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Stability and decay for a class of nonlinear hyperbolic problems. (English) Zbl 0677.35069
Let $$\Omega$$ be a bounded open domain in $$R^ n$$, V a real Hilbert space, and $$L\in {\mathcal L}(V,V')$$ the unique operator such that $$<Lu,v>=a(u,v)$$ for all (u,v)$$\in V\times V$$. Let $$\{B(t,\cdot)\}_{t\geq 0}$$ be a family of nonlinear operator such that $B(t,v)\in V'\quad \forall t\geq 0,\quad \forall v\in V;$ $B(t,v)\in L^{\exists}_{loc}(R^+,V')\quad \forall v\in L^{\infty}_{loc}(R^+,V).$ Consider the evolution equation $(1)\quad u''(t)+Lu(t)+B(t,u'(t))=0\quad for\quad t\geq 0.$ The aim of this paper is to give some sufficient conditions on $$\{B(t,\cdot)\}_{t\geq 0}$$ for the difference of two arbitrary weak solutions of the nonresonant problem (1) to tend to zero in the energy space $$V\times H$$ as $$t\to +\infty$$ by estimating the rate of decay.
Reviewer: J.Tian

##### MSC:
 35L70 Second-order nonlinear hyperbolic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs 35B35 Stability in context of PDEs