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

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 $n\ge 1$. From this, we infer the impact of the nonlinearities of time derivative-type on the critical curve associated with the system.

31 pages