Blow-up in Nonlinear Sobolev Type Equations

Alexander B. Al'shin; Maxim O. Korpusov; Alexey G. Sveshnikov

Published by De Gruyter 2011

Blow-up in Nonlinear Sobolev Type Equations

Alexander B. Al'shin , Maxim O. Korpusov and Alexey G. Sveshnikov

Volume 15 in the series De Gruyter Series in Nonlinear Analysis and Applications

https://doi.org/10.1515/9783110255294

Overview
Contents

About this book

The monograph is devoted to the study of initial-boundary-value problems for multi-dimensional Sobolev-type equations over bounded domains. The authors consider both specific initial-boundary-value problems and abstract Cauchy problems for first-order (in the time variable) differential equations with nonlinear operator coefficients with respect to spatial variables. The main aim of the monograph is to obtain sufficient conditions for global (in time) solvability, to obtain sufficient conditions for blow-up of solutions at finite time, and to derive upper and lower estimates for the blow-up time.

The abstract results apply to a large variety of problems. Thus, the well-known Benjamin-Bona-Mahony-Burgers equation and Rosenau-Burgers equations with sources and many other physical problems are considered as examples. Moreover, the method proposed for studying blow-up phenomena for nonlinear Sobolev-type equations is applied to equations which play an important role in physics. For instance, several examples describe different electrical breakdown mechanisms in crystal semiconductors, as well as the breakdown in the presence of sources of free charges in a self-consistent electric field.

The monograph contains a vast list of references (440 items) and gives an overall view of the contemporary state-of-the-art of the mathematical modeling of various important problems arising in physics. Since the list of references contains many papers which have been published previously only in Russian research journals, it may also serve as a guide to the Russian literature.

Author / Editor information

Alexander B. Al'shin, Maxim O. Korpusov, Alexey G. Sveshnikov, Lomonosov Moscow State University, Russia.

Topics

Frontmatter Requires Authentication Unlicensed Licensed Download PDF	I
Preface Requires Authentication Unlicensed Licensed Download PDF	V
Contents Requires Authentication Unlicensed Licensed Download PDF	VII
Chapter 0 Introduction Requires Authentication Unlicensed Licensed Download PDF	1
Chapter 1 Nonlinear model equations of Sobolev type Requires Authentication Unlicensed Licensed Download PDF	20
Chapter 2 Blow-up of solutions of nonlinear equations of Sobolev type Requires Authentication Unlicensed Licensed Download PDF	69
Chapter 3 Blow-up of solutions of strongly nonlinear Sobolev-type wave equations and equations with linear dissipation Requires Authentication Unlicensed Licensed Download PDF	216
Chapter 4 Blow-up of solutions of strongly nonlinear, dissipative wave Sobolev-type equations with sources Requires Authentication Unlicensed Licensed Download PDF	357
Chapter 5 Special problems for nonlinear equations of Sobolev type Requires Authentication Unlicensed Licensed Download PDF	439
Chapter 6 Numerical methods of solution of initial-boundary-value problems for Sobolev-type equations Requires Authentication Unlicensed Licensed Download PDF	543
Appendix A Some facts of functional analysis Requires Authentication Unlicensed Licensed Download PDF	581
Appendix B To Chapter 6 Requires Authentication Unlicensed Licensed Download PDF	613
Bibliography Requires Authentication Unlicensed Licensed Download PDF	621
Index Requires Authentication Unlicensed Licensed Download PDF	647

(Deutsch)

Language: English
Publisher: De Gruyter
Audience: Graduate and PhD Students, Reseachers; Academic Libraries
Pages
- Front matter: 12
- Main content: 648
Keywords: Blow up; Sobolev; Cauchy problem; Nonlinear equations

eBook

Published: May 26, 2011
ISBN: 9783110255294

Hardcover

Published: May 17, 2011
ISBN: 9783110255270