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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)

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
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