Blow-up in Nonlinear Sobolev Type Equations

1 - Preface [Seite 6]
2 - Contents [Seite 8]
3 - 0 Introduction [Seite 14]
3.1 - 0.1 List of equations [Seite 14]
3.1.1 - 0.1.1 One-dimensional pseudoparabolic equations [Seite 14]
3.1.2 - 0.1.2 One-dimensionalwave dispersive equations [Seite 15]
3.1.3 - 0.1.3 Singular one-dimensional pseudoparabolic equations [Seite 16]
3.1.4 - 0.1.4 Multidimensional pseudoparabolic equations [Seite 16]
3.1.5 - 0.1.5 New nonlinear pseudoparabolic equations with sources [Seite 18]
3.1.6 - 0.1.6 Model nonlinear equations of even order [Seite 19]
3.1.7 - 0.1.7 Multidimensional even-order equations [Seite 20]
3.1.8 - 0.1.8 Results and methods of proving theorems on the nonexistence and blow-up of solutions for pseudoparabolic equations [Seite 23]
3.2 - 0.2 Structure of the monograph [Seite 26]
3.3 - 0.3 Notation [Seite 27]
4 - 1 Nonlinear model equations of Sobolev type [Seite 33]
4.1 - 1.1 Mathematical models of quasi-stationary processes in crystalline semiconductors [Seite 33]
4.2 - 1.2 Model pseudoparabolic equations [Seite 40]
4.2.1 - 1.2.1 Nonlinear waves of Rossby type or drift modes in plasma and appropriate dissipative equations [Seite 40]
4.2.2 - 1.2.2 Nonlinear waves of Oskolkov-Benjamin-Bona-Mahony type [Seite 42]
4.2.3 - 1.2.3 Models of anisotropic semiconductors [Seite 47]
4.2.4 - 1.2.4 Nonlinear singular equations of Sobolev type [Seite 50]
4.2.5 - 1.2.5 Pseudoparabolic equations with a nonlinear operator ontime derivative [Seite 51]
4.2.6 - 1.2.6 Nonlinear nonlocal equations [Seite 52]
4.2.7 - 1.2.7 Boundary-value problems for elliptic equations with pseudoparabolic boundary conditions [Seite 59]
4.3 - 1.3 Disruption of semiconductors as the blow-up of solutions [Seite 61]
4.4 - 1.4 Appearance and propagation of electric domains in semiconductors [Seite 69]
4.5 - 1.5 Mathematical models of quasi-stationary processes in crystalline electromagnetic media with spatial dispersion [Seite 73]
4.6 - 1.6 Model pseudoparabolic equations in electric media with spatial dispersion [Seite 77]
4.7 - 1.7 Model pseudoparabolic equations in magnetic media with spatial dispersion [Seite 79]
5 - 2 Blow-up of solutions of nonlinear equations of Sobolev type [Seite 82]
5.1 - 2.1 Formulation of problems [Seite 82]
5.2 - 2.2 Preliminary definitions, conditions, and auxiliary lemmas [Seite 83]
5.3 - 2.3 Unique solvability of problem (2.1) in the weak generalized sense and blow-up of its solutions [Seite 91]
5.4 - 2.4 Unique solvability of problem (2.1) in the strong generalized sense and blow-up of its solutions [Seite 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 [Seite 124]
5.6 - 2.6 Strong solvability of problem (2.2) in the case where B = 0 [Seite 140]
5.7 - 2.7 Examples [Seite 146]
5.8 - 2.8 Initial-boundary-value problem for a nonlinear equation with double nonlinearity of type (2.1) [Seite 154]
5.8.1 - 2.8.1 Local solvability of problem (2.131)-(2.133)in the weak generalized sense [Seite 155]
5.8.2 - 2.8.2 Blow-up of solutions [Seite 172]
5.9 - 2.9 Problem for a strongly nonlinear equation of type (2.2) with inferior nonlinearity [Seite 177]
5.9.1 - 2.9.1 Unique weak solvability of problem (2.185) [Seite 178]
5.9.2 - 2.9.2 Solvability in a finite cylinder and blow-up for a finite time [Seite 190]
5.9.3 - 2.9.3 Rate of the blow-up of solutions [Seite 196]
5.10 - 2.10 Problem for a semilinear equation of the form (2.2) [Seite 200]
5.10.1 - 2.10.1 Blow-up of classical solutions [Seite 200]
5.11 - 2.11 On sufficient conditions of the blow-up of solutions of the Boussinesq equation with sources and nonlinear dissipation [Seite 209]
5.11.1 - 2.11.1 Local solvability of strong generalized solutions [Seite 210]
5.11.2 - 2.11.2 Blow-up of solutions [Seite 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 [Seite 216]
5.12.1 - 2.12.1 Local solvability of the problem in the strong generalized sense [Seite 216]
5.12.2 - 2.12.2 Blow-up of strong solutions of problem (2.288)-(2.289) and solvability in any finite cylinder [Seite 224]
5.12.3 - 2.12.3 Physical interpretation [Seite 228]
6 - 3 Blow-up of solutions of strongly nonlinear Sobolev-type wave equations and equations with linear dissipation [Seite 229]
6.1 - 3.1 Formulation of problems [Seite 229]
6.2 - 3.2 Preliminary definitions and conditions and auxiliary lemma [Seite 230]
6.3 - 3.3 Unique solvability of problem (3.1) in the weak generalized sense and blow-up of its solutions [Seite 232]
6.4 - 3.4 Unique solvability of problem (3.1) in the strong generalized sense and blow-up of its solutions [Seite 257]
6.5 - 3.5 Unique solvability of problem (3.2) in the weak generalized sense and blow-up of its solutions [Seite 267]
6.6 - 3.6 Unique solvability of problem (3.2) in the strong generalized sense and blow-up of its solutions [Seite 286]
6.7 - 3.7 Examples [Seite 291]
6.8 - 3.8 On certain initial-boundary-value problems for quasilinear wave equations of the form(3.2) [Seite 301]
6.8.1 - 3.8.1 Local solvability in the strong generalized sense of problems (3.141)-(3.143) [Seite 301]
6.8.2 - 3.8.2 Blow-up of solutions [Seite 308]
6.8.3 - 3.8.3 Breakdown of weakened solutions of problem (3.141) [Seite 315]
6.9 - 3.9 On an initial-boundary-value problem for a strongly nonlinear equation of the type (3.1) (generalized Boussinesq equation) [Seite 321]
6.9.1 - 3.9.1 Unique solvability of the problem in the weak sense [Seite 322]
6.9.2 - 3.9.2 Blow-up of solutions and the global solvability of the problem [Seite 328]
6.10 - 3.10 Blow-up of solutions of a class of quasilinear wave dissipative pseudoparabolic equations with sources [Seite 333]
6.10.1 - 3.10.1 Unique local solvability of the problem in the strong sense and blow-up of its solutions [Seite 333]
6.10.2 - 3.10.2 Examples [Seite 340]
6.11 - 3.11 Blow-up of solutions of the Oskolkov-Benjamin-Bona-Mahony-Burgers equation with a cubic source [Seite 342]
6.11.1 - 3.11.1 Unique local solvability of the problem [Seite 343]
6.11.2 - 3.11.2 Global solvability and the blow-up of solutions [Seite 346]
6.11.3 - 3.11.3 Physical interpretation of the obtained results [Seite 350]
6.12 - 3.12 On generalized Benjamin-Bona-Mahony-Burgers equation with pseudo-Laplacian [Seite 350]
6.12.1 - 3.12.1 Blow-up of strong generalized solutions [Seite 350]
6.12.2 - 3.12.2 Physical interpretation of the obtained results [Seite 353]
6.13 - 3.13 Sufficient, close to necessary, conditions of the blow-up of solutions of one problem with pseudo-Laplacian [Seite 354]
6.13.1 - 3.13.1 Blow-up of strong generalized solutions [Seite 354]
6.13.2 - 3.13.2 Physical interpretation of the obtained results [Seite 358]
6.14 - 3.14 Sufficient, close to necessary, conditions of the blow-up of solutions of strongly nonlinear generalized Boussinesq equation [Seite 358]
7 - 4 Blow-up of solutions of strongly nonlinear, dissipative wave Sobolevtype equations with sources [Seite 370]
7.1 - 4.1 Introduction. Statement of problem [Seite 370]
7.2 - 4.2 Unique solvability of problem (4.1) in the weak generalized sense and blow-up of its solutions [Seite 371]
7.3 - 4.3 Unique solvability of problem (4.1) in the strong generalized sense and blow-up of its solutions [Seite 393]
7.4 - 4.4 Examples [Seite 398]
7.5 - 4.5 Blow-up of solutions of a Sobolev-type wave equation with nonlocal sources [Seite 404]
7.5.1 - 4.5.1 Unique local solvability of the problem [Seite 404]
7.5.2 - 4.5.2 Blow-up of strong generalized solutions [Seite 411]
7.6 - 4.6 Blow-up of solutions of a strongly nonlinear equation of spin waves [Seite 415]
7.6.1 - 4.6.1 Unique local solvability in the strong generalized sense [Seite 416]
7.6.2 - 4.6.2 Blow-up of strong generalized solutions and the global solvability [Seite 425]
7.6.3 - 4.6.3 Physical interpretation of the obtained results [Seite 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) [Seite 430]
7.7.1 - 4.7.1 Local unique solvability in the weak generalized sense [Seite 431]
7.7.2 - 4.7.2 Unique solvability of the problem and blow-up of its solution for afinite time [Seite 448]
8 - 5 Special problems for nonlinear equations of Sobolev type [Seite 452]
8.1 - 5.1 Nonlinear nonlocal pseudoparabolic equations [Seite 452]
8.1.1 - 5.1.1 Global-on-time solvability of the problem [Seite 452]
8.1.2 - 5.1.2 Global-on-time solvability of the problem in the strong generalized sense in the case q > 1 [Seite 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 [Seite 484]
8.2 - 5.2 Blow-up of solutions of nonlinear pseudoparabolic equations with sources of the pseudo-Laplacian type [Seite 488]
8.2.1 - 5.2.1 Blow-up ofweakened solutions of problem(5.77) [Seite 489]
8.2.2 - 5.2.2 Blow-up and the global-on-time solvability of problem (5.78) [Seite 490]
8.2.3 - 5.2.3 Blow-up of solutions of problem(5.79) [Seite 492]
8.2.4 - 5.2.4 Blow-up of weakened solutions of problems (5.80) and (5.81) [Seite 495]
8.2.5 - 5.2.5 Interpretation of the obtained results [Seite 497]
8.3 - 5.3 Blow-up of solutions of pseudoparabolic equations with fast increasing nonlinearities [Seite 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) [Seite 498]
8.3.2 - 5.3.2 Local solvability and blow-up for a finite time of solutions of problem (5.114) [Seite 505]
8.4 - 5.4 Blow-up of solutions of nonhomogeneous nonlinear pseudoparabolic equations [Seite 509]
8.4.1 - 5.4.1 Unique local solvability of the problem [Seite 509]
8.4.2 - 5.4.2 Blow-up of strong generalized solutions of problem (5.154)-(5.155) [Seite 512]
8.4.3 - 5.4.3 Blow-up of classical solutions of problem (5.154)-(5.155) [Seite 515]
8.5 - 5.5 Blow-up of solutions of a nonlinear nonlocal pseudoparabolic equation [Seite 516]
8.5.1 - 5.5.1 Unique local solvability of the problem [Seite 517]
8.5.2 - 5.5.2 Blow-up and global solvability of problem (5.177) [Seite 519]
8.5.3 - 5.5.3 Blow-up rate for problem (5.177) under the condition q = 0 [Seite 522]
8.6 - 5.6 Existence of solutions of the Laplace equation with nonlinear dynamic boundary conditions [Seite 524]
8.6.1 - 5.6.1 Reduction the problem to the system of the integral equations [Seite 524]
8.6.2 - 5.6.2 Global-on-time solvability and the blow-up of solutions [Seite 530]
8.7 - 5.7 Conditions of the global-on-time solvability of the Cauchy problem for a semilinear pseudoparabolic equation [Seite 538]
8.7.1 - 5.7.1 Reduction of the problem to an integral equation [Seite 538]
8.7.2 - 5.7.2 Theorems on the existence/nonexistence of global-on-time solutions of the integral equation (5.219) [Seite 540]
8.8 - 5.8 Sufficient conditions of the blow-up of solutions of the Boussinesq equation with nonlinear Neumann boundary condition [Seite 550]
9 - 6 Numerical methods of solution of initial-boundary-value problems for Sobolev-type equations [Seite 556]
9.1 - 6.1 Numerical solution of problems for linear equations [Seite 556]
9.1.1 - 6.1.1 Dynamic potentials for one equation [Seite 557]
9.1.2 - 6.1.2 Solvability of Dirichlet problem [Seite 561]
9.2 - 6.2 Numerical method of solving initial-boundary-value problems for nonlinear pseudoparabolic equations by the Rosenbrock schemes [Seite 567]
9.2.1 - 6.2.1 Stiffmethod of lines [Seite 567]
9.2.2 - 6.2.2 Stiff systems of ODE and methods of solving them [Seite 568]
9.2.3 - 6.2.3 Stiff stability [Seite 568]
9.2.4 - 6.2.4 Schemes ofRosenbrock type [Seite 568]
9.2.5 - 6.2.5 e-embedding method [Seite 570]
9.3 - 6.3 Results of blow-up numerical simulation [Seite 573]
9.3.1 - 6.3.1 Blow-up of pseudoparabolic equations with a linear operator bythe time derivative [Seite 574]
9.3.2 - 6.3.2 Blow-up of strongly nonlinear pseudoparabolic equations [Seite 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) [Seite 588]
10 - Appendix A Some facts of functional analysis [Seite 594]
10.1 - A.1 Sobolev spaces [Seite 594]
10.2 - A.2 Weak and *-weak convergence [Seite 596]
10.3 - A.3 Weak and strong measurability. Bochner integral [Seite 597]
10.4 - A.4 Spaces of integrable functions and distributions [Seite 598]
10.5 - A.5 Nemytskii operator. Krasnoselskii theorem [Seite 599]
10.6 - A.6 Inequalities [Seite 601]
10.7 - A.7 Operator calculus [Seite 602]
10.8 - A.8 Fixed-point theorems [Seite 602]
10.9 - A.9 Weakened solutions of the Poisson equation [Seite 602]
10.10 - A.10 Intersections and sums of Banach spaces [Seite 604]
10.11 - A.11 Classical, weakened, strong generalized, and weak generalized solutions of evolutionary problems [Seite 605]
10.12 - A.12 Two equivalent formulations of weak solutions in L2(0 [Seite 607]
10.13 - A.13 Gâteaux and Fréchet derivatives of nonlinear operators [Seite 609]
10.14 - A.14 On the gradient of a functional [Seite 617]
10.15 - A.15 Lions compactness lemma [Seite 619]
10.16 - A.16 Browder-Minty theorem [Seite 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 [Seite 621]
10.18 - A.18 On the continuity of some inverse matrices [Seite 623]
11 - Appendix B To Chapter 6 [Seite 626]
11.1 - B.1 Convergence of the e-embedding method with the CROS scheme [Seite 626]
12 - Bibliography [Seite 634]
13 - Index [Seite 660]