Global Existence for Coupled 3-D Nonlinear Wave and Klein-Gordon Equations with Large Derivatives of Initial Data

TL;DR

The paper proves global-in-time existence for the 3D coupled wave–Klein-Gordon system with quadratic nonlinearity $Q_0(u,w)$, allowing large derivatives in the wave data. The authors adapt a refined vector-field bootstrap combined with conformal and ghost-weight energy estimates, leveraging the null-structure of $Q_0$, commutator identities, and Sobolev/interpolation tools to balance large wave data against Klein-Gordon decay. They establish precise decay rates: $| abla u(t,x)| \lesssim \langle t+r\rangle^{-1} \langle t-r\rangle^{-1/8}$ and $|w|,|\partial w| \lesssim \epsilon \langle t+r\rangle^{-3/2}$, under explicit initial-data bounds. This extends prior wave–KG results by tolerating large wave-derivative data and clarifies the role of the scaling vector field and null forms in controlling the quadratic nonlinearity.

Abstract

We consider the Cauchy problem of coupled 3-D wave and Klein-Gordon equations with a quadratic form of nonlinearity. We show global existence under several conditions, including large derivative data for wave equations and the null conditions.

Theorems & Definitions (35)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Corollary 2.6
• proof