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

Yuequn Li, Fei Guo

TL;DR

The paper identifies a precise critical curve $\Gamma_m(n,p,q,\beta_1,\beta_2)$ governing global existence versus finite-time blow-up for a weakly coupled system of semilinear Euler-Poisson-Darboux-Tricomi equations with damping and mass terms. It proves blow-up for $\Gamma_m\ge0$ using a novel test-function method adapted to the Gellerstedt operator and establishes global existence for small data when $\Gamma_m<0$ by exploiting $(L^1\cap L^2)-L^2$ estimates of the associated linear problem and a fixed-point argument in high- and low-regularity spaces. The results reveal that damping terms can dominate mass terms, yielding a parabolic-like critical curve that depends on the interplay of $\mu_i,\nu_i$, and $m$, and show explicit decay rates under various regularity assumptions. The work extends classical single-equation and wave-system results to a coupled Euler-Poisson-Darboux-Tricomi setting, highlighting the nontrivial interaction between the two nonlinearities and the PDE's mixed-type nature. Overall, the paper advances understanding of threshold phenomena in weakly coupled, damped-elliptic-hyperbolic systems and provides a robust framework for future refinements and applications.

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*} Γ_m(n,p,q,β_1,β_2)=0, \end{equation*} where $Γ_m$ is defined by (\ref{gammam}). Through the construction of new test functions, the blow-up problem is addressed when $Γ_m(n,p,q,β_1,β_2)\geq0$. 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 $Γ_m(n,p,q,β_1,β_2)<0$, provided that the damping terms prevail over the mass terms.

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

TL;DR

The paper identifies a precise critical curve governing global existence versus finite-time blow-up for a weakly coupled system of semilinear Euler-Poisson-Darboux-Tricomi equations with damping and mass terms. It proves blow-up for using a novel test-function method adapted to the Gellerstedt operator and establishes global existence for small data when by exploiting estimates of the associated linear problem and a fixed-point argument in high- and low-regularity spaces. The results reveal that damping terms can dominate mass terms, yielding a parabolic-like critical curve that depends on the interplay of , and , and show explicit decay rates under various regularity assumptions. The work extends classical single-equation and wave-system results to a coupled Euler-Poisson-Darboux-Tricomi setting, highlighting the nontrivial interaction between the two nonlinearities and the PDE's mixed-type nature. Overall, the paper advances understanding of threshold phenomena in weakly coupled, damped-elliptic-hyperbolic systems and provides a robust framework for future refinements and applications.

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 plane that delineates the threshold between global existence and blow-up for the EPDTS is given by \begin{equation*} Γ_m(n,p,q,β_1,β_2)=0, \end{equation*} where is defined by (\ref{gammam}). Through the construction of new test functions, the blow-up problem is addressed when . Based on the 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 , provided that the damping terms prevail over the mass terms.

Paper Structure

This paper contains 11 sections, 19 theorems, 225 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Blow-up result
  4. Global existence of solutions with high regularity of initial data
  5. The estimates for the corresponding linear equation
  6. The proof of Theorem \ref{['theorem11']}
  7. The proof of (\ref{['key1']})
  8. The proof of (\ref{['key2']})
  9. The proof of Theorem \ref{['theorem12']}
  10. The proof of Theorem \ref{['theorem13']}
  11. Global existence of solutions with low regularity of initial data

Key Result

Proposition 2.1

(Local existence). Let $n\geq1, m>-1$, $\mu_i,\nu_i^2\geq0$, $i=1,2$ such that $\delta_1,\delta_2>0$, where $\delta_i,i=1,2$ are defined by (deltai). Suppose the initial data $(u_0,u_1,v_0,v_1)\in D^\sigma\times D^\sigma$ with some $\sigma>0$ and satisfy the support condition (compact1). Then there exist a $T>1$ and a unique solution to (eqs). Moreover, $(u,v)$ satisfies the support property (co

Figures (1)

  • Figure 1: Diagram 1

Theorems & Definitions (45)

  • Definition 2.1
  • Remark 2.1
  • Proposition 2.1
  • Theorem 2.1
  • Remark 2.2
  • Example 2.1
  • Example 2.2
  • Theorem 2.2
  • Remark 2.3
  • Example 2.3
  • ...and 35 more