Asymptotic behavior of a parabolic problem with terms concentrated in the boundary

Abstract

We analyze the asymptotic behavior of the attractors of a parabolic problem when some reaction and potential terms are concentrated in a neighborhood of a portion Γ of the boundary and this neighborhood shrinks to Γ as a parameter ε goes to zero.We prove that this family of attractors is upper continuous at ε = 0.

A singular perturbation in a linear parabolic equation with terms concentrating on the boundary

“…Some results on the corresponding nonlinear problems have been announced in [7], while very detailed analysis of nonlinear problems can be found in [8].…”

Asymptotic behavior of a reaction-diffusion problem with delay and reaction term concentrated in the boundary

“…As in [3], we compare the solutions of (1.1) and (1.2) with the solutions of the linear parabolic problems without delay (2.6) and (2.7). The advantage of this comparison is that the asymptotic behavior of the attractors of (2.6) and (2.7) was already studied in [11], using the results of [6], on concentrating integrals and elliptic problem associated to parabolic problem, and some previous results of [5]. Some results obtained in [11] and adapted to the case where the space is X α , with X α → C(Ω), are given in the following theorem: Theorem 3.1.…”

“…This kind of problem was initially studied in [6], where linear elliptic equations with terms concentrated were considered and convergence results of the solutions were proved. Later, the asymptotic behavior of the attractors of a parabolic problem without delay was analyzed in [11], where the upper semicontinuity of attractors at = 0 was proved. The same technique of [6] has been used in [3], where some results of [6] were extended to a reaction-diffusion problem with delay.…”

“…The same technique of [6] has been used in [3], where some results of [6] were extended to a reaction-diffusion problem with delay. Thus, our goal is to extend the results of [11] to parabolic problems with delay.…”

Delay nonlinear boundary conditions as limit of reactions concentrating in the boundary

“…They proved that these solutions converge, in certain Bessel potential spaces and in the continuous function space, to the solution of an elliptic problem where the reaction term and the concentrating potential are transformed into a flux condition and a potential on Γ . Later, Jiménez-Casas and Rodríguez-Bernal [9] analyzed the asymptotic behavior of the attractors of a parabolic problem, more precisely, proved that the family of attractors is upper semicontinuous at = 0. The same technique of Arrieta, Jiménez-Casas and Rodríguez-Bernal [6] has been used by Rodríguez-Bernal [14], where the regularity and convergence of the solutions of linear parabolic problems were analyzed.…”