Energy decay of a viscoelastic wave equation with variable exponent logarithmic nonlinearity and weak damping

TL;DR

This paper analyzes the energy decay behavior of solutions to a viscoelastic wave equation with variable exponent logarithmic nonlinearity and weak damping, providing explicit decay rates and extending previous results to broader conditions.

Contribution

The study establishes explicit decay results and refines decay estimates for a viscoelastic wave equation with variable exponents and nonlinear damping, extending prior work to more general conditions.

Findings

Derived explicit general decay rates under mild conditions.

Obtained refined decay estimates improving existing results.

Extended decay rate analysis to cases with $1 \\leq q < 2$, beyond previous limits.

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 $g'(t)\leq-\zeta(t)G(g(t))$ with some suitably given $\zeta$ 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 $g'(t)\leq-\xi(t)g^q(t)$ with $1\leq q<2$, extending earlier studies that were restricted to the case $1\leq q<\frac{3}{2}$.