A blow-up result for a generalized Tricomi equation with nonlinearity of derivative type

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 ${\mathcal{T}}_{\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\mathrm{\ell }}u=\mathrm{\mid }{\mathrm{\partial }}_{t}u{\mathrm{\mid }}^p$, 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 $\frac{\mathcal{Q}}{\mathcal{Q}-2}$, where $\mathcal{Q}=(\mathrm{\ell }+1)n+1$ is the quasi-homogeneous dimension of the generalized Tricomi operator ${\mathcal{T}}_{\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\mathrm{\ell }}$. Furthermore, we get also an upper bound estimate for the lifespan.