Lifespan estimate for the semilinear regular Euler-Poisson-Darboux-Tricomi equation

TL;DR

This paper establishes local well-posedness and lifespan estimates for the semilinear regular Euler-Poisson-Darboux-Tricomi equation, introducing new test functions and analyzing blow-up behavior based on damping and mass parameters.

Contribution

It provides the first lifespan estimates and blow-up results for this class of equations using novel test functions derived from hypergeometric functions.

Findings

Lifespan estimate with Strauss index for the equation

Blow-up result with specific index in the case of δ=1

Extension of analysis to include damping and mass effects

Abstract

In this paper, we begin by establishing local well-posedness for the semilinear regular Euler-Poisson-Darboux-Tricomi equation. Subsequently, we derive a lifespan estimate with the Strauss index given by $p=p_{S}(n+\frac{\mu}{m+1}, m)$ for any $\delta>0$, where $\delta$ is a parameter to describe the interplay between damping and mass. This is achieved through the construction of a new test function derived from the Gaussian hypergeometric function and a second-order ordinary differential inequality, as proven by Zhou \cite{Zhou2014}. Additionally, we extend our analysis to prove a blow-up result with the index $p=\max\{p_{S}(n+\frac{\mu}{m+1}, m), p_{F}((m+1)n+\frac{\mu-1-\sqrt\delta}{2})\}$ by applying Kato$^{\prime}$s Lemma ( i.e., Lemma \ref{katolemma} ), specifically in the case of $\delta=1$.