We study the nonlinear diffusion equation ut=Δϕ(u) on general Euclidean domains, with homogeneous Neumann boundary conditions. We assume that ϕ′(u) is bounded from below by |u|m1−1 for small |u| and by |u|m2−1 for large |u|, the two exponents m1,m2 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 Lq0–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.

Smoothing effects for the filtration equation with different powers

FOTACHE, ALIN RAZVAN;MURATORI, MATTEO
2017

Abstract

We study the nonlinear diffusion equation ut=Δϕ(u) on general Euclidean domains, with homogeneous Neumann boundary conditions. We assume that ϕ′(u) is bounded from below by |u|m1−1 for small |u| and by |u|m2−1 for large |u|, the two exponents m1,m2 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 Lq0–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.
Filtration equation; Gagliardo–Nirenberg inequalities; Neumann problem; Porous medium equation; Smoothing effects; Sobolev inequalities; Analysis
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11311/1031186
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