Nonexistence of global solutions for the semilinear Moore-Gibson-Thompson equation in the conservative case

Abstract

In this work, the Cauchy problem for the semilinear Moore-Gibson-Thompson (MGT) equation with power nonlinearity $|u|^p$ on the right-hand side is studied. Applying $L^2-L^2$ estimates and a fixed point theorem, we obtain local (in time) existence of solutions to the semilinear MGT equation. Then, the blow-up of local in time solutions is proved by using an iteration method, under certain sign assumption for initial data, and providing that the exponent of the power of the nonlinearity fulfills $1 < p \leqslant p_{\mathrm{Str}}(n)$ for $n \geqslant2$ and $p>1$ for $n=1$. Here the Strauss exponent $p_{\mathrm{Str}}(n)$ is the critical exponent for the semilinear wave equation with power nonlinearity. In particular, in the limit case $p=p_{\mathrm{Str}}(n)$ a different approach with a weighted space average of a local in time solution is considered.