Robustness of nonautonomous attractors for a family of nonlocal reaction–diffusion equations without uniqueness

Abstract

In this paper, we consider a non-autonomous nonlocal reactiondiffusion equation with a small perturbation in the nonlocal diffusion term and the non-autonomous force. Under the assumptions imposed on the viscosity function, the uniqueness of weak solutions cannot be guaranteed. In this multi-valued framework, the existence of weak solutions and minimal pullback attractors in the L 2 -norm are analysed. In addition, some relationships between the attractors of the universe of fixed bounded sets and those associated to a universe given by a tempered condition are established. Finally, the upper semicontinuity property of pullback attractors w.r.t. the parameter is proved. Indeed, under suitable assumptions, we prove that the family of pullback attractors converges to the corresponding global compact attractor associated to the autonomous nonlocal limit problem when the parameter goes to zero.Keywords nonlocal diffusion · reaction-diffusion equations without uniqueness · pullback attractors · upper semicontinuity of attractors · multivalued dynamical systems.

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

“…While being an upper-semicontinuous multi-valued process is a property only related with a two-parameter semigroup, the upper-semicontinuous behaviour of attractors takes into account all the processes indexed by the parameter in which it is taken the limit and it analyses an asymptotic property of a whole family of problems.] In this paper we improve the results given in [6], analysing existence of pullback attractors in H 1 0 (Ω) as well as their upper semicontinuous behaviour in H 1 -norm. According to [6,Section 6] we consider the parameterized family (ε ∈ [0, 1]) of non-autonomous nonlocal reaction-diffusion problems…”

“…In this paper we improve the results given in [6], analysing existence of pullback attractors in H 1 0 (Ω) as well as their upper semicontinuous behaviour in H 1 -norm. According to [6,Section 6] we consider the parameterized family (ε ∈ [0, 1]) of non-autonomous nonlocal reaction-diffusion problems…”

“…In addition, in [6] the existence of pullback attractors in L 2 (Ω) is analysed in a multi-valued framework for the non-autonomous nonlocal reaction-diffusion problem where ε ∈ [0, 1] and no locally Lipschitz assumption on the function a or monotonicity on the nonlinearity f are imposed (so lack of uniqueness). Moreover, the upper semicontinuous behaviour of attractors in L 2 -norm is studied.…”

“…Section 2 is devoted to providing abstract results on multi-valued non-autonomous dynamical systems which are essential to prove the existence of pullback attractors in L 2 (Ω) and H 1 0 (Ω) in the last two sections. Namely, in Section 3 we generalise some results of [6], which guarantee the existence of pullback attractors in L 2 (Ω) and will be crucial in what follows. In Section 4, we show the existence of pullback attractors in H 1 -norm and analyse their upper semicontinuity with respect to the parameter.…”

“…When u is a weak solution to (P ε ), making use of the continuity of the function a, (2), (6) and 7, it holds that u ∈ L 2 (τ, T ; H −1 (Ω)) + L q (τ, T ; L q (Ω)) for any T > τ . Therefore, u ∈ C([τ, ∞); L 2 (Ω)) and the initial datum in (P ε ) makes sense.…”