On global attraction to solitary waves. Klein-Gordon equation with mean field interaction at several points
In this article the author considers the \(U(1)\)-invariant Klein-Gordon equation with the mean field self-interaction at \(N\)-points NEWLINE\[NEWLINE {\ddot \psi}(x, t)=\Delta \psi(x, t)-m^2\psi(x, t)+\sum_{I=1}^N\rho_I(x)F_I(\langle\rho_I, \psi(\cdot, t)\rangle),\quad x\in \mathbb R^n, \;n\geq 3, \;t\in \mathbb R, \leqno{(1)} NEWLINE\]NEWLINE where NEWLINE\[NEWLINE \langle\rho_I, \psi(\cdot, t)\rangle=\int_{\mathbb R^n}\overline{\rho}_I(x)\psi(x, t) d^n x, NEWLINE\]NEWLINE \(\rho_I(x)=\rho(x-X_I)\), \(X_I\in \mathbb R^n\), \(1\leq I\leq N\), \(\rho(x)\) is a nonzero smooth real-valued function from the Schwartz class.NEWLINENEWLINEThe author gives conditions for \(\rho\) and \(F\) under which the Cauchy problem for the equation \((1)\) is globally well-posed with global attraction to solitary waves.
scientific article; zbMATH DE number 6033459