Improvement on the blow-up for a weakly coupled wave equations with scale-invariant damping and mass and time derivative nonlinearity

TL;DR

This paper improves the understanding of blow-up regions and lifespan estimates for a coupled wave system with damping, mass, and time-derivative nonlinearity, using refined techniques that also simplify previous proofs.

Contribution

It provides a better characterization of blow-up regions and extends techniques to other systems, simplifying existing proofs in related wave equation studies.

Findings

Enhanced blow-up region characterization

Improved lifespan estimates for solutions

Simplified proof techniques applicable to related systems

Abstract

An improvement of [18] on the blow-up region and the lifespan estimate of a weakly coupled system of wave equations with damping and mass in the scale-invariant case and with time-derivative nonlinearity is obtained in this article. Indeed, thanks to a better understanding of the dynamics of the solutions, we give here a better characterization of the blow-up region. Furthermore, the techniques used in this article may be extended to other systems and interestingly they simplify the proof of the blow-up result in [3] which is concerned with the single wave equation in the same context as in the present work.