Singular limit for a nonlinear parabolic equation with terms concentrating on 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 the family of attractors is upper continuous at the ε = 0.

Attractors for a Nonlinear Parabolic Problem with Terms Concentrating on the Boundary

“…There, the neighborhood is a strip of width and base in a portion of the boundary, without oscillatory behavior. Later, the asymptotic behavior of a parabolic problem of the same type was analyzed in [9,10], where the upper semicontinuity of attractors at = 0 was proved. The same technique of [6] has been used in [2,3], where the results of [6,9] were extended to a reaction-diffusion problem with delay.…”

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].…”

Dynamics of thermoelastic plate system with terms concentrated in the boundary

“…In this work we analyze the asymptotic behavior of the global compact attractors of autonomous thermoelastic plate systems with Neumann boundary conditions when some reaction terms are concentrated in a neighborhood of the boundary and this neighborhood shrinks to boundary as a parameter ε goes to zero. There has been numerous studies to investigate the dynamics, in the sense of attractors, of systems when reaction terms are concentrated in a neighborhood of the boundary and this neighborhood shrinks to boundary as a parameter ε goes to zero, see for instance [2,3,4,5,6,7,8,9,13,14] and references therein.…”