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On the existence of the compact global attractor for semilinear reaction diffusion systems on $$\mathbb{R}^ N$$. (English) Zbl 0867.35045
In this interesting paper, the author considers abstract parabolic systems of the form $\partial_tu_k+A_k(x,\partial)u_k=f_k(x,u),$ where $$u=(u_1,\dots,u_k)$$ is a function of $$t$$ and $$x\in\mathbb{R}^N$$. A typical example is an equation with a potential: $$u_t-\Delta u=m(x)u-u^3$$. The space, the solution is looked for, is the space of bounded uniformly continuous functions. According to the properties of the potential $$m$$, the corresponding semiflow can be compact and, consequently, the global compact attractor exists. Sufficient conditions for the Hausdorff dimension of the attractor to be finite are also discussed.
Reviewer: E.Feireisl (Praha)

##### MSC:
 35K57 Reaction-diffusion equations 37C70 Attractors and repellers of smooth dynamical systems and their topological structure 35K45 Initial value problems for second-order parabolic systems 35B40 Asymptotic behavior of solutions to PDEs
##### Keywords:
abstract parabolic systems; Hausdorff dimension
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