Critical curve for weakly coupled system of semilinear Euler-Poisson-Darboux-Tricomi equations

TL;DR

This paper determines the critical curve for a weakly coupled system of semilinear Euler-Poisson-Darboux-Tricomi equations, identifying conditions for global existence versus blow-up based on damping dominance and nonlinear exponents.

Contribution

It introduces a new critical curve formula for the system and establishes global existence results using novel test functions and previous linear estimates.

Findings

Critical curve for blow-up and global existence identified.

Global solutions exist for small initial data when damping dominates.

Blow-up occurs when the critical function is non-negative.

Abstract

This paper investigates a weakly coupled system of semilinear Euler-Poisson-Darboux-Tricomi equations (EPDTS) with power-type nonlinear terms. More precisely, in the case where the damping terms dominate over the mass terms, the critical curve in the $p-q$ plane that delineates the threshold between global existence and blow-up for the EPDTS is given by \begin{equation*} \Gamma_m(n,p,q,\beta_1,\beta_2)=0, \end{equation*} where $\Gamma_m$ is defined by (\ref{gammam}). Through the construction of new test functions, the blow-up problem is addressed when $\Gamma_m(n,p,q,\beta_1,\beta_2)\geq0$. Based on the $(L^1\cap L^2)-L^2$ estimates of the solution to the corresponding linear equation established in our previous work \cite{LiGuo2025}, we derive the global existence of solutions with small initial data when $\Gamma_m(n,p,q,\beta_1,\beta_2)<0$, provided that the damping terms prevail over the mass terms.