Attractors for Damped Semilinear Wave Equations with a Robin--Acoustic Boundary Perturbation

TL;DR

This paper studies the behavior of solutions to a damped semilinear wave equation with boundary perturbations, establishing existence, uniqueness, and properties of global attractors, including their regularity and finite-dimensionality, as the boundary conditions vary.

Contribution

It introduces a novel analysis of global attractors for damped wave equations with boundary perturbations, including upper-semicontinuity and exponential attractors under minimal assumptions.

Findings

Existence and uniqueness of solutions for all perturbation parameters.

Existence of a family of global attractors with upper-semicontinuity.

Construction of exponential attractors with finite fractal dimension.

Abstract

Under consideration is the damped semilinear wave equation \[ u_{tt}+u_t-\Delta u + u + f(u)=0 \] on a bounded domain $\Omega$ in $\mathbb{R}^3$ with a perturbation parameter $\varepsilon>0$ occurring in an acoustic boundary condition, limiting ($\varepsilon=0$) to a Robin boundary condition. With minimal assumptions on the nonlinear term $f$, the existence and uniqueness of global weak solutions is shown for each $\varepsilon\in[0,1]$. Also, the existence of a family of global attractors is shown to exist. After proving a general result concerning the upper-semicontinuity of a one-parameter family of sets, the result is applied to the family of global attractors. Because of the complicated boundary conditions for the perturbed problem, fractional powers of the Laplacian are not well-defined; moreover, because of the restrictive growth assumptions on $f$, the family of global attractors is obtained from the asymptotic compactness method developed by J. Ball for generalized semiflows. With more relaxed assumptions on the nonlinear term $f$, we are able to show the global attractors possess optimal regularity and prove the existence of an exponential attractor, for each $\varepsilon\in[0,1].$ This result insures that the corresponding global attractor inherits finite (fractal) dimension, however, the dimension is {\em{not}} necessarily uniform in $\varepsilon$.