The interplay of critical regularity of nonlinearities in a weakly coupled system of semi-linear damped wave equations

TL;DR

This paper investigates the critical regularity conditions of nonlinearities in a weakly coupled damped wave system to determine when solutions exist globally or blow up in finite time.

Contribution

It establishes sharp conditions on moduli of continuity that distinguish between global existence and finite-time blow-up for the system.

Findings

Identifies critical thresholds for nonlinear moduli of continuity.

Provides criteria for global existence of solutions.

Characterizes blow-up conditions in the coupled system.

Abstract

We would like to study a weakly coupled system of semi-linear classical damped wave equations with moduli of continuity in nonlinearities whose powers belong to the critical curve in the $p-q$ plane. The main goal of this paper is to find out the sharp conditions of these moduli of continuity which classify between global (in time) existence of small data solutions and finite time blow-up of solutions.