Weak solutions for a system of nonlinear Klein-Gordon equations
Let \(n\geq 3\) be the dimension of \(R^ n\). Let us consider a real number \(\rho\) satisfying the following condition: \[ (1)\quad -1<\rho <4/(n-2). \] Let \(\theta\) and \(\gamma\) be the following real numbers: \[ (2)\quad \theta =2n(\rho +2)/((n-2)(\rho +2)+2n(\rho +1)),\quad \gamma =2n(\rho +2)/((n+2)(\rho +2)-2n(\rho +1)). \] Clearly, \(1/\theta +1/\gamma =1\) and \[ (3)\quad 1<\theta <(\rho +2)/(\rho +1),\quad \gamma >1. \] Then the authors prove: Theorem 1. Let \(\Omega\) be a regular bounded domain of \(R^ n\) and \(\rho\) a real number satisfying the condition (1) if \(n\geq 3\) or \(\rho >- 1\) if \(n=1,2\). Let \[ (4)\quad f_ 1,f_ 2\in L^ 2(0,T;L^ 2(\Omega)), \] \[ (5)\quad u_ 0,v_ 0\in H^ 1_ 0(\Omega)\cap L^ p(\Omega), \] \[ (6)\quad u_ 1,v_ 1\in L^ 2(0,T;L^ 2(\Omega)), \] where \(p=\rho +2\). Then there exists functions u,v: ]0,T[\(\to L^ 2(\Omega)\) such that: \[ (7)\quad u,v\in L^{\infty}(0,T;H^ 1_ 0(\Omega)), \] \[ (8)\quad u',v'\in L^{\infty}(0,T;L^ 2(\Omega))\quad (u'=du/dt), \] \[ (9)\quad uv\in L^{\infty}(0,t;L^{\rho +2}(\Omega)), \] satisfying the nonlinear systems: \[ (10)\quad u''-\Delta u+| v|^{\rho +2} | u|^{\rho} u=f_ 1\quad in\quad L^ 2(0,T;H^{-1}(\Omega)+L^{\theta}(\Omega)), \] \[ (11)\quad v''-\Delta v+| u|^{\rho +2} | v|^{\rho} v=f_ 2\quad in\quad L^ 2(0,T;H^{-1}(\Omega)+L^{\theta}(\Omega)); \] and the initial conditions: \[ (12)\quad u(0)=u_ 0,\quad v(0)=v_ 0;\quad (13)\quad u'(0)=u_ 1,\quad v'(0)=v_ 1. \] Theorem 2. Let u,v: ]0,T[\(\to L^ 2(\Omega)\) be functions in the classes (7), (8) and (9) satisfying from (10) to (13). Then, \(u=v\) provided that \(\rho\geq 0\) in case \(n=1\) or 2; \(u=v\) if \(\rho =0\) in case \(n=3\).
scientific article; zbMATH DE number 4030082