Blow-up and sharp lifespan estimates to the weakly coupled system of structurally damped wave equations with critical nonlinearities

TL;DR

This paper investigates the critical nonlinear damped wave system, establishing sharp conditions for solution existence, blow-up phenomena, and lifespan estimates in Sobolev spaces.

Contribution

It provides new sharp conditions for moduli of continuity and lifespan estimates for solutions to a critical weakly coupled damped wave system.

Findings

Identified sharp conditions for global existence of solutions.

Proved finite-time blow-up under certain conditions.

Derived precise lifespan estimates for solutions in blow-up scenarios.

Abstract

In this paper, we would like to study the weakly coupled system of semilinear structurally damped wave equations with moduli of continuity in nonlinear terms whose powers belong to the critical curve in the $p-q$ plane. Our main purpose is to find a sharp condition for these moduli of continuity by investigating the global (in time) existence of small data Sobolev solutions and the blow-up result for solutions in finite time as well. Furthermore, when the blow-up phenomenon occurs, we are going to achieve the sharp lifespan estimates for the local (in time) Sobolev solution.