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Decay estimates for some semilinear damped hyperbolic problems

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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 1loc (R+; L 2(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 on R +×∂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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Communicated by H. Brezis

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Haraux, A., Zuazua, E. Decay estimates for some semilinear damped hyperbolic problems. Arch. Rational Mech. Anal. 100, 191–206 (1988).

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