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Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities. (English) Zbl 0984.35027
This paper is concerned with the large-time asymptotics of solutions of quasilinear (including degenerate) parabolic systems. The covered scenarios include i) scalar equations with confinement by a uniformly convex potential, ii) scalar equations without confinement, and iii) systems without confinement. The analysis provides some decay rates to equilibria or to self-similar solutions. The authors employ entropy dissipation estimates and, in the degenerate cases, generalized Nash inequalities as tools. Generalized Csizsar-Kullback inequalities provide the link between the decay rates to equilibrium and relative entropy disspation. The analysis provides (as additional results) generalized Sobolev inequalities which are interpolation inequalities between Gross’ logarithmic Sobolev inequalities and the classical Sobolev inequalities. Several classical examples are discussed: The porous media equations; fast diffusion; the \(p\)-Laplace equation; energy transport. And higher-order parabolic equations arising in interface fluctuations, droplet breakup and thin vicsous films. At 82 condensed and technical pages the paper is a tough read, but well written and rich with information.

MSC:
35B40 Asymptotic behavior of solutions to PDEs
35K65 Degenerate parabolic equations
35Q35 PDEs in connection with fluid mechanics
35R45 Partial differential inequalities and systems of partial differential inequalities
35K55 Nonlinear parabolic equations
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