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.
Author(s): Alexander B. Al’shin, Maxim O. Korpusov, Alexey G. Sveshnikov
Series: De Gruyter Series in Nonlinear Analysis and Applications 15
Edition: 1
Publisher: De Gruyter
Year: 2011
Language: English
Pages: 661
Tags: Математика;Математическая физика;
Preface ......Page 6
Contents ......Page 8
0.1.1 One-dimensional pseudoparabolic equations ......Page 14
0.1.2 One-dimensionalwave dispersive equations ......Page 15
0.1.4 Multidimensional pseudoparabolic equations ......Page 16
0.1.5 New nonlinear pseudoparabolic equations with sources ......Page 18
0.1.6 Model nonlinear equations of even order ......Page 19
0.1.7 Multidimensional even-order equations ......Page 20
0.1.8 Results and methods of proving theorems on the nonexistence and blow-up of solutions for pseudoparabolic equations ......Page 23
0.2 Structure of the monograph ......Page 26
0.3 Notation ......Page 27
1.1 Mathematical models of quasi-stationary processes in crystalline semiconductors ......Page 33
1.2.1 Nonlinear waves of Rossby type or drift modes in plasma and appropriate dissipative equations ......Page 40
1.2.2 Nonlinear waves of Oskolkov–Benjamin–Bona–Mahony type ......Page 42
1.2.3 Models of anisotropic semiconductors ......Page 47
1.2.4 Nonlinear singular equations of Sobolev type ......Page 50
1.2.5 Pseudoparabolic equations with a nonlinear operator ontime derivative ......Page 51
1.2.6 Nonlinear nonlocal equations ......Page 52
1.2.7 Boundary-value problems for elliptic equations with pseudoparabolic boundary conditions ......Page 59
1.3 Disruption of semiconductors as the blow-up of solutions ......Page 61
1.4 Appearance and propagation of electric domains in semiconductors ......Page 69
1.5 Mathematical models of quasi-stationary processes in crystalline electromagnetic media with spatial dispersion ......Page 73
1.6 Model pseudoparabolic equations in electric media with spatial dispersion ......Page 77
1.7 Model pseudoparabolic equations in magnetic media with spatial dispersion ......Page 79
2.1 Formulation of problems ......Page 82
2.2 Preliminary definitions, conditions, and auxiliary lemmas ......Page 83
2.3 Unique solvability of problem (2.1) in the weak generalized sense and blow-up of its solutions ......Page 91
2.4 Unique solvability of problem (2.1) in the strong generalized sense and blow-up of its solutions ......Page 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 ......Page 124
2.6 Strong solvability of problem (2.2) in the case where B = 0 ......Page 140
2.7 Examples ......Page 146
2.8 Initial-boundary-value problem for a nonlinear equation with double nonlinearity of type (2.1) ......Page 154
2.8.1 Local solvability of problem (2.131)–(2.133)in the weak generalized sense ......Page 155
2.8.2 Blow-up of solutions ......Page 172
2.9 Problem for a strongly nonlinear equation of type (2.2) with inferior nonlinearity ......Page 177
2.9.1 Unique weak solvability of problem (2.185) ......Page 178
2.9.2 Solvability in a finite cylinder and blow-up for a finite time ......Page 190
2.9.3 Rate of the blow-up of solutions ......Page 196
2.10.1 Blow-up of classical solutions ......Page 200
2.11 On sufficient conditions of the blow-up of solutions of the Boussinesq equation with sources and nonlinear dissipation ......Page 209
2.11.1 Local solvability of strong generalized solutions ......Page 210
2.11.2 Blow-up of solutions ......Page 213
2.12.1 Local solvability of the problem in the strong generalized sense ......Page 216
2.12.2 Blow-up of strong solutions of problem (2.288)–(2.289) and solvability in any finite cylinder ......Page 224
2.12.3 Physical interpretation ......Page 228
3.1 Formulation of problems ......Page 229
3.2 Preliminary definitions and conditions and auxiliary lemma ......Page 230
3.3 Unique solvability of problem (3.1) in the weak generalized sense and blow-up of its solutions ......Page 232
3.4 Unique solvability of problem (3.1) in the strong generalized sense and blow-up of its solutions ......Page 257
3.5 Unique solvability of problem (3.2) in the weak generalized sense and blow-up of its solutions ......Page 267
3.6 Unique solvability of problem (3.2) in the strong generalized sense and blow-up of its solutions......Page 286
3.7 Examples ......Page 291
3.8.1 Local solvability in the strong generalized sense of problems (3.141)–(3.143) ......Page 301
3.8.2 Blow-up of solutions ......Page 308
3.8.3 Breakdown of weakened solutions of problem (3.141) ......Page 315
3.9 On an initial-boundary-value problem for a strongly nonlinear equation of the type (3.1) (generalized Boussinesq equation) ......Page 321
3.9.1 Unique solvability of the problem in the weak sense ......Page 322
3.9.2 Blow-up of solutions and the global solvability of the problem ......Page 328
3.10.1 Unique local solvability of the problem in the strong sense and blow-up of its solutions ......Page 333
3.10.2 Examples ......Page 340
3.11 Blow–up of solutions of the Oskolkov–Benjamin–Bona–Mahony–Burgers equation with a cubic source ......Page 342
3.11.1 Unique local solvability of the problem ......Page 343
3.11.2 Global solvability and the blow-up of solutions ......Page 346
3.12.1 Blow-up of strong generalized solutions ......Page 350
3.12.2 Physical interpretation of the obtained results ......Page 353
3.13.1 Blow-up of strong generalized solutions ......Page 354
3.14 Sufficient, close to necessary, conditions of the blow-up of solutions of strongly nonlinear generalized Boussinesq equation ......Page 358
4.1 Introduction. Statement of problem ......Page 370
4.2 Unique solvability of problem (4.1) in the weak generalized sense and blow-up of its solutions ......Page 371
4.3 Unique solvability of problem (4.1) in the strong generalized sense and blow-up of its solutions ......Page 393
4.4 Examples ......Page 398
4.5.1 Unique local solvability of the problem ......Page 404
4.5.2 Blow-up of strong generalized solutions ......Page 411
4.6 Blow-up of solutions of a strongly nonlinear equation of spin waves ......Page 415
4.6.1 Unique local solvability in the strong generalized sense ......Page 416
4.6.2 Blow-up of strong generalized solutions and the global solvability ......Page 425
4.7 Blow-up of solutions of an initial-boundary-value problem for a strongly nonlinear, dissipative equation of the form (4.1) ......Page 430
4.7.1 Local unique solvability in the weak generalized sense ......Page 431
4.7.2 Unique solvability of the problem and blow-up of its solution for afinite time ......Page 448
5.1.1 Global-on-time solvability of the problem ......Page 452
5.1.2 Global-on-time solvability of the problem in the strong generalized sense in the case q > 1 ......Page 482
5.1.3 Asymptotic behavior of solutions of problem (5.1), (5.2) as t ! C1 in the case q > 0 ......Page 484
5.2 Blow-up of solutions of nonlinear pseudoparabolic equations with sources of the pseudo-Laplacian type ......Page 488
5.2.1 Blow-up ofweakened solutions of problem(5.77) ......Page 489
5.2.2 Blow-up and the global-on-time solvability of problem (5.78) ......Page 490
5.2.3 Blow-up of solutions of problem(5.79) ......Page 492
5.2.4 Blow-up of weakened solutions of problems (5.80) and (5.81) ......Page 495
5.3 Blow-up of solutions of pseudoparabolic equations with fast increasing nonlinearities ......Page 497
5.3.1 Local solvability and blow-up for a finite time of solutions of problems (5.112) and (5.113) ......Page 498
5.3.2 Local solvability and blow-up for a finite time of solutions of problem (5.114) ......Page 505
5.4.1 Unique local solvability of the problem ......Page 509
5.4.2 Blow-up of strong generalized solutions of problem (5.154)–(5.155) ......Page 512
5.4.3 Blow-up of classical solutions of problem (5.154)–(5.155) ......Page 515
5.5 Blow-up of solutions of a nonlinear nonlocal pseudoparabolic equation ......Page 516
5.5.1 Unique local solvability of the problem ......Page 517
5.5.2 Blow-up and global solvability of problem (5.177) ......Page 519
5.5.3 Blow-up rate for problem (5.177) under the condition q D 0 ......Page 522
5.6.1 Reduction the problem to the system of the integral equations ......Page 524
5.6.2 Global-on-time solvability and the blow-up of solutions ......Page 530
5.7.1 Reduction of the problem to an integral equation ......Page 538
5.7.2 Theorems on the existence/nonexistence of global-on-time solutions of the integral equation (5.219) ......Page 540
5.8 Sufficient conditions of the blow-up of solutions of the Boussinesq equation with nonlinear Neumann boundary condition ......Page 550
6.1 Numerical solution of problems for linear equations ......Page 556
6.1.1 Dynamic potentials for one equation ......Page 557
6.1.2 Solvability of Dirichlet problem ......Page 561
6.2.1 Stiffmethod of lines ......Page 567
6.2.4 Schemes ofRosenbrock type ......Page 568
6.2.5 "-embeddingmethod ......Page 570
6.3 Results of blow-up numerical simulation ......Page 573
6.3.1 Blow-up of pseudoparabolic equations with a linear operator bythe time derivative ......Page 574
6.3.2 Blow-up of strongly nonlinear pseudoparabolic equations ......Page 579
6.3.3 Blow-up of equations with nonlocal terms (coefficients of the equation depend on the norm of the function) ......Page 588
A.1 Sobolev spaces Ws;p./, Ws;p 0 ./, and Ws;p./ ......Page 594
A.2 Weak and -weak convergence ......Page 596
A.3 Weak and strong measurability. Bochner integral ......Page 597
A.4 Spaces of integrable functions and distributions ......Page 598
A.5 Nemytskii operator. Krasnoselskii theorem ......Page 599
A.6 Inequalities ......Page 601
A.9 Weakened solutions of the Poisson equation ......Page 602
A.10 Intersections and sums of Banach spaces ......Page 604
A.11 Classical, weakened, strong generalized, and weak generalized solutions of evolutionary problems ......Page 605
A.12 Two equivalent formulations of weak solutions in L2.0; TIB/ ......Page 607
A.13 Gâteaux and Fréchet derivatives of nonlinear operators ......Page 609
A.14 On the gradient of a functional ......Page 617
A.15 Lions compactness lemma ......Page 619
A.16 Browder–Minty theorem ......Page 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 ......Page 621
A.18 On the continuity of some inverse matrices ......Page 623
B.1 Convergence of the "-embedding method with the CROS scheme ......Page 626
Bibliography ......Page 634
Index ......Page 660
