UID:
almahu_9947359867702882
Umfang:
Online-Ressource
,
Online Ressource (4518 KB, 0 S.)
Ausgabe:
1. Aufl.
Ausgabe:
Online-Ausg. 2011 Electronic reproduction; Available via World Wide Web
ISBN:
3110255278
Serie:
De Gruyter series in nonlinear analysis and applications 15
Inhalt:
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 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. Alexander B. Al'shin, Maxim O. Korpusov, Alexey G.Sveshnikov, Lomonosov Moscow State University, Russia.
Anmerkung:
Includes bibliographical references and index
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1 Nonlinear model equations of Sobolev type1.1 Mathematical models of quasi-stationary processes in crystalline semiconductors; 1.2 Model pseudoparabolic equations; 1.2.1 Nonlinear waves of Rossby type or drift modes in plasma and appropriate dissipative equations; 1.2.2 Nonlinear waves of Oskolkov-Benjamin-Bona-Mahony type; 1.2.3 Models of anisotropic semiconductors; 1.2.4 Nonlinear singular equations of Sobolev type; 1.2.5 Pseudoparabolic equations with a nonlinear operator ontime derivative; 1.2.6 Nonlinear nonlocal equations.
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1.2.7 Boundary-value problems for elliptic equations with pseudoparabolic boundary conditions1.3 Disruption of semiconductors as the blow-up of solutions; 1.4 Appearance and propagation of electric domains in semiconductors; 1.5 Mathematical models of quasi-stationary processes in crystalline electromagnetic media with spatial dispersion; 1.6 Model pseudoparabolic equations in electric media with spatial dispersion; 1.7 Model pseudoparabolic equations in magnetic media with spatial dispersion; 2 Blow-up of solutions of nonlinear equations of Sobolev type; 2.1 Formulation of problems.
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2.11.1 Local solvability of strong generalized solutions.
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2.2 Preliminary definitions, conditions, and auxiliary lemmas2.3 Unique solvability of problem (2.1) in the weak generalized sense and blow-up of its solutions; 2.4 Unique solvability of problem (2.1) in the strong generalized sense and blow-up of its solutions; 2.5 Unique solvability of problem (2.2) in the weak generalized sense and estimates of time and rate of the blow-up of its solutions; 2.6 Strong solvability of problem (2.2) in the case where B = 0; 2.7 Examples; 2.8 Initial-boundary-value problem for a nonlinear equation with double nonlinearity of type (2.1).
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2.8.1 Local solvability of problem (2.131)-(2.133)in the weak generalized sense2.8.2 Blow-up of solutions; 2.9 Problem for a strongly nonlinear equation of type (2.2) with inferior nonlinearity; 2.9.1 Unique weak solvability of problem (2.185); 2.9.2 Solvability in a finite cylinder and blow-up for a finite time; 2.9.3 Rate of the blow-up of solutions; 2.10 Problem for a semilinear equation of the form (2.2); 2.10.1 Blow-up of classical solutions; 2.11 On sufficient conditions of the blow-up of solutions of the Boussinesq equation with sources and nonlinear dissipation.
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Preface; Contents; 0 Introduction; 0.1 List of equations; 0.1.1 One-dimensional pseudoparabolic equations; 0.1.2 One-dimensionalwave dispersive equations; 0.1.3 Singular one-dimensional pseudoparabolic equations; 0.1.4 Multidimensional pseudoparabolic equations; 0.1.5 New nonlinear pseudoparabolic equations with sources; 0.1.6 Model nonlinear equations of even order; 0.1.7 Multidimensional even-order equations; 0.1.8 Results and methods of proving theorems on the nonexistence and blow-up of solutions for pseudoparabolic equations; 0.2 Structure of the monograph; 0.3 Notation.
Weitere Ausg.:
ISBN 3110255294
Weitere Ausg.:
ISBN 9783110255294
Sprache:
Englisch
DOI:
10.1515/9783110255294
URL:
http://www.degruyter.com/doi/book/10.1515/9783110255294
