## Abstract

Let *Ω* be a bounded open domain in **R**
^{n}, *g* ∶ **R** → **R** a non-decreasing continuous function such that g(0)=0 and *h* ε *L*
^{1}_{loc}
(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

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). https://doi.org/10.1007/BF00282203

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DOI: https://doi.org/10.1007/BF00282203