Nonexistence of global solutions for the semilinear Moore – Gibson – Thompson equation in the conservative case
Wenhui Chen, Alessandro Palmieri
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.