Blow-up result for a weakly coupled system of two Euler-Poisson-Darboux-Tricomi equations with time derivative nonlinearity

TL;DR

This paper investigates the blow-up behavior of solutions to a coupled system of wave equations with time-dependent propagation speeds, nonlinearities involving time derivatives, and specific damping and mass terms, establishing new blow-up conditions and lifespan estimates.

Contribution

It introduces new blow-up regions and lifespan estimates for a coupled wave system with Tricomi and time-derivative nonlinearities, considering time-dependent propagation speeds.

Findings

New blow-up region identified for the system.

Lifespan estimates for solutions are derived.

Conditions on parameters for blow-up are established.

Abstract

We study in this article the blow-up of solutions to a coupled semilinear wave equations which are characterized by linear damping terms in the \textit{scale-invariant regime}, time-derivative nonlinearities, mass terms and Tricomi terms. The latter are specifically of great interest from both physical and mathematical points of view since they allow the speeds of propagation to be time-dependent ones. However, we assume in this work that both waves are propagating with the same speeds. Employing this fact together with other hypotheses on the aforementioned parameters (mass and damping coefficients), we obtain a new blow-up region for the system under consideration, and we show a lifespan estimate of the maximal existence time.