Finite dimensional attractors for reaction-diffusion equations in $R^n$ with a strong nonlinearity

Messoud Efendiev; Alain Miranville

doi:10.3934/dcds.1999.5.399

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1999, Volume 5, Issue 2: 399-424. Doi: 10.3934/dcds.1999.5.399

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Finite dimensional attractors for reaction-diffusion equations in $R^n$ with a strong nonlinearity

Messoud Efendiev^1, and
Alain Miranville^2,

1.
Freie Universität Berlin, Fachbereich Mathematik und Informatik, Arnimallee 2-6, 14195 Berlin
2.
Université de Poitiers, L3MA, SP2MI, Boulevard 3 - Téléport 2, B. P. 179, 86960 Futuroscope Cedex

Received: March 1998

Revised: June 1998

Published: April 1999

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Abstract

Our aim in this article is to study the long time behavior of a class of reaction-diffusion equations in the whole space for which the nonlinearity depends explicitly on the gradient of the unknown function. We prove the existence of the global attractor and of exponential attractors for the semigroup associated with the equation. We also consider the nonautonomous case, and when the forcing term depends quasiperiodically on the time, we prove the existence of uniform and uniform exponential attractors.

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Mathematics Subject Classification: 35B40, 35B45.

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Finite dimensional attractors for reaction-diffusion equations in $R^n$ with a strong nonlinearity

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