Entropy Dissipation Methods for Degenerate ParabolicProblems and Generalized Sobolev Inequalities |
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Authors: | J A Carrillo A Jüngel P A Markowich G Toscani A Unterreiter |
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Institution: | Universidad de Granada, Spain, ES Universit?t Konstanz, Germany, DE Universit?t Wien, Austria, AT Universitá di Pavia, Italy, IT Universit?t Kaiserslautern, Germany, DE
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Abstract: | We analyse the large-time asymptotics of quasilinear (possibly) degenerate parabolic systems in three cases: 1) scalar problems
with confinement by a uniformly convex potential, 2) unconfined scalar equations and 3) unconfined systems. In particular
we are interested in the rate of decay to equilibrium or self-similar solutions. The main analytical tool is based on the
analysis of the entropy dissipation. In the scalar case this is done by proving decay of the entropy dissipation rate and
bootstrapping back to show convergence of the relative entropy to zero. As by-product, this approach gives generalized Sobolev-inequalities,
which interpolate between the Gross logarithmic Sobolev inequality and the classical Sobolev inequality. The time decay of
the solutions of the degenerate systems is analyzed by means of a generalisation of the Nash inequality. Porous media, fast
diffusion, p-Laplace and energy transport systems are included in the considered class of problems. A generalized Csiszár–Kullback inequality
allows for an estimation of the decay to equilibrium in terms of the relative entropy.
(Received 11 October 2000; in revised form 13 March 2001) |
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Keywords: | 2000 Mathematics Subject Classification: 35B40 35K55 35K65 35Q35 35R45 |
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