c  Birkh¨auser Verlag, Basel, 2006 NoDEA Nonlinear differ. equ. appl. 13 (2006) 435–445 1021–9722/06/040435–11 Decay rate of the range component of solutions to some semilinear evolution equations A. HARAUX Laboratoire Jacques-Louis Lions boite courrier 187 Universit´ e Pierre et Marie Curie 4 place Jussieu 75252 Paris Cedex 05, France e-mail: haraux@ann.jussieu.fr Abstract. We examine the rate of decay to 0, as t → +∞., of the projection on the range of A of the solutions of an equation of the form u  + Au + |u| p−1 u = 0 or u  + u  + Au + |u| p−1 u = 0 in a bounded domain of R N , where A = −∆ with Neumann boundary conditions or A = −∆ − λ1I with Dirichlet boundary conditions. In general this decay is much faster than the decay of the projection on the kernel; it is often exponential, but apparently not always. 2000 Mathematics Subject Classification: 35B35, 34 C40, 34D 23. Key words: Decay rate, range component, evolution equations. 1 Introduction In a recent joint paper [3] the following semilinear heat equation  u t − ∆u + c|u| p−1 u − λu =0 on R + × Ω u =0 on R + × ∂ Ω (1.1) where c> 0,p> 1,λ> 0 and Ω is a bounded smooth domain of R N was studied from the point of view of the decay rate at infinity. We established that any solution u of (1.1) tending to 0 as t → +∞ satisfies the estimate u(t) L ∞ (Ω) ≤ Kt − 1 p−1 In the special case λ = λ 1 (Ω) DOI10.1007/s00030-006-4019-7