Existence and nonexistence of global solutions for a structurally damped wave system with power nonlinearities

TL;DR

This paper investigates the conditions under which solutions to a coupled system of structurally damped wave equations exist globally or blow up, focusing on the impact of nonlinear growth rates.

Contribution

It establishes a threshold criterion for global existence versus nonexistence of solutions in a weakly coupled damped wave system with power nonlinearities.

Findings

Identifies critical exponents for global existence

Provides conditions for blow-up of solutions

Extends previous results to coupled systems

Abstract

Our interest itself of this paper is strongly inspired from an open problem in the paper [1] published by D'Abbicco. In this article, we would like to study the Cauchy problem for a weakly coupled system of semi-linear structurally damped wave equations. Main goal is to find the threshold, which classifies the global (in time) existence of small data solutions or the nonexistence of global solutions under the growth condition of the nonlinearities.