Reduction of infinite dimensional systems to finite dimensions: compact convergence approach (Q2840363)

The authors consider a common class of parameter-dependent semilinear parabolic evolution problems for a second-order elliptic differential operator on a bounded domain with smooth boundary. Assuming that the problem is finite dimensional at the limit value of the parameter, they introduce an abstract functional analytic setting that is applicable to the analysis of concrete problems for which the asymptotic dynamics occur in finite dimensions. The key ingredients in this approach are a notion of compact convergence of the inverse of a parameter-dependent positive sectorial operator as the parameter approaches the limit and invariant manifold techniques. Applications of the abstract results to a cell tissue reaction-diffusion problem and an atmospherics model are given.