A blow-up result for the semilinear Moore-Gibson-Thompson equation with nonlinearity of derivative type in the conservative case

TL;DR

This paper investigates the blow-up behavior of solutions to the semilinear Moore-Gibson-Thompson equation with derivative-type nonlinearity, establishing blow-up conditions in the conservative case for certain ranges of the exponent p.

Contribution

It extends blow-up results to the Moore-Gibson-Thompson equation, matching the blow-up range known for the semilinear wave equation with derivative nonlinearity.

Findings

Blow-up occurs for 1<p≤(n+1)/(n-1) when n≥2.

Blow-up occurs for p>1 when n=1.

The blow-up range matches that of the semilinear wave equation with derivative nonlinearity.

Abstract

In this paper, we study the blow-up of solutions to the semilinear Moore-Gibson-Thompson (MGT) equation with nonlinearity of derivative type $|u_t|^p$ in the conservative case. We apply an iteration method in order to study both the subcritical case and the critical case. Hence, we obtain a blow-up result for the semilinear MGT equation (under suitable assumptions for initial data) when the exponent $p$ for the nonlinear term satisfies $1<p\leqslant (n+1)/(n-1)$ for $n\geqslant2$ and $p>1$ for $n=1$. In particular, we find the same blow-up range for $p$ as in the corresponding semilinear wave equation with nonlinearity of derivative type.