Global existence for a system of quasi-linear wave equations in $3$D satisfying the weak null condition

Abstract

We show global existence of small solutions to the Cauchy problem for a system of quasi-linear wave equations in three space dimensions. The feature of the system lies in that it satisfies the weak null condition, though we permit the presence of some quadratic nonlinear terms which do not satisfy the null condition. Due to the presence of such quadratic terms, the standard argument no longer works for the proof of global existence. To get over this difficulty, we extend the ghost weight method of Alinhac so that it works for the system under consideration. The original theorem of Alinhac for the scalar unknowns is also refined.

Final version. To appear in IMRN

References in corpus (1)

• Combined effects of two nonlinearities in lifespan of small solutions to semi-linear wave equations

Cited by in corpus (3)

• Global nonlinear stability of large dispersive solutions to the Einstein equations
• Global existence for systems of nonlinear wave and Klein-Gordon equations in two space dimensions under a kind of the weak null condition
• Energy decay for small solutions to semilinear wave equations with weakly dissipative structure