frac{2}{N}}ue^{u^2}, \end{aligned}$$

which is one of the candidates for the optimal growth rate of the nonlinearity in the framework of the Orlicz space $\mathrm {exp}L^2({{\mathbb {R}}}^N_+)$ and the optimal decay rate near origin in the framework of $L^2({{\mathbb {R}}}^N_+)$ (see e.g. [18]). Following [10, 14, 28], for problem (1.1) with (1.9), we consider the corresponding integral equation, and prove the existence of global-in-time (mild) solutions under some smallness assumption of the initial data in $\mathrm {exp}L^2({{\mathbb {R}}}^N_+)$. Furthermore, we obtain some decay estimates for the solutions in the following two cases

$\varphi \in \mathrm {exp}L^2({{\mathbb {R}}}^N_+)$ only (slowly decaying case), and $\varphi \in \mathrm {exp}L^2({{\mathbb {R}}}^N_+)\cap L^p({{\mathbb {R}}}^N_+)$ with $p\in [1,2)$ (rapidly decaying case).

In particular, for the rapidly decaying case $p=1$, we show that the global-in-time solutions with some suitable decay estimates behave asymptotically like suitable multiples of the Gauss kernel.

Before treating our main results, we introduce some notation and define a solution to problem (1.1). Throughout this paper we often identify ${{\mathbb {R}}}^{N-1}$ with $\partial {{\mathbb {R}}}^N_+$. Let $g_N=g_N(x,t)$ be the Gauss kernel on ${{\mathbb {R}}}^N$, that is,

$$\begin{aligned} g_N(x,t):=(4\pi t)^{-\frac{N}{2}}\mathrm {exp}\left( -\frac{|x|^2}{4t}\right) ,\quad x\in {{\mathbb {R}}}^N,\quad t>0. \end{aligned}$$

(1.10)

Let $G=G(x,y,t)$ be the Green function for the heat equation on ${{\mathbb {R}}}^N_+$ with the homogenous Neumann boundary condition, that is,

$$\begin{aligned} G(x,y,t):=g_N(x-y,t)+g_N(x-y_*,t),\quad x,y\in {\overline{{{\mathbb {R}}}^N_+}},\quad t>0, \end{aligned}$$

(1.11)

where $y_*=(y',-y_N)$ for $y=(y',y_N)\in {\overline{{{\mathbb {R}}}^N_+}}$. Then, we define a (mild) solution to problem (1.1).

Definition 1.1

Let $\varphi \in \mathrm {exp}L^2({{\mathbb {R}}}^N_+)$, $T\in (0,\infty ]$, and $u\in C({\overline{{{\mathbb {R}}}^N_+}}\times (0,T))\cap L^\infty (0,T;\mathrm {exp}L^2({{\mathbb {R}}}^N_+))$.

• (i) In the case when $N\ge 2$, we call u a solution to problem (1.1) in ${{\mathbb {R}}}^N_+\times (0,T)$ if u satisfies

  $$\begin{aligned} u(x,t)= & {} \int _{{{\mathbb {R}}}^N_+}G(x,y,t)\varphi (y)\,dy\nonumber \\&\quad + \int _0^t\int _{{\mathbb {R}}^{N-1}}G(x,y',0,t-s)f(u(y',0,s))\,dy'\,ds \end{aligned}$$

  (1.12)

  for $(x,t)\in {\overline{{{\mathbb {R}}}^N_+}}\times (0,T)$ and $u(t) \underset{t\rightarrow 0}{\longrightarrow }\varphi$ in the weak$^*$ topology.

• (ii) In the case when $N=1$, we call u a solution to problem (1.1) in $(0,\infty )\times (0,T)$ if u satisfies

  $$\begin{aligned} u(x,t)=\int _0^\infty G(x,y,t)\varphi (y)\,dy + \int _0^tG(x,0,t-s)f(u(0,s))\,ds \end{aligned}$$

  (1.13)

  for $(x,t)\in [0,\infty )\times (0,T)$ and $u(t) \underset{t\rightarrow 0}{\longrightarrow }\varphi$ in the weak$^*$ topology.

In the case when $T=\infty$, we call u a global-in-time solution to problem (1.1).

We recall that $u(t) \underset{t\rightarrow 0}{\longrightarrow }\varphi$ in weak$^*$ topology if and only if

$$\begin{aligned} \lim _{t\rightarrow 0}\int _{{{\mathbb {R}}}^N_+}\bigg (u(x,t)\psi (x) -\varphi (x)\psi (x)\bigg )\,dx=0 \end{aligned}$$

for any $\psi$ belonging to the predual space of $\mathrm {exp}L^2({{\mathbb {R}}}^N_+)$ (see Sect. 2).

In what follows, we denote by $\Vert \cdot \Vert _{\mathrm {exp}L^2}$ the norm of $\mathrm {exp}L^2:=\mathrm {exp}L^2({{\mathbb {R}}}^N_+)$ defined by (2.14), for $r\in [1,\infty ]$, we write $\Vert \cdot \Vert _{L^r}:=\Vert \cdot \Vert _{L^r({{\mathbb {R}}}^N_+)}$ and $|\cdot |_{L^r}:=\Vert \cdot \Vert _{L^r({\mathbb {R}}^{N-1})}$ for simplicity. Furthermore, for a function $\phi (x',x_N)$ with $x' \in {{\mathbb {R}}}^{N-1}$ and $x_N \in [0,\infty )$, we write $| \phi |_{L^r}:=\Vert \phi (x',0)\Vert _{L^r({\mathbb {R}}^{N-1})}$.

Now we are ready to state the main results of this paper. First we show the existence of global-in-time solutions to problem (1.1) under the smallness assumption of the initial data in $\mathrm {exp}L^2$.

Theorem 1.1

Let $N\ge 1$ and $\varphi \in \mathrm {exp}{L^2}$. Suppose that f satisfies (1.9). Then there exist positive constants $\varepsilon =\varepsilon (N) >0$ and $C=C(N)>0$ such that, if $\Vert \varphi \Vert _{\mathrm {exp}{L^2}} <\varepsilon$, then there exists a unique global-in-time solution u to problem (1.1) satisfying

$$\begin{aligned} \sup _{t>0}\bigg (\Vert u(t)\Vert _{\mathrm {exp}{L^2}}+h(t)\Vert u(t)\Vert _{L^\infty }\bigg ) \le C\Vert \varphi \Vert _{\mathrm {exp}{L^2}}, \end{aligned}$$

(1.14)

where $h(t)=\min \{t^{N/4},1\}$, and for any $q\in [2,\infty )$,

$$\begin{aligned} \begin{aligned}&\sup _{t>0}\,t^{\frac{1}{2q}}|u(t)|_{L^q} \le C\bigg \{\Gamma \left( \frac{q}{2}+1\right) \bigg \}^{\frac{1}{q}}\Vert \varphi \Vert _{\mathrm {exp}{L^2}}, \qquad \text{ if }\quad N\ge 2, \\&\sup _{t>0}\,t^{\frac{1}{2q}}|u(0,t)| \le C\bigg \{\Gamma \left( \frac{q}{2}+1\right) \bigg \}^{\frac{1}{q}}\Vert \varphi \Vert _{\mathrm {exp}{L^2}}, \qquad \text{ if }\quad N=1, \end{aligned} \end{aligned}$$

(1.15)

where $\Gamma$ is the gamma function

$$\begin{aligned} \Gamma (q):=\int _0^\infty \xi ^{q-1}e^{-\xi }\,d\xi ,\qquad q>0. \end{aligned}$$

Remark 1.1

• (i) By the definition of ${{\mathbb {R}}}^N_+$, if $N\ge 2$, then the boundary of ${{\mathbb {R}}}^N_+$ is ${\mathbb {R}}^{N-1}$, namely, it is unbounded. On the other hand, if $N=1$, then the boundary of ${\mathbb {R}}_+$ is $x=0$, namely, it is only one point. From these differences, we need to divide the proof into two cases, $N\ge 2$ and $N=1$, and we have two estimates as in (1.15).

• (ii) Following [15], we denote by $\mathrm {exp}L^2_0({{\mathbb {R}}}^N_+)$ the closure of $C_0^\infty ({{\mathbb {R}}}^N_+)$ in $\mathrm {exp}L^2({{\mathbb {R}}}^N_+)$. Then, by an argument similar to that in the proof of [15, Theorem 1.2], it seems likely to obtain the existence of local-in-time solutions to problem (1.1) for any $\varphi \in \mathrm {exp}L^2_0({{\mathbb {R}}}^N_+)$ under the weaker condition

  $$\begin{aligned} |f(u)-f(v)|\le C|u-v|(e^{\lambda u^2}+e^{\lambda v^2})\quad \text{ for } \text{ every }\quad u,v\in {\mathbb {R}}, \qquad f(0)=0, \end{aligned}$$

  where $\lambda >0$ and $C>0$. This has not been fully explored and it is left for further investigation.

From now, we focus on the unique solution u to problem (1.1) satisfying (1.14) and (1.15). The following result gives some decay estimates for the slowly decaying case, that is, $\varphi \in \mathrm {exp}L^2$ only.

Theorem 1.2

Assume the same conditions as in Theorem 1.1. Furthermore, suppose that there exists a unique solution u to problem (1.1) satisfying (1.14) and (1.15). Then there exist some positive constants $\varepsilon = \varepsilon (N) >0$ and $C=C(N)>0$ such that, if $\Vert \varphi \Vert _{\mathrm {exp}L^2} <\varepsilon$, then the solution u satisfies

$$\begin{aligned} \begin{aligned}&\sup _{t\ge 1}\,t^{\frac{N}{2}(\frac{1}{2}-\frac{1}{q})} \bigg (\Vert u(t)\Vert _{L^q}+t^{\frac{1}{2q}}|u(t)|_{L^q}\bigg ) \le C\Vert \varphi \Vert _{\mathrm {exp}L^2}, \qquad \text{ if }\quad N\ge 2, \\&\sup _{t\ge 1}\,t^{\frac{1}{2}(\frac{1}{2}-\frac{1}{q})} \bigg (\Vert u(t)\Vert _{L^q} +t^{\frac{1}{2q}}|u(0,t)|\bigg ) \le C\Vert \varphi \Vert _{\mathrm {exp}L^2}, \qquad \text{ if }\quad N=1, \end{aligned} \end{aligned}$$

(1.16)

for all $q\in [2,\infty ]$.

Remark 1.2

• (i) By Theorem 1.1, if $\Vert \varphi \Vert _{\mathrm {exp}L^2}$ is small enough, then we can show that the assumption of Theorem 1.2 is not empty.

• (ii) We obtain the same decay estimate as the solution to the heat equation in ${{\mathbb {R}}}^N_+$ with the homogeneous Neumann boundary condition and initial data in $L^2$. See $(G_1)$ in Sect. 2.

Next we consider the rapidly decaying case, that is, $\varphi \in \mathrm {exp}L^2 \cap L^p$ with $p\in [1,2)$. We can prove two kinds of results about decay estimates of solutions to problem (1.1). In Theorem 1.3, we only assume the smallness condition of the $\mathrm {exp}L^2$ norm of the initial data. This means that we can allow the $L^p$ norm of the same data to be large. On the other hand, under this mild assumption, we have an additional restriction about the range of $L^q$ spaces for the case $N\ge 3$. In Theorem 1.4, under a stronger assumption, that is, a smallness assumption not only for the $\mathrm {exp}L^2$ but also for the $L^p$ norm of the initial data, we obtain better decay estimates, with no additional restrictions about the range of $L^q$ spaces even for the case $N\ge 3$. In the following we denote for any $r\ge 1$

$$\begin{aligned} \Vert \cdot \Vert _{\mathrm {exp}L^2\cap L^{r}}:=\max \{\Vert \cdot \Vert _{\mathrm {exp}L^2}, \Vert \cdot \Vert _{L^r}\}. \end{aligned}$$

(1.17)

Theorem 1.3

Assume the same conditions as in Theorem 1.2. Furthermore, assume $\varphi \in L^p({{\mathbb {R}}}^N_+)$ for some $p\in [1,2)$. Put

$$\begin{aligned} p_1:=\max \left\{ p,\frac{2N}{N+2}\right\} . \end{aligned}$$

(1.18)

Then there exist some positive constants $\varepsilon =\varepsilon (N)>0$, $C=C(N)>0$ and a positive function $F=F(N, p_1, \Vert \varphi \Vert _{L^{p_1}}, \lambda )$ such that, if

$$\begin{aligned} \Vert \varphi \Vert _{\mathrm {exp}L^2} < \min \left\{ \varepsilon , F\right\} , \end{aligned}$$

(1.19)

then the solution u satisfies

$$\begin{aligned} \begin{aligned}&\sup _{t>0}\,t^{\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})} \bigg ( \Vert u(t)\Vert _{L^q}+t^{\frac{1}{2q}}|u(t)|_{L^q} \bigg ) \le C \Vert \varphi \Vert _{\mathrm {exp}L^2\cap L^{p_1}}, \qquad \text{ if }\quad N\ge 2, \\&\sup _{t>0}\,t^{\frac{1}{2}(\frac{1}{p}-\frac{1}{q})} \bigg (\Vert u(t)\Vert _{L^q} +t^{\frac{1}{2q}}|u(0,t)| \bigg ) \le C \Vert \varphi \Vert _{\mathrm {exp}L^2\cap L^{p}}, \qquad \text{ if }\quad N=1, \end{aligned} \end{aligned}$$

(1.20)

for all $q\in [p_1,\infty ]$. In particular, if $p_1\in (1,2)$, then

$$\begin{aligned} \Vert u(t)\Vert _{L^q} = o\left( t^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})}\right) , \quad t\rightarrow \infty . \end{aligned}$$

(1.21)

Theorem 1.4

Assume the same conditions as in Theorem 1.3. Then there exists a positive constant $\varepsilon =\varepsilon (N)$ such that, if $\Vert \varphi \Vert _{\mathrm {exp}L^2\cap L^p}<\varepsilon$, then (1.20) with $p_1=p$ holds for all $q\in [p,\infty ]$. In particular, for all $q\in [p,\infty )$,

$$\begin{aligned} \begin{aligned}&\sup _{t>0}\,(1+t)^{\frac{N}{2} \left( \frac{1}{p} -\frac{1}{q}\right) } \bigg (\Vert u(t)\Vert _{L^q} +t^{\frac{1}{2q}}|u(t)|_{L^q}\bigg ) \le C\Vert \varphi \Vert _{\mathrm {exp}L^2\cap L^p}, \qquad \text{ if }\quad N\ge 2, \\&\sup _{t>0}\,(1+t)^{\frac{1}{2} \left( \frac{1}{p} -\frac{1}{q}\right) } \bigg (\Vert u(t)\Vert _{L^q} +t^{\frac{1}{2q}}|u(0,t)|\bigg ) \le C\Vert \varphi \Vert _{\mathrm {exp}L^2\cap L^p}, \qquad \text{ if }\quad N=1. \end{aligned} \end{aligned}$$

(1.22)

Furthermore, if $p\in (1,2)$ or $N\ge 3$, then (1.21) with $p_1=p$ holds.

Remark 1.3

By (1.9) the nonlinearity f(u) behaves like $|u|^{1+2/N}$ for $u\rightarrow 0$. So, for the case $N\ge 2$, since it follows from (1.15) that $u\in L^\infty _{\mathrm{loc}}(0,\infty ; L^q(\partial {{\mathbb {R}}}^N_+))$ for $q\ge 2$, the nonlinear term f(u) belongs to $L^p(\partial {{\mathbb {R}}}^N_+)$ for $p \ge (2N)/(N+2)$. For the case $N=2$, this means that $f(u)\in L^p(\partial {{\mathbb {R}}}^N_+)$ for all $p\ge 1$, but this implies a true constraint for the case $N\ge 3$. This is the reason why in Theorem 1.3 we have to introduce some parameters $p_1$ (and $p_2$, $p_3$, and $p_4$ in Lemmata 2.2,  5.1, and  5.6, respectively) which are meaningful only for the case $N\ge 3$.

Finally we address the question of the asymptotic behavior of solutions to problem (1.1) when $\varphi \in \mathrm {exp}L^2 \cap L^1$. We show that global-in-time solutions with suitable decay properties behave asymptotically like suitable multiples of the Gauss kernel.

Theorem 1.5

Let $N\ge 1$ and $\varphi \in \mathrm {exp}L^2 \cap L^1({{\mathbb {R}}}^N_+)$. Furthermore, let u be the global-in-time solution to problem (1.1) satisfying (1.22). Then there exists the limit

$$\begin{aligned} m_*:=\lim _{t\rightarrow \infty } \int _{{{\mathbb {R}}}^N_+}u(x,t)\,dx \end{aligned}$$

such that

$$\begin{aligned} \lim _{t\rightarrow \infty }\,t^{\frac{N}{2}(1-\frac{1}{q})}\Vert u(t)-2m_* g_N(t)\Vert _{L^q}=0,\qquad q\in [1,\infty ]. \end{aligned}$$

(1.23)

Remark 1.4

For the case $N\ge 2$, by (1.12) we see that

$$\begin{aligned} m_*=\int _{{{\mathbb {R}}}^N_+}\varphi (x)\,dx+\int _0^{\infty }\int _{{\mathbb {R}}^{N-1}}f(u(x',0,t))\,dx'\,dt. \end{aligned}$$

On the other hand, for the case $N=1$, by (1.13) we have

$$\begin{aligned} m_*=\int _0^\infty \varphi (x)\,dx+\int _0^{\infty }f(u(0,t))\,dt. \end{aligned}$$

The paper is organized as follows. In Sect. 2 we recall some properties of the kernel G and its associate semigroup. In Sect. 3, applying the Banach contraction map** principle, we prove Theorem 1.1. In Sects. 4 and 5, modifying the arguments of [20], we derive decay estimates on the boundary, and prove Theorems 1.2,  1.3, and  1.4. In Sect. 6 we obtain the asymptotic behavior of solutions to problem (1.1).

2 Preliminaries

In this section we recall some properties of the kernel $G=G(x,y,t)$ and its associate semigroup. Throughout this paper, by the letter C we denote generic positive constants that may have different values also within the same line.

We first recall the following properties of the kernel G (see e.g [13, 20, 22]):

• (i) $\displaystyle {\int _{{{\mathbb {R}}}^N_+}G(x,y,t)dy=1}$ for any $x\in {\overline{{{\mathbb {R}}}^N_+}}$ and $t>0$;

• (ii) for any $(x,t), (z,s)\in {\overline{{{\mathbb {R}}}^N_+}}\times (0,\infty )$, it holds that

  $$\begin{aligned} \int _{{{\mathbb {R}}}^N_+}G(x,y,t)G(y,z,s)\,dy=G(x,z,t+s). \end{aligned}$$

  (2.1)

By (1.11) we have

$$\begin{aligned} g_N(x-y,t)\le G(x,y,t)\le 2g_N(x-y,t),\qquad x,y\in {\overline{{{\mathbb {R}}}^N_+}},\,\,\, t>0. \end{aligned}$$

(2.2)

Furthermore, it follows from (1.10) and (1.11) that

$$\begin{aligned} G(x',0,y,t)=2g_1(y_N,t)g_{N-1}(x'-y',t),\qquad x'\in {\mathbb {R}}^{N-1},\,\,\, y\in {\overline{{{\mathbb {R}}}^N_+}},\,\,\, t>0. \end{aligned}$$

(2.3)

We denote by $S_1(t)\varphi$ the unique bounded solution to the heat equation in ${{\mathbb {R}}}^N_+$ with the homogeneous Neumann boundary condition and the initial datum $\varphi$, that is,

$$\begin{aligned}{}[S_1(t)\varphi ](x):=\int _{{{\mathbb {R}}}^N_+} G(x,y,t)\varphi (y)\,dy,\qquad x\in {\overline{{{\mathbb {R}}}^N_+}},\quad t>0, \end{aligned}$$

(2.4)

and denote by $e^{t\Delta '}\psi$ the unique bounded solution to the heat equation in ${{\mathbb {R}}}^{N-1}$ with the initial datum $\psi$, that is,

$$\begin{aligned}{}[e^{t\Delta '}\psi ](x'):=\int _{{{\mathbb {R}}}^{N-1}}g_{N-1}(x'-y',t)\psi (y')\,dy',\qquad x'\in {{\mathbb {R}}}^{N-1},\quad t>0. \end{aligned}$$

(2.5)

In the case where $N\ge 2$, we put

$$\begin{aligned}{}[S_2(t)\psi ](x):=2g_1(x_N,t)[e^{t\Delta '}\psi ](x'), \qquad x\in {\overline{{{\mathbb {R}}}^N_+}},\,\,\,t>0, \end{aligned}$$

(2.6)

for $\psi \in L^r({\mathbb {R}}^{N-1})$ with some $r\in [1,\infty ]$. Since it holds that, for any $r\in [1,\infty ]$,

$$\begin{aligned} \Vert g_N(t)\Vert _{L^r}\le 4^{-\frac{1}{2r}}(4\pi t)^{-\frac{N}{2}(1-\frac{1}{r})}, \qquad t>0, \end{aligned}$$

(2.7)

by (2.2), (2.3), and applying Young’s inequality to (2.4) and (2.5) we have the following.

$(G_1)$:

There exists a constant $c_1$, which depends only on N, such that

$$\begin{aligned} \Vert S_1(t)\varphi \Vert _{L^r} \le {c_1}t^{-\frac{N}{2}(\frac{1}{q}-\frac{1}{r})}\Vert \varphi \Vert _{L^q}, \qquad t>0, \end{aligned}$$

(2.8)

for $\varphi \in L^q({{\mathbb {R}}}^N_+)$ and $1\le q\le r\le \infty$. Furthermore, there exists a constant $c_2$, which depends only on N, such that, for the case $N\ge 2$,

$$\begin{aligned} |S_1(t)\varphi |_{L^r} \le c_2 t^{-\frac{N}{2}(\frac{1}{q}-\frac{1}{r})-\frac{1}{2r}}\Vert \varphi \Vert _{L^q}, \qquad t>0, \end{aligned}$$

(2.9)

and, for the case $N=1$,

$$\begin{aligned} |[S_1(t)\varphi ](0)| \le c_2t^{-\frac{1}{2q}}\Vert \varphi \Vert _{L^q}, \qquad t>0{.} \end{aligned}$$

(2.10)

$(G_2)$:

For any $\psi \in L^q({\mathbb {R}}^{N-1})$ and $1\le q\le r\le \infty$, it holds that

$$\begin{aligned}&\Vert S_2(t)\psi \Vert _{L^r} \le Ct^{-\frac{N}{2}(\frac{1}{q}-\frac{1}{r})-\frac{1}{2}(1-\frac{1}{q})}|\psi |_{L^q}, \qquad t>0, \end{aligned}$$

(2.11)

$$\begin{aligned}&|S_2(t)\psi |_{L^r} \le Ct^{-\frac{N-1}{2}(\frac{1}{q}-\frac{1}{r})-\frac{1}{2}}|\psi |_{L^q}, \qquad t>0{.} \end{aligned}$$

(2.12)

$(G_3)$:

Let $\varphi \in L^q({{\mathbb {R}}}^N_+)$ with $1\le q\le \infty$. Then, for any $T>0$, $S_1(t)\varphi$ is bounded and smooth in $\overline{{{\mathbb {R}}}^N_+}\times (T,\infty )$.

We recall now the definition and the main properties of the Orlicz space $\mathrm {exp}L^2$.

Definition 2.1

We define the Orlicz space $\mathrm {exp}L^2$ as

$$\begin{aligned} {\mathrm {exp}{L^2}}:=\left\{ u\in L^1_{loc}({{\mathbb {R}}}^N_+); \int _{{{\mathbb {R}}}^N_+}\left( {\mathrm {exp}}\left( \frac{|u(x)|}{\lambda }\right) ^2-1\right) dx<\infty \quad { \mathrm { for\, some}}\ \lambda >0\right\} ,\nonumber \\ \end{aligned}$$

(2.13)

where the norm is given by the Luxemburg type

$$\begin{aligned} \Vert u\Vert _{{\mathrm {exp}{L^2}}}:=\inf \left\{ \lambda >0 \ \mathrm{such \ that}\ \int _{{{\mathbb {R}}}^N_+}\left( {\mathrm {exp}}\left( \frac{|u(x)|}{\lambda }\right) ^2-1\right) dx\le 1 \right\} . \end{aligned}$$

(2.14)

The space $\mathrm {exp}L^2$ endowed with the norm $\Vert u\Vert _{\mathrm {exp}L^2}$ is a Banach space, and admits as predual the Orlicz space defined by the complementary function of $A(t)=e^{t^2}-1$, denoted by ${\tilde{A}}(t)$. This complementary function is a convex function such that ${\tilde{A}}(t)\sim t^2$ as $t\rightarrow 0$ and ${\tilde{A}}(t)\sim t\log ^{1/2}t$ as $t\rightarrow \infty$. (see e.g. [2,  Section 8].) Furthermore, it follows from (2.13) that

$$\begin{aligned} L^2({{\mathbb {R}}}^N_+)\cap L^\infty ({{\mathbb {R}}}^N_+)\subset \mathrm {exp}L^2, \end{aligned}$$

(2.15)

and we have

$$\begin{aligned} \Vert u\Vert _{\mathrm {exp}L^2}\le \frac{1}{\sqrt{\log 2}}(\Vert u\Vert _{L^2}+\Vert u\Vert _{L^\infty }). \end{aligned}$$

(2.16)

(In the case where $\Omega ={\mathbb {R}}^N$, see e.g. [15, 23].) On the other hand, it is well known that, for any $2\le p<\infty$,

$$\begin{aligned} \Vert u\Vert _{L^p}\le \left[ \Gamma \left( \frac{p}{2}+1\right) \right] ^{\frac{1}{p}}\Vert u\Vert _{\mathrm{exp}L^2}. \end{aligned}$$

(2.17)

(See e.g. [15,  Proposition 2.1].) Then, applying the same argument as in the proof of [14,  Lemma 2.2] with (2.17), we have

$$\begin{aligned} \Vert S_1(t)\varphi \Vert _{{\mathrm {exp}}L^2}\le \Vert \varphi \Vert _{\mathrm{exp}L^2},\qquad t>0. \end{aligned}$$

(2.18)

Next we recall the following property of the Gamma function.

Lemma 2.1

[10, Lemma 3.3] For any $q\ge 1$ and $r\ge 1$, there exists a positive constant $C>0$, which is independent of q and r, such that

$$\begin{aligned} \Gamma (rq+1)^{\frac{1}{q}}\le C\Gamma (r+1)q^{r}. \end{aligned}$$

Applying this lemma, we prepare the following estimate for the nonlinear term f for the case $N\ge 2$.

Lemma 2.2

Let $N\ge 2$ and $m>0$. Suppose that, for any $q\in [2,\infty )$, the function $u\in C({\overline{{{\mathbb {R}}}^N_+}}\times (0,\infty ))$ satisfies the condition

$$\begin{aligned} \sup _{t>0}\,t^{\frac{1}{2q}}|u(t)|_{L^q} \le \left\{ \Gamma \left( \frac{q}{2}+1\right) \right\} ^{\frac{1}{q}}m. \end{aligned}$$

(2.19)

Let f be the function satisfying the condition (1.9), and put

$$\begin{aligned} p_2:=\frac{2N}{N+2}. \end{aligned}$$

(2.20)

Then, for all $r\in [p_2,\infty )$, there exists a positive constant $\varepsilon =\varepsilon (r,\lambda ) >0$ such that, if $m<\varepsilon$, then

$$\begin{aligned} \sup _{t>0}\, t^{\frac{1}{2r}} |f(u(t))|_{L^r} \le Crm^{1+\frac{2}{N}}, \end{aligned}$$

(2.21)

where C is independent of r, N, and m.

Proof

For any $k\in {{\mathbb {N}}}\cup \{0\}$, we put

$$\begin{aligned} \ell _k:=2k+1+\frac{2}{N}. \end{aligned}$$

(2.22)

Then, since it holds from $N\ge 2$ and $r\ge p_2$ with (2.20) that

$$\begin{aligned} \ell _kr\ge \left( 1+\frac{2}{N}\right) p_2=\left( 1+\frac{2}{N}\right) \frac{2N}{N+2}= 2,\qquad k\in {{\mathbb {N}}}\cup \{0\}, \end{aligned}$$

by (1.9) and (2.19) we have

$$\begin{aligned} |f(u(t))|_{L^r}\le & {} C\sum _{k=0}^{\infty } \frac{\lambda ^k}{k!}|u(t)|^{\ell _k}_{L^{\ell _kr}}\nonumber \\\le & {} C\sum _{k=0}^{\infty } \frac{\lambda ^k}{k!} \left( \Gamma \left( \frac{\ell _kr}{2}+1\right) ^{\frac{1}{\ell _kr}}t^{-\frac{1}{2\ell _kr}}m \right) ^{\ell _k}, t>0. \end{aligned}$$

(2.23)

Applying Lemma 2.1 with the monotonicity property of the Gamma function $\Gamma (q)$ for $q\ge 3/2$ (see, e.g. [1]), we see that

$$\begin{aligned} \begin{aligned} \Gamma \left( \frac{\ell _kr}{2}+1\right) ^{\frac{1}{r}}&\le C\Gamma \left( \frac{\ell _k}{2}+1\right) r^{\frac{\ell _k}{2}} \\&= C\Gamma \left( k+\frac{3}{2}+\frac{1}{N}\right) r^{\frac{\ell _k}{2}} \\&\le C\Gamma (k+2)r^{\frac{\ell _k}{2}} = C (k+1)! r^{\frac{\ell _k}{2}}. \end{aligned} \end{aligned}$$

This together with (2.23) implies that

$$\begin{aligned} |f(u(t))|_{L^r} \le Ct^{-\frac{1}{2r}} \sum _{k=0}^{\infty } \frac{\lambda ^k(k+1)!}{k!}(rm^2)^{\frac{\ell _k}{2}} = C(rm^2)^{\frac{1}{2}+\frac{1}{N}}t^{-\frac{1}{2r}} \sum _{k=0}^{\infty }(k+1)(\lambda rm^2)^k, \qquad t>0. \end{aligned}$$

Therefore, taking a sufficiently small $m <\varepsilon (r,\lambda )$ if necessary (e.g. $m^2\le 1/(4\lambda r)$), we get

$$\begin{aligned} \sup _{t>0}\,t^{\frac{1}{2r}}|f(u(t))|_{L^r} \le C\frac{(rm^2)^{\frac{1}{2}+\frac{1}{N}}}{(1-\lambda rm^2)^2} \le 2C(rm^2)^{\frac{1}{2}+\frac{1}{N}} \le 2Crm^{1+\frac{2}{N}}. \end{aligned}$$

This implies (2.21), and the proof of Lemma 2.2 is complete. $\square$

Similarly to the case $N\ge 2$, we prepare the following lemma, which is the one dimensional counterpart of Lemma 2.2.

Lemma 2.3

Let $m>0$. Suppose that, for any $q\in [2,\infty )$, the function $u\in C(0,\infty )$ satisfies the condition

$$\begin{aligned} \sup _{t>0}\,t^{\frac{1}{2q}}|u(t)| \le \left\{ \Gamma \left( \frac{q}{2}+1\right) \right\} ^{\frac{1}{q}}m. \end{aligned}$$

(2.24)

Let f be the function satisfying the condition (1.9). Then there exists a positive constant $\varepsilon =\varepsilon (\lambda ) >0$ such that, if $m<\varepsilon$, then

$$\begin{aligned} \sup _{t>0}\, t^{\frac{1}{2}}|f(u(t))| \le Cm^3, \end{aligned}$$

(2.25)

and

$$\begin{aligned} \sup _{t>0}\, t^{\frac{1}{2r}+\frac{1}{2}}|f(u(t))| \le C\left\{ \Gamma \left( \frac{r}{2}+1\right) \right\} ^{\frac{1}{r}}m^3, \qquad r\in [2,\infty ), \end{aligned}$$

(2.26)

where C is independent of m and r.

Proof

We first prove (2.25). For any $k\in {{\mathbb {N}}}\cup \{0\}$, let $\ell _k$ be the constant defined by (2.22) with $N=1$, namely, $\ell _k=2k+3$. Then, by (1.9) and (2.24) with $q=\ell _k$ we have

$$\begin{aligned} |f(u(t))| \le C\sum _{k=0}^{\infty } \frac{\lambda ^k}{k!}|u(t)|^{\ell _k} \le C\sum _{k=0}^{\infty } \frac{\lambda ^k}{k!} \left( \Gamma \left( \frac{\ell _k}{2}+1\right) ^{\frac{1}{\ell _k}}t^{-\frac{1}{2\ell _k}}m \right) ^{\ell _k}, \quad t>0. \end{aligned}$$

(2.27)

Since it holds from the monotonicity property of the Gamma function that

$$\begin{aligned} \Gamma \left( \frac{\ell _k}{2}+1\right) = \Gamma \left( k+\frac{5}{2}\right) \le \Gamma (k+3) = (k+2)!, \end{aligned}$$

by (2.27) we have

$$\begin{aligned} |f(u(t))| \le Ct^{-\frac{1}{2}} \sum _{k=0}^{\infty } \frac{\lambda ^k(k+2)!}{k!}m^{\ell _k} = Ct^{-\frac{1}{2}}m^3 \sum _{k=0}^{\infty }(k+2)(k+1)(\lambda m^2)^k, \qquad t>0. \end{aligned}$$

Therefore, taking a sufficiently small $m <\varepsilon (\lambda )$ if necessary, we get

$$\begin{aligned} \sup _{t>0}\,t^{\frac{1}{2}}|f(u(t))| \le C\frac{m^3}{(1-\lambda m^2)^3} \le 2Cm^3. \end{aligned}$$

This implies (2.25).

Next we prove (2.26). For any $k\in {\mathbb {N}}\cup \{0\}$, put ${\tilde{\ell }}_k=2k+2$. Then, similarly to the proof of (2.25), we have

$$\begin{aligned} |f(u(t))| \le C|u(t)|\sum _{k=0}^{\infty } \frac{\lambda ^k}{k!}|u(t)|^{{\tilde{\ell }}_k} \end{aligned}$$

and then, by (2.24) with $q=r$ and also $q={\tilde{\ell }}_k$ and taking a sufficiently small $m <\varepsilon (\lambda )$ if necessary, we have

$$\begin{aligned} \begin{aligned} |u(t)|\sum _{k=0}^{\infty } \frac{\lambda ^k}{k!}|u(t)|^{{\tilde{\ell }}_k}&\le m\left\{ \Gamma \left( \frac{r}{2}+1\right) \right\} ^{\frac{1}{r}}t^{-\frac{1}{2r}}\sum _{k=0}^{\infty } \frac{\lambda ^k}{k!} \left( \Gamma \left( \frac{{\tilde{\ell }}_k}{2}+1\right) ^{\frac{1}{{\tilde{\ell }}_k}} t^{-\frac{1}{2{\tilde{\ell }}_k}}m \right) ^{{\tilde{\ell }}_k} \\&\le m\left\{ \Gamma \left( \frac{r}{2}+1\right) \right\} ^{\frac{1}{r}}t^{-\frac{1}{2r}-\frac{1}{2}} \sum _{k=0}^\infty \frac{\lambda ^k(k+1)!}{k!}m^{{\tilde{\ell }}_k} \\&\le Cm^3\left\{ \Gamma \left( \frac{r}{2}+1\right) \right\} ^{\frac{1}{r}}t^{-\frac{1}{2r}-\frac{1}{2}}, \quad t>0. \end{aligned} \end{aligned}$$

This implies (2.26), and the proof of Lemma 2.3 is complete. $\square$

3 Existence

In this section we prove Theorem 1.1. We first consider the case $N\ge 2$. We introduce some notation. Let $M>0$. Set

$$\begin{aligned} X_M:=\left\{ \begin{array}{l} u\in C({\overline{{{\mathbb {R}}}^N_+}}\times (0,\infty ))\cap L^\infty (0,\infty ;\mathrm {exp}L^2({{\mathbb {R}}}^N_+)): \\ \displaystyle {\sup _{t>0}\,\Vert u(t)\Vert _{\mathrm {exp}L^2}\le M},\qquad \displaystyle {\sup _{t>0}\,h(t)\Vert u(t)\Vert _{L^\infty }\le M} \quad \text{ with }\quad h(t)=\min \{t^{\frac{N}{4}},1\}, \\ \displaystyle {\sup _{t>0}\, t^{\frac{1}{2q}}|u(t)|_{L^q} \le \left\{ \Gamma \left( \frac{q}{2}+1\right) \right\} ^{\frac{1}{q}}M} \quad \text{ with }\quad q\in [2,\infty ) \end{array} \right\} , \end{aligned}$$

equipped with the metric

$$\begin{aligned} d_X(u,v):=\sup _{t>0}\bigg (h(t)\Vert u(t)-v(t)\Vert _{L^\infty }+t^{\frac{1}{4N}}|u(t)-v(t)|_{L^{2N}}\bigg ). \end{aligned}$$

(3.1)

Then $(X_M,d_X)$ is a complete metric space. For the proof of Theorem 1.1 we apply the Banach contraction map** principle in $X_M$ to find a fixed point of

$$\begin{aligned} \Phi [u](t):=S_1(t)\varphi +D[u](t), \end{aligned}$$

(3.2)

where $S_1(t)$ is as in (2.4) and

$$\begin{aligned} D[u](t):=\int _0^t S_2(t-s)f(u(s))\,ds. \end{aligned}$$

(3.3)

Here $S_2(t)$ is as in (2.6) and f satisfies (1.9). We remark that, for $u\in X_M$, the function f(u) belongs to $C({\overline{{{\mathbb {R}}}^N_+}}\times (0,\infty ))$. Therefore, by Lemma 2.2 we have that $f(u(\cdot ,0,s))\in L^r({\mathbb {R}}^{N-1})$ with $r\in [p_2,\infty )$, and we can define $S_2(t-s)f(u(s))$ for $t>s>0$. More precisely, with an abuse of notation we denote by $S_2(t-s)f(u(s))$ the operator $S_2(t-s)$ applied to the function $f(u(x',0,s))$. In particular, we have

$$\begin{aligned} \begin{aligned} D[u](t) =&\int _0^t S_2(t-s)f(u(s))\,ds \\ =&\int _0^t\int _{{\mathbb {R}}^{N-1}}2g_1(x_N,t-s) g_{N-1}(x'-y',t-s)f(u(y',0,s))\,dy' \,ds \\ =&\int _0^t\int _{{\mathbb {R}}^{N-1}}G(x,y',0,t-s)f(u(y',0,s))\,dy' \,ds. \end{aligned} \end{aligned}$$

Hence any fixed point of the integral operator $\Phi$ satisfies the equation (1.12).

Furthermore, we have the following estimates for the function D[u].

Lemma 3.1

Let $N\ge 2$ and $u\in X_M$. Then there exists a positive constant $\varepsilon _*=\varepsilon _*(N, \lambda )>0$ such that, if $M<\varepsilon _*$, then, for any $q\in [2,\infty )$,

$$\begin{aligned} \sup _{t>0}\,\bigg (\Vert D[u](t)\Vert _{L^2}+\Vert D[u](t)\Vert _{L^\infty }+t^{\frac{1}{2q}}|D[u](t)|_{L^q}\bigg ) \le C M^{1+\frac{2}{N}}, \end{aligned}$$

(3.4)

where C is independent of q and M. Furthermore, D[u] is continuous in $\overline{{{\mathbb {R}}}^N_+}\times (0,\infty )$.

Proof

We first prove (3.4). Let $p_2$ be the constant given in (2.20). Then, it holds that

$$\begin{aligned} 1-\frac{N}{2}\left( \frac{1}{p_2}-\frac{1}{2}\right) -\frac{1}{2}\bigg (1-\frac{1}{p_2}\bigg )-\frac{1}{2p_2} =\frac{N+2}{4}-\frac{N}{2p_2}=0. \end{aligned}$$

By (2.11) with $(q,r)=(p_2,2)$ and (3.3) we have

$$\begin{aligned} \begin{aligned} \Vert D[u](t)\Vert _{L^2}&\le \int _0^t\Vert S_2(t-s)f(u(s))\Vert _{L^2}\,ds \\&\le C\int _0^t(t-s)^{-\frac{N}{2}(\frac{1}{p_2}-\frac{1}{2})-\frac{1}{2}(1-\frac{1}{p_2})}|f(u(s))|_{L^{p_2}}\,ds, \qquad t>0. \end{aligned} \end{aligned}$$

(3.5)

Since $u\in X_M$, taking a sufficiently small $\varepsilon _1=\varepsilon _1(p_2, \lambda )>0$ such that, for $M<\varepsilon _1$, we can apply Lemma 2.2, and it holds that

$$\begin{aligned} |f(u(t))|_{L^{p_2}}\le Cp_2M^{1+\frac{2}{N}}t^{-\frac{1}{2p_2}},\qquad t>0. \end{aligned}$$

(3.6)

Substituting (3.6) to (3.5), we see that

$$\begin{aligned} \begin{aligned} \Vert D[u](t)\Vert _{L^2}&\le Cp_2M^{1+\frac{2}{N}} \int _0^t(t-s)^{-\frac{N}{2}(\frac{1}{p_2}-\frac{1}{2})-\frac{1}{2}(1-\frac{1}{p_2})}s^{-\frac{1}{2p_2}}\,ds \\&\le CM^{1+\frac{2}{N}}B\left( \frac{1}{2p_2},1-\frac{1}{2p_2}\right) ,\qquad t>0, \end{aligned} \end{aligned}$$

(3.7)

where B is the beta function, namely

$$\begin{aligned} B(p,q)=\Gamma (p)\Gamma (q)/\Gamma (p+q),\quad p,q>0. \end{aligned}$$

Furthermore, similarly to (3.5), by (2.11) with $(q,r)=(N,\infty )$ and (3.3) we have

$$\begin{aligned} \Vert D[u](t)\Vert _{L^\infty }\le C\int _0^t(t-s)^{-\frac{N-1}{2N}-\frac{1}{2}}|f(u(s))|_{L^N}\,ds,\qquad t>0. \end{aligned}$$

(3.8)

Since $N\ge 2\ge p_2$, similarly to (3.6), taking a sufficiently small $\varepsilon _2=\varepsilon _2(N, \lambda )>0$ such that, for $M<\varepsilon _2$, we get

$$\begin{aligned} |f(u(t))|_{L^N}\le CNM^{1+\frac{2}{N}}t^{-\frac{1}{2N}},\qquad t>0. \end{aligned}$$

(3.9)

Substituting (3.9) to (3.8), we see that

$$\begin{aligned} \begin{aligned} \Vert D[u](t)\Vert _{L^\infty }&\le CNM^{1+\frac{2}{N}}\int _0^t(t-s)^{-1+\frac{1}{2N}}s^{-\frac{1}{2N}}\,ds \\&\le CM^{1+\frac{2}{N}}B\left( \frac{1}{2N},1-\frac{1}{2N}\right) ,\qquad t>0. \end{aligned} \end{aligned}$$

(3.10)

On the other hand, for fixed $q\in [2,\infty )$, we put

$$\begin{aligned} q_*:=\frac{Nq}{N+q}. \end{aligned}$$

Then, it holds that $p_2\le q_*< q$ and

$$\begin{aligned} -\frac{N-1}{2}\left( \frac{1}{q_*}-\frac{1}{q}\right) -\frac{1}{2}=-1+\frac{1}{2N}, \qquad \frac{1}{2N}-\frac{1}{2q_*}=-\frac{1}{2q}. \end{aligned}$$

(3.11)

By (2.12) with $(q,r)=(q_*,q)$ and (3.3) we have

$$\begin{aligned} \begin{aligned} |D[u](t)|_{L^q}&\le \int _0^t|S_2(t-s)f(u(s))|_{L^q}\,ds \\&\le C\int _0^t(t-s)^{-\frac{N-1}{2}(\frac{1}{q_*}-\frac{1}{q})-\frac{1}{2}}|f(u(s))|_{L^{q_*}}\,ds, \qquad t>0. \end{aligned} \end{aligned}$$

(3.12)

Since $p_2\le q_*\le N$, similarly to (3.6) again, taking a sufficiently small $\varepsilon _3=\varepsilon _3(N, \lambda )>0$ such that, for $M<\varepsilon _3$, we have

$$\begin{aligned} |f(u(t))|_{L^{q_*}}\le Cq_*M^{1+\frac{2}{N}}t^{-\frac{1}{2q_*}}\le C N M^{1+\frac{2}{N}}t^{-\frac{1}{2q_*}}, \qquad t>0. \end{aligned}$$

This together with (3.11) and (3.12) yields that

$$\begin{aligned} \begin{aligned} |D[u](t)|_{L^q}&\le CM^{1+\frac{2}{N}}\int _0^t (t-s)^{-1+\frac{1}{2N}}s^{-\frac{1}{2q_*}}\,ds \\&\le CM^{1+\frac{2}{N}}t^{-\frac{1}{2q}} B\left( \frac{1}{2N}, 1-\frac{1}{2q_*}\right) \le CM^{1+\frac{2}{N}}t^{-\frac{1}{2q}},\qquad t>0, \end{aligned} \end{aligned}$$

(3.13)

where the constant C depends only on N since $p_2\le q_*\le N$. Thus, taking $\varepsilon _*=\min \{\varepsilon _1, \varepsilon _2, \varepsilon _3\}$ with (3.7), (3.10), and (3.13), we obtain (3.4).

Next we prove the continuity of D[u](x, t). Let T be an arbitrary positive constant. Then, it follows from (2.1) that

$$\begin{aligned} \begin{aligned} D[u](x,t)&=\int _0^t[S_2(t-s)f(u(s))](x)\,ds \\&=[S_1(t-T/2)D[u](T/2)](x)+\int _{T/2}^t[S_2(t-s)f(u(s))](x)\,ds \end{aligned} \end{aligned}$$

for $x\in {\overline{{{\mathbb {R}}}^N_+}}$ and $0<T<t<\infty$. Then, by (3.4) and $(G_3)$ we see that

$$\begin{aligned}{}[S_1(t-T/2)D[u](T/2)](x) \end{aligned}$$

is continuous in ${\overline{{{\mathbb {R}}}^N_+}}\times (T,\infty )$. Furthermore, since it follows from $u(t)\in L^\infty ({\overline{{{\mathbb {R}}}^N_+}})$ for $t\ge T/2$ that $f(u(t))\in L^\infty (\partial {{\mathbb {R}}}^N_+)$ for $t\ge T/2$, we apply the same argument as in [9, Section 3, Chapter 1] to see that

$$\begin{aligned} \int _{T/2}^t[S_2(t-s)f(u(s))](x)\,ds \end{aligned}$$

is also continuous in ${\overline{{{\mathbb {R}}}^N_+}}\times (T,\infty )$. (See also [7, Proposition 5.2] and [16, Lemma 2.1].) Therefore we deduce that D[u] is continuous in $\overline{{{\mathbb {R}}}^N_+}\times (T,\infty )$. Thus Lemma 3.1 follows from arbitrariness of T. $\square$

Lemma 3.2

Let $N\ge 2$ and $u,v\in X_M$. Then there exist some positive constants $C=C(N)$ and $\varepsilon ^*=\varepsilon ^*(N,\lambda )>0$ such that, if $M<\varepsilon ^*$, then

$$\begin{aligned} d_X(D[u],D[v])\le CM^{\frac{2}{N}}d_X(u,v). \end{aligned}$$

(3.14)

Proof

For any $k\in {\mathbb {N}}\cup \{0\}$, we put

$$\begin{aligned} {\tilde{\ell }}_k:=2k+\frac{2}{N}. \end{aligned}$$

(3.15)

Then, by (1.9) we recall that

$$\begin{aligned} |f(u)-f(v)| \le C|u-v|\sum _{k=0}^\infty \frac{\lambda ^k}{k!}(|u|^{{\tilde{\ell }}_k}+|v|^{{\tilde{\ell }}_k}). \end{aligned}$$

(3.16)

Since $h(t)\le 1$, by (2.11) with $(q,r)=(N,\infty )$, (3.3), and (3.16), for any $t>0$, we have

$$\begin{aligned} \begin{aligned}&h(t)\Vert D[u](t)-D[v](t)\Vert _{L^\infty } \\&\le \int _0^t\Vert S_2(t-s)(f(u(s)-f(v(s)))\Vert _{L^\infty }\,ds \\&\le C\sum _{k=0}^\infty \frac{\lambda ^k}{k!} \int _0^t(t-s)^{-1+\frac{1}{2N}} \bigg ||u(s)-v(s)|\bigg (|u(s)|^{{\tilde{\ell }}_k}+|v(s)|^{{\tilde{\ell }}_k}\bigg )\bigg |_{L^{N}}\,ds,\qquad t>0. \end{aligned} \end{aligned}$$

(3.17)

Since it follows from Hölder’s inequality that

$$\begin{aligned} \bigg ||u(s)-v(s)|\bigg (|u(s)|^{{\tilde{\ell }}_k}+|v(s)|^{{\tilde{\ell }}_k}\bigg )\bigg |_{L^{N}} \le |u(s)-v(s)|_{L^{2N}}\bigg (|u(s)|_{L^{2{\tilde{\ell }}_kN}}^{{\tilde{\ell }}_k} +|v(s)|_{L^{2{\tilde{\ell }}_kN}}^{{\tilde{\ell }}_k}\bigg ), \end{aligned}$$

by (3.1), (3.15), and (3.17) we see that, for $u,v\in X_M$,

$$\begin{aligned} \begin{aligned}&h(t)\Vert D[u](t)-D[v](t)\Vert _{L^\infty } \\&\le C\sum _{k=0}^\infty \frac{\lambda ^k}{k!} \int _0^t(t-s)^{-1+\frac{1}{2N}} |u(s)-v(s)|_{L^{2N}}\bigg (|u(s)|_{L^{2{\tilde{\ell }}_kN}}^{{\tilde{\ell }}_k} +|v(s)|_{L^{2{\tilde{\ell }}_kN}}^{{\tilde{\ell }}_k}\bigg )\,ds \\&\le C\sum _{k=0}^\infty \frac{\lambda ^k}{k!} \int _0^t(t-s)^{-1+\frac{1}{2N}}s^{-\frac{1}{4N}-\frac{{\tilde{\ell }}_k}{4{\tilde{\ell }}_kN}} \bigg (\sup _{s>0}\,s^{\frac{1}{4N}}|u(s)-v(s)|_{L^{2N}}\bigg ) \times \\&\qquad \qquad \qquad \qquad \times \bigg \{\bigg (\sup _{s>0}\,s^{\frac{1}{4{\tilde{\ell }}_kN}} |u(s)|_{L^{2{\tilde{\ell }}_kN}}\bigg )^{{\tilde{\ell }}_k} +\bigg (\sup _{s>0}\,s^{\frac{1}{4{\tilde{\ell }}_kN}}|v(s)|_{L^{2{\tilde{\ell }}_kN}}\bigg )^{{\tilde{\ell }}_k}\bigg \}\,ds \\&\le Cd_X(u,v)\sum _{k=0}^\infty \frac{\lambda ^k}{k!} \bigg (\Gamma \bigg ({{\tilde{\ell }}_k N}+1\bigg )^{\frac{1}{2{\tilde{\ell }}_kN}}M\bigg )^{{\tilde{\ell }}_k} \int _0^t(t-s)^{-1+\frac{1}{2N}}s^{-\frac{1}{2N}}\,ds \\&\le CM^{\frac{2}{N}}d_X(u,v)B\bigg (\frac{1}{2N},1-\frac{1}{2N}\bigg ) \sum _{k=0}^\infty \frac{(\lambda M^2)^k}{k!} \Gamma \bigg ({{\tilde{\ell }}_k N}+1\bigg )^{\frac{1}{2N}},\qquad t>0. \end{aligned} \nonumber \\ \end{aligned}$$

(3.18)

For $k=0$, by (3.15) we have $\Gamma ({{\tilde{\ell }}_0 N}+1)=\Gamma (3)$. Furthermore, applying Lemma 2.1 with (3.15) and by the monotonicity property of the Gamma function, for $k\ge 1$, we see that

$$\begin{aligned} \begin{aligned} \Gamma \bigg ({{\tilde{\ell }}_k N}+1\bigg )^{\frac{1}{2{N}}}&\le C\Gamma \bigg (\frac{{\tilde{\ell }}_k}{2}+1\bigg )(2N)^{\frac{{\tilde{\ell }}_k}{2}} \\&=C\Gamma \bigg (k+\frac{1}{N}+1\bigg )(2N)^{\frac{{\tilde{\ell }}_k}{2}} \\&\le C\Gamma (k+2)(2N)^{\frac{{\tilde{\ell }}_k}{2}}=C(k+1)!(2N)^{\frac{{\tilde{\ell }}_k}{2}}. \end{aligned} \end{aligned}$$

These together with (3.18) implies that

$$\begin{aligned} \begin{aligned}&h(t)\Vert D[u](t)-D[v](t)\Vert _{L^\infty } \\&\le CM^{\frac{2}{N}}d_X(u,v) \sum _{k=0}^\infty \frac{(\lambda M^2)^k}{k!}(k+1)!(2N)^{\frac{{\tilde{\ell }}_k}{2}} \\&\le C(NM^2)^{\frac{1}{N}}d_X(u,v) \sum _{k=0}^\infty (k+1)(2\lambda NM^2)^k, \qquad t>0. \end{aligned} \end{aligned}$$

Then, taking a sufficiently small $\varepsilon ^*=\varepsilon ^*(N,\lambda )>0$ such that, for $M<\varepsilon ^*$, in a similar way as in Lemma 2.2, it holds that

$$\begin{aligned} \sup _{t>0}h(t)\Vert D[u](t)-D[v](t)\Vert _{L^\infty }\le & {} C\frac{(NM)^{\frac{2}{N}}}{(1-2\lambda NM^2)^2}d_X(u,v)\nonumber \\\le & {} CM^{\frac{2}{N}}d_X(u,v). \end{aligned}$$

(3.19)

On the other hand, similarly to (3.12), by (2.12) with $(q,r)=((2N)/3,2N)$, (3.3), and (3.16) we have

$$\begin{aligned} \begin{aligned}&t^{\frac{1}{4N}}|D[u](t)-D[v](t)|_{L^{2N}} \\&\le t^{\frac{1}{4N}} \int _0^t|S_2(t-s)(f(u(s))-f(v(s)))|_{L^{2N}}\,ds \\&\le C\sum _{k=0}^\infty \frac{\lambda ^k}{k!} t^{\frac{1}{4N}}\int _0^t(t-s)^{-\frac{N-1}{2N}-\frac{1}{2}} \bigg ||u(s)-v(s)|\bigg (|u(s)|^{{\tilde{\ell }}_k}+|v(s)|^{{\tilde{\ell }}_k}\bigg )\bigg |_{L^{\frac{2N}{3}}}\,ds,\qquad t>0. \end{aligned} \end{aligned}$$

Therefore, applying the same argument as in the proof of (3.19), for $M<\varepsilon ^*$, it holds that

$$\begin{aligned} \begin{aligned}&t^{\frac{1}{4N}}|D[u](t)-D[v](t)|_{L^{2N}} \\&\le Ct^{\frac{1}{4N}}\sum _{k=0}^\infty \frac{\lambda ^k}{k!} \int _0^t(t-s)^{-1+\frac{1}{2N}} |u(s)-v(s)|_{L^{2N}}\bigg (|u(s)|_{L^{{\tilde{\ell }}_kN}}^{{\tilde{\ell }}_k} +|v(s)|_{L^{{\tilde{\ell }}_kN}}^{{\tilde{\ell }}_k}\bigg )\,ds \\&\le Ct^{\frac{1}{4N}}d_X(u,v)\sum _{k=0}^\infty \frac{\lambda ^k}{k!} \bigg (\Gamma \bigg (\frac{{\tilde{\ell }}_k N}{2}+1\bigg )^{\frac{1}{{\tilde{\ell }}_kN}}M\bigg )^{{\tilde{\ell }}_k} \int _0^t(t-s)^{-1+\frac{1}{2N}}s^{-\frac{3}{4N}}\,ds \\&\le C(NM^2)^{\frac{1}{N}}d_X(u,v)B\bigg (\frac{1}{2N},1-\frac{3}{4N}\bigg ) \sum _{k=0}^\infty (k+1)(\lambda NM^2)^k \\&\le CM^\frac{2}{N}d_X(u,v),\qquad t>0. \end{aligned} \end{aligned}$$

This implies that

$$\begin{aligned} \sup _{t>0}\,t^{\frac{1}{4N}}|D[u](t)-D[v](t)|_{L^{2N}} \le CM^{\frac{2}{N}}d_X(u,v). \end{aligned}$$

(3.20)

Combining (3.19) and (3.20), we have (3.14), thus Lemma 3.2 follows. $\square$

Remark 3.1

In the proof of Lemma 3.2, the estimate for $\sup _{t>0}t^{1/(4N)}|\cdot |_{L^{2N}}$ is closed by itself. We need the term $\sup _{t>0} h(t)\Vert \cdot \Vert _{L^\infty }$ in the definition of the metric $d_X$ in order to ensure the uniform convergence of the Cauchy sequence so that the solution is continuous.

Now we are ready to complete the proof of Theorem 1.1 for the case $N\ge 2$.

Proof of Theorem 1.1

($N\ge 2$). Let

$$\begin{aligned} M:=6\,\max \{1,c_1,c_2\}\Vert \varphi \Vert _{\mathrm {exp}L^2}, \end{aligned}$$

where $c_1$ and $c_2$ are constant given in $(G_1)$. Then, by (2.8), (2.9), (2.17), and (2.18) we see that

$$\begin{aligned} \begin{aligned}&\sup _{t>0}\,\Vert S_1(t)\varphi \Vert _{\mathrm {exp}L^2}\le \frac{M}{2}, \qquad \sup _{t>0}\, t^{\frac{N}{4}}\Vert S_1(t)\varphi \Vert _{L^\infty }\le \frac{M}{2}, \\&\sup _{t>0}\, t^{\frac{1}{2q}}|S_1(t)\varphi |_{L^q} \le \left\{ \Gamma \left( \frac{q}{2}+1\right) \right\} ^{\frac{1}{q}}\frac{M}{2},\qquad q\in [2,\infty ). \end{aligned} \end{aligned}$$

(3.21)

Let $u\in X_M$. Then, by Lemma 3.1 with (2.16) we can take a sufficiently small $\varepsilon _4=\varepsilon _4(N, \lambda )>0$ such that, for $M<\varepsilon _4$, it holds $CM^{2/N}<1/2$ and so

$$\begin{aligned} \begin{aligned}&\sup _{t>0}\,\Vert D[u](t)\Vert _{\mathrm {exp}L^2}\le \frac{M}{2}, \qquad \sup _{t>0}\,\Vert D[u](t)\Vert _{L^\infty }\le \frac{M}{2}, \\&\sup _{t>0}\, t^{\frac{1}{2q}}|D[u](t)|_{L^q} \le \left\{ \Gamma \left( \frac{q}{2}+1\right) \right\} ^{\frac{1}{q}}\frac{M}{2},\qquad q\in [2,\infty ). \end{aligned} \end{aligned}$$

This together with property ($G_3$), Lemma 3.1, (3.2), and (3.21) yields that $\Phi$ is a map on $X_M$ to itself. Furthermore, since it follows from (3.1) and (3.2) that

$$\begin{aligned} d_X(\Phi [u],\Phi [v])=d_X(D[u],D[v]) \end{aligned}$$

for $u,v\in X_M$, taking a sufficiently small $\varepsilon _5=\varepsilon _5(N)>0$ if necessary, for $M<\varepsilon _5$, we can apply Lemma 3.2, and it holds that

$$\begin{aligned} d_X(\Phi [u],\Phi [v])\le \frac{1}{4}d_X(u,v). \end{aligned}$$

Then, applying the contraction map** theorem ensures that there exists a unique $u\in X_M$ with

$$\begin{aligned} u=\Phi [u](t)=S_1(t)\varphi +D[u](t)\qquad \text{ in }\qquad X_M. \end{aligned}$$

Thus we see that u is the unique global-in-time solution of problem (1.12) satisfying (1.14) and (1.15). Furthermore, by the same argument as in the proof of [14, (1.7)] with Lemma 3.1, we can prove that $u(t) \underset{t\rightarrow 0}{\longrightarrow }\varphi$ in the weak$^*$ topology, and the proof of Theorem 1.1 for the case $N\ge 2$ is complete. $\square$

Next we consider the case $N=1$. Similarly to the case $N\ge 2$, let $M>0$, and we set

$$\begin{aligned} Y_M:=\left\{ \begin{array}{l} u\in C([0,\infty )\times (0,\infty ))\cap L^\infty (0,\infty ;\mathrm {exp}L^2(0,\infty ))\,:\, \\ \displaystyle {\sup _{t>0}\,\Vert u(t)\Vert _{\mathrm {exp}L^2}\le M}, \qquad \displaystyle {\sup _{t>0}\,h(t)\Vert u(t)\Vert _{L^\infty }\le M} \quad \text{ with }\quad h(t)=\min \{t^{\frac{1}{4}},1\}, \\ \displaystyle {\sup _{t>0}\, t^{\frac{1}{2q}}|u(0,t)| \le \left\{ \Gamma \left( \frac{q}{2}+1\right) \right\} ^{\frac{1}{q}}M} \quad \text{ with }\quad q\in [2,\infty ) \end{array} \right\} , \end{aligned}$$

equipped with the metric

$$\begin{aligned} d_Y(u,v):=\sup _{t>0}\bigg (h(t)\Vert u(t)-v(t)\Vert _{L^\infty }+t^{\frac{1}{4}}|u(0,t)-v(0,t)|\bigg ). \end{aligned}$$

(3.22)

Then $(Y_M,d_Y)$ is a complete metric space. Similarly to the proof of Theorem 1.1 for the case $N\ge 2$, we apply the Banach contraction map** principle in $Y_M$ to find a fixed point of

$$\begin{aligned} \Psi [u](t):=S_1(t)\varphi +{\tilde{D}}[u](t), \end{aligned}$$

where

$$\begin{aligned} {\tilde{D}}[u](x,t):=2\int _0^t g_1(x,t-s)f(u(0,s))\,ds,\quad x\in [0,\infty ). \end{aligned}$$

(3.23)

Here $g_1$ is as in (1.10) and f satisfies (1.9).

Applying Lemma 2.3, we have the following.

Lemma 3.3

Let $u\in Y_M$. Then there exists a positive constant $\varepsilon _*=\varepsilon _*(\lambda )>0$ such that, if $M<\varepsilon _*$, then, for any $q\in [2,\infty )$,

$$\begin{aligned}&\sup _{t>0}\,\bigg (\Vert {\tilde{D}}[u](t)\Vert _{L^2}+h(t)\Vert {\tilde{D}}[u](t)\Vert _{L^\infty }\bigg )\le C M^3, \end{aligned}$$

(3.24)

$$\begin{aligned}&\sup _{t>0}\,t^{\frac{1}{2q}}|{\tilde{D}}[u](0,t)| \le C \left\{ \Gamma \left( \frac{q}{2}+1\right) \right\} ^{\frac{1}{q}}M^3, \end{aligned}$$

(3.25)

where C is independent of q and M. Furthermore, ${\tilde{D}}[u]$ is continuous in $[0,\infty )\times (0,\infty )$.

Proof

By (2.7) with $(N,r)=(1,2)$ and (3.23) we have

$$\begin{aligned} \begin{aligned} \Vert {\tilde{D}}[u](t)\Vert _{L^2}&\le 2\int _0^t\Vert g_1(t-s)\Vert _{L^2}|f(u(0,s))|\,ds \\&\le C\int _0^t(t-s)^{-\frac{1}{4}}|f(u(0,s))|\,ds, \qquad t>0. \end{aligned} \end{aligned}$$

(3.26)

Since $u\in Y_M$, taking a sufficiently small $\varepsilon _*=\varepsilon _*(\lambda )>0$ such that, for $M<\varepsilon _*$, we can apply Lemma 2.3, and it holds from (2.26) with $r=2$ and (3.26) that

$$\begin{aligned} \Vert {\tilde{D}}[u](t)\Vert _{L^2} \le CM^3\int _0^t(t-s)^{-\frac{1}{4}}s^{-\frac{3}{4}}\,ds \le CM^3B\left( \frac{3}{4},\frac{1}{4}\right) ,\qquad t>0. \end{aligned}$$

(3.27)

Similarly, by (2.7) with $(N,r)=(1,\infty )$, (2.26), and (3.23), for any $q\in [2,\infty )$, it holds that

$$\begin{aligned} \begin{aligned} |{\tilde{D}}[u](x,t)|&\le C\int _0^t(t-s)^{-\frac{1}{2}}|f(u(0,s))|\,ds \\&\le C\left\{ \Gamma \left( \frac{q}{2}+1\right) \right\} ^{\frac{1}{q}}M^3 \int _0^t(t-s)^{-\frac{1}{2}}s^{-\frac{1}{2q}-\frac{1}{2}}\,ds \\&\le C\left\{ \Gamma \left( \frac{q}{2}+1\right) \right\} ^{\frac{1}{q}}M^3 \left\{ \left( \frac{t}{2}\right) ^{-\frac{1}{2}}\int _0^{t/2}s^{-\frac{1}{2q}-\frac{1}{2}}\,ds +\left( \frac{t}{2}\right) ^{-\frac{1}{2q}-\frac{1}{2}}\int _{t/2}^t(t-s)^{-\frac{1}{2}}\,ds\right\} \\&\le C\left\{ \Gamma \left( \frac{q}{2}+1\right) \right\} ^{\frac{1}{q}}M^3t^{-\frac{1}{2q}} \left( \frac{2^{\frac{1}{2q}+1}}{1-\frac{1}{q}}+2^{\frac{1}{2q}+1}\right) \\&\le C\left\{ \Gamma \left( \frac{q}{2}+1\right) \right\} ^{\frac{1}{q}}M^3t^{-\frac{1}{2q}}, \qquad x\in [0,\infty ),\quad t>0, \end{aligned} \end{aligned}$$

where C is independent of q and M. This implies that

$$\begin{aligned}&h(t)\Vert {\tilde{D}}[u](t)\Vert _{L^\infty }\le CM^3,\nonumber \\&\quad |{\tilde{D}}[u](0,t)| \le C\left\{ \Gamma \left( \frac{q}{2}+1\right) \right\} ^{\frac{1}{q}}M^3t^{-\frac{1}{2q}}, \qquad t>0. \end{aligned}$$

(3.28)

Thus, by (3.27) and (3.28) we obtain (3.24) and (3.25). Furthermore, applying the same argument as in the proof of Lemma 3.1, we see that ${\tilde{D}}[u]$ is continuous in $[0,\infty )\times (0,\infty )$, and the proof of Lemma 3.3 is complete. $\square$

Lemma 3.4

Let $u,v\in Y_M$. Then there exists a positive constant $\varepsilon ^*=\varepsilon ^*(\lambda )>0$ such that, if $M<\varepsilon ^*$, then

$$\begin{aligned} d_Y({\tilde{D}}[u],{\tilde{D}}[v])\le CM^2d_Y(u,v), \end{aligned}$$

(3.29)

where C is independent of M.

Proof

For any $k\in {\mathbb {N}}\cup \{0\}$, let ${\tilde{\ell }}_k$ be the constant defined by (3.15) with $N=1$. Then, similarly to (3.18), by (2.7) with $(N,r)=(1,\infty )$, (3.16), (3.22), and (3.23), for $u,v\in Y_M$, we have

$$\begin{aligned}&|{\tilde{D}}[u](x,t)-{\tilde{D}}[v](x,t)| \\&\le C \int _0^t(t-s)^{-\frac{1}{2}}|f(u(0,s))-f(v(0,s))|\,ds \\&\le C \sum _{k=0}^\infty \frac{\lambda ^k}{k!} \int _0^t(t-s)^{-\frac{1}{2}} |u(0,s)-v(0,s)|\bigg (|u(0,s)|^{{\tilde{\ell }}_k}+|v(0,s)|^{{\tilde{\ell }}_k}\bigg )\,ds \\&\le C\sum _{k=0}^\infty \frac{\lambda ^k}{k!} \int _0^t(t-s)^{-\frac{1}{2}}s^{-\frac{1}{4}-\frac{{\tilde{\ell }}_k}{2{\tilde{\ell }}_k}} \bigg (\sup _{s>0}\,s^{\frac{1}{4}}|u(0,s)-v(0,s)|\bigg ) \times \\&\qquad \times \bigg \{\bigg (\sup _{s>0}\,s^{\frac{1}{2{\tilde{\ell }}_k}}|u(0,s)|\bigg )^{{\tilde{\ell }}_k}+ \bigg (\sup _{s>0}\,s^{\frac{1}{2{\tilde{\ell }}_k}}|v(0,s)|\bigg )^{{\tilde{\ell }}_k}\bigg \}\,ds \\&\le Cd_Y(u,v)\sum _{k=0}^\infty \frac{\lambda ^k}{k!} \bigg (\,\Gamma \bigg (\frac{{\tilde{\ell }}_k}{2}+1\bigg )^\frac{1}{{\tilde{\ell }}_k}M\bigg )^{{\tilde{\ell }}_k} \int _0^t(t-s)^{-\frac{1}{2}}s^{-\frac{3}{4}}\,ds \\&\le CM^2t^{-\frac{1}{4}}d_Y(u,v)B\bigg (\frac{1}{2},\frac{1}{4}\bigg ) \sum _{k=0}^\infty \frac{(\lambda M^2)^k}{k!}\Gamma (k+2) \\&\le CM^2t^{-\frac{1}{4}}d_Y(u,v)\sum _{k=0}^\infty (k+1)(\lambda M^2)^k, \qquad x\in [0,\infty ),\quad t>0. \end{aligned}$$

Then, we can take a sufficiently small $\varepsilon ^*=\varepsilon ^*(\lambda )>0$ such that, for $M<\varepsilon ^*$, it holds that

$$\begin{aligned} \begin{aligned}&\sup _{t>0}\,h(t)\Vert {\tilde{D}}[u](t)-{\tilde{D}}[v](t)\Vert _{L^\infty } \le CM^2d_Y(u,v), \\&\sup _{t>0}\,t^{\frac{1}{4}}|{\tilde{D}}[u](0,t)-{\tilde{D}}[v](0,t)| \le CM^2d_Y(u,v). \end{aligned} \end{aligned}$$

This implies (3.29), thus Lemma 3.4 follows. $\square$

Proof of Theorem 1.1

($N=1$). By Lemmata 3.3, 3.4, and applying the same arguments as in the proof of Theorem 1.1 for the case $N\ge 2$, we can prove Theorem 1.1 for the case $N=1$. $\square$

4 Slowly decaying initial data

In this section we prove Theorem 1.2. Similarly to Sect. 3, we first consider the case $N\ge 2$. Let u be the unique solution to problem (1.1) satisfying (1.14) and (1.15). Put

$$\begin{aligned} v(x,t):= u(x,t+1). \end{aligned}$$

(4.1)

Then, it follows from (1.12) and (2.1) that the function v satisfies

$$\begin{aligned} v(t)=S_1(t)u(1)+D[v](t),\qquad t>0, \end{aligned}$$

(4.2)

where D[v] is the function defined by (3.3). Since it follows from (1.14) and (2.17) that

$$\begin{aligned} \Vert u(1)\Vert _{L^q} \le c_*\Vert \varphi \Vert _{\mathrm {exp}L^2},\qquad q\in [2,\infty ], \end{aligned}$$

(4.3)

by (2.9) with $q=r$, for any $q\in [2,\infty ]$, we have

$$\begin{aligned} |S_1(t)u(1)|_{L^q}\le c_2t^{-\frac{1}{2q}}\Vert u(1)\Vert _{L^q} \le c_2c_*t^{-\frac{1}{2q}}\Vert \varphi \Vert _{\mathrm {exp}L^2},\qquad t>0. \end{aligned}$$

(4.4)

Here $c_*$ is a constant independent of q and $\Vert \varphi \Vert _{\mathrm {exp}L^2}$. Furthermore, since it follows from the continuity of the function D[u](x, t) that $|D[u](t)|_{L^\infty }\le \Vert D[u](t)\Vert _{L^\infty }$, applying the same argument as in the proof of Lemma 3.1 with (1.15) and (4.1), we see that, for any $q\in [2,\infty ]$,

$$\begin{aligned} |D[v](t)|_{L^q}\le Ct^{-\frac{1}{2q}}\Vert \varphi \Vert _{\mathrm {exp}L^2}^{1+\frac{2}{N}},\qquad t>0. \end{aligned}$$

(4.5)

Then, we can take a sufficiently small $\varepsilon >0$ such that, for $\Vert \varphi \Vert _{\mathrm {exp}L^2}<\varepsilon$, it follows from (4.2), (4.4), and (4.5) that

$$\begin{aligned} |v(t)|_{L^q}\le 2c_2c_*t^{-\frac{1}{2q}}\Vert \varphi \Vert _{\mathrm {exp}L^2},\qquad t>0. \end{aligned}$$

(4.6)

On the other hand, we have the following.

Lemma 4.1

Let $N\ge 2$, $T>0$, and $A>0$. Suppose that, for any $q\in [2,\infty ]$, the function $v\in C({\overline{{{\mathbb {R}}}^N_+}}\times (0,\infty ))$ satisfies

$$\begin{aligned} \sup _{0<t\le T} (1+t)^{\frac{N}{2}(\frac{1}{2}-\frac{1}{q})}t^{\frac{1}{2q}}|v(t)|_{L^q} \le A. \end{aligned}$$

(4.7)

Let f be a function satisfying (1.9). Then, there exists $\varepsilon _*>0$, independent of T, such that, if $A<\varepsilon _*$, then, for any $r\in [p_2,\infty ]$,

$$\begin{aligned} \sup _{0<t\le T} (1+t)^{\frac{N}{2}(\frac{1}{2}-\frac{1}{r})+\frac{1}{2}}t^{\frac{1}{2r}}|f(v(t))|_{L^r} \le 2C_fA^{1+\frac{2}{N}}, \end{aligned}$$

(4.8)

where $C_f$ and $p_2$ are given in (1.9) and (2.20), respectively.

Proof

Let $k\in {{\mathbb {N}}}\cup \{0\}$ and $\ell _k$ be the constant given in (2.22). Then, for any $r\in [p_2,\infty ]$, by (1.9) and (4.7) we have

$$\begin{aligned} \begin{aligned} |f(v(t))|_{L^r}&\le C_f\sum _{k=0}^{\infty } \frac{\lambda ^k}{k!}|v(t)|_{L^{\ell _kr}}^{\ell _k} \\&\le C_f\sum _{k=0}^{\infty } \frac{\lambda ^k}{k!} \left( (1+t)^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{\ell _kr})}t^{-\frac{1}{2\ell _kr}} A\right) ^{\ell _k} \\&\le C_fA^{1+\frac{2}{N}}(1+t)^{\frac{N}{2r}-\frac{N}{4}(1+\frac{2}{N})}t^{-\frac{1}{2r}} \sum _{k=0}^{\infty } \frac{\lambda ^k}{k!} \left( (1+t)^{-\frac{N}{4}}A\right) ^{2k} \\&\le C_fA^{1+\frac{2}{N}}(1+t)^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{r})-\frac{1}{2}}t^{-\frac{1}{2r}} \sum _{k=0}^{\infty } \frac{\lambda ^k}{k!} A^{2k},\qquad t>0. \end{aligned} \end{aligned}$$

(4.9)

We can take a sufficiently small $\varepsilon _*=\varepsilon _*(\lambda )>0$ so that, for $A< \varepsilon _*$, it holds that

$$\begin{aligned} \sum _{k=0}^{\infty } \frac{\lambda ^k}{k!} A^{2k}=e^{\lambda A^2}\le 2. \end{aligned}$$

(4.10)

This together with (4.9) implies (4.8). Thus Lemma 4.1 follows. $\square$

Now we are in position to prove Theorem 1.2 for the case $N\ge 2$.

Proof of Theorem 1.2

($N\ge 2$). Following the idea of the proof of [20, Lemma 2.4], we prove this theorem.

Let u be a unique solution to problem (1.1) satisfying (1.14) and (1.15), and let v be the function defined by (4.2). Then, applying arguments similar to that in the proof of [20, Lemma 2.1] with (4.6), we see that

$$\begin{aligned} v\in C((0,\infty ); L^q(\partial {{\mathbb {R}}}^N_+)),\qquad q\in [2,\infty ]. \end{aligned}$$

(4.11)

Let $\Vert \varphi \Vert _{\mathrm {exp}L^2}$ be a sufficiently small to be chosen later. Put

$$\begin{aligned} \delta =2^{\frac{N}{2}(\frac{1}{2}-\frac{1}{q})}c_*\Vert \varphi \Vert _{\mathrm {exp}L^2}, \end{aligned}$$

(4.12)

and

$$\begin{aligned} T=\sup \left\{ 0<s<\infty \,;\,|v(t)|_{L^q}\le 2{c_2}\delta (1+t)^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{q})}t^{-\frac{1}{2q}} \,\,\,\hbox {for all} q \in [2,\infty ] \hbox {and} 0<t<s\right\} , \end{aligned}$$

where $c_*$ and $c_2$ are given in (4.3) and (2.9), respectively. Then, by (4.6) and (4.12) we have $T\ge 1$.

We prove $T=\infty$. The proof is by contradiction. We assume that $T<\infty$. Then, by (4.11) we see that

$$\begin{aligned} |v(T)|_{L^q}=2c_2\delta (1+T)^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{q})}T^{-\frac{1}{2q}}. \end{aligned}$$

(4.13)

On the other hand, by (2.9) with $(q,r)=(2,q)$, (4.3) and (4.12) we have

$$\begin{aligned} \begin{aligned} |S_1(T)u(1)|_{L^q}&\le c_2T^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{q})-\frac{1}{2q}}\Vert u(1)\Vert _{L^2} \\&\le 2^{\frac{N}{2}(\frac{1}{2}-\frac{1}{q})} c_2(1+T)^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{q})}T^{-\frac{1}{2q}}c_*\Vert \varphi \Vert _{\mathrm {exp}L^2} \\&\le c_2\delta (1+T)^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{q})}T^{-\frac{1}{2q}}. \end{aligned} \end{aligned}$$

(4.14)

Furthermore, by the definition of T, taking a sufficiently small $\Vert \varphi \Vert _{\mathrm {exp}L^2}$ if necessary, we can apply Lemma 4.1, and it holds that, for any $r\in [p_2,\infty ]$,

$$\begin{aligned} \sup _{0<t\le T}\, (1+t)^{\frac{N}{2}(\frac{1}{2}-\frac{1}{r})+\frac{1}{2}}t^{\frac{1}{2r}}|f(v(t))|_{L^r} \le 2C_f(2c_2\delta )^{1+\frac{2}{N}}, \end{aligned}$$

(4.15)

where $C_f$ and $p_2$ are given in (1.9) and (2.20), respectively. Let D[v] be the function defined by (3.3). Then, we put

$$\begin{aligned} \begin{aligned} |D[v](T)|_{L^q}&\le \bigg (\int _0^{T/2}+\int _{T/2}^T\bigg )|S_2(T-s)f(v(s))|_{L^q}\,ds \\&=:I_1(T)+I_2(T). \end{aligned} \end{aligned}$$

(4.16)

For the term $I_1$, since $T\ge 1$ and $N(1/2-1/p_2)=-1$, by (2.12) with $(q,r)=(p_2,q)$ and (4.15) we obtain

$$\begin{aligned} \begin{aligned} I_1(T)&\le C\int _0^{T/2}(T-s)^{-\frac{N-1}{2}(\frac{1}{p_2}-\frac{1}{q})-\frac{1}{2}}|f(v(s))|_{L^{p_2}}\,ds \\&\le C \delta ^{1+\frac{2}{N}}T^{-\frac{N-1}{2}(\frac{1}{p_2}-\frac{1}{q})-\frac{1}{2}} \int _0^{T/2}(1+s)^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{p_2})-\frac{1}{2}}s^{-\frac{1}{2p_2}}\,ds \\&\le C \delta ^{1+\frac{2}{N}}T^{-\frac{N-1}{2}(\frac{1}{p_2}-\frac{1}{q})-\frac{1}{2}} \int _0^{T/2}s^{-\frac{1}{2p_2}}\,ds \\&\le C\delta ^{1+\frac{2}{N}}T^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{q})-\frac{1}{2q}} \\&\le D_1\delta ^{1+\frac{2}{N}}(1+T)^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{q})}T^{-\frac{1}{2q}}, \end{aligned} \end{aligned}$$

(4.17)

where $D_1$ is a positive constant independent of q and $\delta$. Furthermore, for the term $I_2$, since $T\ge 1$, by (2.12) with $q=r$ and (4.15) we have

$$\begin{aligned} \begin{aligned} I_2(T)&\le C\int _{T/2}^T(T-s)^{-\frac{1}{2}}|f(v(s))|_{L^q}\,ds \\&\le C\delta ^{1+\frac{2}{N}} \int _{T/2}^T(T-s)^{-\frac{1}{2}} (1+s)^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{q})-\frac{1}{2}}s^{-\frac{1}{2q}}\,ds \\&\le C\delta ^{1+\frac{2}{N}}(1+T)^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{q})}T^{-\frac{1}{2q}} \int _0^T(T-s)^{-\frac{1}{2}}s^{-\frac{1}{2}}\,ds \\&\le D_2\delta ^{1+\frac{2}{N}}(1+T)^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{q})}T^{-\frac{1}{2q}}, \end{aligned} \end{aligned}$$

(4.18)

where $D_2$ is a positive constant independent of q and $\delta$. Then, combining (4.17) and (4.18), we see that

$$\begin{aligned} |D[v](T)|_{L^q}\le (D_1+D_2)\delta ^{1+\frac{2}{N}}(1+T)^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{q})}T^{-\frac{1}{2q}}. \end{aligned}$$

(4.19)

Taking a sufficiently small $\Vert \varphi \Vert _{\mathrm {exp}L^2}$ if necessary, we have

$$\begin{aligned} (D_1+D_2)\delta ^{\frac{2}{N}}<c_2. \end{aligned}$$

This together with (4.2), (4.14), and (4.19) implies that

$$\begin{aligned} |v(T)|_{L^q} \le |S_1(T)u(1)|_{L^q}+|D[v(T)]|_{L^q} <2c_2\delta (1+T)^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{q})}T^{-\frac{1}{2q}}. \end{aligned}$$

This contradicts (4.13), and we see $T=\infty$. Therefore, for any $q\in [2,\infty ]$, it holds that

$$\begin{aligned} |v(t)|_{L^q}\le 2c_2\delta (1+t)^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{q})}t^{-\frac{1}{2q}},\qquad t>0. \end{aligned}$$

(4.20)

It remains to show that, for any $q\in [2,\infty ]$,

$$\begin{aligned} \Vert v(t)\Vert _{L^q}\le C\delta t^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{q})},\qquad t>0. \end{aligned}$$

(4.21)

By (2.8), (4.3), and (4.12) we see that

$$\begin{aligned} \Vert S_1(t)u(1)\Vert _{L^q} \le Ct^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{q})}\Vert u(1)\Vert _{L^2} \le C\delta t^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{q})},\qquad t>0. \end{aligned}$$

(4.22)

On the other hand, by (4.20), similarly to (4.15), it holds that, for any $r\in [p_2,\infty ]$,

$$\begin{aligned} |f(v(t))|_{L^r} \le C\delta (1+t)^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{r})-\frac{1}{2}}t^{-\frac{1}{2r}}, \qquad t>0. \end{aligned}$$

(4.23)

Similarly to (4.16), by (3.3) we put

$$\begin{aligned} \begin{aligned} \Vert D[v](t)\Vert _{L^q}&\le \int _0^{t/2}\Vert S_2(t-s)f(v(s))\Vert _{L^q}\,ds+\int _{t/2}^t\Vert S_2(t-s)f(v(s))\Vert _{L^q}\,ds \\&=:J_1(t)+J_2(t), \qquad t>0. \end{aligned} \end{aligned}$$

(4.24)

Then, for the term $J_1$, it holds from (2.11) with $(q,r)=(p_2,q)$ and (4.23) that

$$\begin{aligned} \begin{aligned} J_1(t)&\le C\int _0^{t/2}(t-s)^{-\frac{N}{2}(\frac{1}{p_2}-\frac{1}{q})-\frac{1}{2}(1-\frac{1}{p_2})} |f(v(s))|_{L^{p_2}}\,ds \\&\le C\delta t^{-\frac{N}{2}(\frac{1}{p_2}-\frac{1}{q})-\frac{1}{2}(1-\frac{1}{p_2})} \int _0^{t/2}(1+s)^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{p_2})-\frac{1}{2}}s^{-\frac{1}{2p_2}}\,ds \\&\le C\delta t^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{q})-1+\frac{1}{2p_2}} \int _0^ts^{-\frac{1}{2p_2}}\,ds \le C\delta t^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{q})}, \qquad t>0. \end{aligned}\nonumber \\ \end{aligned}$$

(4.25)

Furthermore, for the term $J_2$, by (2.11) with $q=r$ and (4.23) we have

$$\begin{aligned} \begin{aligned} J_2(t)&\le C\int _{t/2}^t(t-s)^{-\frac{1}{2}(1-\frac{1}{q})}|f(v(s))|_{L^q}\,ds \\&\le C\delta \int _{t/2}^t(t-s)^{-\frac{1}{2}(1-\frac{1}{q})} (1+s)^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{q})-\frac{1}{2}}s^{-\frac{1}{2q}}\,ds \\&\le C\delta (1+t)^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{q})}t^{-\frac{1}{2}-\frac{1}{2q}} \int _0^ts^{-\frac{1}{2}(1-\frac{1}{q})}\,ds \le C\delta (1+t)^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{q})}, \qquad t>0. \end{aligned} \nonumber \\ \end{aligned}$$

(4.26)

Then, combining (4.24), (4.25), and (4.26), we see that

$$\begin{aligned} \Vert D[v]\Vert _{L^q}\le C\delta t^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{q})},\qquad t>0. \end{aligned}$$

This together with (4.2) and (4.22) yields (4.21). Therefore, by (4.1), (4.20), and (4.21) we have (1.16), and the proof of Theorem 1.2 for the case $N\ge 2$ is complete. $\square$

We next consider the case $N=1$. Let v be the function defined by (4.1). Then, it follows from (1.13) and (2.1) that the function v satisfies

$$\begin{aligned} v(t)=S_1(t)u(1)+{\tilde{D}}[v](t),\qquad t>0, \end{aligned}$$

(4.27)

where ${\tilde{D}}[v]$ is the function defined by (3.23). Then, by (2.10) and (4.3) we have

$$\begin{aligned} |[S_1(t)u(1)](0)|\le c_2t^{-\frac{1}{4}}\Vert u(1)\Vert _{L^2} \le d_*t^{-\frac{1}{4}}\Vert \varphi \Vert _{\mathrm {exp}L^2},\qquad t>0. \end{aligned}$$

(4.28)

Here $d_*$ is a constant independent of $\Vert \varphi \Vert _{\mathrm {exp}L^2}$. Furthermore, similarly to (4.5), applying the same argument as in the proof of Lemma 3.3 with (1.15) and (4.1), we see that

$$\begin{aligned} |{\tilde{D}}[v](0,t)|\le Ct^{-\frac{1}{4}}\Vert \varphi \Vert _{\mathrm {exp}L^2}^3,\qquad t>0. \end{aligned}$$

(4.29)

Then, we can take a sufficiently small $\varepsilon >0$ such that, for $\Vert \varphi \Vert _{\mathrm {exp}L^2}<\varepsilon$, it follows from (4.27), (4.28), and (4.29) that

$$\begin{aligned} |v(0,t)|\le 2d_*t^{-\frac{1}{4}}\Vert \varphi \Vert _{\mathrm {exp}L^2},\qquad t>0. \end{aligned}$$

(4.30)

On the other hand, we have the following, which is the one dimensional counterpart of Lemma 4.1.

Lemma 4.2

Let $N=1$, $T>0$, and $A>0$. Suppose that the function $v\in C(0,\infty )$ satisfying

$$\begin{aligned} \sup _{0<t\le T} (1+t)^{\frac{1}{4}}|v(t)|\le A. \end{aligned}$$

(4.31)

Let f be a function satisfying (1.9). Then, there exists $\varepsilon _*>0$, independent of T, such that, if $A<\varepsilon _*$, then

$$\begin{aligned} \sup _{0<t\le T} (1+t)^{\frac{3}{4}}|f(v(t))|\le 2C_fA^3, \end{aligned}$$

(4.32)

where $C_f$ is constant given in (1.9).

Proof

Let $k\in {{\mathbb {N}}}\cup \{0\}$ and $\ell _k$ be the constant given in (2.22) with $N=1$, namely, $\ell _k=2k+3$. Then, by (1.9) and (4.31) we have

$$\begin{aligned} \begin{aligned} |f(v(t))| \le C_f\sum _{k=0}^{\infty } \frac{\lambda ^k}{k!}|v(t)|^{\ell _k}&\le C_f\sum _{k=0}^{\infty } \frac{\lambda ^k}{k!} \left( (1+t)^{-\frac{1}{4}}A\right) ^{\ell _k} \\&\le C_fA^3(1+t)^{-\frac{3}{4}} \sum _{k=0}^{\infty } \frac{\lambda ^k}{k!} \left( (1+t)^{-\frac{1}{4}}A\right) ^{2k} \\&\le C_fA^3(1+t)^{-\frac{3}{4}} \sum _{k=0}^{\infty } \frac{\lambda ^k}{k!} A^{2k}, \qquad t>0. \end{aligned} \end{aligned}$$

(4.33)

This together with (4.10) implies (4.32). Thus Lemma 4.2 follows. $\square$

Proof of Theorem 1.2

($N=1$). Let v be the function defined by (4.27). Then, since $\Vert \varphi \Vert _{\mathrm {exp}L^2}$ is sufficiently small, by (4.30), Lemma 4.2, and applying the same argument as in the proof of Theorem 1.2 for the case $N\ge 2$, we can prove that

$$\begin{aligned} |v(0,t)|\le C(1+t)^{-\frac{1}{4}}\Vert \varphi \Vert _{\mathrm {exp}L^2},\qquad t>0, \end{aligned}$$

(4.34)

and

$$\begin{aligned} |f(v(0,t))|\le C(1+t)^{-\frac{3}{4}}\Vert \varphi \Vert _{\mathrm {exp}L^2},\qquad t>0. \end{aligned}$$

(4.35)

Let $q\in [2,\infty ]$. Then, by (2.7) with $(N,r)=(1,q)$, (3.23), and (4.35) we have

$$\begin{aligned} \begin{aligned} \Vert {\tilde{D}}[v](t)\Vert _{L^q}&\le 2\int _0^t\Vert g_1(t-s)\Vert _{L^q}|f(v(0,s))|\,ds \\&\le C\Vert \varphi \Vert _{\mathrm {exp}L^2}\int _0^t(t-s)^{-\frac{1}{2}(1-\frac{1}{q})}s^{-\frac{3}{4}}\,ds \\&\le Ct^{-\frac{1}{2}(\frac{1}{2}-\frac{1}{q})} B\bigg (\frac{1}{2}+\frac{1}{2q},\frac{1}{4}\bigg )\Vert \varphi \Vert _{\mathrm {exp}L^2} \\&\le Ct^{-\frac{1}{2}(\frac{1}{2}-\frac{1}{q})}\Vert \varphi \Vert _{\mathrm {exp}L^2} ,\qquad t>0. \end{aligned} \end{aligned}$$

This together with (4.22) and (4.27) implies

$$\begin{aligned} \Vert v(t)\Vert _{L^q}\le Ct^{-\frac{1}{2}(\frac{1}{2}-\frac{1}{q})}\Vert \varphi \Vert _{\mathrm {exp}L^2} ,\qquad t>0. \end{aligned}$$

(4.36)

Therefore, by (4.1), (4.34), and (4.36) we have (1.16), and the proof of Theorem 1.2 for the case $N=1$ is complete. $\square$

5 Rapidly decaying initial data

In this section we prove Theorems 1.3 and 1.4. Let

$$\begin{aligned} L :=\Vert \varphi \Vert _{\mathrm {exp}L^2} \end{aligned}$$

(5.1)

We can assume, without loss of generality, that $L<1$. Let $p_1$ be the constant given in (1.18). For $\Vert \varphi \Vert _{L^{p_1}}>0$, we denote

$$\begin{aligned} K:=2\,\max \{1,c_1,c_2\} \Vert \varphi \Vert _{\mathrm {exp}L^2\cap L^{p_1}} \end{aligned}$$

(5.2)

and

$$\begin{aligned} {\tilde{K}}:=2\,\max \{1,c_1,c_2\} \max \{1,\Vert \varphi \Vert _{L^{p_1}}\}, \end{aligned}$$

(5.3)

where $c_1$ and $c_2$ are given in $(G_1)$. Since we assume $L<1$ and thanks to (1.17) we have

$$\begin{aligned} L\le K\le {\tilde{K}}. \end{aligned}$$

(5.4)

Then we first show the following lemma, which is analogous to Lemma 4.1.

Lemma 5.1

Let $N\ge 2$, $T>0$, and $p\in [1,2)$. Furthermore let $p_1$ be the constant given in (1.18). Suppose that, for any $q\in [p_1,\infty ]$, the function $u\in C({\overline{{{\mathbb {R}}}^N_+}}\times (0,\infty ))$ satisfies

$$\begin{aligned} \sup _{0<t\le T}\,t^{\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})+\frac{1}{2q}}|u(t)|_{L^q} \le D K, \end{aligned}$$

(5.5)

where D is independent of q and K is the constant given in (5.2). Let f be a function satisfying (1.9). Then, for ${\tilde{K}}$ as in (5.3), there exists a sufficiently large constant $T_1=T_1({\tilde{K}},p_1,\lambda ,D)\ge 1$ such that if $T\ge T_1$ it follows that, for any $r\in [p_3,\infty ]$,

$$\begin{aligned} \sup _{ T_1\le t\le T}\,t^{\frac{N}{2}(\frac{1}{p_1}-\frac{1}{r})+\frac{1}{p_1}+\frac{1}{2r}}|f(u(t))|_{L^r} \le 2C_f(DK)^{1+\frac{2}{N}}, \end{aligned}$$

(5.6)

where $C_f$ is given in (1.9) and

$$\begin{aligned} p_3:=\max \left\{ 1,\frac{p_1N}{N+2}\right\} . \end{aligned}$$

(5.7)

Proof

Let $k\in {{\mathbb {N}}}\cup \{0\}$ and $\ell _k$ be the constant given in (2.22). Since

$$\begin{aligned} \ell _k r\ge \bigg (1+\frac{2}{N}\bigg )p_3\ge p_1, \end{aligned}$$

for any $r\in [p_3,\infty ]$, by (1.9) and (5.5) we have

$$\begin{aligned} \begin{aligned} |f(u(t))|_{L^r}&\le C_f\sum _{k=0}^{\infty } \frac{\lambda ^k}{k!}|u(t)|_{L^{\ell _kr}}^{\ell _k} \\&\le C_f\sum _{k=0}^{\infty } \frac{\lambda ^k}{k!} \left( DKt^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{\ell _kr})-\frac{1}{2\ell _kr}}\right) ^{\ell _k} \\&\le C_f(DK)^{1+\frac{2}{N}}t^{\frac{N}{2r}-\frac{N}{2p_1}(1+\frac{2}{N})-\frac{1}{2r}} \sum _{k=0}^{\infty } \frac{\lambda ^k}{k!} \left( DKt^{-\frac{N}{2p_1}}\right) ^{2k} \\&\le C_f(DK)^{1+\frac{2}{N}}t^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{r})-\frac{1}{p_1}-\frac{1}{2r}} \mathrm {exp}\bigg (\lambda (DK)^2t^{-\frac{N}{p_1}}\bigg ),\qquad t>0. \end{aligned} \end{aligned}$$

(5.8)

for all $t>0$. We can take a sufficiently large constant $T_1\ge 1$ such that, for all $t>T_1$, it holds that

$$\begin{aligned} \mathrm {exp}\bigg (\lambda (DK)^2t^{-\frac{N}{p_1}}\bigg )\le 2. \end{aligned}$$

It is enough to choose

$$\begin{aligned} T_1\ge \bigg (\frac{\lambda (D{\tilde{K}})^2}{\log 2}\bigg )^{\frac{p_1}{N}}\ge \bigg (\frac{\lambda (D K)^2}{\log 2}\bigg )^{\frac{p_1}{N}}. \end{aligned}$$

(5.9)

This together with (5.8) implies (5.6). Thus Lemma 5.1 follows. $\square$

Similarly, for the case $N=1$, we have the following.

Lemma 5.2

Let $N=1$, $T>0$, and $p\in [1,2)$. Suppose that the function $u\in C(0,\infty )$ satisfies

$$\begin{aligned} \sup _{0<t\le T}\,t^{\frac{1}{2p}}|u(t)| \le DK, \end{aligned}$$

(5.10)

where $D>0$ and K is the constant given in (5.2). Let f be a function satisfying (1.9). Then, for ${\tilde{K}}$ as in (5.3), there exists a sufficiently large constant ${\tilde{T}}_1={\tilde{T}}_1({\tilde{K}},p,\lambda ,D)$ such that, if $T\ge {\tilde{T}}_1$, then it follows that

$$\begin{aligned} \sup _{ {\tilde{T}}_1\le t\le T}\,t^{\frac{3}{2p}}|f(u(t))| \le 2C_f(DK)^3, \end{aligned}$$

(5.11)

where $C_f$ is given in (1.9).

Proof

Let $k\in {{\mathbb {N}}}\cup \{0\}$ and $\ell _k$ be the constant given in (2.22) with $N=1$, namely, $\ell _k=2k+3$. Furthermore, let ${\tilde{T}}_1$ be a sufficiently large constant satisfying (5.9) with $(N,p_1)=(1,p)$. Then, by (1.9) and (5.10) we have

$$\begin{aligned} \begin{aligned} |f(u(t))| \le C_f\sum _{k=0}^{\infty } \frac{\lambda ^k}{k!}|u(t)|^{\ell _k}&\le C_f\sum _{k=0}^{\infty } \frac{\lambda ^k}{k!} \left( DKt^{-\frac{1}{2p}}\right) ^{\ell _k} \\&\le C_f(DK)^3t^{-\frac{3}{2p}} \sum _{k=0}^{\infty } \frac{\lambda ^k}{k!} \left( DKt^{-\frac{1}{2p}}\right) ^{2k} \\&\le C_f(DK)^3t^{-\frac{3}{2p}} \mathrm {exp}\bigg (\lambda (DK)^2t^{-\frac{1}{p}}\bigg ) \\&\le 2C_f(DK)^3t^{-\frac{3}{2p}},\qquad t\ge {\tilde{T}}_1. \end{aligned} \end{aligned}$$

This implies (5.11). Thus Lemma 5.2 follows. $\square$

Next we prove (1.20) for small times.

Lemma 5.3

Let $N\ge 1$ and u be the unique solution to problem (1.1) satisfying (1.14) and (1.15). Suppose $\varphi \in L^p$ for $p\in [1,2)$. Let $p_1$ and K be the constants given in (1.18) and (5.2), respectively. Then, for any fixed $T_*\ge 1$, there exists a constant $\varepsilon = \varepsilon (p_1, T_*) >0$ such that, if $L <\varepsilon$ (where L is the constant given in (5.1)) then, for any $q\in [p_1,\infty ]$,

$$\begin{aligned}&\sup _{0<t\le 2 T_*}\, t^{\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})} \bigg (\Vert u(t)\Vert _{L^q}+t^{\frac{1}{2q}}|u(t)|_{L^q}\bigg ) \le C_*K,\qquad \text{ if }\quad N\ge 2, \end{aligned}$$

(5.12)

$$\begin{aligned}&\sup _{0<t\le {2 T_*}}\, t^{\frac{1}{2}(\frac{1}{p}-\frac{1}{q})} \bigg ( \Vert u(t)\Vert _{L^q}+t^{\frac{1}{2q}}|u(0,t)|\bigg ) \le C_*K,\qquad \text{ if }\quad N=1, \end{aligned}$$

(5.13)

where $C_*$ is independent of q, K, and $T_*$.

Proof

We first prove (5.12). Let $N\ge 2$. By (1.12) we consider

$$\begin{aligned} u(t)= S_1(t)\varphi +D[u](t), \end{aligned}$$

(5.14)

where D[u] is the function defined by (3.3). For the linear part, by (2.8), (2.9), and (5.2), for any $q\in [p_1,\infty ]$, we have

$$\begin{aligned} \Vert S_1(t)\varphi \Vert _{L^q} +t^{\frac{1}{2q}}|S_1(t)\varphi |_{L^q}\le & {} (c_1+c_2)t^{-\frac{N}{2}(\frac{1}{p_1} -\frac{1}{q})}\Vert \varphi \Vert _{L^{p_1}}\nonumber \\\le & {} Kt^{-\frac{N}{2}(\frac{1}{p_1} -\frac{1}{q})},\quad t>0. \end{aligned}$$

(5.15)

For the nonlinear part D[u], let $p_2$ be the constant given in (2.20). Then, by (2.11) with $(q,r)=(2N,\infty )$ and (3.3) we see that

$$\begin{aligned} \Vert D[u](t)\Vert _{L^\infty } \le C\int _0^t(t-s)^{-\frac{3}{4}+\frac{1}{4N}}|f(u(s))|_{L^{2N}}\,ds, \qquad t>0. \end{aligned}$$

(5.16)

On the other hand, for $r\in [p_2,\infty )$, by (1.15) and taking a sufficiently small $\varepsilon =\varepsilon (r)>0$, for $L<\varepsilon$, we can apply Lemma 2.2, and it holds that

$$\begin{aligned} |f(u(t))|_{L^r}\le CrL^{1+\frac{2}{N}}t^{-\frac{1}{2r}},\qquad t>0, \end{aligned}$$

(5.17)

where $C>0$ is independent of r, N, and L. Since $2N\ge p_2$, by (5.16) and (5.17) we obtain

$$\begin{aligned} \Vert D[u](t)\Vert _{L^\infty } \le CL^{1+\frac{2}{N}}\int _0^t(t-s)^{-\frac{3}{4}+\frac{1}{4N}}s^{-\frac{1}{4N}}\,ds \le CL^{1+\frac{2}{N}}T_*^{\frac{1}{4}}, \qquad t\le 2T_*. \end{aligned}$$

(5.18)

Furthermore, since it follows from $p\in [1,2)$ with (1.18) and (2.20) that $N(1/p_2-1/p_1)<1$, by (2.11) with $(q,r)=(p_2,p_1)$, (3.3), and (5.17) we have

$$\begin{aligned} \begin{aligned} \Vert D[u](t)\Vert _{L^{p_1}}&\le C\int _0^t(t-s)^{-\frac{N}{2}(\frac{1}{p_2}-\frac{1}{p_1})-\frac{1}{2}(1-\frac{1}{p_2})} |f(u(s))|_{L^{p_2}}\,ds \\&\le CL^{1+\frac{2}{N}} \int _0^t(t-s)^{-\frac{N}{2}(\frac{1}{p_2}-\frac{1}{p_1})-\frac{1}{2}(1-\frac{1}{p_2})}s^{-\frac{1}{2p_2}}\,ds \\&\le CL^{1+\frac{2}{N}}t^{\frac{1}{2}-\frac{N}{2}(\frac{1}{p_2}-\frac{1}{p_1})} \\&\le CL^{1+\frac{2}{N}}T_*^{\frac{1}{2}-\frac{N}{2}(\frac{1}{p_2}-\frac{1}{p_1})}, \qquad t\le 2T_*. \end{aligned} \end{aligned}$$

(5.19)

Similarly, by (2.12) with $(q,r)=(p_2,p_1)$ we obtain

$$\begin{aligned} \begin{aligned} |D[u](t)|_{L^{p_1}}&\le \int _0^t(t-s)^{-\frac{N-1}{2}(\frac{1}{p_2}-\frac{1}{p_1})-\frac{1}{2}}|f(u(s))|_{L^{p_2}}\,ds \\&\le CL^{1+\frac{2}{N}} \int _0^t(t-s)^{-\frac{N-1}{2}(\frac{1}{p_2}-\frac{1}{p_1})-\frac{1}{2}}s^{-\frac{1}{2p_2}}\,ds \\&\le CL^{1+\frac{2}{N}}t^{\frac{1}{2}-\frac{N-1}{2}(\frac{1}{p_2}-\frac{1}{p_1})-\frac{1}{2p_1}} \\&\le CL^{1+\frac{2}{N}}t^{-\frac{1}{2p_1}}T_*^{\frac{1}{2}-\frac{N-1}{2}(\frac{1}{p_2}-\frac{1}{p_1})}, \qquad t\le 2T_*. \end{aligned}\nonumber \\ \end{aligned}$$

(5.20)

If we choose L small enough such that

$$\begin{aligned} \max \bigg (T_*^{\frac{1}{4}}, T_*^{\frac{1}{2}-\frac{N-1}{2}(\frac{1}{p_2}-\frac{1}{p_1})}\bigg ) L^{\frac{2}{N}}<T_*^{-\frac{N}{2p_1}}, \end{aligned}$$

then, by (5.4), (5.18), (5.19), and (5.20), for any $q\in [p_1,\infty ]$, we get

$$\begin{aligned} \Vert D[u](t)\Vert _{L^q}+t^{\frac{1}{2q}}|D[u](t)|_{L^q} \le CLT_*^{-\frac{N}{2p_1}} \le CKT_*^{-\frac{N}{2p_1}},\qquad t\le 2T_*. \end{aligned}$$

(5.21)

Since $T_*\ge 1$, by (5.15) and (5.21), for any $q\in [p_1,\infty ]$, we obtain

$$\begin{aligned} \begin{aligned} \Vert u(t)\Vert _{L^q}+t^{\frac{1}{2q}}|u(t)|_{L^q}&\le C_*K \bigg ( t^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})} + T_*^{-\frac{N}{2p_1}} \bigg ) \\&\le C_*K \bigg ( t^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})} + T_*^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})} \bigg ) \\&\le C_*K t^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})} \quad t\le 2T_*. \end{aligned} \end{aligned}$$

where $C_*$ is independent of q, K, and $T_*$. This implies (5.12).

Next we prove (5.13). Let $N=1$. Then, we recall that $p_1=p$. By (1.13) we consider

$$\begin{aligned} u(t)= S_1(t)\varphi +{\tilde{D}}[u](t), \end{aligned}$$

(5.22)

where ${\tilde{D}}[u]$ is the function defined by (3.23). For the linear part, by (2.8), (2.10), and (5.2), for any $q\in [p,\infty ]$, we have

$$\begin{aligned} \Vert S_1(t)\varphi \Vert _{L^q} \le c_1t^{-\frac{1}{2}(\frac{1}{p} -\frac{1}{q})}\Vert \varphi \Vert _{L^p} \le Kt^{-\frac{1}{2}(\frac{1}{p} -\frac{1}{q})},\quad t>0 \end{aligned}$$

(5.23)

and

$$\begin{aligned} |[S_1(t)\varphi ](0)| \le c_2t^{-\frac{1}{2p}}\Vert \varphi \Vert _{L^p} \le Kt^{-\frac{1}{2p}},\quad t>0. \end{aligned}$$

(5.24)

On the other hand, by (1.15) and taking a sufficiently small $\varepsilon >0$, for $L<\varepsilon$, we can apply Lemma 2.3, and we have

$$\begin{aligned} |f(u(0,t))|\le CL^3t^{-\frac{1}{2}},\qquad t>0. \end{aligned}$$

(5.25)

Then, for the nonlinear part ${\tilde{D}}[u]$, it holds from (2.7), (3.23), and (5.25) that, for any $q\in [p,\infty ]$,

$$\begin{aligned} \begin{aligned} \Vert {\tilde{D}}[u](t)\Vert _{L^q}&\le C\int _0^t(t-s)^{-\frac{1}{2}(1-\frac{1}{q})}|f(u(0,s))|\,ds \\&\le CL^3\int _0^t(t-s)^{-\frac{1}{2}(1-\frac{1}{q})}s^{-\frac{1}{2}}\,ds \\&\le CL^3t^{\frac{1}{2q}}B\bigg (\frac{1}{2}+\frac{1}{2q},\frac{1}{2}\bigg ) \le CL^3T_*^{\frac{1}{2q}} \qquad t\le 2T_*. \end{aligned} \end{aligned}$$

(5.26)

Similarly, we have

$$\begin{aligned} \begin{aligned} |{\tilde{D}}[u](0,t)|&\le C\int _0^t(t-s)^{-\frac{1}{2}}|f(u(0,s))|\,ds \\&\le CL^3\int _0^t(t-s)^{-\frac{1}{2}}s^{-\frac{1}{2}}\,ds \le CL^3, \qquad t\le 2T_*. \end{aligned} \end{aligned}$$

(5.27)

If we choose $L<T_*^{-1/(4p)}$, then, by (5.4) and (5.26), for any $q\in [p,\infty ]$, we get

$$\begin{aligned} \Vert {\tilde{D}}[u](t)\Vert _{L^q} \le CLT_*^{-\frac{1}{2}(\frac{1}{p}-\frac{1}{q})} \le CKT_*^{-\frac{1}{2}(\frac{1}{p}-\frac{1}{q})},\qquad t\le 2T_*. \end{aligned}$$

(5.28)

Furthermore, by (5.4) and (5.27), it holds that

$$\begin{aligned} |{\tilde{D}}[u](0,t)|\le CLT_*^{-\frac{1}{2p}}\le CKT_*^{-\frac{1}{2p}},\qquad t\le 2T_*. \end{aligned}$$

(5.29)

Combining (5.23) and (5.28), we have

$$\begin{aligned} \Vert u(t)\Vert _{L^q}\le CK\bigg (t^{-\frac{1}{2}(\frac{1}{p}-\frac{1}{q})} +T_*^{-\frac{1}{2}(\frac{1}{p}-\frac{1}{q})}\bigg ) \le CKt^{-\frac{1}{2}(\frac{1}{p}-\frac{1}{q})}, \qquad t\le 2T_*. \end{aligned}$$

Similarly, by (5.24) and (5.29), we obtain

$$\begin{aligned} |u(0,t)| \le CK\bigg (t^{-\frac{1}{2p}}+T_*^{-\frac{1}{2p}}\bigg ) \le CKt^{-\frac{1}{2p}}, \qquad t\le 2T_*. \end{aligned}$$

These imply (5.13), thus Lemma 5.3 follows. $\square$

For the case $N\ge 2$, applying Lemmata 5.1 and 5.3, we show the decay estimate of $|u(t)|_{L^q}$.

Lemma 5.4

Assume the same conditions as in Lemma 5.3 for the case $N\ge 2$. Then, for ${\tilde{K}}$ as in (5.3), there exists a positive function $F=F(N,p_1,{\tilde{K}}, \lambda )$ such that, if $L<F$ and L is small enough, then, for any $q\in [p_1,\infty ]$,

$$\begin{aligned} \sup _{t>0}\,t^{\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})+\frac{1}{2q}}|u(t)|_{L^q} \le CK, \end{aligned}$$

(5.30)

where C depends only on N.

Proof

Let u be a unique solution to problem (1.1) satisfying (1.14) and (1.15). Then, similarly to (4.11), applying arguments similar to that in the proof of [20, Lemma 2.1] with (1.15), we see that

$$\begin{aligned} u\in C((0,\infty ); L^q(\partial {{\mathbb {R}}}^N_+)),\qquad q\in [p_1,\infty ]. \end{aligned}$$

(5.31)

By Lemma 5.3, for any $T_*\ge 1$, there exists $\varepsilon =\varepsilon (p_1,T_*)$ such that, if $L<\varepsilon$, then

$$\begin{aligned} |u(t)|_{L^q}\le C_*Kt^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\frac{1}{2q}},\qquad 0<t\le 2T_*, \end{aligned}$$

(5.32)

where $C_*\ge 1$ is independent of q,  K and $T_*$. Let us fix $T_*$ large enough to be chosen later, put

$$\begin{aligned} T=\sup \bigg \{0<s<\infty \,;\,|u(t)|_{L^q}\le 2C_*Kt^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\frac{1}{2q}} \quad \hbox {for all}\, q\in [p_1,\infty ] \, \hbox {and}\, 0<t<s\bigg \}. \end{aligned}$$

Then, since $T_*\ge 1$, by (5.32) we have $T\ge 2T_*\ge 2$.

We prove $T=\infty$. The proof is by contradiction. We assume that $T<\infty$. Then, by (5.31) we see that

$$\begin{aligned} |u(T)|_{L^q}=2C_*KT^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\frac{1}{2q}}. \end{aligned}$$

(5.33)

On the other hand, by (2.9) with $(q,r)=(p_1,q)$ and (5.2) we have

$$\begin{aligned} |S_1(T)\varphi |_{L^q} \le c_2T^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\frac{1}{2q}}\Vert \varphi \Vert _{L^{p_1}} \le C_*KT^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\frac{1}{2q}}. \end{aligned}$$

(5.34)

Let $T_1$ be the constant given in Lemma 5.1 with $D=2C_*$, and let us assume that

$$\begin{aligned} T_*\ge T_1\ge 1. \end{aligned}$$

(5.35)

Furthermore, let $I_1$ and $I_2$ be functions given in (4.16), and let $p_2$ be the constant given in (2.20). Then, for the term $I_1$, since $T\ge 2T_*$, by (2.12) with $(q,r)=(p_2,q)$ we get

$$\begin{aligned} \begin{aligned} I_1(T)&\le C\bigg (\int _0^{T_*}+\int _{T_*}^{T/2}\bigg ) (T-s)^{-\frac{N-1}{2}(\frac{1}{p_2}-\frac{1}{q})-\frac{1}{2}}|f(u(s))|_{L^{p_2}}\,ds \\&=: A(T) + B(T). \end{aligned}\nonumber \\ \end{aligned}$$

(5.36)

Since $p_1\ge p_2\ge 1$ and $T\ge 1$, due to (1.15) and taking a sufficiently small L if necessary, we can apply Lemma 2.2 to the term A(T), and we obtain

$$\begin{aligned} \begin{aligned} A(T)&\le CT^{-\frac{N-1}{2}(\frac{1}{p_2}-\frac{1}{q})-\frac{1}{2}} \int _0^{T_*}L^{1+\frac{2}{N}}s^{-\frac{1}{2p_2}}\,ds \\&\le CL^{1+\frac{2}{N}}T^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\frac{1}{2q}} T^{-\frac{1}{2}(1-\frac{1}{p_2})-\frac{N}{2}(\frac{1}{p_2}-\frac{1}{p_1})} T_*^{1-\frac{1}{2p_2}} \\&\le CL^{1+\frac{2}{N}}T^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\frac{1}{2q}} T_*^{1-\frac{1}{2p_2}}. \end{aligned}\nonumber \\ \end{aligned}$$

(5.37)

Furthermore, let $p_3$ be the constant given in (5.7). Then, since $T_*\ge T_1$ and it follows from $p_1<2$ that $p_2\ge p_3$ we can apply Lemma 5.1 to the term B(T), and we have

$$\begin{aligned} \begin{aligned} B(T)&\le CT^{-\frac{N-1}{2}(\frac{1}{p_2}-\frac{1}{q})-\frac{1}{2}} \int _{T_*}^{T/2}K^{1+\frac{2}{N}} s^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{p_2})-\frac{1}{p_1}-\frac{1}{2p_2}}\,ds \\&\le CK^{1+\frac{2}{N}}T^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\frac{1}{2q}} \int _{T_*}^{T/2}T^{-\frac{1}{2}(1-\frac{1}{p_2})-\frac{N}{2}(\frac{1}{p_2}-\frac{1}{p_1})} s^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{p_2})-\frac{1}{p_1}-\frac{1}{2p_2}}\,ds \\&\le CK^{1+\frac{2}{N}}T^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\frac{1}{2q}} \int _{T_*}^{T/2}s^{-\frac{1}{2}-\frac{1}{p_1}}\,ds \\&\le CK^{1+\frac{2}{N}}T^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\frac{1}{2q}} \int _{T_*}^\infty s^{-\frac{1}{2}-\frac{1}{p_1}}\,ds \\&\le CK^{1+\frac{2}{N}}T^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\frac{1}{2q}} T_*^{-\frac{1}{p_1}+\frac{1}{2}}, \end{aligned} \end{aligned}$$

(5.38)

where C is independent of q, L, K, and $T_*$. Moreover, for the term $I_2$, since $q\ge p_1\ge p_3$ and $p_1<2$, by (2.12) with $q=r$ and (5.6) we see that

$$\begin{aligned} \begin{aligned} I_2(T)&\le \int _{T/2}^T(T-s)^{-\frac{1}{2}}|f(u(s))|_{L^q}\,ds \\&\le CK^{1+\frac{2}{N}} \int _{T/2}^T(T-s)^{-\frac{1}{2}} s^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\frac{1}{p_1}-\frac{1}{2q}}\,ds \\&\le CK^{1+\frac{2}{N}}T^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\frac{1}{p_1}-\frac{1}{2q}} \int _{T/2}^T(T-s)^{-\frac{1}{2}}\,ds \\&\le CK^{1+\frac{2}{N}}T^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\frac{1}{2q}} T_*^{-\frac{1}{p_1}+\frac{1}{2}}, \end{aligned} \end{aligned}$$

where C is independent of q, L, K, and $T_*$. This together with (4.16), (5.36), (5.37), and (5.38) implies that

$$\begin{aligned} |D[u](T)|_{L^q}\le & {} I_1(T)+I_2(T)\nonumber \\\le & {} D_*T^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\frac{1}{2q}} \bigg (L^{1+\frac{2}{N}}T_*^{1-\frac{1}{2p_2}} +K^{1+\frac{2}{N}}T_*^{-\frac{1}{p_1}+\frac{1}{2}} \bigg ), \end{aligned}$$

(5.39)

where $D_*$ is a constant independent of L, K, and $T_*$. Since $p_1<2$, we can take a sufficiently large constant $T_*\ge 1$ so that

$$\begin{aligned} D_*T_*^{-\frac{1}{p_1}+\frac{1}{2}}K^{\frac{2}{N}}\le D_*T_*^{-\frac{1}{p_1}+\frac{1}{2}}{{\tilde{K}}}^{\frac{2}{N}} \le \frac{C_*}{4} \end{aligned}$$

(5.40)

which means

$$\begin{aligned} T_*\ge \bigg (\frac{4D_*{{\tilde{K}}}^{\frac{2}{N}}}{C_*}\bigg )^{\frac{1}{\frac{1}{p_1}-\frac{1}{2}}}. \end{aligned}$$

(5.41)

This together with (5.35) implies that $T_*$ depends on $\lambda$, ${\tilde{K}}$, and $p_1$ but not on L. Then we can also take a sufficiently small constant L so that

$$\begin{aligned} D_*T_*^{1-\frac{1}{2p_2}}L^{\frac{2}{N}}\le \frac{C_*}{4} \end{aligned}$$

(5.42)

and this means

$$\begin{aligned} L\le \bigg (\frac{4D_*T_*^{1-\frac{1}{2p_2}}}{C_*}\bigg )^{-\frac{N}{2}}. \end{aligned}$$

(5.43)

By (5.4), (5.39), (5.40), and (5.42) we have

$$\begin{aligned} |D[u](T)|_{L^q} \le \frac{1}{2} C_*KT^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\frac{1}{2q}}. \end{aligned}$$

(5.44)

Combining (5.14), (5.34), and (5.44), we see that

$$\begin{aligned} \begin{aligned} |u(T)|_{L^q}&\le |S_1(T)\varphi |_{L^q}+|D[u](t)|_{L^q}\\&\le {\bigg (C_*+\frac{C_*}{2}\bigg )KT^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\frac{1}{2q}}} <2C_*KT^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\frac{1}{2q}}. \end{aligned} \end{aligned}$$

This contradicts (5.33), and we see $T=\infty$. In order to make clear the dependence of the choice we made on $T_*$ and L, we collect below all the conditions (5.35), (5.41), and (5.43)

$$\begin{aligned} T_*\ge T_1, \quad T_*\ge \bigg (\frac{4D_*{{{\tilde{K}}}}^{\frac{2}{N}}}{C_*}\bigg )^{\frac{1}{\frac{1}{p_1}-\frac{1}{2}}}, \quad L\le \bigg (\frac{4D_*T_*^{1-\frac{1}{2p_2}}}{C_*}\bigg )^{-\frac{N}{2}}, \end{aligned}$$

where $T_1$ satisfies (5.9) with $D=2C_*$, namely

$$\begin{aligned} T_1\ge \bigg (\frac{\lambda (2C_*{ {{\tilde{K}}}})^2}{\log 2}\bigg )^{\frac{p_1}{N}}. \end{aligned}$$

Here $C_*$ and $D_*$ are constants depending at most on N and $p_1$. Then we can find a function F depending on N, $p_1$, ${\tilde{K}}$, and $\lambda$ such that the conditions on L can be written as $L< F(N, p_1, { {{\tilde{K}}}}, \lambda )$ and L small enough. Thus Lemma 5.4 follows. $\square$

Similarly, for the case $N=1$, applying Lemmata 5.2 and 5.3, we have the following.

Lemma 5.5

Assume the same conditions as in Lemma 5.3 for the case $N=1$. Then, for ${\tilde{K}}$ as in (5.3), there exists a positive function $F=F(p,{\tilde{K}}, \lambda )$ such that, if $L<F$ and L is small enough, then,

$$\begin{aligned} \sup _{t>0}\,t^{\frac{1}{2p}}|u(0,t)| \le CK, \end{aligned}$$

where C is independent of p and K.

Proof

Applying the same argument as in the proof of Lemma 5.4, we can prove this lemma. For reader’s convenience, we give it here.

Let u be a unique solution to problem (1.1) satisfying (1.14) and (1.15). Then, similarly to (5.31), we can easily show that

$$\begin{aligned} u(0,t)\in C((0,\infty )). \end{aligned}$$

(5.45)

By Lemma 5.3, for any $T_*\ge 1$, there exists $\varepsilon =\varepsilon (p,T_*)$ such that, if $L<\varepsilon$, then

$$\begin{aligned} |u(0,t)|\le C_*Kt^{-\frac{1}{2p}},\qquad 0<t\le 2T_*, \end{aligned}$$

(5.46)

where $C_*\ge 1$ is independent of K and $T_*$. Let us fix $T_*$ large enough to be chosen later, put

$$\begin{aligned} T=\sup \bigg \{0<s<\infty \,;\, |u(0,t)|\le 2C_*Kt^{-\frac{1}{2p}}\quad \hbox { for all}\ 0<t<s\bigg \}. \end{aligned}$$

Then, since $T_*\ge 1$, by (5.46) we have $T\ge 2T_*\ge 2$.

We prove $T=\infty$. The proof is by contradiction. We assume that $T<\infty$. Then, by (5.45) we see that

$$\begin{aligned} |u(0,T)|=2C_*KT^{-\frac{1}{2p}}. \end{aligned}$$

(5.47)

On the other hand, by (2.10) with $q=p$ and (5.2) we have

$$\begin{aligned} |[S_1(T)\varphi ](0)| \le c_2T^{-\frac{1}{2p}}\Vert \varphi \Vert _{L^p} \le C_*KT^{-\frac{1}{2p}}. \end{aligned}$$

(5.48)

Let ${\tilde{T}}_1$ be the constant given in Lemma 5.2 with $D=2C_*$, and let us assume

$$\begin{aligned} T_*\ge {\tilde{T}}_1\ge 1. \end{aligned}$$

(5.49)

Furthermore, since $T\ge 2T_*$, by (3.23) we put

$$\begin{aligned} \begin{aligned} |{\tilde{D}}[u](0,T)|&\le C\bigg (\int _0^{T_*}+\int _{T_*}^{T/2}+\int _{T/2}^T\bigg )(T-s)^{-\frac{1}{2}}|f(u(0,s))|\,ds \\&=:{\tilde{I}}_1(T)+{\tilde{I}}_2(T)+{\tilde{I}}_3(T). \end{aligned} \end{aligned}$$

(5.50)

Since $p\ge 1$ and $T\ge 2T_*$, due to (1.15) and taking a sufficiently small L if necessary, we can apply Lemma 2.3 to the term ${\tilde{I}}_1(T)$, and we obtain

$$\begin{aligned} \begin{aligned} {\tilde{I}}_1(T)&\le CT^{-\frac{1}{2}} \int _0^{T_*}L^3s^{-\frac{1}{2}}\,ds \\&\le CL^3T^{-\frac{1}{2p}} T^{-\frac{1}{2}(1-\frac{1}{p})} T_*^{\frac{1}{2}} \le CL^3T^{-\frac{1}{2p}} T_*^{\frac{1}{2p}}. \end{aligned} \end{aligned}$$

(5.51)

Furthermore, since $T_*\ge {\tilde{T}}_1$ and $p<2$, for the terms ${\tilde{I}}_2(T)$ and ${\tilde{I}}_3(T)$, we can apply Lemma 5.2, and we have

$$\begin{aligned} \begin{aligned} {\tilde{I}}_2(T)&\le CT^{-\frac{1}{2}} \int _{T_*}^{T/2}K^3s^{-\frac{3}{2p}}\,ds \\&\le CK^3T^{-\frac{1}{2p}} \int _{T_*}^{T/2}T^{-\frac{1}{2}(1-\frac{1}{p})} s^{-\frac{3}{2p}}\,ds \\&\le CK^3T^{-\frac{1}{2p}} \int _{T_*}^{T/2}s^{-\frac{1}{2}-\frac{1}{p}}\,ds \le CK^3T^{-\frac{1}{2p}} T_*^{-\frac{1}{p}+\frac{1}{2}}, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} {\tilde{I}}_3(T)&\le CK^3 \int _{T/2}^T(T-s)^{-\frac{1}{2}} s^{-\frac{3}{2p}}\,ds \\&\le CK^3T^{-\frac{3}{2p}} \int _{T/2}^T(T-s)^{-\frac{1}{2}}\,ds \le CK^3T^{-\frac{1}{2p}} T_*^{-\frac{1}{p}+\frac{1}{2}}, \end{aligned} \end{aligned}$$

where C is a constant independent of p, L, K, and $T_*$. These together with (5.50) and (5.51) imply that

$$\begin{aligned} |{\tilde{D}}[u](0,T)|\le D_*T^{-\frac{1}{2p}} \bigg (L^3T_*^{\frac{1}{2p}} +K^3T_*^{-\frac{1}{p}+\frac{1}{2}} \bigg ), \end{aligned}$$

(5.52)

where $D_*$ is a constant independent of L, K, and $T_*$. Since $p<2$, we can take a sufficiently large constant $T_*\ge 1$ so that

$$\begin{aligned} D_*T_*^{-\frac{1}{p}+\frac{1}{2}}K^2\le D_*T_*^{-\frac{1}{p}+\frac{1}{2}}{{\tilde{K}}}^2 \le \frac{C_*}{4} \end{aligned}$$

(5.53)

which means

$$\begin{aligned} T_*\ge \bigg (\frac{4D_*{{\tilde{K}}}^2}{C_*}\bigg )^{\frac{1}{\frac{1}{p}-\frac{1}{2}}} \ge \bigg (\frac{4D_*K^2}{C_*}\bigg )^{\frac{1}{\frac{1}{p}-\frac{1}{2}}}. \end{aligned}$$

(5.54)

This together with (5.49) implies that $T_*$ depends on $\lambda$, ${\tilde{K}}$, and p but not on L. Then we can also take a sufficiently small constant L so that

$$\begin{aligned} D_*T_*^{\frac{1}{2p}}L^2\le \frac{C_*}{4} \end{aligned}$$

(5.55)

and this means

$$\begin{aligned} L\le \bigg (\frac{4D_*T_*^{\frac{1}{2p}}}{C_*}\bigg )^{-\frac{1}{2}}. \end{aligned}$$

(5.56)

By (5.4), (5.52), (5.53), and (5.55) we have

$$\begin{aligned} |{\tilde{D}}[u](0,T)|\le \frac{1}{2}C_*KT^{-\frac{1}{2p}}. \end{aligned}$$

This together with (5.22) and (5.48) implies

$$\begin{aligned} |u(0,T)| \le |[S_1(T)\varphi ](0)|+|{\tilde{D}}[u](0,T)|\le \left( C_*+\frac{C_*}{2}\right) KT^{-\frac{1}{2p}} <2C_*KT^{-\frac{1}{2p}}. \end{aligned}$$

This contradicts (5.47), and we see $T=\infty$. In order to make clear the dependence of the choice we made on $T_*$ and L, we collect below all the conditions (5.49), (5.54), and (5.56)

$$\begin{aligned} T_*\ge {\tilde{T}}_1, \quad T_*\ge \bigg (\frac{4D_*{\tilde{K}}^2}{C_*}\bigg )^{\frac{1}{\frac{1}{p}-\frac{1}{2}}}, \quad L\le \bigg (\frac{4D_*T_*^{\frac{1}{2p}}}{C_*}\bigg )^{-\frac{1}{2}}, \end{aligned}$$

where ${\tilde{T}}_1$ satisfies (5.9) with $(N,p_1)=(1,p)$ and $D=2C_*$, namely

$$\begin{aligned} {\tilde{T}}_1\ge \bigg (\frac{\lambda (2C_*{\tilde{K}})^2}{\log 2}\bigg )^p. \end{aligned}$$

Here $C_*$ and $D_*$ are constants depending at most on p. Then we can find a function F depending on p, ${\tilde{K}}$, and $\lambda$ such that the condition L can be written as $L< F(p, {\tilde{K}}, \lambda )$ and L small enough. Thus Lemma 5.5 follows. $\square$

Now we ready to prove Theorem 1.3. We first prove it for the case $N\ge 2$.

Proof of Theorem 1.3

($N\ge 2$). Let u be a unique solution to problem (1.1) satisfying (1.14) and (1.15). Let T be a sufficiently large constant to be chosen later, which satisfies $T\ge T_1$, where $T_1$ is the constant given in Lemma 5.1 with $D=C_*$. Suppose that L is small enough such that Lemmata 5.3 and 5.4 hold. Then, by (5.12) and (5.30), in order to prove (1.20), it suffices to prove the decay estimate of $\Vert u(t)\Vert _{L^q}$ for $t\ge 2T$.

Let $p_1$ be the constant given in (1.18) and $q\in [p_1,\infty ]$. For the linear part, by (2.8) with $(q,r)=(p_1,q)$ and (5.2) we have

$$\begin{aligned} \Vert S_1(t)\varphi \Vert _{L^q} \le c_1t^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})}\Vert \varphi \Vert _{L^{p_1}} \le Kt^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})},\qquad t>0. \end{aligned}$$

(5.57)

For the nonlinear part, let $J_1$ and $J_2$ be functions given in (4.24), and let $p_2$ be the constant given in (2.20). Then, for the term $J_1$, similarly to (5.36), by (2.11) with $(q,r)=(p_2,q)$ we put

$$\begin{aligned} \begin{aligned} J_1(t)&\le C\bigg (\int _0^T+\int _T^{t/2}\bigg ) (t-s)^{-\frac{N}{2}(\frac{1}{p_2}-\frac{1}{q})-\frac{1}{2}(1-\frac{1}{p_2})}|f(u(s))|_{L^{p_2}}\,ds \\&=: {\tilde{A}}(t)+{\tilde{B}}(t), \qquad t\ge 2T. \end{aligned} \end{aligned}$$

(5.58)

For the term ${\tilde{A}}(t)$, since $p_1\ge p_2\ge 1$, by (1.15) and taking a sufficiently small L if necessary, we can apply Lemma 2.2, and we have

$$\begin{aligned} \begin{aligned} {\tilde{A}}(t)&\le CL^{1+\frac{2}{N}}t^{-\frac{N}{2}(\frac{1}{p_2}-\frac{1}{q})-\frac{1}{2}(1-\frac{1}{p_2})} \int _0^Ts^{-\frac{1}{2p_2}}\,ds \\&\le CL^{1+\frac{2}{N}}t^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})} t^{-\frac{N}{2}(\frac{1}{p_2}-\frac{1}{p_1})-\frac{1}{2}(1-\frac{1}{p_2})}T^{1-\frac{1}{2p_2}}, \qquad t\ge 2T. \end{aligned} \end{aligned}$$

(5.59)

Furthermore, let $p_3$ be the constant given in (5.7). Then, for the term ${\tilde{B}}(t)$, since $T\ge T_1$ , and $p_2\ge p_3$, we can apply Lemma 5.1, and it follows from $p_2\ge 1$ that

$$\begin{aligned} \begin{aligned} {\tilde{B}}(t)&\le CK^{1+\frac{2}{N}}t^{-\frac{N}{2}(\frac{1}{p_2}-\frac{1}{q})-\frac{1}{2}(1-\frac{1}{p_2})} \int _T^{t/2}s^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{p_2})-\frac{1}{p_1}-\frac{1}{2p_2}}\,ds \\&\le CK^{1+\frac{2}{N}}t^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})} t^{-\frac{N}{2}(\frac{1}{p_2}-\frac{1}{p_1})-\frac{1}{2}(1-\frac{1}{p_2})} \int _T^{t/2}s^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{p_2})-\frac{1}{p_1}-\frac{1}{2p_2}}\,ds, \quad t\ge 2T. \end{aligned} \end{aligned}$$

(5.60)

For $p\in (p_2,2)$ (which implies $p_1=p$), since $p_1<2$, we can choose $\sigma _1\in (0,1)$ satisfying

$$\begin{aligned} 0<\sigma _1 < \min \left\{ \frac{1}{p_1}-\frac{1}{2}, \frac{N}{2}\left( \frac{1}{p_2}-\frac{1}{p_1}\right) \right\} . \end{aligned}$$

Then, by (5.59) and (5.60) we have

$$\begin{aligned} {\tilde{A}}(t) \le CL^{1+\frac{2}{N}} t^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\frac{N}{2}(\frac{1}{p_2}-\frac{1}{p_1})}T^{1-\frac{1}{2p_2}} \le CL^{1+\frac{2}{N}} t^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\sigma _1}T^{1-\frac{1}{2p_2}}, \qquad t\ge 2T, \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} {\tilde{B}}(t)&\le CK^{1+\frac{2}{N}}t^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\sigma _1} \int _T^{t/2}t^{-\frac{N}{2}(\frac{1}{p_2}-\frac{1}{p_1})-\frac{1}{2}(1-\frac{1}{p_2})+\sigma _1} s^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{p_2})-\frac{1}{p_1}-\frac{1}{2p_2}}\,ds \\&\le CK^{1+\frac{2}{N}}t^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\sigma _1} \int _{T}^{t/2} s^{-\frac{1}{p_1}-\frac{1}{2}+\sigma _1 }\,ds \\&\le CK^{1+\frac{2}{N}}t^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\sigma _1} \int _{T}^\infty s^{-\frac{1}{p_1}-\frac{1}{2}+\sigma _1 }\,ds \\&\le CK^{1+\frac{2}{N}}t^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\sigma _1}T^{\frac{1}{2}-\frac{1}{p_1}+\sigma _1}, \qquad t\ge 2T. \end{aligned} \end{aligned}$$

This together with (5.4) and (5.58) implies that

$$\begin{aligned} J_1(t) \le CL^{1+\frac{2}{N}}t^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\sigma _1}T^{1-\frac{1}{2p_2}} + CK^{1+\frac{2}{N}}t^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\sigma _1}T^{\frac{1}{2}-\frac{1}{p_1}+\sigma _1}, \qquad t\ge 2T. \end{aligned}$$

Choosing T large enough such that

$$\begin{aligned} K^{\frac{2}{N}}T^{\frac{1}{2}-\frac{1}{p_1}+\sigma _1}\le {{\tilde{K}}}^{\frac{2}{N}}T^{\frac{1}{2}-\frac{1}{p_1}+\sigma _1}\le 1 \end{aligned}$$

namely

$$\begin{aligned} T\ge ({{\tilde{K}}}^{\frac{2}{N}})^{\frac{1}{\frac{1}{p_1}-\frac{1}{2}-\sigma _1}} \end{aligned}$$

and L small enough such that

$$\begin{aligned} L^{\frac{2}{N}}T^{1-\frac{1}{2p_2}}\le 1, \end{aligned}$$

thanks to (5.4) we get

$$\begin{aligned} J_1(t)\le & {} CL t^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\sigma _1}+ CKt^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\sigma _1}\nonumber \\\le & {} CKt^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\sigma _1}, \qquad t\ge 2T. \end{aligned}$$

(5.61)

On the other hand, for $p\le p_2$, namely $p_1=p_2$, we consider two cases, $N=2$ and $N\ge 3$. For the case $N\ge 3$, since $p_1\in (1,2)$, we can choose $\sigma _2\in (0,1)$ satisfying

$$\begin{aligned} 0<\sigma _2<\min \bigg \{\frac{1}{p_1}-\frac{1}{2}, \frac{1}{2}\bigg (1-\frac{1}{p_1}\bigg )\bigg \}. \end{aligned}$$

Then, by (5.59) and (5.60) we see that

$$\begin{aligned} {\tilde{A}}(t) \le CL^{1+\frac{2}{N}} t^{{-\frac{N}{2}}(\frac{1}{p_1}-\frac{1}{q})-\sigma _2}T^{1-\frac{1}{2p_2}}, \qquad t\ge 2T, \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} {\tilde{B}}(t)&\le CK^{1+\frac{2}{N}} t^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\sigma _2} \int _T^{t/2}t^{-\frac{1}{2}(1-\frac{1}{p_1})+\sigma _2} s^{-\frac{3}{2p_1}}\,ds \\&\le CK^{1+\frac{2}{N}} t^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\sigma _2} \int _{T}^{t/2} s^{-\frac{1}{p_1}-\frac{1}{2}+\sigma _2}\,ds \\&\le CK^{1+\frac{2}{N}} t^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\sigma _2} \int _{T}^\infty s^{-\frac{1}{p_1}-\frac{1}{2}+\sigma _2}\,ds \\&\le CK^{1+\frac{2}{N}} t^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\sigma _2}T^{\frac{1}{2}-\frac{1}{p_1}+\sigma _2}, \qquad t\ge 2T. \end{aligned} \end{aligned}$$

This together with (5.4) and (5.58) implies that

$$\begin{aligned} J_1(t) \le CL^{1+\frac{2}{N}} t^{{-\frac{N}{2}}(\frac{1}{p_1}-\frac{1}{q})-\sigma _2}T^{1-\frac{1}{2p_2}} +CK^{1+\frac{2}{N}}t^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\sigma _2}T^{\frac{1}{2}-\frac{1}{p_1}+\sigma _2} , \qquad t\ge 2T. \end{aligned}$$

Choosing T large enough such that

$$\begin{aligned} K^{\frac{2}{N}}T^{\frac{1}{2}-\frac{1}{p_1}+\sigma _2} \le {{\tilde{K}}}^{\frac{2}{N}}T^{\frac{1}{2}-\frac{1}{p_1}+\sigma _2}\le 1 \end{aligned}$$

and L small enough such that

$$\begin{aligned} L^{\frac{2}{N}}T^{1-\frac{1}{2p_2}}\le 1, \end{aligned}$$

thanks to (5.4) we get

$$\begin{aligned} J_1(t)\le & {} CL t^{{-\frac{N}{2}}(\frac{1}{p_1}-\frac{1}{q})-\sigma _2} +CKt^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})- \sigma _2}\nonumber \\\le & {} CKt^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\sigma _2}, \qquad t\ge 2T. \end{aligned}$$

(5.62)

For the case $N=2$, since $p_1=p_2=1$ (which implies $p=1$), by (5.59) and (5.60) again we see that

$$\begin{aligned} {\tilde{A}}(t)\le C L^{2} t^{-(\frac{1}{p_1}-\frac{1}{q})}T^{\frac{1}{2}}, \qquad t\ge 2T, \end{aligned}$$

and

$$\begin{aligned} {\tilde{B}}(t) \le CK^2t^{-(\frac{1}{p_1}-\frac{1}{q})} \int _T^{t/2}s^{-\frac{3}{2}}\,ds \le CK^2t^{-(\frac{1}{p_1}-\frac{1}{q})}T^{-\frac{1}{2}}, \qquad t\ge 2T. \end{aligned}$$

This together with (5.4) and (5.58) implies that

$$\begin{aligned} J_1(t)\le CL^2t^{-(\frac{1}{p_1}-\frac{1}{q})}T^{\frac{1}{2}} + CK^2t^{-(\frac{1}{p_1}-\frac{1}{q})}T^{-\frac{1}{2}}, \qquad t\ge 2T. \end{aligned}$$

Choosing T large enough such that

$$\begin{aligned} KT^{-\frac{1}{2}} \le {{\tilde{K}}}T^{-\frac{1}{2}}\le 1 \end{aligned}$$

and L small enough such that

$$\begin{aligned} LT^{\frac{1}{2}}\le 1, \end{aligned}$$

thanks to (5.4) we get

$$\begin{aligned} J_1(t) \le CL t^{-(\frac{1}{p_1}-\frac{1}{q})}+C Kt^{-(\frac{1}{p_1}-\frac{1}{q})} \le C Kt^{-(\frac{1}{p_1}-\frac{1}{q})}, \qquad t\ge 2T. \end{aligned}$$

(5.63)

Therefore, by (5.61), (5.62), and (5.63), for $N\ge 2$, we have

$$\begin{aligned} J_1(t)\le CKt^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})}, \qquad t\ge 2T. \end{aligned}$$

(5.64)

Let us come back to the $J_2(t)$ term. Since $T\ge T_1$ and $q\ge p_1\ge p_3$, we can apply Lemma 5.1, and by (2.11) with $q=r$ and (5.6) we have

$$\begin{aligned} \begin{aligned} J_2(t)&\le CK^{1+\frac{2}{N}}\int _{t/2}^{t}(t-s)^{-\frac{1}{2}(1-\frac{1}{q})} s^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\frac{1}{p_1}-\frac{1}{2q}}\, ds \\&\le CK^{1+\frac{2}{N}} t^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\frac{1}{p_1}-\frac{1}{2q}} \int _{t/2}^{t}(t-s)^{-\frac{1}{2}(1-\frac{1}{q})}\, ds \\&\le CK^{1+\frac{2}{N}}t^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\frac{1}{p_1}+\frac{1}{2}}, \qquad t\ge 2T. \end{aligned} \end{aligned}$$

Since $p_1<2$, we can choose $\sigma _3 >0$ satisfying $0<\sigma _3 <1/p_1-1/2$, and we get

$$\begin{aligned} J_2(t) \le CK^{1+\frac{2}{N}} t^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\sigma _3} \le CK^{1+\frac{2}{N}}t^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\frac{\sigma _3}{2}} T ^{-\frac{\sigma _3}{2}}, \qquad t\ge 2T. \end{aligned}$$

Choosing T large enough such that

$$\begin{aligned} K^{\frac{2}{N}} T^{-\frac{\sigma _3}{2}} \le {{\tilde{K}}}^{\frac{2}{N}}T^{-\frac{\sigma _3}{2}}\le 1, \end{aligned}$$

we have

$$\begin{aligned} J_2(t)\le C Kt^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})-\frac{\sigma _3}{2}}, \qquad t\ge 2T. \end{aligned}$$

(5.65)

Combining (5.57), (5.64), and (5.65), we obtain

$$\begin{aligned} \Vert u(t)\Vert _{L^q}\le CKt^{-\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})}, \qquad t\ge 2T, \end{aligned}$$

thus (1.20) follows.

Finally we prove (1.21) by the same arguments as in the proof of [10, Theorem 2.2]. Indeed, let $p_1\in (1,2)$. By (5.61), (5.62), and (5.65) we have

$$\begin{aligned} t^{\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})}\Vert u(t)-S_1(t)\varphi \Vert _{L^q}=o(1),\qquad t\rightarrow \infty . \end{aligned}$$

Now, by density, let $\{\varphi _n\}\subset C_0^\infty$ such that $\varphi _n\rightarrow \varphi$ in $L^{p_1}$. Then, by (2.8), it holds that

$$\begin{aligned} \begin{aligned} t^{\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})}\Vert S_1(t)\varphi \Vert _{L^q}&\le t^{\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})}\Vert S_1(t)(\varphi -\varphi _n)\Vert _{L^q} +t^{\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})} \Vert S_1(t)\varphi _n\Vert _{L^q} \\&\le C\Vert \varphi -\varphi _n\Vert _{L^{p_1}} + Ct^{\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q}) }t^{-\frac{N}{2}(1-\frac{1}{q}) }\Vert \varphi _n\Vert _{L^1} \\&\le C\Vert \varphi -\varphi _n\Vert _{L^{p_1}} +Ct^{-\frac{N}{2}(1-\frac{1}{p_1})}\Vert \varphi _n\Vert _{L^1},\qquad t>0. \end{aligned} \end{aligned}$$

Since $p_1>1$, this proves that

$$\begin{aligned} t^{\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})}\Vert S_1(t)\varphi \Vert _{L^q}= o(1), \quad t\rightarrow \infty , \end{aligned}$$

and so

$$\begin{aligned} t^{\frac{N}{2}(\frac{1}{p_1}-\frac{1}{q})}\Vert u(t)\Vert _{L^q} = o(1), \quad t\rightarrow \infty . \end{aligned}$$

Thus the proof of Theorem 1.3 for the case $N\ge 2$ is complete. $\square$

Next, applying the same argument as in the prof of Theorem 1.3 for the case $N\ge 2$, we prove Theorem 1.3 for the case $N=1$.

Proof of Theorem 1.3

($N=1$). Let u be a unique solution to problem (1.1) satisfying (1.14) and (1.15). Let T be a sufficiently large constant to be chosen later, which satisfies $T\ge {\tilde{T}}_1$, where ${\tilde{T}}_1$ is the constant given in Lemma 5.2 with $D=C_*$. Suppose that L is sufficiently small so that Lemmata 5.3 and 5.5 hold. Then, it is enough to prove the decay estimate of $\Vert {\tilde{D}}(t)\Vert _{L^q}$ for $t\ge 2T$ in order to obtain (1.20).

Let $q\in [p,\infty ]$. Then, similarly to (5.50), by (2.7) and (3.23) we put

$$\begin{aligned} \begin{aligned} \Vert {\tilde{D}}[u](t)\Vert _{L^q}&\le C\bigg (\int _0^T+\int _T^{t/2}+\int _{t/2}^t\bigg )(t-s)^{-\frac{1}{2}(1-\frac{1}{q})}|f(u(0,s))|\,ds \\&=:{\tilde{J}}_1(t)+{\tilde{J}}_2(t)+{\tilde{J}}_3(t),\qquad t\ge 2T. \end{aligned} \end{aligned}$$

(5.66)

For the term ${\tilde{J}}_1$, by (1.15) and taking a sufficiently small L if necessary, we can apply Lemma 2.3, and we have

$$\begin{aligned} \begin{aligned} {\tilde{J}}_1(t)&\le CL^3t^{-\frac{1}{2}(1-\frac{1}{q})} \int _0^Ts^{-\frac{1}{2}}\,ds \\&\le CL^3 t^{-\frac{1}{2}(\frac{1}{p}-\frac{1}{q})} t^{-\frac{1}{2}(1-\frac{1}{p})}T^{\frac{1}{2}}, \qquad t\ge 2T. \end{aligned} \end{aligned}$$

(5.67)

Furthermore, for the term ${\tilde{J}}_2(t)$, since $T\ge {\tilde{T}}_1$, we can apply Lemma 5.2, and it holds that

$$\begin{aligned} \begin{aligned} {\tilde{J}}_2(t)&\le C K^3t^{-\frac{1}{2}(1-\frac{1}{q})} \int _T^{t/2}s^{-\frac{3}{2p}}\,ds, \\&\le C K^3t^{-\frac{1}{2}(\frac{1}{p}-\frac{1}{q})} t^{-\frac{1}{2}(1-\frac{1}{p})} \int _T^{t/2}s^{-\frac{3}{2p}}\,ds, \qquad t\ge 2T. \end{aligned} \end{aligned}$$

For $p\in (1,2)$, we can choose ${\tilde{\sigma }}_1\in (0,1)$ satisfying

$$\begin{aligned} 0<{\tilde{\sigma }}_1< \min \left( \frac{1}{p}-\frac{1}{2}, \frac{1}{2} \left( 1-\frac{1}{p}\right) \right) . \end{aligned}$$

Then, for $t\ge 2T$ we have

$$\begin{aligned} \begin{aligned}&CK^3t^{-\frac{1}{2}(\frac{1}{p}-\frac{1}{q})} t^{-\frac{1}{2}(1-\frac{1}{p})} \int _T^{t/2}s^{-\frac{3}{2p}}\,ds\\&= CK^3t^{-\frac{1}{2}(\frac{1}{p}-\frac{1}{q})-{\tilde{\sigma }}_1} \int _T^{t/2} t^{-\frac{1}{2}(1-\frac{1}{p})+{\tilde{\sigma }}_1} s^{-\frac{3}{2p}}\,ds\\&\le CK^3t^{-\frac{1}{2}(\frac{1}{p}-\frac{1}{q})-{\tilde{\sigma }}_1} \int _T^{t/2} s^{-\frac{3}{2p}- \frac{1}{2}(1-\frac{1}{p})+{\tilde{\sigma }}_1} \,ds\\&\le CK^3t^{-\frac{1}{2}(\frac{1}{p}-\frac{1}{q})-{\tilde{\sigma }}_1} T^{\frac{1}{2} -\frac{1}{p}+{\tilde{\sigma }}_1}. \end{aligned} \end{aligned}$$

(5.68)

Then, by (5.67) and (5.68) we have

$$\begin{aligned} \begin{aligned} {\tilde{J}}_1(t)+{\tilde{J}}_2(t)&\le CL^3 t^{-\frac{1}{2}(\frac{1}{p}-\frac{1}{q})} t^{-\frac{1}{2}(1-\frac{1}{p})}T^{\frac{1}{2}} + CK^3t^{-\frac{1}{2}(\frac{1}{p}-\frac{1}{q})-{\tilde{\sigma }}_1} T^{\frac{1}{2} -\frac{1}{p}+{\tilde{\sigma }}_1}\\&\le C t^{-\frac{1}{2}(\frac{1}{p}-\frac{1}{q})-{\tilde{\sigma }}_1} \left( L^3T^{\frac{1}{2}}+K^3T^{\frac{1}{2} -\frac{1}{p}+{\tilde{\sigma }}_1}\right) . \end{aligned} \end{aligned}$$

Now, choosing T large enough such that

$$\begin{aligned} K^2 T^{\frac{1}{2} -\frac{1}{p}+{\tilde{\sigma }}_1} \le {\tilde{K}}^2 T^{\frac{1}{2} -\frac{1}{p}+{\tilde{\sigma }}_1} \le 1 \end{aligned}$$

and then L small enough so that

$$\begin{aligned} L^2 T^{\frac{1}{2}} \le 1, \end{aligned}$$

thanks to (5.4) we get

$$\begin{aligned} {\tilde{J}}_1(t)+{\tilde{J}}_2(t) \le C K t^{-\frac{1}{2}(\frac{1}{p}-\frac{1}{q})-{\tilde{\sigma }}_1}. \end{aligned}$$

(5.69)

On the other hand, for $p=1$, by (5.67) and (5.68) again we see that

$$\begin{aligned} {\tilde{J}}_1(t)+{\tilde{J}}_2(t)\le CK t^{-\frac{1}{2}(\frac{1}{p}-\frac{1}{q})}, \qquad t\ge 2T. \end{aligned}$$

This together with (5.69) implies for all $p\in [1,2)$

$$\begin{aligned} {\tilde{J}}_1(t)+{\tilde{J}}_2(t)\le CKt^{-\frac{1}{2}(\frac{1}{p}-\frac{1}{q})}, \qquad t\ge 2T. \end{aligned}$$

(5.70)

For the ${\tilde{J}}_3(t)$ term, since $T\ge {\tilde{T}}_1$, we can apply Lemma 5.2, and it holds that

$$\begin{aligned} \begin{aligned} {\tilde{J}}_3(t)&\le CK^3\int _{t/2}^{t}(t-s)^{-\frac{1}{2}(1-\frac{1}{q})} s^{-\frac{3}{2p}}\, ds \\&\le C K^3 t^{-\frac{3}{2p}} \int _{t/2}^{t}(t-s)^{-\frac{1}{2}(1-\frac{1}{q})}\, ds \le C K^3 t^{-\frac{1}{2}(\frac{1}{p}-\frac{1}{q})-\frac{1}{p}+\frac{1}{2}}, \qquad t\ge 2T. \end{aligned} \end{aligned}$$

Since $p<2$, we can choose ${\tilde{\sigma }}_2 >0$ satisfying $0<{\tilde{\sigma }}_2 <1/p-1/2$, and we get

$$\begin{aligned} {\tilde{J}}_3(t) \le CK^3 t^{-\frac{1}{2}(\frac{1}{p}-\frac{1}{q})-{\tilde{\sigma }}_2} \le CK^3 t^{-\frac{1}{2}(\frac{1}{p}-\frac{1}{q})-\frac{{\tilde{\sigma }}_2}{2}}T^{-\frac{{\tilde{\sigma }}_2}{2}}, \qquad t\ge 2T. \end{aligned}$$

Now, choosing T large enough such that

$$\begin{aligned} K^2 T^{-\frac{{\tilde{\sigma }}_2}{2}}\le {\tilde{K}}^2 T^{-\frac{{\tilde{\sigma }}_2}{2}}\le 1, \end{aligned}$$

we get

$$\begin{aligned} {\tilde{J}}_3(t) \le C K t^{-\frac{1}{2}(\frac{1}{p}-\frac{1}{q})-\frac{{\tilde{\sigma }}_2}{2}}. \end{aligned}$$

(5.71)

Combining (5.66), (5.70), and (5.71), we see that

$$\begin{aligned} \Vert {\tilde{D}}[u](t)\Vert _{L^q}\le CKt^{-\frac{1}{2}(\frac{1}{p}-\frac{1}{q})}, \qquad t\ge 2T, \end{aligned}$$

thus (1.20) follows. Furthermore, applying the same arguments as in the proof of Theorem 1.3 for the case $N\ge 2$ with (5.69) and (5.71), we obtain (1.21). Thus the proof of Theorem 1.3 for the case $N=1$ is complete. $\square$

Remark 5.1

Similarly to the case of the Cauchy problem for the semilinear heat equation with (1.7), the nonlinear boundary problem (1.1) with (1.9) has no scaling invariance and the $L^p$ and $\mathrm {exp}L^2$ norms have no relationship between each other. In order to have initial data which fulfill condition (1.19), let us choose a function $\varphi \in L^p({{\mathbb {R}}}^N_+) \cap L^\infty ({{\mathbb {R}}}^N_+)$ with $p\in [1,2)$. Then, by (2.15) we see that $\varphi \in \mathrm {exp}L^2$. Then, let us consider a dilation $\varphi _\lambda (x)= \lambda ^{N/p} \varphi (\lambda x)$ so that $\Vert \varphi _\lambda \Vert _{L^p}=\Vert \varphi \Vert _{L^p}$. Since $\Vert \varphi _\lambda \Vert _{L^2}= \lambda ^{N(1/p -1/2)}\Vert \varphi \Vert _{L^2}$ and $\Vert \varphi _\lambda \Vert _{L^\infty }= \lambda ^{N/p}\Vert \varphi \Vert _{L^\infty }$, it follows

$$\begin{aligned} \limsup _{\lambda \rightarrow 0} \Vert \varphi _\lambda \Vert _{\mathrm {exp}L^2} \le \lim _{\lambda \rightarrow 0}\left( \Vert \varphi _\lambda \Vert _{L^2} +\Vert \varphi _\lambda \Vert _{L^\infty }\right) =0. \end{aligned}$$

This implies that there is $\lambda >0$ so that $\varphi _\lambda$ fulfills condition (2.4), even though its $L^p$ norm might be large.

In the end of this section we prove Theorem 1.4. In the following Lemmata, we assume $\Vert u(t)\Vert _{L^q}$ bounded at the origin and decaying at infinity, and we can deduce that also $\Vert f(u(t))\Vert _{L^r}$ is bounded and decays at infinity for $r\ge p_3$, where $p_3$ is given in (5.7).

Lemma 5.6

Let $N\ge 2$, $p\in [1,2)$, and $K>0$. Suppose that $u\in C({\overline{{{\mathbb {R}}}^N_+}}\times (0,\infty ))$ and for any $q\in [p,\infty ]$,

$$\begin{aligned} \sup _{t>0}\,(1+t)^{\frac{N}{2}(\frac{1}{p}-\frac{1}{q})}t^{\frac{1}{2q}}|u(t)|_{L^q} \le CK, \end{aligned}$$

(5.72)

where C is independent of q and K. Let f be a function satisfying (1.9). Then, there is $\varepsilon >0$ depending only on $\lambda$ such that, if $K<\varepsilon$, then, for any $r\in [p_4,\infty ]$,

$$\begin{aligned} \sup _{t>0}\,(1+t)^{\frac{N}{2}(\frac{1}{p}-\frac{1}{r})+\frac{1}{p}}t^{\frac{1}{2r}}|f(u(t))|_{L^r} \le 2C_f (CK)^{1+\frac{2}{N}}, \end{aligned}$$

(5.73)

where $C_f$ is given in (1.9) and

$$\begin{aligned} p_4:=\max \bigg \{1,\frac{pN}{N+2}\bigg \}. \end{aligned}$$

(5.74)

Proof

Let $k\in {\mathbb {N}}\cup \{0\}$ and $\ell _k$ be the constant given in (2.22). Then, since it follows from (5.74) that

$$\begin{aligned} \ell _k r\ge \bigg (1+\frac{2}{N}\bigg )p_4\ge p, \end{aligned}$$

similarly to (4.9), for any $r\in [p_4,\infty ]$, it follows from (1.9) and (5.72) that

$$\begin{aligned} \begin{aligned} |f(u(t))|_{L^r}&\le C_f\sum _{k=0}^{\infty } \frac{\lambda ^k}{k!}|u(t)|_{L^{\ell _kr}}^{\ell _k} \\&\le C_f\sum _{k=0}^{\infty } \frac{\lambda ^k}{k!} \left( (1+t)^{-\frac{N}{2}(\frac{1}{p}-\frac{1}{\ell _kr})}t^{-\frac{1}{2\ell _kr}}(CK)\right) ^{\ell _k} \\&\le C_f(CK)^{1+\frac{2}{N}}(1+t)^{\frac{N}{2r}-\frac{N}{2p}(1+\frac{2}{N})}t^{-\frac{1}{2r}} \sum _{k=0}^{\infty } \frac{\lambda ^k}{k!} \left( (1+t)^{-\frac{N}{2p}}(CK)\right) ^{2k} \\&\le C_f(CK)^{1+\frac{2}{N}}(1+t)^{-\frac{N}{2}(\frac{1}{p}-\frac{1}{r})-\frac{1}{p}}t^{-\frac{1}{2r}} \sum _{k=0}^{\infty } \frac{\lambda ^k}{k!} (CK)^{2k}, \qquad t>0. \end{aligned}\nonumber \\ \end{aligned}$$

(5.75)

We can take a sufficiently small $\varepsilon =\varepsilon (\lambda )>0$ so that, for $K\le \varepsilon$, it holds that

$$\begin{aligned} \sum _{k=0}^\infty \frac{\lambda ^k}{k!}(C K)^{2k} =e^{\lambda (C K)^2}\le 2. \end{aligned}$$

(5.76)

This together with (5.75) implies (5.73). Thus Lemma 5.6 follows. $\square$

Lemma 5.7

Let $N=1$, $p\in [1,2)$, and $K>0$. Suppose $u\in C((0,\infty ))$ and

$$\begin{aligned} \sup _{t>0}\,(1+t)^{\frac{1}{2p}}|u(t)| \le CK, \end{aligned}$$

(5.77)

where C is independent of K. Let f be a function satisfying (1.9). Then, there is $\varepsilon >0$ such that, if $K<\varepsilon$, then,

$$\begin{aligned} \sup _{t>0}\,(1+t)^{\frac{3}{2p}}|f(u(t))| \le 2C_f(C K)^3, \end{aligned}$$

(5.78)

where $C_f$ is given in (1.9).

Proof

Let $k\in {\mathbb {N}}\cup \{0\}$ and $\ell _k$ be the constant given in (2.22) with $N=1$, namely, $\ell _k=2k+3$. Furthermore, let $\varepsilon$ be a sufficiently small constant given in Lemma 5.6. Then, similarly to (4.33), it follows from (1.9), (5.76), and (5.77) that

$$\begin{aligned} \begin{aligned} |f(u(t))|&\le C_f\sum _{k=0}^{\infty } \frac{\lambda ^k}{k!}|u(t)|^{\ell _k} \\&\le C_f\sum _{k=0}^{\infty } \frac{\lambda ^k}{k!} \left( (1+t)^{-\frac{1}{2p}}(C K)\right) ^{\ell _k} \\&\le C_f(C K)^3(1+t)^{-\frac{3}{2p}} \sum _{k=0}^{\infty } \frac{\lambda ^k}{k!}(C K)^{2k} \le 2C_f(C K)^3(1+t)^{-\frac{3}{2p}}, \qquad t>0. \end{aligned} \end{aligned}$$

This implies (5.78), thus Lemma 5.7 follows. $\square$

Proof of Theorem 1.4

Put $K=\Vert \varphi \Vert _{\mathrm {exp}L^2\cap L^p}$. Applying the same arguments as in the proofs of Theorems 1.2 and 1.3 with Lemmata 5.6 and  5.7, we can prove Theorem 1.4. $\square$

6 Asymptotic behavior

Let us come to the asymptotic behavior of the solution u as stated in Theorem 1.5.

Proof of Theorem 1.5

Let u be the global-in-time solution to problem (1.1) satisfying (1.22). Furthermore, let $\varepsilon >0$ be a sufficiently small constant chosen later. Then, by (1.22) and (2.16) we can take a sufficiently large $T=T(\varepsilon ,N)>0$ so that

$$\begin{aligned} \Vert u(T)\Vert _{\mathrm {exp}L^2} \le C(\Vert u(T)\Vert _{L^2}+\Vert u(T)\Vert _{L^\infty }) \le C(1+T)^{-\frac{N}{4}}<\varepsilon . \end{aligned}$$

Therefore, applying the semigroup property of the kernel G, namely (2.1), we can assume, without loss of generality, that $\Vert \varphi \Vert _{\mathrm {exp}L^2\cap L^1}<\varepsilon$.

We first consider the case $N\ge 2$. By (1.22), taking a sufficiently small $\varepsilon >0$ if necessary, and applying the same argument as in the proof of Lemmata 2.2 and 5.1 with $p_1=p_2=p_3=1$, we have

$$\begin{aligned} \sup _{t>0}\,t^{\frac{1}{2}}(1+t)|f(u(t))|_{L^1}<\infty . \end{aligned}$$

Therefore we can define a mass of u(t) denote by m(t), that is,

$$\begin{aligned} m(t):=\int _{{{\mathbb {R}}}^N_+}\varphi (x)\,dx+\int _0^t\int _{{\mathbb {R}}^{N-1}}f(u(x',0,s))\,dx'\,ds,\qquad t\ge 0. \end{aligned}$$

Furthermore, it holds that

$$\begin{aligned} \begin{aligned} \int _0^t \int _{{\mathbb {R}}^{N-1}} f(u(x',0,s))\,dx'\,ds&= \bigg (\int _0^1+\int _1^t\bigg )|f(u(s))|_{L^1}\,ds \\&\le C\int _0^1s^{-\frac{1}{2}}\,ds + C \int _1^\infty s^{-\frac{3}{2}}\,ds \le C,\qquad t\ge 1. \end{aligned} \end{aligned}$$

(6.1)

This implies that there exists the limit of m(t), which we denote by $m_*$, such that

$$\begin{aligned} m_*:=\lim _{t\rightarrow \infty }m(t) = \int _{{{\mathbb {R}}}^N_+} \varphi (x)\,dx +\int _0^{\infty }\int _{{\mathbb {R}}^{N-1}}f(u(x',0,s))\,dx'\,ds. \end{aligned}$$

Furthermore, similarly to (6.1), we obtain

$$\begin{aligned} m_*-m(t) \le C\int _t^\infty \int _{{\mathbb {R}}^{N-1}}f(u(x',0,s))\,dx'\,ds \le Ct^{-\frac{1}{2}},\qquad t\ge 1. \end{aligned}$$

Therefore, applying an argument similar to the proof of [20, Theorem 1.1] (see also [22]) with (1.22), we have (1.23) for the case $N\ge 2$.

Next we consider the case $N=1$. By (1.22) and taking a sufficiently small $\varepsilon >0$ if necessary, we can apply Lemmata 2.3 and  5.2, and we have

$$\begin{aligned} \sup _{t>0}\,t^{\frac{1}{2}}(1+t)|f(u(0,t))|<\infty . \end{aligned}$$

Therefore we can define a mass of u(t) denote by m(t), that is,

$$\begin{aligned} m(t):=\int _0^\infty \varphi (x)\,dx+\int _0^tf(u(0,s))\,ds,\qquad t\ge 0. \end{aligned}$$

Furthermore, it holds that

$$\begin{aligned} \int _0^t f(u(0,s))\,ds = \bigg (\int _0^1+\int _1^t\bigg )|f(u(0,s))|\,ds \le C\int _0^1s^{-\frac{1}{2}}\,ds + C \int _1^t s^{-\frac{3}{2}}\,ds \le C,\quad t\ge 1. \end{aligned}$$

This implies that there exists the limit of m(t), which we denote by $m_*$, such that

$$\begin{aligned} m_*:=\lim _{t\rightarrow \infty }m(t) = \int _0^\infty \varphi (x)\,dx +\int _0^\infty f(u(0,s))\,ds, \end{aligned}$$

and it holds that

$$\begin{aligned} m_*-m(t) \le C\int _t^\infty f(u(0,s))\,ds \le Ct^{-\frac{1}{2}},\qquad t\ge 1. \end{aligned}$$

Therefore, applying the same argument as in the proof of (1.23) for the case $N\ge 2$, we have (1.23) for the case $N=1$. Thus the proof of Theorem 1.5 is complete. $\square$