Attractors for Damped Semilinear Wave Equations with Singularly Perturbed Acoustic Boundary Conditions

TL;DR

This paper studies the long-term behavior of solutions to a damped semilinear wave equation with singularly perturbed acoustic boundary conditions, proving the existence and regularity of global and exponential attractors and analyzing their dependence on the perturbation parameter.

Contribution

It introduces a new approach to establish the existence, regularity, and upper-semicontinuity of attractors for the wave equation with singular perturbations and weaker nonlinear assumptions.

Findings

Existence of a family of global attractors for each perturbation parameter.

Optimal regularity and finite fractal dimension of the attractors.

Upper-semicontinuity of attractors with respect to the perturbation parameter.

Abstract

Under consideration is the damped semilinear wave equation \[ u_{tt}+u_t-\Delta u+u+f(u)=0 \] in a bounded domain $\Omega$ in $\mathbb{R}^3$ subject to an acoustic boundary condition with a singular perturbation, which we term "massless acoustic perturbation," \[ \ep\delta_{tt}+\delta_t+\delta = -u_t\quad\text{for}\quad \ep\in[0,1]. \] By adapting earlier work by S. Frigeri, we prove the existence of a family of global attractors for each $\ep\in[0,1]$. We also establish the optimal regularity for the global attractors, as well as the existence of an exponential attractor, for each $\ep\in[0,1].$ The later result insures the global attractors possess finite (fractal) dimension, however, we cannot yet guarantee that this dimension is independent of the perturbation parameter $\ep.$ The family of global attractors are upper-semicontinuous with respect to the perturbation parameter $\ep$, a result which follows by an application of a new abstract result also contained in this article. Finally, we show that it is possible to obtain the global attractors using weaker assumptions on the nonlinear term $f$, however, in that case, the optimal regularity, the finite dimensionality, and the upper-semicontinuity of the global attractors does not necessarily hold.