Long-Time Behavior for Semilinear Equation with Time-Dependent and Almost Sectorial Linear Operator

Abstract

In this paper we study the solvability and asymptotic dynamics of a nonautonomous semilinear reaction–diffusion equation in a domain with a handle $\Omega _0 = \Omega \cup R_0$, formed by an open subset $\Omega \subset \mathbb {R}^{N}$ connected to a line segment $R_0$ at the ending points of the segment. We also assume that the linear part of this equation (the diffusion term) is time-dependent and the growth condition on the nonlinearity F is more general than linear growth. o obtain existence of local solution, the uniformly almost sectoriality of the family of linear operator associated to the evolution equation is explored. An abstract result on existence of mild solution for semilinear problems of the form

$$\begin{aligned} \begin{aligned}&u_t +A(t) u = F(u), t>\tau ; \quad u(\tau ) = u_0, \end{aligned} \end{aligned}$$

where A(t) is uniformly almost sectorial, is proved and we analyze its application to the equation in $\Omega _{0}$. Through an iterative procedure we obtain estimates of the solution in the spaces $L^{2^{k}}$, for any $k\in \mathbb {N}$, resulting in global well-posedness of the solution and existence of pullback attractor. We also explore how the line segment $R_0$ impacts in the pullback attractor obtained. Those results are obtained without requiring monotonicity or asymptotic assumption on A(t) as $t\rightarrow \infty$.