Global strong solutions of the coupled Klein-Gordon-Schr\"{o}dinger equations

TL;DR

This paper proves the existence and uniqueness of global strong solutions for the coupled Klein-Gordon-Schrödinger equations in dimensions up to four, using a novel approximation and compactness method.

Contribution

It introduces a new approach to construct global strong solutions without relying on the Brezis-Gallouet technique or compactness arguments.

Findings

Established global existence and uniqueness of solutions in specified Sobolev spaces.

Developed a method based on Yosida approximation for solution construction.

Proved solutions form bounded and convergent sequences in relevant function spaces.

Abstract

We study the initial-boundary value problem for the coupled Klein-Gordon-Schr\"{o}dinger equations in a domain in $\mathbb R^N$ with $N \leq 4$. Under natural assumptions on the initial data, we prove the existence and uniqueness of global solutions in $H^2 \oplus H^2 \oplus H^1$. The method of the construction of global strong solutions depends on the proof that solutions of regularized systems by the Yosida approximation form a bounded sequence in $H^2 \oplus H^2 \oplus H^1$ and a convergent sequence in $H^1 \oplus H^1 \oplus L^2$. The method of proof is independent of the Brezis-Gallouet technique and a compactness argument.