Nonlinear partial differential equations with applications
type: Item
instance of: scholarly article
title: Nonlinear partial differential equations with applications
zbMATH Open document ID: 1270.35005
DOI: 10.1007/978-3-0348-0513-1
author: Tomáş Roubíçek
published in: ISNM. International Series of Numerical Mathematics
publication date: 2012-07-31T00:00:00Z
review text: The monograph is concerned with the study of various classes of nonlinear elliptic and parabolic partial differential equations with many applications. The book is organized in twelve chapters, the bibliography and an index. The author presents in the first chapter the auxiliary results used in the rest of the book, related to functional analysis, function spaces, Nemytskii mappings, Green formula, Bochner spaces and some ordinary differential equations. Then, he divides the monograph into two parts: the first one ``Steady-state problems'' which contains five chapters and the second one ``Evolution problems'' with six chapters. The prototype tasks addressed in Part I are boundary value problems for quasilinear equations of the type \[ -\text{div}(a(u,\nabla u))+c(u,\nabla u)=g, \] various generalizations of it, in particular variational inequalities, and some systems of such equations. The author discusses the pseudomonotone and weakly continuous mappings (Chapter 2), accretive mappings (Chapter 3), potential problems -- the smooth case (Chapter 4), nonsmooth problems and variational inequalities (Chapter 5) and some systems of equations with applications in (thermo)mechanics of solids and fluids, in electrical devices, engineering, chemistry and biology (Chapter 6). The second part of the book is devoted to the evolution variants of the boundary-value problems studied in Part I, of the form \[ \displaystyle\frac{\partial u}{\partial t}-\text{div}(a(u,\nabla u))+c(u,\nabla u)=g, \] subject to boundary conditions and initial or periodic conditions. The author studies some special auxiliary tools (Chapter 7), evolution by pseudomonotone or weakly continuous mappings (Chapter 8), some evolution problems governed by accretive mappings (Chapter 9), evolution governed by certain set-valued mappings (Chapter 10), doubly nonlinear problems (Chapter 11), and some particular examples of systems of equations which are real-world applications (Chapter 12). There are investigate…
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35J87
Mathematics Subject Classification ID: 35K86
Mathematics Subject Classification ID: 47H04
Mathematics Subject Classification ID: 47H05
Mathematics Subject Classification ID: 47H06
zbMATH DE Number: 6062356
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: regularity
zbMATH Keywords: continuous dependence
zbMATH Keywords: pseudomonotone mappings
zbMATH Keywords: accretive mappings
zbMATH Keywords: nonlinear semigroups
zbMATH Keywords: set-valued mappings
zbMATH Keywords: compactness methods
Wikidata QID: Q55880638
reviewed by: Rodica Luca
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-0348-0513-1
OpenAlex ID: W1525219438
description: scientific article; zbMATH DE number 6062356