On the global strong solutions of coupled Klein-Gordon-Schrödinger equations
We consider the following system of equations in three space dimensions: \[ (1)\quad id\psi /dt-A_ 1\psi =-\psi \phi,\quad d^ 2\phi /dt^ 2+A_ 2\phi =| \psi |^ 2. \] Where \(A_ 1\) and \(A_ 2\) denote positive selfadjoint elliptic operators of order 2 with Dirichlet- zero conditions over a bounded or unbounded domain \(\Omega \subset {\mathbb{R}}^ 3\). If \(A_ 1=-\Delta\) and \(A_ 2=-\Delta +I\), where \(\Delta\) denotes the spatial Laplacian, (1) is the so called Klein- Gordon-Schrödinger (K-G-S) equation with Yukawa coupling in which \(\psi\) describes complex scalar neucleon field and \(\phi\) describes real scalar meson field. Our purpose in this paper is to show the existence of global strong solutions to (1) which include the K-G-S equations, under the same initial conditions as in the work by \textit{J. B. Baillon} and \textit{J. M. Chadam} [North-Holland Math. Stud. 30, 37-44 (1978; Zbl 0404.35084)] and the same boundary conditions as in the work by \textit{I. Fukuda} and \textit{M. Tsutsumi} [J. Math. Anal. Appl. 66, 358-378 (1978; Zbl 0396.35082)]. We get the result by using estimates of the nonlinearity in fractional order Besov spaces, nonlinear interpolation theorem obtained by W. von Wahl and the inequality of H. Brezis and T. Gallouet.
scientific article; zbMATH DE number 4073533