Blow-up and lifespan estimate for the generalized Tricomi equation with mixed nonlinearities

TL;DR

This paper investigates the blow-up behavior and lifespan of solutions to a generalized Tricomi equation with mixed nonlinearities, revealing new blow-up regions and deriving lifespan estimates depending on the Tricomi parameter.

Contribution

It extends previous blow-up results to the Tricomi equation with both nonlinearities and derives lifespan estimates that depend on the parameter m.

Findings

New blow-up regions identified for the generalized Tricomi equation with mixed nonlinearities.

Lifespan estimates derived in terms of the Tricomi parameter m.

Method applied also reproduces known blow-up results for single nonlinearity cases.

Abstract

We study in this article the blow-up of the solution of the generalized Tricomi equation in the presence of two mixed nonlinearities, namely we consider $$ (Tr) \hspace{1cm} u_{tt}-t^{2m}\Delta u=|u_t|^p+|u|^q, \quad \mbox{in}\ \mathbb{R}^N\times[0,\infty),$$ with small initial data, where $m\ge0$.\\ For the problem $(Tr)$ with $m=0$, which corresponds to the uniform wave speed of propagation, it is known that the presence of mixed nonlinearities generates a new blow-up region in comparison with the case of a one nonlinearity ($|u_t|^p$ or $|u|^q$). We show in the present work that the competition between the two nonlinearities still yields a new blow region for the Tricomi equation $(Tr)$ with $m\ge0$, and we derive an estimate of the lifespan in terms of the Tricomi parameter $m$. As an application of the method developed for the study of the equation $(Tr)$ we obtain with a different approach the same blow-up result as in \cite{Lai2020} when we consider only one time-derivative nonlinearity, namely we keep only $|u_t|^p$ in the right-hand side of $(Tr)$.