Monotone flows and rapid convergence for nonlinear partial differential equations.

*(English)*Zbl 1017.35001
Series in Mathematical Analysis and Applications. 7. London: Taylor and Francis. x, 318 p. (2003).

This book is a treatise on methods for proving the existence of solutions to nonlinear partial differential equations. Under some monotonicity assumptions on \(F\) \((Lu=F(x,u))\), one obtains the existence of maximal and minimal solutions. This is combined with an iterative technique \((Lu_{n+1}= F(x,u_n))\). Using quasilinearization one gets the rapid convergence of monotone sequences. Here, an improvement over previous results is the relaxation of a convexity assumption. The book is divided in two parts where the first one deals with classical cases and the second with a variational approach and generalized solutions. Each part deals in a unified fashion with elliptic, parabolic (including impulsive parabolic in the first part) and hyperbolic equations. The appendices remind basics on Sobolev spaces and PDEs. One finds at the end a bibliography (85 entries) and a small index.

Reviewer: A.Akutowicz (Berlin)

##### MSC:

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

35A35 | Theoretical approximation in context of PDEs |

35A15 | Variational methods applied to PDEs |

35D05 | Existence of generalized solutions of PDE (MSC2000) |

35R12 | Impulsive partial differential equations |

35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |