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

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

References in corpus (1)

• Fundamental Solutions for the Klein-Gordon Equation in de Sitter Spacetime

Cited by in corpus (6)

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• A note on the nonexistence of global solutions to the semilinear wave equation with nonlinearity of derivative-type in the generalized Einstein-de Sitter spacetime
• Lifespan estimates for the compressible Euler equations with damping via Orlicz spaces techniques
• On the the critical exponent for the semilinear Euler-Poisson-Darboux-Tricomi equation with power nonlinearity
• A blow-up result for a Nakao-type weakly coupled system with nonlinearities of derivative-type
• Blow-up and lifespan estimates for a damped wave equation in the Einstein-de Sitter spacetime with nonlinearity of derivative type