In this note, we prove a blow-up result for a semilinear generalized Tricomi equation with nonlinear term of derivative type, i.e., for the equation $\mathscr{ T}_{ \!\!\ell} u = |\partial_t u|^p$, where $ \mathscr{ T}_{ \!\!\ell} = \partial_t^2-t^{ 2\ell}\Delta$. Smooth solutions blow up in finite time for positive Cauchy data when the exponent $p$ of the nonlinear term is below $\frac{ \mathscr{ Q}}{ \mathscr{ Q}-2}$, where $\mathscr{ Q}=(\ell+1)n+1$ is the quasi-homogeneous dimension of the generalized Tricomi operator $\mathscr{ T}_{ \!\!\ell}$. Furthermore, we get also an upper bound estimate for the lifespan.
Categories: math.AP