On the existence of the compact global attractor for semilinear reaction diffusion systems on \(\mathbb{R}^ N\)
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.
scientific article; zbMATH DE number 952551