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

TL;DR

This paper proves finite-time blow-up for solutions of a generalized Tricomi equation with derivative nonlinearity when the exponent is below a critical threshold, and provides lifespan estimates.

Contribution

It establishes a blow-up result for a class of generalized Tricomi equations with derivative nonlinearities, including lifespan bounds, extending previous blow-up theories.

Findings

Solutions blow up in finite time for subcritical exponents.

Derived an upper bound for the lifespan of solutions.

Identified the critical exponent related to the quasi-homogeneous dimension.

Abstract

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.