Uniform Bounds for Solutions to Quasilinear Parabolic Equations

We consider a class of quasilinear parabolic equations whose model is the heat equation corresponding to the p-Laplacian operator, u=Δpu ∑di=1 ∂i( up−2 ∂iu) with p [2, d), on a domain D d of finite measure. We prove that u(t, x) c Dα t−β u0 γr for all t>0,x D and for all initial data u0 Lr(D), provided r is not smaller than a suitable r0, where α, β, γ 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, +∞].