Asymptotic behaviour of nonlocal reaction–diffusion equations

The existence of a global attractor in L 2 ( Ω ) is established for a reaction–diffusion equation on a bounded domain Ω in R d with Dirichlet boundary conditions, where the reaction term contains an operator F : L 2 ( Ω ) → L 2 ( Ω ) which is nonlocal and possibly nonlinear. Existence of weak soluti...

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Bibliographic Details
Published in:Nonlinear analysis 2010-11, Vol.73 (9), p.3044-3057
Main Authors: Anguiano, M., Kloeden, P.E., Lorenz, T.
Format: Article
Language:English
Subjects:
Bounded absorbing sets
Exact sciences and technology
Global attractor
Mathematical analysis
Mathematics
Multivalued flow
Nonlocal reaction term
Partial differential equations
Reaction–diffusion equation
Sciences and techniques of general use
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Summary:The existence of a global attractor in L 2 ( Ω ) is established for a reaction–diffusion equation on a bounded domain Ω in R d with Dirichlet boundary conditions, where the reaction term contains an operator F : L 2 ( Ω ) → L 2 ( Ω ) which is nonlocal and possibly nonlinear. Existence of weak solutions is established, but uniqueness is not required. Compactness of the multivalued flow is obtained via estimates obtained from limits of Galerkin approximations. In contrast with the usual situation, these limits apply for all and not just for almost all time instants.
ISSN:0362-546X
1873-5215
DOI:10.1016/j.na.2010.06.073