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Weak solutions for a system of nonlinear Klein-Gordon equations. (English) Zbl 0633.35053
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 $$n=3$$.
Reviewer: Y.Ebihara

##### MSC:
 35L70 Second-order nonlinear hyperbolic equations 35L20 Initial-boundary value problems for second-order hyperbolic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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##### References:
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