Critical exponent for damped wave equations with nonlinear memory

TL;DR

This paper investigates the global existence, asymptotic behavior, and blow-up phenomena of solutions to a semilinear damped wave equation with nonlinear memory across various dimensions, revealing critical exponents and solution dynamics.

Contribution

It extends the analysis of damped wave equations with nonlinear memory to all dimensions, establishing conditions for global existence, asymptotic behavior, and blow-up, including the derivation of a critical exponent.

Findings

Global existence for small data in dimensions 1 to 3

Asymptotic behavior of solutions as time approaches infinity

Blow-up results for positive data in all dimensions

Abstract

We consider the Cauchy problem in $\mathbb{R}^n,$ $n\geq 1,$ for a semilinear damped wave equation with nonlinear memory. Global existence and asymptotic behavior as $t\rightarrow\infty$ of small data solutions have been established in the case when $1\leq n\leq3.$ Moreover, we derive a blow-up result under some positive data in any dimensional space.