ebook ebooks e-book e-books downloaden bei MyEbooks.ch downloaden

Blow-up in Nonlinear Sobolev Type Equations

:	Alexander B. Al'shin, Maxim O. Korpusov, Alexey G. Sveshnikov
:	Blow-up in Nonlinear Sobolev Type Equations
:	Walter de Gruyter GmbH& Co.KG
:	9783110255294
:	De Gruyter Series in Nonlinear Analysis and ApplicationsISSN
:	1
:	CHF 221.80
:

:	Analysis
:	English

:	660
:	Wasserzeichen/DRM
:	PC/MAC/eReader/Tablet
:	PDF

< PAN lang=EN>

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.

/html>

< >Alexander B. Al'shin,Maxi O. Korpusov,Ale ey G.Sveshnikov, Lomonosov Moscow State University, Russia.

	Preface	6
	Contents	8
	0 Introduction	14
	0.1 List of equations	14
	0.1.1 One-dimensional pseudoparabolic equations	14
	0.1.2 One-dimensionalwave dispersive equations	15
	0.1.3 Singular one-dimensional pseudoparabolic equations	16
	0.1.4 Multidimensional pseudoparabolic equations	16
	0.1.5 New nonlinear pseudoparabolic equations with sources	18
	0.1.6 Model nonlinear equations of even order	19
	0.1.7 Multidimensional even-order equations	20
	0.1.8 Results and methods of proving theorems on the nonexistence and blow-up of solutions for pseudoparabolic equations	23
	0.2 Structure of the monograph	26
	0.3 Notation	27
	1 Nonlinear model equations of Sobolev type	33
	1.1 Mathematical models of quasi-stationary processes in crystalline semiconductors	33
	1.2 Model pseudoparabolic equations	40
	1.2.1 Nonlinear waves of Rossby type or drift modes in plasma and appropriate dissipative equations	40
	1.2.2 Nonlinear waves of Oskolkov–Benjamin–Bona–Mahony type	42
	1.2.3 Models of anisotropic semiconductors	47
	1.2.4 Nonlinear singular equations of Sobolev type	50
	1.2.5 Pseudoparabolic equations with a nonlinear operator ontime derivative	51
	1.2.6 Nonlinear nonlocal equations	52
	1.2.7 Boundary-value problems for elliptic equations with pseudoparabolic boundary conditions	59
	1.3 Disruption of semiconductors as the blow-up of solutions	61
	1.4 Appearance and propagation of electric domains in semiconductors	69
	1.5 Mathematical models of quasi-stationary processes in crystalline electromagnetic media with spatial dispersion	73
	1.6 Model pseudoparabolic equations in electric media with spatial dispersion	77
	1.7 Model pseudoparabolic equations in magnetic media with spatial dispersion	79
	2 Blow-up of solutions of nonlinear equations of Sobolev type	82
	2.1 Formulation of problems	82
	2.2 Preliminary definitions, conditions, and auxiliary lemmas	83
	2.3 Unique solvability of problem (2.1) in the weak generalized sense and blow-up of its solutions	91
	2.4 Unique solvability of problem (2.1) in the strong generalized sense and blow-up of its solutions	114
	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	124
	2.6 Strong solvability of problem (2.2) in the case where B = 0	140
	2.7 Examples	146
	2.8 Initial-boundary-value problem for a nonlinear equation with double nonlinearity of type (2.1)	154
	2.8.1 Local solvability of problem (2.131)–(2.133)in the weak generalized sense	155
	2.8.2 Blow-up of solutions	172
	2.9 Problem for a strongly nonlinear equation of type (2.2) with inferior nonlinearity	177
	2.9.1 Unique weak solvability of problem (2.185)	178
	2.9.2 Solvability in a finite cylinder and blow-up for a finite time	190
	2.9.3 Rate of the blow-up of solutions	196
	2.10 Problem for a semilinear equation of the form (2.2)	200
	2.10.1 Blow-up of classical solutions	200
	2.11 On sufficient conditions of the blow-up of solutions of the Boussinesq equation with sources and nonlinear dissipation	209
	2.11.1 Local solvability of strong generalized solutions	210
	2.11.2 Blow-up of solutions	213
	2.12 Sufficient conditions of the blow-up of solutions of initial-boundaryvalue problems for a strongly nonlinear pseudoparabolic equation of Rosenau type	216
	2.12.1 Local solvability of the problem in the strong generalized sense	216
	2.12.2 Blow-up of strong solutions of problem (2.288)–(2.289) and solvability in any finite cylinder	224
	2.12.3 Physical interpretation	228
	3 Blow-up of solutions of strongly nonlinear Sobolev-type wave equations and equations with linear dissipation	229
	3.1 Formulation of problems	229
	3.2 Preliminary definitions and conditions and auxiliary lemma	230
	3.3 Unique solvability of problem (3.1) in the weak generalized sense and blow-up of its solutions	232
	3.4 Unique solvability of problem (3.1) in the strong generalized sense and blow-up of its solutions	257
	3.5 Unique solvability of problem (3.2) in the weak generalized sense and blow-up of its solutions	267
	3.6 Unique solvability of problem (3.2) in the strong generalized sense and blow-up of its solutions	286
	3.7 Examples	291
	3.8 On certain initial-b