Stability Properties of Distributional Solutions for Nonlinear Viscoelastic Wave Equations with Variable Exponents

A system of nonlinear wave equations in viscoelasticity with variable exponents is considered. It is assumed that the kernel included in the integral term of the equations depends on both the time and the spatial variables. Using the Faedo–Galerkin method and the contraction mapping principle, a theorem of unique solvability of the problem is proved. In addition, under appropriate variable assumptions, an estimate of the stability of the solution to the problem of determining the kernel is obtained. The study is based on Komornik’s inequality. We expand the class of nonlinear boundary value problems that can be investigated by well-known methods.