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Finite-dimensional behavior in dissipative partial differential equations. (English) Zbl 1055.35501
Summary: Dissipative partial differential equations have applications throughout the sciences: models of turbulence in fluids, chemical reactions, and morphogenesis in biology can all be written in a general form which allows them to be subjected to a unified analysis. Recent results on these equations show that in many cases they are not as complex as they initially appear, and can be converted into a set of ordinary differential equations. However, most of the relevant references present a bewildering array of terms which can obscure the simple underlying ideas. The main purpose of this paper is to introduce this terminology, motivated by several major results, slowly and by example. Detailed proofs are omitted, but it is hoped that this approach will give a good understanding of and intuitive feel for the subject without recourse to technicalities. Nevertheless, sufficient mathematical detail is included to allow application of these results to many examples.

MSC:
35B40 Asymptotic behavior of solutions to PDEs
34G20 Nonlinear differential equations in abstract spaces
35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
35K57 Reaction-diffusion equations
37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
47H20 Semigroups of nonlinear operators
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