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

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 L² - L² 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 1 $ for $ n = 1 $. Here the Strauss exponent pStr(n) is the critical exponent for the semilinear wave equation with power nonlinearity. In particular, in the limit case p = pStr(n) a different approach with a weighted space average of a local in time solution is considered.