## 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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### References:

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