Compactness methods for nonlinear evolutions.
2nd ed.

*(English)*Zbl 0842.47040
Pitman Monographs and Surveys in Pure and Applied Mathematics. 75. Harlow, Essex: Longman Group. New York, NY: Wiley & Sons, xvi, 241 p. (1995).

The author presents a good show of most fundamental results concerning compactness methods for strongly nonlinear evolution equations, specially those related to the existence of mild solutions, of equations involving perturbations of \(m\)-accretive operators by nonaccretive ones, including some ill-posed problems.

From my point of view, a relevant fact of the book is that the abstract results are illustrated by a lot of detailed examples and remarks, which are very useful to clarify many of the stated aspects. These examples treat about nonlinear diffusion, propagation of forced waves, heat conduction in materials with memory, fluid dynamics, optimal control, Navier-Stokes equations, nonlinear Volterra integrodifferential equations, functional differential equations with delay, etc.

In this second edition, some new results are included with respect to the first one; for instance, those about compactness, in \(L^p ([a,b]; X)\), of subsets of mild solutions to nonlinear evolution equations, governed by \(m\)-accretive operators; also, reaction-diffusion systems and closed loop systems. Moreover, some proofs have been simplified and the references have been updated.

The book is divided into five chapters. The first one, “Elements of nonlinear functional analysis”, is a good and clear survey (without proofs), of those concepts and results from the mentioned discipline which are used in subsequent chapters. In chapter II, “Fundamental compactness results”, the author shows several general sufficient conditions for that some subsets of mild solutions of a nonlinear evolution equation be relatively compact in different function spaces. The results of this chapter are used very often in the remainder of the book. Also, compact semigroups are briefly studied. Chapter III, “Nonlinear perturbations of accretive operators”, is dedicated to develop a general and abstract theory about existence results for nonlinear evolution equations which are non-accretive perturbations of \(m\)-accretive operators. In chapter IV, “Demiclosed perturbations of subdifferentials”, the author studies three local existence theorems about equations of the type \[ F(t, u(t))\in {{du(t)} \over {dt}}+ \partial \varphi =(u (t)), \quad 0\leq t\leq T, \qquad u(0)= 3Du_0, \] where \(\varphi\) is a proper, l.s.c., convex function defined on a real Hilbert space \(H\), \(F:[0,T ]\times D(\partial \varphi)\to H\), \(\partial \varphi (u (t))\) is the subdifferential of \(\varphi\), calculated at \(u(t)\) and the perturbing term is allowed to be discontinuous and defined on \(D(\partial \varphi)\).

The last chapter contains several local existence results for some nonlinear Volterra integrodifferential equations and functional differential equations with delay.

The book is completed with some interesting bibliographical and historical notes, comments and open problems. The theorems are carefully proved and a reader with some knowledges about topology, functional analysis and ordinary differential equations, to undergraduate level, may understand the book. I am sure that this monograph will be very useful for those interested in nonlinear abstract evolution equations.

From my point of view, a relevant fact of the book is that the abstract results are illustrated by a lot of detailed examples and remarks, which are very useful to clarify many of the stated aspects. These examples treat about nonlinear diffusion, propagation of forced waves, heat conduction in materials with memory, fluid dynamics, optimal control, Navier-Stokes equations, nonlinear Volterra integrodifferential equations, functional differential equations with delay, etc.

In this second edition, some new results are included with respect to the first one; for instance, those about compactness, in \(L^p ([a,b]; X)\), of subsets of mild solutions to nonlinear evolution equations, governed by \(m\)-accretive operators; also, reaction-diffusion systems and closed loop systems. Moreover, some proofs have been simplified and the references have been updated.

The book is divided into five chapters. The first one, “Elements of nonlinear functional analysis”, is a good and clear survey (without proofs), of those concepts and results from the mentioned discipline which are used in subsequent chapters. In chapter II, “Fundamental compactness results”, the author shows several general sufficient conditions for that some subsets of mild solutions of a nonlinear evolution equation be relatively compact in different function spaces. The results of this chapter are used very often in the remainder of the book. Also, compact semigroups are briefly studied. Chapter III, “Nonlinear perturbations of accretive operators”, is dedicated to develop a general and abstract theory about existence results for nonlinear evolution equations which are non-accretive perturbations of \(m\)-accretive operators. In chapter IV, “Demiclosed perturbations of subdifferentials”, the author studies three local existence theorems about equations of the type \[ F(t, u(t))\in {{du(t)} \over {dt}}+ \partial \varphi =(u (t)), \quad 0\leq t\leq T, \qquad u(0)= 3Du_0, \] where \(\varphi\) is a proper, l.s.c., convex function defined on a real Hilbert space \(H\), \(F:[0,T ]\times D(\partial \varphi)\to H\), \(\partial \varphi (u (t))\) is the subdifferential of \(\varphi\), calculated at \(u(t)\) and the perturbing term is allowed to be discontinuous and defined on \(D(\partial \varphi)\).

The last chapter contains several local existence results for some nonlinear Volterra integrodifferential equations and functional differential equations with delay.

The book is completed with some interesting bibliographical and historical notes, comments and open problems. The theorems are carefully proved and a reader with some knowledges about topology, functional analysis and ordinary differential equations, to undergraduate level, may understand the book. I am sure that this monograph will be very useful for those interested in nonlinear abstract evolution equations.

Reviewer: A.Cañada (Granada)

##### MSC:

47H20 | Semigroups of nonlinear operators |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

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

35L05 | Wave equation |

45K05 | Integro-partial differential equations |

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

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

34G20 | Nonlinear differential equations in abstract spaces |

35K55 | Nonlinear parabolic equations |