Blow-up in Nonlinear Sobolev Type Equations (eBook)

Alexander B. Al'shin, Maxim O. Korpusov, Alexey G. Sveshnikov (Autoren)

eBook Download: PDF

2011
660 Seiten
De Gruyter (Verlag)
978-3-11-025529-4 (ISBN)

Infos

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.

Autor

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

Inhalt

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-boundary-value problems for quasilinear wave equations of the form(3.2) 301
3.8.1 Local solvability in the strong generalized sense of problems (3.141)–(3.143) 301
3.8.2 Blow-up of solutions 308
3.8.3 Breakdown of weakened solutions of problem (3.141) 315
3.9 On an initial-boundary-value problem for a strongly nonlinear equation of the type (3.1) (generalized Boussinesq equation) 321
3.9.1 Unique solvability of the problem in the weak sense 322
3.9.2 Blow-up of solutions and the global solvability of the problem 328
3.10 Blow-up of solutions of a class of quasilinear wave dissipative pseudoparabolic equations with sources 333
3.10.1 Unique local solvability of the problem in the strong sense and blow-up of its solutions 333
3.10.2 Examples 340
3.11 Blow–up of solutions of the Oskolkov–Benjamin–Bona–Mahony–Burgers equation with a cubic source 342
3.11.1 Unique local solvability of the problem 343
3.11.2 Global solvability and the blow-up of solutions 346
3.11.3 Physical interpretation of the obtained results 350
3.12 On generalized Benjamin–Bona–Mahony–Burgers equation with pseudo-Laplacian 350
3.12.1 Blow-up of strong generalized solutions 350
3.12.2 Physical interpretation of the obtained results 353
3.13 Sufficient, close to necessary, conditions of the blow-up of solutions of one problem with pseudo-Laplacian 354
3.13.1 Blow-up of strong generalized solutions 354
3.13.2 Physical interpretation of the obtained results 358
3.14 Sufficient, close to necessary, conditions of the blow-up of solutions of strongly nonlinear generalized Boussinesq equation 358
4 Blow-up of solutions of strongly nonlinear, dissipative wave Sobolevtype equations with sources 370
4.1 Introduction. Statement of problem 370
4.2 Unique solvability of problem (4.1) in the weak generalized sense and blow-up of its solutions 371
4.3 Unique solvability of problem (4.1) in the strong generalized sense and blow-up of its solutions 393
4.4 Examples 398
4.5 Blow-up of solutions of a Sobolev-type wave equation with nonlocal sources 404
4.5.1 Unique local solvability of the problem 404
4.5.2 Blow-up of strong generalized solutions 411
4.6 Blow-up of solutions of a strongly nonlinear equation of spin waves 415
4.6.1 Unique local solvability in the strong generalized sense 416
4.6.2 Blow-up of strong generalized solutions and the global solvability 425
4.6.3 Physical interpretation of the obtained results 430
4.7 Blow-up of solutions of an initial-boundary-value problem for a strongly nonlinear, dissipative equation of the form (4.1) 430
4.7.1 Local unique solvability in the weak generalized sense 431
4.7.2 Unique solvability of the problem and blow-up of its solution for afinite time 448
5 Special problems for nonlinear equations of Sobolev type 452
5.1 Nonlinear nonlocal pseudoparabolic equations 452
5.1.1 Global-on-time solvability of the problem 452
5.1.2 Global-on-time solvability of the problem in the strong generalized sense in the case q > 1
5.1.3 Asymptotic behavior of solutions of problem (5.1), (5.2) as t -> C1 in the case q >
5.2 Blow-up of solutions of nonlinear pseudoparabolic equations with sources of the pseudo-Laplacian type 488
5.2.1 Blow-up ofweakened solutions of problem(5.77) 489
5.2.2 Blow-up and the global-on-time solvability of problem (5.78) 490
5.2.3 Blow-up of solutions of problem(5.79) 492
5.2.4 Blow-up of weakened solutions of problems (5.80) and (5.81) 495
5.2.5 Interpretation of the obtained results 497
5.3 Blow-up of solutions of pseudoparabolic equations with fast increasing nonlinearities 497
5.3.1 Local solvability and blow-up for a finite time of solutions of problems (5.112) and (5.113) 498
5.3.2 Local solvability and blow-up for a finite time of solutions of problem (5.114) 505
5.4 Blow-up of solutions of nonhomogeneous nonlinear pseudoparabolic equations 509
5.4.1 Unique local solvability of the problem 509
5.4.2 Blow-up of strong generalized solutions of problem (5.154)–(5.155) 512
5.4.3 Blow-up of classical solutions of problem (5.154)–(5.155) 515
5.5 Blow-up of solutions of a nonlinear nonlocal pseudoparabolic equation 516
5.5.1 Unique local solvability of the problem 517
5.5.2 Blow-up and global solvability of problem (5.177) 519
5.5.3 Blow-up rate for problem (5.177) under the condition q = 0 522
5.6 Existence of solutions of the Laplace equation with nonlinear dynamic boundary conditions 524
5.6.1 Reduction the problem to the system of the integral equations 524
5.6.2 Global-on-time solvability and the blow-up of solutions 530
5.7 Conditions of the global-on-time solvability of the Cauchy problem for a semilinear pseudoparabolic equation 538
5.7.1 Reduction of the problem to an integral equation 538
5.7.2 Theorems on the existence/nonexistence of global-on-time solutions of the integral equation (5.219) 540
5.8 Sufficient conditions of the blow-up of solutions of the Boussinesq equation with nonlinear Neumann boundary condition 550
6 Numerical methods of solution of initial-boundary-value problems for Sobolev-type equations 556
6.1 Numerical solution of problems for linear equations 556
6.1.1 Dynamic potentials for one equation 557
6.1.2 Solvability of Dirichlet problem 561
6.2 Numerical method of solving initial-boundary-value problems for nonlinear pseudoparabolic equations by the Rosenbrock schemes 567
6.2.1 Stiffmethod of lines 567
6.2.2 Stiff systems of ODE and methods of solving them 568
6.2.3 Stiff stability 568
6.2.4 Schemes ofRosenbrock type 568
6.2.5 e-embedding method 570
6.3 Results of blow-up numerical simulation 573
6.3.1 Blow-up of pseudoparabolic equations with a linear operator bythe time derivative 574
6.3.2 Blow-up of strongly nonlinear pseudoparabolic equations 579
6.3.3 Blow-up of equations with nonlocal terms (coefficients of the equation depend on the norm of the function) 588
Appendix A Some facts of functional analysis 594
A.1 Sobolev spaces 594
A.2 Weak and *-weak convergence 596
A.3 Weak and strong measurability. Bochner integral 597
A.4 Spaces of integrable functions and distributions 598
A.5 Nemytskii operator. Krasnoselskii theorem 599
A.6 Inequalities 601
A.7 Operator calculus 602
A.8 Fixed-point theorems 602
A.9 Weakened solutions of the Poisson equation 602
A.10 Intersections and sums of Banach spaces 604
A.11 Classical, weakened, strong generalized, and weak generalized solutions of evolutionary problems 605
A.12 Two equivalent formulations of weak solutions in L2(0 T
A.13 Gâteaux and Fréchet derivatives of nonlinear operators 609
A.14 On the gradient of a functional 617
A.15 Lions compactness lemma 619
A.16 Browder–Minty theorem 620
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
A.18 On the continuity of some inverse matrices 623
Appendix B To Chapter 6 626
B.1 Convergence of the e-embedding method with the CROS scheme 626
Bibliography 634
Index 660

Produkt-Details

• Erscheint lt. Verlag: 26.5.2011
• Reihe/Serie: De Gruyter Series in Nonlinear Analysis and Applications
• Reihe/Serie: De Gruyter Series in Nonlinear Analysis and Applications
• Reihe/Serie: ISSN
• Reihe/Serie: ISSN
• Verlagsort: Berlin/Boston
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Technik
• Schlagworte: Blow up • Cauchy problem • Nonlinear Equations • Sobolev
• ISBN-10: 3-11-025529-4 / 3110255294
• ISBN-13: 978-3-11-025529-4 / 9783110255294