USA-Chile Workshop on Nonlinear Analysis,
Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 189-202.

Decay rates for solutions of degenerate parabolic systems

Ansgar Jungel, Peter A. Markowich, & Giuseppe Toscani

Explicit decay rates for solutions of systems of degenerate parabolic equations in the whole space or in bounded domains subject to homogeneous Dirichlet boundary conditions are proven. These systems include the scalar porous medium, fast diffusion and p-Laplace equation and strongly coupled systems of these equations. For the whole space problem, the (algebraic) decay rates turn out to be optimal. In the case of bounded domains, algebraic and exponential decay rates are shown to hold depending on the nonlinearities. The proofs of these results rely on the use of the entropy functional together with generalized Nash inequalities (for the whole space problem) or Poincare inequalities (for the bounded domain case).

Published January 8, 2001.
Math Subject Classifications: 35K65, 35K55, 35B40.
Key Words: Explicit decay rates, long-time behavior of solutions, algebraic decay, exponential decay, degenerate parabolic equations, Nash inequality.

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Ansgar Jungel
Fachbereich Mathematik und Statistik,
Universit\"at Konstanz,
78457 Konstanz, Germany
Peter A. Markowich
Institut f\"ur Mathematik
Universitat Wien
1090 Wien, Austria
Giuseppe Toscani
Dipartimento di Matematica
Universita di Pavia
27100 Pavia, Italy

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