Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 53, pp. 1-17.

Title: Optimal energy decay rates for viscoelastic  wave equations with nonlinearity of variable exponent
       
Author: Muhammad I. Mustafa (Univ. of Sharjah, Sharjah, UAE)
         
Abstract: 
In this article, we consider the  viscoelastic wave equation
$$
u_{tt}-\Delta u+\int_0^{t}g(t-s)\Delta u(s)ds+a| u_t| ^{m(.)-2}u_t=0
$$
with a nonlinear feedback having a variable exponent m(x). We investigate
the interaction between the two types of damping and establish an optimal
decay result under very general assumptions on the relaxation function g.
We construct explicit formulae which provide faster energy decay rates than
the ones already existing in the literature.

Submitted March 22, 2023. Published August 28, 2023.
Math Subject Classifications: 35B40, 74D99, 93D15, 93D20.
Key Words: Viscoelasticity; frictional damping; variable exponent; energy decay.