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

Abstract

Delay in the boundary Convergence of solutionsIn this work, we are interested in the dynamic behavior of a parabolic problem with nonlinear boundary conditions and delay in the boundary. We construct a reaction-diffusion problem with delay in the interior, where the reaction term is concentrated in a neighborhood of the boundary and this neighborhood shrinks to boundary, as a parameter goes to zero. We analyze the limit of the solutions of this concentrated problem and prove that these solutions converge in certain continuous function spaces to the unique solution of the parabolic problem with delay in the boundary. This convergence result allows us to approximate the solution of equations with delay acting on the boundary by solutions of equations with delay acting in the interior and it may contribute to analyze the dynamic behavior of delay equations when the delay is at the boundary.

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

“…In order to prove the results of this paper, besides of the hypotheses (H1) and (H2) previously assumed in [3] and given by:…”

“…This analysis of the asymptotic behavior uses, as a main tool, the convergence result found in [3]. Here, we prove the existence of a family of global attractors and that this family is upper semicontinuous at = 0.…”

“…In [3] was proved that, under certain conditions, the limit problem of (1.1) is the following parabolic problem with delay in the boundary…”

“…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. Thus, our goal is to extend the results of [11] to parabolic problems with delay.…”

“…The paper will proceed as follows: in Section 2, assuming the hypotheses (H1) and (H2), we will give the notation that it will be used in this paper and we will remember some results obtained in [3]. We will be interested only in nonnegative solutions of (1.1) and (1.2), thus we will consider only nonnegative initial conditions of (1.1) and (1.2), since this will implicate the positiveness of the solutions, as we already saw in [3].…”