Critical curve for the weakly coupled system of damped wave equations with mixed nonlinearities

TL;DR

This paper establishes a new critical curve for a weakly coupled system of damped wave equations with mixed nonlinearities, using harmonic analysis to determine conditions for global existence and blow-up of solutions.

Contribution

It introduces a novel critical curve $pq = 1+ \frac{2}{n}$ for the system, highlighting the influence of nonlinearities of time derivative type.

Findings

Global existence of small Sobolev solutions when $pq > 1+\frac{2}{n}$

Finite-time blow-up of solutions when $pq < 1+\frac{2}{n}$

Impact of time derivative nonlinearities on the critical curve

Abstract

In this paper, we would like to consider the Cauchy problem for a weakly coupled system of semi-linear damped wave equations with mixed nonlinear terms. Our main objective is to draw conclusions about the critical curve of this problem using tools from Harmonic Analysis. Precisely, we obtain a new critical curve $pq = 1+ \frac{2}{n}$ for $n =1,2$ by proving global (in time) existence of small data Sobolev solutions when $pq > 1 +\frac{2}{n}$ and blow-up of weak solutions in finite time even for small data when $pq < 1+ \frac{2}{n}$ for $n \geq 1$. From this, we infer the impact of the nonlinearities of time derivative-type on the critical curve associated with the system.