Al'shin / Korpusov / Sveshnikov Blow-up in Nonlinear Sobolev Type Equations

1. Auflage 2011
ISBN: 978-3-11-025529-4
Verlag: De Gruyter
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, Band 15, 660 Seiten

Reihe: De Gruyter Series in Nonlinear Analysis and ApplicationsISSN

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.

Zielgruppe

Graduate and PhD Students, Reseachers; Academic Libraries

Autoren/Hrsg.

Al'shin, Alexander B.

Korpusov, Maxim O.

Sveshnikov, Alexey G.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Variationsrechnung
• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik

Inhaltsverzeichnis

1;Preface;6
2;Contents;8
3;0 Introduction;14
3.1;0.1 List of equations;14
3.1.1;0.1.1 One-dimensional pseudoparabolic equations;14
3.1.2;0.1.2 One-dimensionalwave dispersive equations;15
3.1.3;0.1.3 Singular one-dimensional pseudoparabolic equations;16
3.1.4;0.1.4 Multidimensional pseudoparabolic equations;16
3.1.5;0.1.5 New nonlinear pseudoparabolic equations with sources;18
3.1.6;0.1.6 Model nonlinear equations of even order;19
3.1.7;0.1.7 Multidimensional even-order equations;20
3.1.8;0.1.8 Results and methods of proving theorems on the nonexistence and blow-up of solutions for pseudoparabolic equations;23
3.2;0.2 Structure of the monograph;26
3.3;0.3 Notation;27
4;1 Nonlinear model equations of Sobolev type;33
4.1;1.1 Mathematical models of quasi-stationary processes in crystalline semiconductors;33
4.2;1.2 Model pseudoparabolic equations;40
4.2.1;1.2.1 Nonlinear waves of Rossby type or drift modes in plasma and appropriate dissipative equations;40
4.2.2;1.2.2 Nonlinear waves of Oskolkov–Benjamin–Bona–Mahony type;42
4.2.3;1.2.3 Models of anisotropic semiconductors;47
4.2.4;1.2.4 Nonlinear singular equations of Sobolev type;50
4.2.5;1.2.5 Pseudoparabolic equations with a nonlinear operator ontime derivative;51
4.2.6;1.2.6 Nonlinear nonlocal equations;52
4.2.7;1.2.7 Boundary-value problems for elliptic equations with pseudoparabolic boundary conditions;59
4.3;1.3 Disruption of semiconductors as the blow-up of solutions;61
4.4;1.4 Appearance and propagation of electric domains in semiconductors;69
4.5;1.5 Mathematical models of quasi-stationary processes in crystalline electromagnetic media with spatial dispersion;73
4.6;1.6 Model pseudoparabolic equations in electric media with spatial dispersion;77
4.7;1.7 Model pseudoparabolic equations in magnetic media with spatial dispersion;79
5;2 Blow-up of solutions of nonlinear equations of Sobolev type;82
5.1;2.1 Formulation of problems;82
5.2;2.2 Preliminary definitions, conditions, and auxiliary lemmas;83
5.3;2.3 Unique solvability of problem (2.1) in the weak generalized sense and blow-up of its solutions;91
5.4;2.4 Unique solvability of problem (2.1) in the strong generalized sense and blow-up of its solutions;114
5.5;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
5.6;2.6 Strong solvability of problem (2.2) in the case where B = 0;140
5.7;2.7 Examples;146
5.8;2.8 Initial-boundary-value problem for a nonlinear equation with double nonlinearity of type (2.1);154
5.8.1;2.8.1 Local solvability of problem (2.131)–(2.133)in the weak generalized sense;155
5.8.2;2.8.2 Blow-up of solutions;172
5.9;2.9 Problem for a strongly nonlinear equation of type (2.2) with inferior nonlinearity;177
5.9.1;2.9.1 Unique weak solvability of problem (2.185);178
5.9.2;2.9.2 Solvability in a finite cylinder and blow-up for a finite time;190
5.9.3;2.9.3 Rate of the blow-up of solutions;196
5.10;2.10 Problem for a semilinear equation of the form (2.2);200
5.10.1;2.10.1 Blow-up of classical solutions;200
5.11;2.11 On sufficient conditions of the blow-up of solutions of the Boussinesq equation with sources and nonlinear dissipation;209
5.11.1;2.11.1 Local solvability of strong generalized solutions;210
5.11.2;2.11.2 Blow-up of solutions;213
5.12;2.12 Sufficient conditions of the blow-up of solutions of initial-boundaryvalue problems for a strongly nonlinear pseudoparabolic equation of Rosenau type;216
5.12.1;2.12.1 Local solvability of the problem in the strong generalized sense;216
5.12.2;2.12.2 Blow-up of strong solutions of problem (2.288)–(2.289) and solvability in any finite cylinder;224
5.12.3;2.12.3 Physical interpretation;228
6;3 Blow-up of solutions of strongly nonlinear Sobolev-type wave equations and equations with linear dissipation;229
6.1;3.1 Formulation of problems;229
6.2;3.2 Preliminary definitions and conditions and auxiliary lemma;230
6.3;3.3 Unique solvability of problem (3.1) in the weak generalized sense and blow-up of its solutions;232
6.4;3.4 Unique solvability of problem (3.1) in the strong generalized sense and blow-up of its solutions;257
6.5;3.5 Unique solvability of problem (3.2) in the weak generalized sense and blow-up of its solutions;267
6.6;3.6 Unique solvability of problem (3.2) in the strong generalized sense and blow-up of its solutions;286
6.7;3.7 Examples;291
6.8;3.8 On certain initial-boundary-value problems for quasilinear wave equations of the form(3.2);301
6.8.1;3.8.1 Local solvability in the strong generalized sense of problems (3.141)–(3.143);301
6.8.2;3.8.2 Blow-up of solutions;308
6.8.3;3.8.3 Breakdown of weakened solutions of problem (3.141);315
6.9;3.9 On an initial-boundary-value problem for a strongly nonlinear equation of the type (3.1) (generalized Boussinesq equation);321
6.9.1;3.9.1 Unique solvability of the problem in the weak sense;322
6.9.2;3.9.2 Blow-up of solutions and the global solvability of the problem;328
6.10;3.10 Blow-up of solutions of a class of quasilinear wave dissipative pseudoparabolic equations with sources;333
6.10.1;3.10.1 Unique local solvability of the problem in the strong sense and blow-up of its solutions;333
6.10.2;3.10.2 Examples;340
6.11;3.11 Blow–up of solutions of the Oskolkov–Benjamin–Bona–Mahony–Burgers equation with a cubic source;342
6.11.1;3.11.1 Unique local solvability of the problem;343
6.11.2;3.11.2 Global solvability and the blow-up of solutions;346
6.11.3;3.11.3 Physical interpretation of the obtained results;350
6.12;3.12 On generalized Benjamin–Bona–Mahony–Burgers equation with pseudo-Laplacian;350
6.12.1;3.12.1 Blow-up of strong generalized solutions;350
6.12.2;3.12.2 Physical interpretation of the obtained results;353
6.13;3.13 Sufficient, close to necessary, conditions of the blow-up of solutions of one problem with pseudo-Laplacian;354
6.13.1;3.13.1 Blow-up of strong generalized solutions;354
6.13.2;3.13.2 Physical interpretation of the obtained results;358
6.14;3.14 Sufficient, close to necessary, conditions of the blow-up of solutions of strongly nonlinear generalized Boussinesq equation;358
7;4 Blow-up of solutions of strongly nonlinear, dissipative wave Sobolevtype equations with sources;370
7.1;4.1 Introduction. Statement of problem;370
7.2;4.2 Unique solvability of problem (4.1) in the weak generalized sense and blow-up of its solutions;371
7.3;4.3 Unique solvability of problem (4.1) in the strong generalized sense and blow-up of its solutions;393
7.4;4.4 Examples;398
7.5;4.5 Blow-up of solutions of a Sobolev-type wave equation with nonlocal sources;404
7.5.1;4.5.1 Unique local solvability of the problem;404
7.5.2;4.5.2 Blow-up of strong generalized solutions;411
7.6;4.6 Blow-up of solutions of a strongly nonlinear equation of spin waves;415
7.6.1;4.6.1 Unique local solvability in the strong generalized sense;416
7.6.2;4.6.2 Blow-up of strong generalized solutions and the global solvability;425
7.6.3;4.6.3 Physical interpretation of the obtained results;430
7.7;4.7 Blow-up of solutions of an initial-boundary-value problem for a strongly nonlinear, dissipative equation of the form (4.1);430
7.7.1;4.7.1 Local unique solvability in the weak generalized sense;431
7.7.2;4.7.2 Unique solvability of the problem and blow-up of its solution for afinite time;448
8;5 Special problems for nonlinear equations of Sobolev type;452
8.1;5.1 Nonlinear nonlocal pseudoparabolic equations;452
8.1.1;5.1.1 Global-on-time solvability of the problem;452
8.1.2;5.1.2 Global-on-time solvability of the problem in the strong generalized sense in the case q > 1;482
8.1.3;5.1.3 Asymptotic behavior of solutions of problem (5.1), (5.2) as t -> C1 in the case q > 0;484
8.2;5.2 Blow-up of solutions of nonlinear pseudoparabolic equations with sources of the pseudo-Laplacian type;488
8.2.1;5.2.1 Blow-up ofweakened solutions of problem(5.77);489
8.2.2;5.2.2 Blow-up and the global-on-time solvability of problem (5.78);490
8.2.3;5.2.3 Blow-up of solutions of problem(5.79);492
8.2.4;5.2.4 Blow-up of weakened solutions of problems (5.80) and (5.81);495
8.2.5;5.2.5 Interpretation of the obtained results;497
8.3;5.3 Blow-up of solutions of pseudoparabolic equations with fast increasing nonlinearities;497
8.3.1;5.3.1 Local solvability and blow-up for a finite time of solutions of problems (5.112) and (5.113);498
8.3.2;5.3.2 Local solvability and blow-up for a finite time of solutions of problem (5.114);505
8.4;5.4 Blow-up of solutions of nonhomogeneous nonlinear pseudoparabolic equations;509
8.4.1;5.4.1 Unique local solvability of the problem;509
8.4.2;5.4.2 Blow-up of strong generalized solutions of problem (5.154)–(5.155);512
8.4.3;5.4.3 Blow-up of classical solutions of problem (5.154)–(5.155);515
8.5;5.5 Blow-up of solutions of a nonlinear nonlocal pseudoparabolic equation;516
8.5.1;5.5.1 Unique local solvability of the problem;517
8.5.2;5.5.2 Blow-up and global solvability of problem (5.177);519
8.5.3;5.5.3 Blow-up rate for problem (5.177) under the condition q = 0;522
8.6;5.6 Existence of solutions of the Laplace equation with nonlinear dynamic boundary conditions;524
8.6.1;5.6.1 Reduction the problem to the system of the integral equations;524
8.6.2;5.6.2 Global-on-time solvability and the blow-up of solutions;530
8.7;5.7 Conditions of the global-on-time solvability of the Cauchy problem for a semilinear pseudoparabolic equation;538
8.7.1;5.7.1 Reduction of the problem to an integral equation;538
8.7.2;5.7.2 Theorems on the existence/nonexistence of global-on-time solutions of the integral equation (5.219);540
8.8;5.8 Sufficient conditions of the blow-up of solutions of the Boussinesq equation with nonlinear Neumann boundary condition;550
9;6 Numerical methods of solution of initial-boundary-value problems for Sobolev-type equations;556
9.1;6.1 Numerical solution of problems for linear equations;556
9.1.1;6.1.1 Dynamic potentials for one equation;557
9.1.2;6.1.2 Solvability of Dirichlet problem;561
9.2;6.2 Numerical method of solving initial-boundary-value problems for nonlinear pseudoparabolic equations by the Rosenbrock schemes;567
9.2.1;6.2.1 Stiffmethod of lines;567
9.2.2;6.2.2 Stiff systems of ODE and methods of solving them;568
9.2.3;6.2.3 Stiff stability;568
9.2.4;6.2.4 Schemes ofRosenbrock type;568
9.2.5;6.2.5 e-embedding method;570
9.3;6.3 Results of blow-up numerical simulation;573
9.3.1;6.3.1 Blow-up of pseudoparabolic equations with a linear operator bythe time derivative;574
9.3.2;6.3.2 Blow-up of strongly nonlinear pseudoparabolic equations;579
9.3.3;6.3.3 Blow-up of equations with nonlocal terms (coefficients of the equation depend on the norm of the function);588
10;Appendix A Some facts of functional analysis;594
10.1;A.1 Sobolev spaces ;594
10.2;A.2 Weak and *-weak convergence;596
10.3;A.3 Weak and strong measurability. Bochner integral;597
10.4;A.4 Spaces of integrable functions and distributions;598
10.5;A.5 Nemytskii operator. Krasnoselskii theorem;599
10.6;A.6 Inequalities;601
10.7;A.7 Operator calculus;602
10.8;A.8 Fixed-point theorems;602
10.9;A.9 Weakened solutions of the Poisson equation;602
10.10;A.10 Intersections and sums of Banach spaces;604
10.11;A.11 Classical, weakened, strong generalized, and weak generalized solutions of evolutionary problems;605
10.12;A.12 Two equivalent formulations of weak solutions in L2(0; T;B);607
10.13;A.13 Gâteaux and Fréchet derivatives of nonlinear operators;609
10.14;A.14 On the gradient of a functional;617
10.15;A.15 Lions compactness lemma;619
10.16;A.16 Browder–Minty theorem;620
10.17;A.17 Sufficient conditions of the independence of the interval, on which a solution of a system of differential equations exists, of the order of this system;621
10.18;A.18 On the continuity of some inverse matrices;623
11;Appendix B To Chapter 6;626
11.1;B.1 Convergence of the e-embedding method with the CROS scheme;626
12;Bibliography;634
13;Index;660

Über Autor(innen)

Korpusov, Maxim O.
Lomonosov Moscow State University, Russia

Sveshnikov, Alexey G.
Lomonosov Moscow State University, Russia

Al'shin, Alexander B.
Lomonosov Moscow State University, Russia

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