Preface -- Contents -- 0 Introduction -- 0.1 List of equations -- 0.1.1 One-dimensional pseudoparabolic equations -- 0.1.2 One-dimensionalwave dispersive equations -- 0.1.3 Singular one-dimensional pseudoparabolic equations -- 0.1.4 Multidimensional pseudoparabolic equations -- 0.1.5 New nonlinear pseudoparabolic equations with sources -- 0.1.6 Model nonlinear equations of even order -- 0.1.7 Multidimensional even-order equations -- 0.1.8 Results and methods of proving theorems on the nonexistence and blow-up of solutions for pseudoparabolic equations -- 0.2 Structure of the monograph -- 0.3 Notation -- 1 Nonlinear model equations of Sobolev type -- 1.1 Mathematical models of quasi-stationary processes in crystalline semiconductors -- 1.2 Model pseudoparabolic equations -- 1.2.1 Nonlinear waves of Rossby type or drift modes in plasma and appropriate dissipative equations -- 1.2.2 Nonlinear waves of Oskolkov-Benjamin-Bona-Mahony type -- 1.2.3 Models of anisotropic semiconductors -- 1.2.4 Nonlinear singular equations of Sobolev type -- 1.2.5 Pseudoparabolic equations with a nonlinear operator ontime derivative -- 1.2.6 Nonlinear nonlocal equations -- 1.2.7 Boundary-value problems for elliptic equations with pseudoparabolic boundary conditions -- 1.3 Disruption of semiconductors as the blow-up of solutions -- 1.4 Appearance and propagation of electric domains in semiconductors -- 1.5 Mathematical models of quasi-stationary processes in crystalline electromagnetic media with spatial dispersion -- 1.6 Model pseudoparabolic equations in electric media with spatial dispersion -- 1.7 Model pseudoparabolic equations in magnetic media with spatial dispersion -- 2 Blow-up of solutions of nonlinear equations of Sobolev type -- 2.1 Formulation of problems.

2.2 Preliminary definitions, conditions, and auxiliary lemmas -- 2.3 Unique solvability of problem (2.1) in the weak generalized sense and blow-up of its solutions -- 2.4 Unique solvability of problem (2.1) in the strong generalized sense and blow-up of its solutions -- 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 -- 2.6 Strong solvability of problem (2.2) in the case where B = 0 -- 2.7 Examples -- 2.8 Initial-boundary-value problem for a nonlinear equation with double nonlinearity of type (2.1) -- 2.8.1 Local solvability of problem (2.131)-(2.133)in the weak generalized sense -- 2.8.2 Blow-up of solutions -- 2.9 Problem for a strongly nonlinear equation of type (2.2) with inferior nonlinearity -- 2.9.1 Unique weak solvability of problem (2.185) -- 2.9.2 Solvability in a finite cylinder and blow-up for a finite time -- 2.9.3 Rate of the blow-up of solutions -- 2.10 Problem for a semilinear equation of the form (2.2) -- 2.10.1 Blow-up of classical solutions -- 2.11 On sufficient conditions of the blow-up of solutions of the Boussinesq equation with sources and nonlinear dissipation -- 2.11.1 Local solvability of strong generalized solutions -- 2.11.2 Blow-up of solutions -- 2.12 Sufficient conditions of the blow-up of solutions of initial-boundaryvalue problems for a strongly nonlinear pseudoparabolic equation of Rosenau type -- 2.12.1 Local solvability of the problem in the strong generalized sense -- 2.12.2 Blow-up of strong solutions of problem (2.288)-(2.289) and solvability in any finite cylinder -- 2.12.3 Physical interpretation -- 3 Blow-up of solutions of strongly nonlinear Sobolev-type wave equations and equations with linear dissipation -- 3.1 Formulation of problems.

3.2 Preliminary definitions and conditions and auxiliary lemma -- 3.3 Unique solvability of problem (3.1) in the weak generalized sense and blow-up of its solutions -- 3.4 Unique solvability of problem (3.1) in the strong generalized sense and blow-up of its solutions -- 3.5 Unique solvability of problem (3.2) in the weak generalized sense and blow-up of its solutions -- 3.6 Unique solvability of problem (3.2) in the strong generalized sense and blow-up of its solutions -- 3.7 Examples -- 3.8 On certain initial-boundary-value problems for quasilinear wave equations of the form(3.2) -- 3.8.1 Local solvability in the strong generalized sense of problems (3.141)-(3.143) -- 3.8.2 Blow-up of solutions -- 3.8.3 Breakdown of weakened solutions of problem (3.141) -- 3.9 On an initial-boundary-value problem for a strongly nonlinear equation of the type (3.1) (generalized Boussinesq equation) -- 3.9.1 Unique solvability of the problem in the weak sense -- 3.9.2 Blow-up of solutions and the global solvability of the problem -- 3.10 Blow-up of solutions of a class of quasilinear wave dissipative pseudoparabolic equations with sources -- 3.10.1 Unique local solvability of the problem in the strong sense and blow-up of its solutions -- 3.10.2 Examples -- 3.11 Blow-up of solutions of the Oskolkov-Benjamin-Bona-Mahony-Burgers equation with a cubic source -- 3.11.1 Unique local solvability of the problem -- 3.11.2 Global solvability and the blow-up of solutions -- 3.11.3 Physical interpretation of the obtained results -- 3.12 On generalized Benjamin-Bona-Mahony-Burgers equation with pseudo-Laplacian -- 3.12.1 Blow-up of strong generalized solutions -- 3.12.2 Physical interpretation of the obtained results.

3.13 Sufficient, close to necessary, conditions of the blow-up of solutions of one problem with pseudo-Laplacian -- 3.13.1 Blow-up of strong generalized solutions -- 3.13.2 Physical interpretation of the obtained results -- 3.14 Sufficient, close to necessary, conditions of the blow-up of solutions of strongly nonlinear generalized Boussinesq equation -- 4 Blow-up of solutions of strongly nonlinear, dissipative wave Sobolevtype equations with sources -- 4.1 Introduction. Statement of problem -- 4.2 Unique solvability of problem (4.1) in the weak generalized sense and blow-up of its solutions -- 4.3 Unique solvability of problem (4.1) in the strong generalized sense and blow-up of its solutions -- 4.4 Examples -- 4.5 Blow-up of solutions of a Sobolev-type wave equation with nonlocal sources -- 4.5.1 Unique local solvability of the problem -- 4.5.2 Blow-up of strong generalized solutions -- 4.6 Blow-up of solutions of a strongly nonlinear equation of spin waves -- 4.6.1 Unique local solvability in the strong generalized sense -- 4.6.2 Blow-up of strong generalized solutions and the global solvability -- 4.6.3 Physical interpretation of the obtained results -- 4.7 Blow-up of solutions of an initial-boundary-value problem for a strongly nonlinear, dissipative equation of the form (4.1) -- 4.7.1 Local unique solvability in the weak generalized sense -- 4.7.2 Unique solvability of the problem and blow-up of its solution for afinite time -- 5 Special problems for nonlinear equations of Sobolev type -- 5.1 Nonlinear nonlocal pseudoparabolic equations -- 5.1.1 Global-on-time solvability of the problem -- 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 > 0 -- 5.2 Blow-up of solutions of nonlinear pseudoparabolic equations with sources of the pseudo-Laplacian type -- 5.2.1 Blow-up ofweakened solutions of problem(5.77) -- 5.2.2 Blow-up and the global-on-time solvability of problem (5.78) -- 5.2.3 Blow-up of solutions of problem(5.79) -- 5.2.4 Blow-up of weakened solutions of problems (5.80) and (5.81) -- 5.2.5 Interpretation of the obtained results -- 5.3 Blow-up of solutions of pseudoparabolic equations with fast increasing nonlinearities -- 5.3.1 Local solvability and blow-up for a finite time of solutions of problems (5.112) and (5.113) -- 5.3.2 Local solvability and blow-up for a finite time of solutions of problem (5.114) -- 5.4 Blow-up of solutions of nonhomogeneous nonlinear pseudoparabolic equations -- 5.4.1 Unique local solvability of the problem -- 5.4.2 Blow-up of strong generalized solutions of problem (5.154)-(5.155) -- 5.4.3 Blow-up of classical solutions of problem (5.154)-(5.155) -- 5.5 Blow-up of solutions of a nonlinear nonlocal pseudoparabolic equation -- 5.5.1 Unique local solvability of the problem -- 5.5.2 Blow-up and global solvability of problem (5.177) -- 5.5.3 Blow-up rate for problem (5.177) under the condition q D 0 -- 5.6 Existence of solutions of the Laplace equation with nonlinear dynamic boundary conditions -- 5.6.1 Reduction the problem to the system of the integral equations -- 5.6.2 Global-on-time solvability and the blow-up of solutions -- 5.7 Conditions of the global-on-time solvability of the Cauchy problem for a semilinear pseudoparabolic equation -- 5.7.1 Reduction of the problem to an integral equation.

5.7.2 Theorems on the existence/nonexistence of global-on-time solutions of the integral equation (5.219).