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Nonlinear evolution operators and semigroups. Applications to partial differential equations. (English) Zbl 0626.35003

Lecture Notes in Mathematics. 1260. Berlin etc.: Springer-Verlag. VI, 285 p.; DM 42.50 (1987).
The author provides an up-to-date treatment of the theory of nonlinear evolution equations and nonlinear semigroups of operators. This theory has been developed by many researchers over the past two decades. Fundamental results of the theory are presented in a complete fashion. Recent results, including many of the author, are discussed, and many open problems are posed. A number of interesting applications of the general theory to nonlinear partial differential equations are considered in detail.
Chapter 1 is concerned with the basic properties of nonlinear evolution operators U(t,s), \(t\geq s\geq 0\) and their connection to the abstract nonlinear evolution equation \(x'(t)=A(t)x(t)\). The construction and convergence of (difference scheme) DS-approximate solutions due to K. Kobayasi, Y. Kobayashi, and S. Oharu is developed. Various notions of solutions are discussed including the integral solution of Ph. Benilan. The time dependence of A(t) in t is explored through results of T. Kato and M. Crandall and A. Pazy. An interesting discussion is given of compact evolution operators and a theorem of H. Brezis is extended to yield necessary and sufficient conditions for compactness of the evolution operators.
Chapter 2 is concerned with the basic theory of nonlinear semigroups of operators \(S_ A(t)\), \(t\geq 0\) and their generation by (possibly multivalued) nonlinear accretive operators A. The approach again uses DS- limit solutions and integral solutions. The exponential formula \(S_ A(t)x=\lim_{n\to \infty}(I-(t/n)A)^{-n}x\) of M. Crandall and T. Liggett is established. The generalized domain of the generator due to M. Crandall is discussed as well as the idea of strong solutions. Results due to R. Martin concerning flow invariant sets (that is, sets invariant under the operators \(S_ A(t)\), \(t\geq 0)\) are given. A variety of perturbation results provide sufficient conditions for \(A+B\) to be a generator given that A is a generator and B belongs to certain classes of nonlinear operators. There is also a discussion of the asymptotic behavior of solutions, that is, their properties as time t approaches infinity.
Chapter 3 is devoted to applications to nonlinear partial differential equations. The Benjamin, Bona, and Mahoney equation for modelling long water waves of small amplitude is treated from the point of view of nonlinear semigroups. Also treated is the porous medium equation, semilinear and nonlinear parabolic equations, semilinear wave equations, and semilinear Schrödinger equations.
This monograph will be valuable to researchers in the theory of abstract nonlinear differential equations in Banach spaces. It provides a complete treatment of many of the basic results in this theory and a detailed discussion of many useful applications to partial differential equations. This monograph will also be valuable to newcomers to this subject. It provides in book form a unified introduction to the theory and applications of nonlinear evolution operators and nonlinear semigroups in Banach spaces.
Reviewer: G.F.Webb

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
47A20 Dilations, extensions, compressions of linear operators
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
34G20 Nonlinear differential equations in abstract spaces
35B20 Perturbations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
35L70 Second-order nonlinear hyperbolic equations