Archive for Rational Mechanics and Analysis

, Volume 100, Issue 2, pp 191–206

Decay estimates for some semilinear damped hyperbolic problems


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

DOI: 10.1007/BF00282203

Cite this article as:
Haraux, A. & Zuazua, E. Arch. Rational Mech. Anal. (1988) 100: 191. doi:10.1007/BF00282203


Let Ω be a bounded open domain in Rn, gRR a non-decreasing continuous function such that g(0)=0 and h ε Lloc1(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 → + ∞.

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© Springer-Verlag GmbH & Co 1988