Uniform Bounds for Solutions to Quasilinear Parabolic Equations

By F. Cipriani, G. Grillo. 2001

72 citations recorded by the source corpus.

Abstract 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(|∇u|p−2 ∂iu) with p∈[2, d), on a domain D⊂ R d of finite measure. We prove that |u(t, x)|⩽c |D|α 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, +∞].

Subfields: Applied Mathematics, Computational Theory and Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/f1454efe64d2973cbfa35f9fe1b354ff2121e371

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In Applied Mathematics

In Computational Theory and Mathematics

In Control and Systems Engineering

Citations in the corpus

Fast diffusion flow on manifolds of nonpositive curvature (2008, background): “Certain analogues of this kind of correspondence have been investigated in several nonlinear setting in the papers [10], [3], [4] [5].”
On decay estimates (2009, background): “As noticed in Remark 5.2 ,t he decay estimate (5.34) can be found in [14] when p = 2, r0 ≥ 2 and � is a set of finite measure of R N , N ≥ 2. Moreover, again in [ 14 ]i t is proved that ifp > 2,”
Asymptotics of the porous media equation via Sobolev inequalities (2005, background): “Later on in [12] it was shown that a new family of logarithmic Sobolev inequalities involving the p–energy functional can be applied to study the L–L smoothing and the asymptotics of the nonlinear parabolic equation driven by the p–Laplacian: this is not surprising in view of the fact that the p–ene”