Regularity results and exponential growth for pullback attractors of a non-autonomous reaction–diffusion model with dynamical boundary conditions

Abstract

In this paper, we prove some regularity results for pullback attractors of a nonautonomous reaction-diffusion model with dynamical boundary conditions considered in [4]. Under certain assumptions of the nonlinear terms we show a regularity result for the unique solution of the problem. We establish a general result about boundedness of invariant sets for the associated evolution process in the norm of the domain of the spatial linear operator appearing in the equation. As a consequence, we deduce that the pullback attractors of the model are bounded in this domain norm. After that, under additional assumptions, some exponential growth results for pullback attractors when time goes to −∞ are proved.

H2-boundedness of the pullback attractor for the non-autonomous SIR equations with diffusion

“…], we establish that the process U t,t 0 has a unique pullback attractor. However, as far as we know, there are no results in the literature concerning regularity of the pullback attractor as we will consider in this paper (for similar results for the reaction-diffusion equations see [2,4], and for the Navier-Stokes equations see [10]). The regularity results on the solutions and the attractors (that we obtain here) might be useful in the future in order to implement new methods to seek solutions of more general problems by different arguments, to gain attraction in higher norms, or for numerical purposes.…”

Asymptotic Behavior of Solutions to Reaction-Diffusion Equations with Dynamic Boundary Conditions and Irregular Data

“…Then in [11][12][13] the existence of global attractors and their fractal dimensions were further studied for certain semilinear reaction-diffusion equations with dynamic boundary conditions, while in [9,14,15], the existence of global attractors was obtained for more general quasilinear parabolic equations with dynamic boundary conditions. For the nonautonomous case, the existence of pullback attractors for parabolic equations with dynamic boundary conditions was first obtained in [16], and then in [17][18][19], while the existence of uniform attractors for linear and quasilinear parabolic equations with dynamic boundary conditions was investigated in [12,20].…”