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**Nonlinear differential equations of monotone types in Banach spaces.**
*(English)*
Zbl 1197.35002

Springer Monographs in Mathematics. Berlin: Springer (ISBN 978-1-4419-5541-8/hbk; 978-1-4614-2557-1/pbk; 978-1-4419-5542-5/ebook). x, 272 p. (2010).

This book presents the main results concerning monotone and accretive operators as well as a lot of applications associated with such operators. It comes after some previous books by the author on the topic, such as [Nonlinear semigroups and differential equations in Banach spaces. Bucuresti: Editura Academiei; Leyden, The Netherlands: Noordhoff International Publishing (1976; Zbl 0328.47035) and Analysis and Control of nonlinear infinite dimensional systems. Mathematics in Science and Engineering. 190. Boston: Academic Press, Inc. (1993; Zbl 0776.49005)]. This time the author pays more attention to applications, with the aim to show the power of functional tools in exploring a broad range of problems described by partial differential equations. It is worth pointing out that the book relies on the author’s long experience in research and teaching at different universities. It was his intention to avoid details and offer only key theoretical results and significant applications.

The first chapter, entitled “Fundamental functional analysis”, is an introductory one. It provides some basic facts on Banach spaces, convex functions, Sobolev spaces, and the variational theory of linear elliptic boundary value problems (BVPs).

The second chapter is called “Maximal monotone operators in Banach spaces”. It presents the theory of such operators, including the subpotential ones, as well as elliptic variational inequalities, and nonlinear elliptic problems of divergence type.

Chapter 3 starts with a section on the basic theory of quasi-\(m\)-accretive operators in Banach spaces. The next two sections are concerned with applications to nonlinear elliptic BVPs in \(L^p\), and quasilinear partial differential operators of first order.

Chapter 4 is devoted to the Cauchy problem for first order differential equations associated with nonlinear quasi-accretive or monotone operators. This chapter collects the main theoretical results on global existence of mild or strong solutions for both autonomous and time-dependent evolution equations, including a class of stochastic equations. Approximation results, such as the Trotter-Kato theorem for nonlinear evolutions and the nonlinear version of the Chernoff theorem, are also included.

The last chapter (Chapter 5: “Existence theory of nonlinear dissipative dynamics”) is fully concerned with applications of the theory developed in Chapter 4, namely: semilinear parabolic equations, parabolic variational inequalities, the porous media diffusion equation, the phase field system, conservation laws, semilinear wave equations, Navier-Stokes equations.

Bibliographical comments and references are provided at the end of each chapter.

The book is useful for both graduate students and researchers.

The first chapter, entitled “Fundamental functional analysis”, is an introductory one. It provides some basic facts on Banach spaces, convex functions, Sobolev spaces, and the variational theory of linear elliptic boundary value problems (BVPs).

The second chapter is called “Maximal monotone operators in Banach spaces”. It presents the theory of such operators, including the subpotential ones, as well as elliptic variational inequalities, and nonlinear elliptic problems of divergence type.

Chapter 3 starts with a section on the basic theory of quasi-\(m\)-accretive operators in Banach spaces. The next two sections are concerned with applications to nonlinear elliptic BVPs in \(L^p\), and quasilinear partial differential operators of first order.

Chapter 4 is devoted to the Cauchy problem for first order differential equations associated with nonlinear quasi-accretive or monotone operators. This chapter collects the main theoretical results on global existence of mild or strong solutions for both autonomous and time-dependent evolution equations, including a class of stochastic equations. Approximation results, such as the Trotter-Kato theorem for nonlinear evolutions and the nonlinear version of the Chernoff theorem, are also included.

The last chapter (Chapter 5: “Existence theory of nonlinear dissipative dynamics”) is fully concerned with applications of the theory developed in Chapter 4, namely: semilinear parabolic equations, parabolic variational inequalities, the porous media diffusion equation, the phase field system, conservation laws, semilinear wave equations, Navier-Stokes equations.

Bibliographical comments and references are provided at the end of each chapter.

The book is useful for both graduate students and researchers.

Reviewer: Gheorghe Moroşanu (Budapest)

### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

34G20 | Nonlinear differential equations in abstract spaces |

34G25 | Evolution inclusions |

47H05 | Monotone operators and generalizations |

47H06 | Nonlinear accretive operators, dissipative operators, etc. |

47H20 | Semigroups of nonlinear operators |

47J35 | Nonlinear evolution equations |

35K58 | Semilinear parabolic equations |

35L71 | Second-order semilinear hyperbolic equations |

35J87 | Unilateral problems for nonlinear elliptic equations and variational inequalities with nonlinear elliptic operators |

35K86 | Unilateral problems for nonlinear parabolic equations and variational inequalities with nonlinear parabolic operators |