Nonexistence of global weak solutions for a wave equation with nonlinear memory and damping terms

TL;DR

This paper proves the nonexistence of global weak solutions for a wave equation with nonlinear memory and damping, addressing an open problem and removing initial value positivity constraints through analysis of integral inequalities.

Contribution

It provides a novel proof demonstrating nonexistence of solutions without positivity assumptions, advancing understanding of wave equations with nonlinear memory and damping.

Findings

Nonexistence of global weak solutions under certain conditions

Results do not require initial value positivity

Uses asymptotic analysis of integral inequalities

Abstract

In this paper, we study the nonexistence of global weak solutions for a wave equation with nonlinear memory and damping terms. We give an answer to an open problem posed in [M. D'Abbicco, A wave equation with structural damping and nonlinear memory, Nonlinear Differ. Equ. Appl. 21 (2014), 751-773]. Moreover, comparing with the existing results, our results do not require any positivity condition of the initial values. The proof of our results is based on the asymptotic properties of solutions for an integral inequality.