Invariants, conservation laws and time decay for a nonlinear system of Klein–Gordon equations with Hamiltonian structure

Changxing Miao ¹   ; Youbin Zhu ²

¹ Institute of Applied Physics and Computational Mathematics P.O. Box 8009 Beijing 100088, China
² Department of Mathematics Xidian University P.O. Box 245 Xi'an, Shanxi 710071, China

Applicationes Mathematicae, Tome 33 (2006) no. 3-4, pp. 323-344

Résumé

We discuss invariants and conservation laws for a nonlinear system of Klein–Gordon equations with Hamiltonian structure $$\cases{u_{tt}-{\mit\Delta} u+m^2u=-F_1(|u|^2,|v|^2)u,\cr v_{tt}-{\mit\Delta} v+m^2v=-F_2(|u|^2, |v|^2)v}$$ for which there exists a function $F(\lambda, \mu)$ such that $$\frac{\partial F(\lambda,\mu)}{\partial \lambda}=F_1(\lambda,\mu),\quad \frac{\partial F(\lambda,\mu)}{\partial \mu}=F_2(\lambda,\mu).$$ Based on Morawetz-type identity, we prove that solutions to the above system decay to zero in local $L^2$-norm, and local energy also decays to zero if the initial energy satisfies $$\displaylines{ E(u, v, \Bbb R^n, 0)={}\frac12\int_{\Bbb R^n}(|\nabla u(0)|^2+|u_t(0)|^2+m^2|u(0)|^2 +|\nabla v(0)|^2\cr {} +|v_t(0)|^2+m^2|v(0)|^2+F(|u(0)|^2, |v(0)|^2))\,dx\infty,\cr}$$ and $$\displaylines{ F_1(|u|^2, |v|^2)|u|^2+F_2(|u|^2, |v|^2)|v|^2-F(|u|^2, |v|^2)\cr \ge a F(|u|^2, |v|^2)\ge 0,\ \quad a>0.}$$

Détail

DOI : 10.4064/am33-3-7

Keywords: discuss invariants conservation laws nonlinear system klein gordon equations hamiltonian structure cases mit delta f mit delta f which there exists function lambda frac partial lambda partial lambda lambda quad frac partial lambda partial lambda based morawetz type identity prove solutions above system decay zero local norm local energy decays zero initial energy satisfies displaylines bbb frac int bbb nabla nabla infty displaylines f quad