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Weak solutions for a system of nonlinear Klein-Gordon equations. (English) Zbl 0633.35053

Let n3 be the dimension of R n . Let us consider a real number ρ satisfying the following condition:

(1)-1<ρ<4/(n-2)·

Let θ and γ be the following real numbers:

(2)θ=2n(ρ+2)/((n-2)(ρ+2)+2n(ρ+1)),γ=2n(ρ+2)/((n+2)(ρ+2)-2n(ρ+1))·

Clearly, 1/θ+1/γ=1 and

(3)1<θ<(ρ+2)/(ρ+1),γ>1·

Then the authors prove:

Theorem 1. Let Ω be a regular bounded domain of R n and ρ a real number satisfying the condition (1) if n3 or ρ>-1 if n=1,2. Let

(4)f 1 ,f 2 L 2 (0,T;L 2 (Ω)),
(5)u 0 ,v 0 H 0 1 (Ω)L p (Ω),
(6)u 1 ,v 1 L 2 (0,T;L 2 (Ω)),

where p=ρ+2. Then there exists functions u,v: ]0,T[L 2 (Ω) such that:

(7)u,vL (0,T;H 0 1 (Ω)),
(8)u ' ,v ' L (0,T;L 2 (Ω))(u ' =du/dt),
(9)uvL (0,t;L ρ+2 (Ω)),

satisfying the nonlinear systems:

(10)u '' -Δu+|v| ρ+2 |u| ρ u=f 1 inL 2 (0,T;H -1 (Ω)+L θ (Ω)),
(11)v '' -Δv+|u| ρ+2 |v| ρ v=f 2 inL 2 (0,T;H -1 (Ω)+L θ (Ω));

and the initial conditions:

(12)u(0)=u 0 ,v(0)=v 0 ;(13)u ' (0)=u 1 ,v ' (0)=v 1 ·

Theorem 2. Let u,v: ]0,T[L 2 (Ω) be functions in the classes (7), (8) and (9) satisfying from (10) to (13). Then, u=v provided that ρ0 in case n=1 or 2; u=v if ρ=0 in case n=3.

Reviewer: Y.Ebihara

MSC:
35L70Nonlinear second-order hyperbolic equations
35L20Second order hyperbolic equations, boundary value problems
35A05General existence and uniqueness theorems (PDE) (MSC2000)
References:
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