Smoothing effects for the filtration equation with different powers

Abstract

Abstract. We study the nonlinear diffusion equation ut = ∆φ(u) on general Euclidean domains, with homogeneous Neumann boundary conditions. We assume that φ ′ (u) is bounded from below by |u| m 1 −1 for small |u| and by |u| m 2 −1 for large |u|, the two exponents m 1 , m 2 being possibly different and larger than one. The equality case corresponds to the well-known porous medium equation. We establish sharp short-and long-time L q 0 -L ∞ smoothing estimates: similar issues have widely been investigated in the literature in the last few years, but the Neumann problem with different powers had not been addressed yet. This work extends some previous results in many directions.

Wasserstein stability of porous medium-type equations on manifolds with Ricci curvature bounded below

“…As concerns L 1 -continuity, as a first step we point out that it could be proved by means of an alternative construction of weak energy solutions that takes advantage of time-discretization and the Crandall-Liggett Theorem: see e.g. [16,Remark 3.7]. More comments on such a construction will be made in Remark 4.8 at the end of this section.…”

“…To this end we need to ask some crucial extra assumptions: the validity of the Sobolev-type inequality (H2) and a bound from below on the degeneracy of P given by the left-hand side of (H4). The proof is largely inspired from [16,Section 4], where a Moser-type iteration is exploited (see also references quoted therein); nevertheless, here we are also interested in keeping track of the dependence of the multiplying constants on m as m ↓ 1. Proposition 4.3 (Smoothing effect).…”

Sobolev-type inequalities on Cartan-Hadamard manifolds and applications to some nonlinear diffusion equations

“…In the last decades, several results that connect the validity of functional inequalities of Sobolev, log-Sobolev or Poincaré type with smoothing effects for nonlinear diffusion equations (mostly modeled on the porous medium equation) have been established: see [25] for weighted porous medium equations in the presence of weights and Poincaré inequalities, [23,18] for optimal short and long-time smoothing estimates for porous medium equations (or the more general filtration equation) on Euclidean domains in the case of homogeneous Neumann problems, and the above-mentioned paper [24] for similar analyses focused on the consequences of the validity of families of Sobolev-type inequalities of the type of (1.4). Previous results in this direction, having already in mind the manifold case, can be found in [10].…”