Robustness of time-dependent attractors in <i>H</i>¹-norm for nonlocal problems

Abstract

In this paper, the existence of regular pullback attractors as well as their upper semicontinuous behaviour in H 1-norm are analysed for a parameterized family of non-autonomous nonlocal reaction-diffusion equations without uniqueness, improving previous results [Nonlinear Dyn. 84 (2016), 35-50].

About the Structure of Attractors for a Nonlocal Chafee-Infante Problem

“…The dynamical analysis of problem (1) and in particular the existence of global attractors has been established till now in several papers (cf. [17][18][19][20][21]). Other differential operators such as the p-Laplacian coupled with nonlocal viscosity has also been considered (cf.…”

“…Other differential operators such as the p-Laplacian coupled with nonlocal viscosity has also been considered (cf. [21][22][23]). However, in general little is known about the internal structure of the attractor, which is very important as it gives us a deep insight into the long-term dynamics of the problem.…”

Structure of the attractor for a non-local Chafee-Infante problem

“…, where E is the Lyapunov function (10) (see [5,Lemma 7] for the details), so by the properties of Lyapunov functions the heteroclinic connection is not possible.…”

“…The asymptotic behavior of solutions as times goes to infinity plays an important role in the study of non-linear partial differential equations. In the last years, several authors have studied the existence and properties of global attractors for such kind of equations in both the autonomous and nonautonomous frameworks (see [2], [7], [11], [10], [9], [30]). Also, non-local equations without uniquenes (see [8], [6]) and with delay [35] have been considered.…”