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

TL;DR

This paper investigates the nonexistence of global solutions for the semilinear Moore-Gibson-Thompson equation in the conservative case, establishing blow-up results for certain nonlinear exponents using fixed point and iteration methods.

Contribution

It provides new blow-up results for the semilinear MGT equation with power nonlinearity, extending understanding of solution behavior in the critical and subcritical regimes.

Findings

Solutions blow up for $1 < p \,\leq p_{Str}(n)$ in the conservative case.

Local existence of solutions is established via $L^2-L^2$ estimates and fixed point theorem.

Different methods are used at the critical exponent $p=p_{Str}(n)$ to analyze solution behavior.

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.