  We study in this article the blow-up of the solution of the generalized
Tricomi equation in the presence of two mixed nonlinearities, namely we
consider $$ (Tr) \hspace{1cm} u_{tt}-t^{2m}\Delta u=|u_t|^p+|u|^q, \quad
\mbox{in}\ \mathbb{R}^N\times[0,\infty),$$ with small initial data, where
$m\ge0$.\\ For the problem $(Tr)$ with $m=0$, which corresponds to the uniform
wave speed of propagation, it is known that the presence of mixed
nonlinearities generates a new blow-up region in comparison with the case of a
one nonlinearity ($|u_t|^p$ or $|u|^q$). We show in the present work that the
competition between the two nonlinearities still yields a new blow region for
the Tricomi equation $(Tr)$ with $m\ge0$, and we derive an estimate of the
lifespan in terms of the Tricomi parameter $m$. As an application of the method
developed for the study of the equation $(Tr)$ we obtain with a different
approach the same blow-up result as in \cite{Lai2020} when we consider only one
time-derivative nonlinearity, namely we keep only $|u_t|^p$ in the right-hand
side of $(Tr)$.
