Invariants, conservation laws and time decay for a nonlinear system of Klein–Gordon equations with Hamiltonian structure
Volume 33 / 2006
Abstract
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 $$\displaylines{ F_1(|u|^2, |v|^2)|u|^2+F_2(|u|^2, |v|^2)|v|^2-F(|u|^2, |v|^2)\cr \ge a F(|u|^2, |v|^2)\ge 0,\ \quad a>0.}$$
