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Zbl 0953.35022
Wang, Bixiang
Attractors for reaction-diffusion equations in unbounded domains. (English)
[J] Physica D 128, No.1, 41-52 (1999). ISSN 0167-2789

The asymptotic behaviour of solutions for the following reaction-diffusion equation $${\partial u\over \partial t}=\nu \Delta u-\lambda u-f(u)+ g\quad \text{in }\bbfR^n \times\bbfR^+ \tag 1$$ is investigated. There $\nu$ and $\lambda$ are positive constants, $g$ is a given function from $L^2(\bbfR^n)$, and the nonlinear function $f$ satisfies the following conditions: $f(u)u\ge 0$, $f(0)=0$, $f'(u)\ge-C$, $|f'(u)|\le C(1+|u|^r)$ with $r\ge 0$ for $n\le 2$ and $r\le\min \{4/n;2/(n-2)\}$ for $n\ge 3$.\par 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 $\{S(t)\}_{t\ge 0}$ such that $S(t)u_0=u(t)$ in $H=L^2 (\bbfR^2)$. Later the asymptotic compactness of $S(t)$ is shown. To overcome the difficulty of the lack of compactness of the Sobolev embeddings in $\bbfR^n$, the author approaches $\bbfR^n$ 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.
[A.Cichocka (Katowice)]

MSC 2000:: *35B41 Attractors
35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions of PDE

Keywords: asymptotic compactness; lack of compactness of Sobolev embeddings

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Zentralblatt MATH Berlin [Germany]