Infinite dimensional dynamics described by ordinary differential equations
The author develops an abstract theory applicable to the reduction of the study of reaction diffusion problems where the diffusion coefficient is large in parts of the region and small in some other parts of the region to the study of the large time dynamics of an ordinary differential equation. The use of fractional power of sectorial operators and of the corresponding associated spaces is appropriated. Energy spaces are the natural framework to derive conclusions on the global asymptotic dynamics of parabolic systems. On the other hand, the bifurcation questions included in reaction diffusion problems are decisive to establish invariant manifolds. Results on existence, stability and smoothness of invariant manifolds are obtained following the ideas close to those in of the classical book of Dan Henry. The author refers to an example justifying these reductions the difficulties of the hard question of finding equilibrium points of a parabolic problem with a polynomial nonlinearity less meaningful in the case of ordinary differential equations.
global asymptotic dynamics of parabolic systems
scientific article; zbMATH DE number 736408