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

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 $1\leq q<\frac{3}{2}$.

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