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

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 $\delta =1$.

24 pages