标题： 具有导数型非线性的广义Tricomi方程的爆破结果 显示英文标题

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

摘要： 在本文中，我们证明了一个带有导数型非线性项的半线性广义Tricomi方程的爆破结果，即对于方程 $\mathscr{T}_{\!\!\ell} u = |\partial_t u|^p$，其中 $ \mathscr{T}_{\!\!\ell} = \partial_t^2-t^{2\ell}\Delta$。 当非线性项的指数 $p$低于 $\frac{\mathscr{Q}}{\mathscr{Q}-2}$时，对于正的柯西数据，光滑解在有限时间内爆破，其中 $\mathscr{Q}=(\ell+1)n+1$是广义Tricomi算子的准齐次维数 $\mathscr{T}_{\!\!\ell}$。 此外，我们还得到了寿命的上界估计。 显示英文摘要

摘要： 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.