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

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=pS(n+μm+1,m) for any δ>0, where δ 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 and Han [Life-span of solutions to critical semilinear wave equations. Comm Partial Differ Equations. 2014;39:439–451. doi: 10.1080/03605302.2013.863914]. Additionally, we extend our analysis to prove a blow-up result with the index p=max{pS(n+μm+1,m),pF((m+1)n+μ−1−δ2)} by applying Kato's Lemma (i.e. Lemma 4.1), specifically in the case of δ=1.