Energy decay of a viscoelastic wave equation with variable exponent logarithmic nonlinearity and weak damping
Qingqing Peng, Yikan Liu
TL;DR
This work analyzes energy decay for a viscoelastic wave equation with memory in a bounded domain, incorporating a weak damping term and a variable-exponent logarithmic nonlinearity. By developing a Lyapunov functional framework and detailed kernel estimates, the authors establish global existence and explicit decay rates under general memory kernels. They extend decay results to the full range $1\le q<2$ for the damping law $g'(t)\le -\xi(t)g(t)^q$, obtaining either exponential or polynomial decay depending on the case, and illustrate with concrete kernels such as $g(t)=a e^{-t^p}$. The results generalize prior work by relaxing smallness and structural assumptions, and by incorporating variable exponents and logarithmic nonlinearities into the viscoelastic setting, enhancing understanding of long-time behavior in complex damping-memory systems.
Abstract
In this paper, we investigate the energy decay of the solution to a viscoelastic wave equation with variable exponents logarithmic nonlinearity and weak damping in a bounded domain. We establish an explicit general decay result under mild conditions on the relaxation function $g$. Furthermore, under the general assumption $g'(t)\leq-ζ(t)G(g(t))$ with some suitably given $ζ$ and $G$, we derive a refined decay estimate improving existing results. In particular, uniform exponential and polynomial decay rates are obtained under a further special situation $g'(t)\leq-ξ(t)g^q(t)$ with $1\leq q<2$, extending earlier studies that were restricted to the case $1\leq q<\frac{3}{2}$.