Abstract
This paper is devoted to the singular limit of a model for the regulation of growth and patterning in develo** tissues by diffusing morphogens. The model is governed by a system of nonlinear PDEs. The arguments are based on energy estimates and a change of variable that reduces the system into a nonlinear PDE with singular diffusion.
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1 Introduction
The differentiation and growth of embryonic cells are mainly regulated by morphogens (see [1, 8, 9, 12]). Experimental evidences show that morphogens develop from a localized source spreading in concentration gradients that control the behavior of surrounding cells as a function of their distance from the source, see Wartlick, Mumcu, Kicheva, Bitting, Seum, Jülicher, and González-Gaitán in [10, 11].
The experimental observations mentioned in [10, 11] have been implemented in the mathematical model proposed by Averbukh, Ben-Zvi, Mishra and Barkai in [2], in which a growth law based on a parameter \(\theta \) is formulated. It takes into account the fact that a cell divides when it detects that the relative morphogens concentration increases by a factor of \(1+ \theta \).
The model developed in [2] is the following one
where the unknowns are
and
Here \(M_\theta (t,x)\) is the morphogen concentration in the one-dimensional growing tissue \([0,L_\theta (t)]\), \(L_\theta (t)\) is the length of the tissue, \(u_\theta (t,x)\) is the (local) flow rate of the growing tissue with \(\partial _xu_\theta (t,x)\) being the cell proliferation rate, and \(\alpha ,\,D, \,\eta \) are positive parameters that correspond to the morphogen degradation rate, diffusion rate and incoming morphogen flux rate, respectively. The evolution of the morphogens concentration in (1.1) is described by the first equation, which is a nonlinear advection–reaction–diffusion PDE. The second equation gives the expression of the cell division rule due to morphogens proliferation and flow rates. Finally, the tissue length L(t) obeys an ODE flow type. The two PDEs are augmented with suitable initial data and non-homogeneous Neumann-type boundary conditions.
Here we are interested in the analysis of \((M_\theta (t,x), u_\theta (t,x), L_\theta (t))\) as
as a consequence in the following we will always assume
Under this condition we have established in [4] the well-posedness (existence, uniqueness, and stability) and in [6] the asymptotic behavior as \(t\rightarrow \infty \) of \((M_\theta (t,x),\) \(u_\theta (t,x),\) \(L_\theta (t))\). We will often recall some results of these papers, and therefore, we will assume that the hypotheses assumed therein are satisfied, also here. Before stating them explicitly, we point out that the hypothesis in [6]:
in this paper it is assumed by formulating two alternative conditions:
The results in the two cases are different while retaining a certain “symmetry”. Having said that, in this paper we assume that the following hypotheses are satisfied
and one within the following
The difference between the two cases is further highlighted by the different initial mean morphogens concentrations, indeed
Key tool for the analysis of the well-posedness (see [3, 4, 7]) and of the asymptotic behavior as \(t\rightarrow \infty \) (see [5, 6]) is the definition of a suitable family of “characteristic” curves which start at the points of \( [0, L_0] \) and “cover” \(\{(t, x) \,| \,t \ge 0,0 \le x \le L_\theta (t) \} \). Let us briefly recall them because they are also useful in this paper.
Let \((M_\theta (t,x), u_\theta (t,x), L_\theta (t))\) be the solution of (1.1), for every \(y\in [0,L_0],\) let \(X_\theta (t,y)\) be the solution of
Thanks to (1.1), it is clear that 0 solves (1.5) in correspondence of \(y=0\) and \(L_\theta (t)\) solves (1.5) in correspondence of \(y=L_0.\) The image of \(X_\theta (t,\cdot )\) is \([0, L_\theta (t)]\). \(X_\theta (t,\cdot )\) is invertible; its inverse \(Y_\theta (t,\cdot )\) is defined on \([0, L_\theta (t)]\) and its image is \([0,L_0]\) (see [3, 4, 7]).
The main results of this paper are the following.
Theorem 1.1
If \(\displaystyle \left\| M_0 \right\| _{L^1(0,L_0)}\le \frac{\eta }{\alpha }\) and the assumptions (1.2), (1.3) hold, we have that
- i):
-
\(\displaystyle \lim \limits _{\theta \rightarrow 0}\int _0^{L_0} |M_\theta (t,X_\theta (t,y))-M_0(y)|^r\textrm{d}y=0,\quad 1\le r<\infty ,\) uniformly with respect to t on every compact set [0, T];
- ii):
-
\(0\le u_\theta (t,x);\quad \displaystyle \limsup \limits _{\theta \rightarrow 0}u_\theta (t,x)\le e^{-\alpha t}\,\,\frac{\eta -\alpha \left\| M_0 \right\| _{L^1(0,L_0)}}{M_0(L_0)};\)
- iii):
-
\(\displaystyle L_0e^{-\alpha t}+\sqrt{\alpha D}\tanh \Big ({L_0\sqrt{\frac{\alpha }{D}}}\Big ) \frac{1-e^{-\alpha t}}{\alpha }\le \liminf \limits _{\theta \rightarrow 0} L_{\theta }(t)\) \(\qquad \quad \displaystyle \le \limsup \limits _{\theta \rightarrow 0} L_{\theta }(t)\le L_0e^{-\alpha t}+\frac{\eta }{M_0(L_0)}\frac{1-e^{-\alpha t}}{\alpha }.\)
Theorem 1.2
If \(\displaystyle \left\| M_0 \right\| _{L^1(0,L_0)}\ge \frac{\eta }{\alpha }\) and the assumptions (1.2), (1.4) hold, we have that
- i):
-
\(\displaystyle \lim \limits _{\theta \rightarrow 0}\int _0^{L(t)} |M_\theta (t,x)-M_0(Y_\theta (t,x)|^r\textrm{d}y=0,\quad 1\le r<\infty ,\) uniformly with respect to t on every compact set [0, T];
- ii):
-
\(\displaystyle -e^{-\alpha t}\,\, \frac{\alpha \left\| M_0 \right\| _{L^1(0,L_0)}-\eta }{M_0(L_0)}0\) \(\le \displaystyle \liminf \limits _{\theta \rightarrow 0}u_\theta (t,x);\quad u_\theta (t,x)\le 0;\)
- iii):
-
\(\displaystyle L_0e^{-\alpha t}+\frac{\eta }{M_0(0)} \frac{1-e^{-\alpha t}}{\alpha }\) \(\le \liminf \limits _{\theta \rightarrow 0} L_{\theta }(t)\le \limsup \limits _{\theta \rightarrow 0} L_{\theta }(t)\le \) \(\qquad \quad \le \displaystyle L_0e^{-\alpha t}+\frac{\eta }{M_0(0)}\frac{1-e^{-\alpha t}}{\alpha }+\sqrt{\alpha D}\,\Big (\log \frac{M_0(0)}{M_0(L_0)}\Big ) \frac{1-e^{-\alpha t}}{\alpha }.\)
The paper is organized as follows. In Sect. 2 we recall some preliminary results. Section 3 is devoted to some a priori estimates on the sign of the derivatives of the unknowns. Theorems 1.1 and 1.2 are proved in Sects. 4 and 5, respectively.
2 Preliminary results
We transform (1.1) into a problem equivalent to it, in the sense that the well-posedness of one of them implies the well-posedness of the other one and from the solution of one of them we obtain at the solution of the other one. Defining
(1.1) is equivalent to the following problem
where
For every \(0\le T<\infty \) we will use the following notation
We define in an analogous way \(E_\infty \) and \(\overline{E}_\infty .\)
Let us recall some properties of \(N_\beta (t, y) \) useful in the next sections.
Theorem 2.1
(Existence, uniqueness, and regularity of \(N_\beta (t,y)\) [4, Theorem 2.1], [6, Theorem 2.3]) If (1.2) holds, then, for every \(\beta \) and \(T>0\), (2.1) admits a unique solution \(N_\beta (t,y)\) such that
- i):
-
\(c_* e^{-\frac{\alpha }{\beta +1}t}\le N_\beta (t,y), \,\,\, t\ge 0,\,\, 0\le y\le L_0\,\,\,\) and \(N_\beta \in L^\infty (E_\infty );\)
- ii):
-
\(\displaystyle \partial _tN_\beta ,\,\,\,\partial _yN_\beta ,\,\,\,\partial _y\left( \frac{M_0^\beta }{N_\beta ^\beta }\partial _yN_\beta \right) \in L^2(E_T),\) \(\partial ^2_{yy}N\in L^1(E_T);\)
- iii):
-
here exists \(c(T)>0\) such that for every \((t_1,y_1), (t_2,y_2)\in E_T\) \(|N_\beta (t_1,y_1)- N_\beta (t_2,y_2)|\le c(T)\big (\sqrt{|t_1-t_2|}+|y_1-y_2|\big )^\frac{1}{4}\);
- iv):
-
\(\partial _yN_\beta \in C(]0,\infty [\times [0,L_0]);\,\,\,\partial _tN_\beta ,\,\partial ^2_{yy}N_\beta \in C(E_\infty ).\)
Let us show how to pass from \(N_\beta (t,y)\) to \((M_\theta (t,x),\,u_\theta (t,x),\,L_\theta (t))\) and vice versa. Thanks to the properties of \(N_\beta (t,y)\), the function \((t,y)\mapsto \Big (\frac{M_0(y)}{N_\beta (t,y)}\Big )^\beta \) is positive and Hölder continuous in every \(\overline{E}_T\) (see [7, Theorem 2.1], [3, Theorem 2.1], and [6, Theorem 2.3]). As a consequence
(Footnote 1) admits a unique (maximal) solution \(Y_\theta (t,\cdot ).\)
Let \([0,L_\theta (t)]\) be the (maximal) existence interval of \(Y_\theta (t,\cdot )\). We have \(Y_\theta (t,L_\theta (t))=L_0,\) and defining
\((M_\theta (t,x),\,u_\theta (t,x)\,L_\theta (t))\) is a solution of (1.1). (Footnote 2) As a first step in our analysis, we begin by studying the behavior of \(N_\beta (t,y)\) as
from now on we assume that
\(\beta >1\) (Footnote 3).
Let us also briefly recall the results on the asymptotic behavior for \(t\rightarrow \infty \) of \(N_\beta (t,y)\) (see [6, Theorem 2.1]) and of \((M_\theta (t, x), \, u_\theta (t, x), \, L_\theta (t)) \) (see [6, Theorem1.1]). If the assumptions (1.2) and (1.3) or (1.2) and (1.4) hold, then the function \(N_\beta (\cdot ,y)\) is monotone and its limit
belongs to \(C^2([0,L_0])\), is positive, decreasing and solves the stationary problem
Moreover, \(y\mapsto N_\beta (t,y)\) converges to \(\overline{N}_\beta (y)\) uniformly with respect to y as \(t\rightarrow \infty \).
The triplet \((M_\theta (t,x),\,u_\theta (t,x),\,L_\theta (t))\) satisfies the following statements.
- i):
-
\(L_\theta (t)\) converges to \(\overline{L}_\theta \) as \(t\rightarrow \infty ,\) and
$$\begin{aligned} |L_\theta (t)-\overline{L}_\theta |\le ce^{-\alpha t}, \end{aligned}$$for some constant c independent on t.
- ii):
-
\(\lim \limits _{t\rightarrow \infty }u_\theta (t,x)= 0\) uniformly with respect to x.
- iii):
-
\(M_\theta (t,x)\) converges to \(\overline{M}_\theta \in C^2([0,\overline{L}_\theta ])\) as \(t\rightarrow \infty ,\) \(M_\theta (t,\xi L_\theta (t))\) converges to \(\overline{M}_\theta (\xi \overline{L}_\theta )\) uniformly with respect to \(0\le \xi \le 1.\) Moreover, \(\overline{M}_\theta (x)\) satisfies
$$\begin{aligned} {\left\{ \begin{array}{ll} \alpha \overline{M}_\theta =D\overline{M}_\theta '',\,\, \text { in } [0,\overline{L}_\theta ],\\ \overline{M}_\theta '(0)=-\frac{\eta }{\overline{D}};\,\,\, \overline{M}_\theta '(\overline{L}_\theta )=0, \end{array}\right. } \end{aligned}$$and its explicit expression is
$$\begin{aligned} \overline{M}_\theta (x)=\frac{\eta }{\sqrt{\alpha D}}\frac{\cosh {\big [(x-\overline{L}_\theta )}\sqrt{\frac{\alpha }{D}}\big ]}{\sinh \big [{\overline{L}_\theta }\sqrt{\frac{\alpha }{D}}\big ]},\quad 0\le x\le \overline{L}_\theta . \end{aligned}$$
We conclude this section recalling that
3 On the signs of \(\partial _tN_\beta (t,y)\) and \(\partial _yN_\beta (t,y)\)
On the sign of \(\partial _tN_\beta (t,y)\), we proved the following result.
Theorem 3.1
(Sign of \(\partial _tN_\beta (t,y)\) [6, Theorem 2.5]) For every \(t> 0,\,\,0\le y\le L_0\), we have that
- i):
-
\(DM_0^{''}(y)-\alpha M_0(y)\ge 0\,\,\Rightarrow \,\, \partial _tN_\beta (t,y)\ge 0,\)
- ii):
-
\(DM_0^{''}(y)-\alpha M_0(y)\le 0\,\,\Rightarrow \,\, \partial _tN_\beta (t,y)\le 0.\)
To clarify the link between the hypotheses (1.3), (1.4) and the initial mean morphogens concentration, i.e., \(\left\| M_0 \right\| _{L^1(0, L_0)}\), the following lemma is needed.
Lemma 3.1
([6, Theorem 2.1. ii]) For every \(\beta \ge 1\) and \(t\ge 0\), we have that
Proof
Let us quickly sketch the proof of [6, Theorem2.1.ii)]. It is not difficult to rewrite the equation of (2.1) as follows
We integrate both sides in y on \([0,L_0]\), thanks to the boundary and initial data in (2.1),
and then
that gives the claim. \(\square \)
The relation between the assumptions (1.3), (1.4) and \(\left\| M_0 \right\| _{L^1(0,L_0)}\) is clarified in the next statements
We prove only (3.1), because the same argument works also for (3.2)
To determine the sign of \(\partial _yN_\beta (t, y)\), it is convenient to consider a reformulation of (2.1) useful for partially camouflaging the cumbersome initial datum \(M_0 (y)\). We will use the following notations, given \( 0 \le T <\infty , \)
Similarly, we define \(Q_\infty \) and \(\overline{Q}_\infty .\)
Due to the assumptions on \(M_0\), we can consider the function
where
If \(Y_0(z)\) is the inverse of \(Z_0(y)\), we define
Passing from the unknown \(N_\beta (t,y)\) to \(n_\beta (t,z)\), we simplify (2.1) in the following way
where
Theorem 3.2
(Sign of \(\partial _yN_\beta (t,y)\)) Let \(\beta >0\) be given. If
then
Proof
In order to keep the presentation simple and clear, we start considering (3.3) and proving
(Footnote 4) We begin by assuming \(\beta \not =1.\) Consider the functions
\(\omega \) satisfies in the weak sense the following identity
with
.
(Footnote 5) Moreover, the following are satisfied in the sense of traces
Let us distinguish two cases \(0<\beta <1\), \(1<\beta \).
\(\underline{0<\beta <1}.\) We multiply (3.6) by
and integrate over \(Q_t,\,\,t>0.\) Being
we have
Since
we have
and using (3.7)
Being \(\beta <1\)
and then
Since \(\omega \omega ^+\ge 0\) the two sides of the identity have different sings. As a consequence, they must vanish and
that gives \(\omega (t,z)^+=0,\) namely \(\omega (t,z)\le 0.\) In light of the definition of \(\omega (t,z)\) we have
\(\underline{\beta >1}.\) We argue as before and multiply (3.6) by
Being \(\beta >1\) we have
and then
Since \(\omega \omega ^-\le 0\), the two sides of the identity have different sings. As a consequence, they must vanish and
that gives \(\omega (t,z)^-=0,\) namely \(\omega (t,z)\ge 0.\) As in the previous case
\(\underline{\beta =1}.\) Define
\(\omega \) satisfies in the weak sense the identity
and in the sense of traces
We multiply (3.8) by \(\omega ^+\) and integrate over \(Q_t.\) Arguing as in the previous cases we obtain
that implies \(\omega (t,z)^+=0,\) namely \(\omega (t,z)\le 0.\) Therefore,
In this way we have proved (3.5).
Finally, being \(N_\beta (t,y)=n_\beta (t,Z_0(y)), \) we have
that concludes the proof. \(\square \)
4 Proof of Theorem 1.1
We begin this section by proving some a priori estimates on \(N_\beta (t,y)\) and \(\partial _yN_\beta (t,y)\) independent on \(\beta \).
Lemma 4.1
We have that
Proof
We prove (4.1). The lower bound on \(N_\beta (\cdot ,y)\) follows from the monotonicity of \(N_\beta (\cdot ,y)\) (see (1.3) and Theorem 3.1) and the identity \(N_\beta (0,y)=M_0(y)\). We have to prove the upper bound on \(N_\beta (\cdot ,y)\). Since
the monotonicity of \(\overline{N}_\beta (y)\) (see [6, Theorem 2.1]) and (3.8) guarantee
Moreover, by observing
we must have
Since \(\coth \) is nonincreasing,
that proves (4.1).
We have to prove (4.2). Thanks to the assumption (1.3) and Theorem 3.1, we know that \(\partial _tN_\beta (t,y)\ge 0.\) Using the equation in (2.1) we have that \(\Big (\frac{M_0(y)}{N_\beta (t,y)}\Big )^\beta \partial _yN_\beta (t,y)\) is nondecreasing with respect to y, for every \(t>0\) and \(\beta > 1.\) Using the boundary conditions, we gain
Employing (1.2), Theorem 2.1 and the fact
we conclude \(\partial _yN_\beta (t,y)\le 0, \,\, (t,y)\in ]0,\infty [\times [0,L_0],\) that proves (4.2). \(\square \)
We continue with the following result on the limit of \(\partial _tN_\beta (t,y)\) as \(\beta \rightarrow \infty \).
Theorem 4.1
We have that
The following lemma is needed
Lemma 4.2
For every \(T>0\)
Proof
We multiply the equation in (2.1) by \(\big (\frac{N_\beta }{M_0}\big )^\beta \partial _tN_\beta (t,y):\)
and integrate over \(E_T:\)
Rearranging the terms in the following way
we get the claim. \(\square \)
Proof
(Proof of Theorem 4.1) Since \(\partial _tN_\beta (t,y)\ge 0,\) by Lemma 4.2,
Being \(M_0(y)\le N_\beta (t,y)\) (see (4.1))
that proves the claim. \(\square \)
We continue with the behavior of \(N_\beta (t,y)\) as \(\beta \rightarrow \infty \).
Theorem 4.2
For every \(T>0\) and \(1\le r<\infty \)
uniformly with respect to \(t\in ]0,T[.\)
Proof
Since
thanks to Theorem 4.1
uniformly with respect to \(t\in ]0,T[.\)
The boundedness of \((N_\beta )_{\beta > 1}\) in \(L^\infty (E_T)\) (see (4.1)) and the boundedness of \(M_0(y)\) (see (1.2)) imply the claim. \(\square \)
We are finally ready for the proof of Theorem 1.1.
Proof
(Proof of Theorem 1.1) Since \(M_\theta (t,X_\theta (t,y))=N_\beta (t,y)\) and \(\beta =\frac{\log 2}{\theta }\), for every , we have
In light of Theorem 4.2, we have i).
Since \(\partial _tN_\beta (t,y)\ge 0,\,\,(t,y)\in E_T,\) (see Theorem 3.1.i)) and \(\beta >0\),
The monotonicity of \(N_\beta (t,y)\) with respect to t, \(0\le Y_\theta (t,x)\le L_0\), \(N_\beta (0,y)=M_0(y)\) and the definition of \(u_\theta (t,x)\) guarantee
The monotonicity assumption on \(M_0(y)\) gives
and using the equation in (2.1)
The boundary conditions in (2.1) and Lemma 3.1:
As a consequence,
that proves ii)
The equation in (2.1) gives
and then
where
Integrating with respect to y on \([0, L_0]\)
Integrating with respect to t on [0, T]
and then
Thanks to Theorem 3.2
Since \(\partial _yN_\beta (t,y)\le 0,\,\, (t,y)\in E_\infty ,\) (see Theorem 3.2), we observe
Using (4.5)
Thanks to (4.1)
Since \(a^*=\frac{\alpha \beta }{\beta +1}\)
Sending \(\theta \rightarrow 0^+\), namely \(\beta \rightarrow \infty \), we obtain iii). \(\square \)
5 Proof of Theorem 1.2
We begin by proving some a priori estimates on \(N_\beta (t,y)\) and \(\partial _yN_\beta (t,y)\) independent on \(\beta \).
Lemma 5.1
We have that
Proof
The lower bound in (5.1) follows from Theorem 2.1.i). For the upper bound in (5.1), we observe that \(N_\beta (\cdot ,y)\) and \(M_0(y)\) are nonincreasing (see (1.4) and Theorem 3.1); therefore,
We multiply the equation in (2.1) by \(\partial _yN_\beta (t,y)\)
Since \(\partial _yN_\beta (t,y)\le 0\) (see Theorem 3.2) and \(\partial _tN_\beta (t,y)\le 0\) (see Theorem 3.1), thanks to (5.3),
Integrating with respect to y over \([\xi ,L_0],\,\,0\le \xi \le L_0,\)
and using the boundary conditions in (2.1) and (2.2)
namely
Since \(N_\beta (\cdot ,\xi )\) is nonincreasing
using \(\partial _yN_\beta (t,\xi )\le 0\), we have (5.2). \(\square \)
We continue with the analysis of the behavior of \(\partial _tN_\beta (t,y)\) as \(\beta \rightarrow \infty \).
Theorem 5.1
For every \(T>0\)
The following lemma is needed.
Lemma 5.2
We have that
Proof
We multiply the equation in (2.1) by \(\partial _yN_\beta (t,y)\)
Integrating with respect to y on \([0,L_0]\) and using Theorem 2.1 and the boundary conditions in (2.1)
Thanks to (5.1)
and the (5.4).
Lemma 4.2 holds independently on the sign of \(DM_0''(y)-\alpha M_(y)\), (1.3) and (1.4). Indeed, its proof uses only (1.2) and Theorem 2.1. Therefore
Since \(N_\beta (\cdot ,y)\) is nonincreasing, the second term is negative; as a consequence in order to prove (5.5), it is enough to prove that the second term vanishes as \(\beta \rightarrow \infty \).
Thanks to Lemma 5.1
since \(\partial _yN_\beta (t,y)\le 0\) (see Theorem 3.2) and \(\partial _tN_\beta (t,y)\le 0\) (see Theorem 3.1)
By (5.4)
that gives (5.5). \(\square \)
Proof of Theorem 5.1
Since \(N_\beta (\cdot ,y)\) in nonincreasing
and then
The claim follows from (5.5). \(\square \)
We study the behavior of \(N_\beta (t,y)\) as \(\beta \rightarrow \infty \).
Theorem 5.2
For every \(0\le T<\infty \)
uniformly with respect to \(t\in [0,T].\)
Proof
Consider
Since \(\Big (\frac{N_\beta (\cdot ,y)}{M_0(y)}\Big )^\beta \) is nonincreasing, for every \(0\le \tau \le t\)
and then
Given \(0\le T<\infty \)
Using Theorem 5.1
uniformly with respect to \(t\in [0,T].\) Finally, since \((N_\beta )_{\beta >1}\) is bounded in \(L^\infty (E_T)\) (see Lemma 5.1.i)) and \(M_0\in L^\infty (0,L_0)\) (see (1.2)),
that gives the claim. \(\square \)
We are finally ready for the proof of Theorem 1.2.
Proof of Theorem 1.2
and consider the change of variable \(y=Y_\theta (t,x).\) For every t, \(x=X_\theta (t,y)\) is the inverse of \(y=Y_\theta (t,x),\) therefore
Thanks to the definition of \(N_\beta (t,y)\),
and, using Theorem 5.2, we get i).
Thanks to (1.4) and Theorem 3.1.ii), we have \(\partial _tN_\beta (t,y)\le 0,\,\,(t,y)\in E_\infty \). Moreover, since \(0\le Y_\theta (t,x)\le L_0\),
and
Due to (5.1)
where \(c_*\) is defined in (1.2). Since \(M_0(y)\) is nonincreasing (see (1.4))
Being \(|\partial _tN_\beta (t,y)|=-\partial _tN_\beta (t,y)\), thanks to the equation in (2.1) and (2.2)
Lemma 3.1 and the boundary conditions in (2.1) imply
In light of (5.7) we get
that gives ii).
We observe that
Being \(N_\beta (\cdot ,y)\) and \(M_0(y)\) nonincreasing, we have
and using Theorem 5.2
From the equation in (2.1), we get
Multiplying by \(1/{M_0(y)}\) and integrating over \([0,L_0]\)
and then
Using (1.4) and Theorem 3.2, we have \(\partial _yN_\beta (t,y)M'_0(y)\ge 0,\) that gives
Integrating with respect to t on [0, T]
Since
we have
Sending \(\beta \rightarrow \infty \) we get the claim. \(\square \)
Notes
t is a parameter.
\(Y_\theta (t,\cdot )\) is the inverse of \(X_\theta (t,\cdot )\) for every \(t\ge 0\).
The assumption \(\theta < \log (2)\) is equivalent to \(\beta >1\).
The differentiation in the weak sense of the function \(v^m\partial _z\omega \) and the role of the test function \(\omega ^-\) can be justified by the following observations
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Acknowledgements
GMC is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
Funding
Open access funding provided by Politecnico di Bari within the CRUI-CARE Agreement. GMC has been partially supported by the Project funded under the National Recovery and Resilience Plan (NRRP), Mission 4 Component 2 Investment 1.4 -Call for tender No. 3138 of 16/12/2021 of Italian Ministry of University and Research funded by the European Union -NextGenerationEUoAward Number: CN000023, Concession Decree No. 1033 of 17/06/2022 adopted by the Italian Ministry of University and Research, CUP: D93C22000410001, Centro Nazionale per la Mobilita’ Sostenibile and the Italian Ministry of Education, University and Research under the Programme Department of Excellence Legge 232/2016 (Grant No. CUP - D93C23000100001). GMC expresses its gratitude to the HIAS - Hamburg Institute for Advanced Study for their warm hospitality.
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Coclite, G.M., Coclite, M. On a singular limit as \(\theta \rightarrow 0\) for a model for the evolution of morphogens in a growing tissue. Z. Angew. Math. Phys. 74, 97 (2023). https://doi.org/10.1007/s00033-023-01993-z
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DOI: https://doi.org/10.1007/s00033-023-01993-z