and

We consider a class of quasilinear parabolic equations whose model is the heat equation corresponding to the p-Laplacian operator, u̇=Dpu:=;di=1 “i(|Nu|p−2 “iu) with p ¥ [2, d), on a domain D … Rd of finite measure. We prove that |u(t, x) | [ c |D|a t−b ||u0 || c r for all t> 0, x ¥ D and for all initial data u0 ¥ L r(D), provided r is not smaller than a suitable r0, where a, b, c are positive constants explicitly computed in terms of d, p, r. The nonlinear cases associated with the case p=2 display exactly the same contractivity properties which hold for the linear heat equation. We also show that the nonlinear evolution considered is contractive on any Lq space for any q ¥ [2,+.]. © 2001 Elsevier Science Key Words: logarithmic Sobolev inequalities; quasilinear parabolic equations