On the global strong solutions of coupled Klein-Gordon-Schrödinger equations
type: Item
instance of: scholarly article
title: On the global strong solutions of coupled Klein-Gordon-Schrödinger equations
zbMATH Open document ID: 0657.35034
author: Nakao Hayashi
author: Wolf von Wahl
published in: Journal of the Mathematical Society of Japan
publication date: 1987-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 35G30
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 4073533
zbMATH Keywords: positive selfadjoint
zbMATH Keywords: order 2
zbMATH Keywords: Dirichlet-zero conditions
zbMATH Keywords: Klein-Gordon- Schrödinger
zbMATH Keywords: Yukawa coupling
zbMATH Keywords: complex scalar neucleon field
zbMATH Keywords: real scalar meson field
zbMATH Keywords: existence
zbMATH Keywords: global strong solutions
zbMATH Keywords: fractional order Besov spaces
zbMATH Keywords: nonlinear interpolation
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/03930489
OpenAlex ID: W1982828055
DOI: 10.2969/JMSJ/03930489
description: scientific article; zbMATH DE number 4073533