Uniform bounds for solutions to quasilinear parabolic equations

We consider a class of quasilinear parabolic equations whose model is the h eat equation corresponding to the p-Laplacian operator, u = Delta (p)u: = S igma (d)(i=1) partial derivative (i)(\ delu \ (p similar to2) partial deriv ative (i)u) with p epsilon [2, d), on a domain D subset of R-d of finite me asure. We prove that \u(t, x)\ less than or equal to c \D \ (alpha) t(-beta ) parallel tou(0)parallel to (7)(r) for all t > 0, x epsilon D and for all initial data u(0) epsilon L'(D), provided r is not smaller than a suitable r(0) where alpha, beta, gamma are positive constants explicitly computed in terms of d, p, r. The nonlinear cases associated with the case p = 2 displ ay exactly the same contractivity properties which hold for the linear heat equation. We also show that the nonlinear evolution considered is contract ive on any L-q space for any q epsilon [2, + infinity]. (C) 2001 Academic P ress.