Attractors for Nonautonomous Parabolic Equations without Uniqueness

Cung The Anh,Nguyen Dinh Binh

doi:10.1155/2010/103510

Abstract

Using the theory of uniform global attractors of multivalued semiprocesses, we prove the existence of a uniform global attractor for a nonautonomous semilinear degenerate parabolic equation in which the conditions imposed on the nonlinearity provide the global existence of a weak solution, but not uniqueness. The Kneser property of solutions is also studied, and as a result we obtain the connectedness of the uniform global attractor.

Highlights

The understanding of the asymptotic behavior of dynamical systems is one of the most important problems of modern mathematical physics
Using the theory of uniform global attractors of multivalued semiprocesses, we prove the existence of a uniform global attractor for a nonautonomous semilinear degenerate parabolic equation in which the conditions imposed on the nonlinearity provide the global existence of a weak solution, but not uniqueness
One way to attack the problem for a dissipative dynamical system is to consider its global attractor

Summary

Introduction

The understanding of the asymptotic behavior of dynamical systems is one of the most important problems of modern mathematical physics. The existence of the global attractor has been derived for a large class of PDEs see 1, 2 and references therein , for both autonomous and nonautonomous equations These researches may not be applied to a wide class of problems, in which solutions may not be unique. The existence of global attractor and the Kneser property have been derived for some classes of partial differential equations without uniqueness, to the best of our knowledge, little seems to be known for nonautonomous degenerate equations. Under conditions (H1)–(H3), problem 1.1 defines a family of strict multivalued semiprocesses {Uσ}σ∈Σ, which possesses a uniform global compact connected attractor A in L2 Ω. We obtain the connectedness of the uniform global attractor

Function Space and Operator

Existence of Uniform Global Attractors

L2 Ω un

The Kneser Property and Connectedness of the Attractors