Hadamard well-posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping

TL;DR

This paper investigates the well-posedness of a viscoelastic wave equation with supercritical sources and damping, establishing existence, uniqueness, and continuous dependence of solutions using advanced mathematical tools.

Contribution

It introduces a novel analysis of a viscoelastic wave equation with supercritical sources, proving well-posedness and global existence under damping dominance.

Findings

Existence of a unique local weak solution

Continuous dependence on initial data

Global solutions when damping dominates source

Abstract

Presented here is a study of a viscoelastic wave equation with supercritical source and damping terms. We employ the theory of monotone operators and nonlinear semigroups, combined with energy methods to establish the existence of a unique local weak solution. In addition, it is shown that the solution depends continuously on the initial data and is global provided the damping dominates the source in an appropriate sense.