Abstract. In this paper we investigate the behavior of a family of steady state solutions of a nonlinear reaction diffusion equation when some reaction and potential terms are concentrated in a -neighborhood of a portion Γ of the boundary. We assume that this -neighborhood shrinks to Γ as the small parameter goes to zero. Also, we suppose the upper boundary of this -strip presents a highly oscillatory behavior. Our main goal here is to show that this family of solutions converges to the solutions of a limit problem, a nonlinear elliptic equation that captures the oscillatory behavior. Indeed, the reaction term and concentrating potential are transformed into a flux condition and a potential on Γ, which depends on the oscillating neighborhood.

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.

We analyze the dynamics of the flow generated by a nonlinear parabolic problem when some reaction and potential terms are concentrated in a neighborhood of the boundary. We assume that this neighborhood shrinks to the boundary as a parameter goes to zero. Also, we suppose that the "inner boundary" of this neighborhood presents a highly oscillatory behavior. Our main goal here is to show the continuity of the family of attractors with respect to . Indeed, we prove upper semicontinuity under the usual properties of regularity and dissipativeness and, assuming hyperbolicity of the equilibria, we also show the lower semicontinuity of the attractors at = 0.2010 Mathematics Subject Classification: 35R15, 35B40, 35B41, 35B25. Key words and phrases: Partial differential equations on infinite-dimensional spaces, asymptotic behavior of solutions, attractors, singular perturbations, concentrating terms, oscillatory behavior, lower semicontinuity.

In this paper, we analyze the behavior of a family of solutions of a nonlinear elliptic equation with nonlinear boundary conditions, when the boundary of the domain presents a highly oscillatory behavior, which is uniformly Lipschitz and nonlinear terms, are concentrated in a region, which neighbors the boundary of domain. We prove that this family of solutions converges to the solutions of a limit problem in H1an elliptic equation with nonlinear boundary conditions which captures the oscillatory behavior of the boundary and whose nonlinear terms are transformed into a flux condition on the boundary. Indeed, we show the upper semicontinuity of this family of solutions.Copyright © 2015 John Wiley & Sons, Ltd.

Abstract.In this work we analyze the asymptotic behavior of the solutions of a reaction-diffusion problem with delay when the reaction term is concentrated in a neighborhood of the boundary and this neighborhood shrinks to boundary, as a parameter goes to zero. 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. We also prove the continuity of the set of equilibria at = 0.