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Archive for Rational Mechanics and Analysis

, Volume 100, Issue 2, pp 191–206 | Cite as

Decay estimates for some semilinear damped hyperbolic problems

  • A. Haraux
  • E. Zuazua
Article

Abstract

Let Ω be a bounded open domain in R n , gRR a non-decreasing continuous function such that g(0)=0 and h ε L loc 1 (R+; L2(Ω)). Under suitable assumptions on g and h, the rate of decay of the difference of two solutions is studied for some abstract evolution equations of the general form u′′ + Lu + g(u) = h(t,x) as t → + ∞. The results, obtained by use of differential inequalities, can be applied to the case of the semilinear wave equation
$$u_u - \Delta u + g{\text{(}}u_t {\text{) = }}h{\text{ in }}R^ + \times \Omega ,{\text{ }}u = {\text{0 on }}R^ + \times \partial \Omega$$
in R+×Ω, u=0 onR+×∂Ω. For instance if \(g(s) = c\left| s \right|^{p - 1} s + d\left| s \right|^{q - 1} s\) with c, d>0 and 1 < p≦q, (n−2)q≦n+2, then if \(h \in L^\infty (R + ;L^2 (\Omega ))\), all solutions are bounded in the energy space for t≧0 and if u, v are two such solutions, the energy norm of u(t) − v(t) decays like t−1/p−1 as t → + ∞.

Keywords

Neural Network Continuous Function Complex System Wave Equation Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag GmbH & Co 1988

Authors and Affiliations

  • A. Haraux
    • 1
  • E. Zuazua
    • 1
  1. 1.Laboratoire d'Analyse NumériqueUniversité Pierre et Marie CurieParis

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