Exponential stability for an infinite memory wave equation with frictional damping and logarithmic nonlinear terms

TL;DR

This paper proves exponential energy decay for an infinite memory wave equation with logarithmic nonlinearity and frictional damping, using advanced analysis techniques, and shows the infinite memory effect alone can ensure stability under certain conditions.

Contribution

It establishes exponential stability for a wave equation with infinite memory and nonlinear terms, even without damping if the material density is suitably chosen.

Findings

Exponential decay of energy under general conditions.

Stability achieved without damping for specific density forms.

Application of microlocal analysis techniques.

Abstract

This article is concerned with the energy decay of an infinite memory wave equation with a logarithmic nonlinear term and a frictional damping term. The problem is formulated in a bounded domain in $\mathbb R^d$ ($d\ge3$) with a smooth boundary, on which we prescribe a mixed boundary condition of the Dirichlet and the acoustic types. We establish an exponential decay result for the energy with a general material density $\rho(x)$ under certain assumptions on the involved coefficients. The proof is based on a contradiction argument, the multiplier method and some microlocal analysis techniques. In addition, if $\rho(x)$ takes a special form, our result even holds without the damping effect, that is, the infinite memory effect alone is strong enough to guarantee the exponential stability of the system.