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Zbl 0654.35070
Haraux, A.; Zuazua, E.
Decay estimates for some semilinear damped hyperbolic problems. (English)
[J] Arch. Ration. Mech. Anal. 100, No.2, 191-206 (1988). ISSN 0003-9527; ISSN 1432-0673/e

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 $$ u\sb{tt}-\Delta u+g(u\sb t)=h\quad in\quad (t,x)\in {\bbfR}\quad +\times \Omega, $$ where $\Omega$ $\subset {\bbfR}\sp n $is a bounded domain and $u=0$ on ${\bbfR}$ $+\times \partial \Omega$. If e.g. $g(s)=c\vert s\vert\sp{p-1}s+d\vert s\vert\sp{q-1}s$ with $c,d>0$ and $1<p\le q\le (n+2)/(n-2)$ then the difference of two solutions is shown to decay like $t\sp{-1/(p-1)}$ 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.
[H.Pecher]

MSC 2000:: *35L70 Second order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions of PDE
35L75 Nonlinear hyperbolic PDE of higher $(>2)$ order
35B05 General behavior of solutions of PDE

Keywords: asymptotic behaviour; damped; semilinear; decay; energy functional

Cited in: Zbl 1053.35021

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