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Weak solutions for a system of nonlinear Klein-Gordon equations. (English) Zbl 0633.35053
Let $n\ge 3$ be the dimension of $R\sp n$. 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 $R\sp n$ and $\rho$ a real number satisfying the condition (1) if $n\ge 3$ or $\rho >- 1$ if $n=1,2$. Let $$ (4)\quad f\sb 1,f\sb 2\in L\sp 2(0,T;L\sp 2(\Omega)), $$ $$ (5)\quad u\sb 0,v\sb 0\in H\sp 1\sb 0(\Omega)\cap L\sp p(\Omega), $$ $$ (6)\quad u\sb 1,v\sb 1\in L\sp 2(0,T;L\sp 2(\Omega)), $$ where $p=\rho +2$. Then there exists functions u,v: ]0,T[$\to L\sp 2(\Omega)$ such that: $$ (7)\quad u,v\in L\sp{\infty}(0,T;H\sp 1\sb 0(\Omega)), $$ $$ (8)\quad u',v'\in L\sp{\infty}(0,T;L\sp 2(\Omega))\quad (u'=du/dt), $$ $$ (9)\quad uv\in L\sp{\infty}(0,t;L\sp{\rho +2}(\Omega)), $$ satisfying the nonlinear systems: $$ (10)\quad u''-\Delta u+\vert v\vert\sp{\rho +2} \vert u\vert\sp{\rho} u=f\sb 1\quad in\quad L\sp 2(0,T;H\sp{-1}(\Omega)+L\sp{\theta}(\Omega)), $$ $$ (11)\quad v''-\Delta v+\vert u\vert\sp{\rho +2} \vert v\vert\sp{\rho} v=f\sb 2\quad in\quad L\sp 2(0,T;H\sp{-1}(\Omega)+L\sp{\theta}(\Omega)); $$ and the initial conditions: $$ (12)\quad u(0)=u\sb 0,\quad v(0)=v\sb 0;\quad (13)\quad u'(0)=u\sb 1,\quad v'(0)=v\sb 1. $$ Theorem 2. Let u,v: ]0,T[$\to L\sp 2(\Omega)$ be functions in the classes (7), (8) and (9) satisfying from (10) to (13). Then, $u=v$ provided that $\rho\ge 0$ in case $n=1$ or 2; $u=v$ if $\rho =0$ in case $n=3$.
Reviewer: Y.Ebihara

35L70Nonlinear second-order hyperbolic equations
35L20Second order hyperbolic equations, boundary value problems
35A05General existence and uniqueness theorems (PDE) (MSC2000)
Full Text: DOI
[1] J.Ferreira - G.Perla Menzala,Decay of solutions of a system of nonlinear Klein-Gordon equations (to appear). · Zbl 0617.35073
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