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Decay estimates for some semilinear damped hyperbolic problems. (English) Zbl 0654.35070

The authors consider the asymptotic behaviour of solutions to a class of nonlinear damped hyperbolic problems at $t\to +\infty$. A typical example is the semilinear wave equation

${u}_{tt}-{\Delta }u+g\left({u}_{t}\right)=h\phantom{\rule{1.em}{0ex}}in\phantom{\rule{1.em}{0ex}}\left(t,x\right)\in ℝ\phantom{\rule{1.em}{0ex}}+×{\Omega },$

where ${\Omega }$ $\subset {ℝ}^{n}$is a bounded domain and $u=0$ on $ℝ$ $+×\partial {\Omega }$. If e.g. $g\left(s\right)={c|s|}^{p-1}s+d{|s|}^{q-1}s$ with $c,d>0$ and $1 then the difference of two solutions is shown to decay like ${t}^{-1/\left(p-1\right)}$ as $t\to +\infty$. The method of proof is to use suitable functionals related to the energy functional and to show that they fulfill an ordinary differential inequality the solution of which has the desired asymptotic property. This method works in the autonomous as well as in the nonautonomous case. If g is a single power nonlinearity these results were partly known before by M. Nakao.

Reviewer: H.Pecher

##### MSC:
 35L70 Nonlinear second-order hyperbolic equations 35B40 Asymptotic behavior of solutions of PDE 35L75 Nonlinear hyperbolic PDE of higher $\left(>2\right)$ order 35B05 Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
##### Keywords:
asymptotic behaviour; damped; semilinear; decay; energy functional
##### References:
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