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

Abstract

In this paper we consider linear parabolic problems when some reaction and potential terms are concentrated in a neighborhood of a portion of the boundary. This neighborhood shrinks to as a parameter ε goes to zero. Then we derive the limit equation which has some new terms on . We also analyze the regularity and convergence of the solutions.

Singular limit for a nonlinear parabolic equation with terms concentrating on the boundary

“…Now for problems (2.1) and (2.2), the following results have been proved in [12] and will be used in a crucial way in the rest of the paper.…”

“…Then we must prove that the nonlinear semigroups defined by (1.2) converge as ε → 0 to the one defined by (1.3). This in turn, will be obtained from the convergence of solutions for the corresponding linear equations, see [12].…”

“…Some related nonlinear problems have been analyzed in [5]. Second, linear parabolic equations have been considered in [12]. All these results are the starting point for the present paper.…”

“…In Section 2 we recall previous results in [12] about linear parabolic equations with concentrated terms. These include results about the setting for the solvability of the linear equations, Theorem 2.2, and results about the convergence of solutions, Theorem 2.3.…”

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

“…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. Since we hope to use this technique in order to get the dynamic behavior in the boundary, then we first need to extend the results of Arrieta, Jiménez-Casas and Rodríguez-Bernal [6] and Jiménez-Casas and Rodríguez-Bernal [9] to reaction-diffusion problems with delay, in the case where Γ is the whole boundary.…”