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Exponential decay of solutions of damped nonlinear hyperbolic equations. (English) Zbl 0656.35091
We consider the asymptotic behaviour of solutions to some nonlinear hyperbolic equations with linear damping of the following type \[ (1)\quad u''(t)+Au'(t)+(\alpha +m(| A^{1/2}u(t)|^ 2))Au(t)+\beta A^ 2u(t)=0. \] Here \(\alpha\),\(\beta\geq 0\), A is a selfadjoint and positive operator in a (real) Hilbert space H. The nonnegative function m defined on [0,\(\infty)\) describes the nonlinear effects; m is nondecreasing and \(m(0)=0\).

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
35R20 Operator partial differential equations (= PDEs on finite-dimensional spaces for abstract space valued functions)
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