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

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.