A non-linear damping structure and global stability of wave-Klein-Gordon coupled system in $\mathbb{R}^{3+1}$

TL;DR

This paper proves the global existence of solutions for a wave-Klein-Gordon coupled system in 3+1 dimensions, showing that specific nonlinear damping effects are key to stability, using hyperboloidal foliation and vector field methods.

Contribution

It introduces conditions on nonlinear coefficients that induce damping, enabling the proof of global solutions for the coupled system in Minkowski spacetime.

Findings

Established global existence of solutions in 3+1 dimensions.

Identified damping effects from nonlinear coefficients as crucial for stability.

Applied hyperboloidal foliation and vector field methods for the proof.

Abstract

This paper establishes the global existence of solutions for a class of wave-Klein-Gordon coupled systems with specific nonlinearities in 3+1-dimensional Minkowski spacetime. The study demonstrates that imposing certain constraints on the coefficients of these specific nonlinear terms induces a damping effect within the system, which is crucial for proving the global existence of solutions. The proof is conducted within the framework of a bootstrap argument, primarily employing the hyperboloidal foliation method and the vector field method.