Decay Estimates for some Semilinear Damped Hyperbolic Problems A. HARAUX & E. ZUAZUA Communicated by H. BREZIS Abstract Let ~ be a bounded open domain in R n, g: R -+ R a non-decreasing continuous function such that g(o) = 0 and h E L~o~(R+;L2(g2)). Under suitable assump- tions 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-+ -k co. The results, obtained by use of differential inequalities, can be applied to the case of the semilinear wave equation utt-Au+g(ut)=h in R+• f2, u=0 on R+• For instance if g(s)=clsE,-~s+dlslq-ls with c,d>O and l<p<~q, (n - 2) q ~ n -k 2, then if h E L~(R +; L2(X2)), 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 -~/(p-I) as t-+ + co. Introduction and notation Let ~2 be a bounded open domain in R n and H = L2(~Q) with norm and inner product respectively denoted by 1. ] and (., .). Let V be a real Hilbert space such that V Q H with dense and continuous imbedding. We denote by II'll the norm on V, by a(., .) the inner product on V and by L E .s V') the unique operator such that <Lu, v> = a(u, v) for all (u, v) E V• V. Let g: R ~ R be a non-decreasing continuous function such that g(0) = 0 and h E L~or H). We consider the nonlinear partial differential equation of evolution u'" + Lu + g(u') = h in R + • ~, (o.1) uE C(R+; V)/~ C'(R+; H).