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Global attractors in abstract parabolic problems. (English) Zbl 0954.35002
London Mathematical Society Lecture Note Series. 278. Cambridge: Cambridge University Press. xii, 235 p. (2000).
The monograph is devoted to autonomous parabolic equations with emphasis on the existence of attractors. It starts with a preliminary chapter, recalling in particular elements of the stability theory in abstract dynamical systems (including theorems on the existence of attractors for dissipative systems), and discussing analytic semigroups generated by elliptic differential operators. Chapters 2 and 3 contain a systematic treatment of abstract semilinear parabolic equations in fractional power spaces. Local existence and regularity theorems, together with sufficient conditions for global existence are proved in detail, variation of the constants formula being at their grounds. These results are followed by a short Chapter 4, where the dissipativity properties of parabolic equations are examined. Both compact and noncompact semigroups are discussed here. A significant portion of the book, encompassing Chapters 5 and 6, is devoted to applications of the abstract theory in specific parabolic problems. Reaction diffusion systems, Cahn-Hilliard equations, Burgers equations and Navier-Stokes equations in 2D are treated. The last three chapters extend and complement the main topics discussed above. Sections on the backward uniqueness, linear stability, abstract form of the maximum principle and on various notions of solutions are included.
The material selected in the monograph has a very clear-cut ending – it stops at the existence of the attractor. Qualitative properties of attractors and solutions on them are not discussed, if we do not count a few superficial remarks at the end. Even such an intimately close topic as the proof of the finite dimensionality of attractors is not included. As the authors say “... the book was devoted rather to the introductory studies of the global attractors related to abstract parabolic problems than to the detailed properties of such objects.”
The authors have striven to make the book accessible to a large audience. They have included a lot of helpful explanatory comments throughout the account of the theory. Reading the book will certainly be beneficial to students and researchers interested in parabolic equations and their semigroups.

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35B41 Attractors
35K90 Abstract parabolic equations
35Kxx Parabolic equations and parabolic systems
37Lxx Infinite-dimensional dissipative dynamical systems