  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.
