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Attractors for reaction-diffusion equations in unbounded domains. (English) Zbl 0953.35022

The asymptotic behaviour of solutions for the following reaction-diffusion equation

u t=νΔu-λu-f(u)+gin n × + (1)

is investigated. There ν and λ are positive constants, g is a given function from L 2 ( n ), and the nonlinear function f satisfies the following conditions: f(u)u0, f(0)=0, f ' (u)-C, |f ' (u)|C(1+|u| r ) with r0 for n2 and rmin{4/n;2/(n-2)} for n3.

First the existence of a unique solution u(t) of equation (1) with the initial data (2) u(0)=u 0 is shown, which establishes the existence of a dynamical system {S(t)} t0 such that S(t)u 0 =u(t) in H=L 2 ( 2 ). Later the asymptotic compactness of S(t) is shown. To overcome the difficulty of the lack of compactness of the Sobolev embeddings in n , the author approaches n by a bounded domain and uses the compactness of the embeddings in bounded domains. The main result states that problem (1) (2) has a global attractor in H. At the end the finite dimensionality of this global attractor is studied.

MSC:
35B41Attractors (PDE)
35K57Reaction-diffusion equations
35B40Asymptotic behavior of solutions of PDE