Blow-up in nonlinear Sobolev type equations / Alexander B. Alʹshin, Maxim O. Korpusov, Alexey G. Sveshnikov.

Format:

Book

Author/Creator:

Alʹshin, A. B.

Contributor:

Korpusov, M. O.

Sveshnikov, A. G. (Alekseĭ Georgievich), 1924-

Series:

De Gruyter series in nonlinear analysis and applications ; 15.

De Gruyter series in nonlinear analysis and applications, 0941-8183X ; 15

Language:

English

Subjects (All):

Initial value problems--Numerical solutions.

Initial value problems.

Nonlinear difference equations.

Mathematical physics.

Physical Description:

1 online resource (660 p.)

Edition:

1st ed.

Place of Publication:

Berlin ; New York : De Gruyter, c2011.

Language Note:

English

Summary:

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.

Contents:

Frontmatter

Preface

Contents

Chapter 0 Introduction

Chapter 1 Nonlinear model equations of Sobolev type

Chapter 2 Blow-up of solutions of nonlinear equations of Sobolev type

Chapter 3 Blow-up of solutions of strongly nonlinear Sobolev-type wave equations and equations with linear dissipation

Chapter 4 Blow-up of solutions of strongly nonlinear, dissipative wave Sobolev-type equations with sources

Chapter 5 Special problems for nonlinear equations of Sobolev type

Chapter 6 Numerical methods of solution of initial-boundary-value problems for Sobolev-type equations

Appendix A Some facts of functional analysis

Appendix B To Chapter 6

Bibliography

Index

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

ISBN:

9786613166821

9781283166829

1283166828

9783110255294

3110255294

OCLC:

749781836