Semilinear hyperbolic systems violating the null condition

TL;DR

This paper proves global existence for certain semilinear wave systems in three dimensions that violate the null condition, introducing a new structural condition and revealing unique asymptotic behaviors.

Contribution

It introduces a novel structural condition related to the weak null condition for systems violating the null condition, and analyzes their long-term behavior.

Findings

Small data global existence under the new condition

Asymptotic dissipation of only one component in two-component systems

One component behaves like a free wave at large times

Abstract

We consider systems of semilinear wave equations in three space dimensions with quadratic nonlinear terms not satisfying the null condition. We prove small data global existence of the classical solution under a new structural condition related to the weak null condition. For two-component systems satisfying this condition, we also observe a new kind of asymptotic behavior: Only one component is dissipated and the other one behaves like a free solution in the large time.