zbMATH — the first resource for mathematics

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

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