Lifespan estimate for the semilinear regular Euler-Poisson-Darboux-Tricomi equation
Yuequn Li, Fei Guo
TL;DR
This work analyzes the semilinear regular Euler-Poisson-Darboux-Tricomi equation with damping and mass terms, establishing local well-posedness and deriving sharp lifespan estimates at the Strauss-type exponent $p_S(n+\frac{\mu}{m+1},m)$. By constructing a Gaussian-hypergeometric test function and proving key nonlinear estimates, the authors obtain an upper bound on the lifespan $T(\varepsilon)\le e^{C\varepsilon^{-p(p-1)}}$ for $\delta\in(0,(m+1)^2n^2)$ and $p=p_S(n+\frac{\mu}{m+1},m)$. In the special case $\delta=1$, they further prove blow-up at $p=\max\{p_S(n+\frac{\mu}{m+1},m), p_F((m+1)n+\frac{\mu-1-\sqrt{\delta}}{2})\}$, using a reduction to a second-order ODE and Kato’s lemma. Overall, the paper advances the critical-exponent theory for EPDT-type equations by clarifying the interplay between damping and mass and by extending blow-up results to the regular problem across dimensions.
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 $p=p_{S}(n+\fracμ{m+1}, m)$ 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 \cite{Zhou2014}. Additionally, we extend our analysis to prove a blow-up result with the index $p=\max\{p_{S}(n+\fracμ{m+1}, m), p_{F}((m+1)n+\frac{μ-1-\sqrtδ}{2})\}$ by applying Kato$^{\prime}$s Lemma ( i.e., Lemma \ref{katolemma} ), specifically in the case of $δ=1$.