This monograph focuses on initial-boundary-value problems for multi-dimensional Sobolev-type equations in bounded domains. It examines specific initial-boundary-value problems and abstract Cauchy problems for first-order differential equations with nonlinear operator coefficients related to spatial variables. The primary objective is to establish sufficient conditions for global solvability, conditions for finite-time blow-up of solutions, and to derive estimates for blow-up time. The findings are applicable to a wide range of problems, including the Benjamin-Bona-Mahony-Burgers equation and Rosenau-Burgers equations with sources, among others. Additionally, the methodology for analyzing blow-up phenomena in nonlinear Sobolev-type equations is extended to equations significant in physics, such as those describing electrical breakdown mechanisms in crystal semiconductors and breakdown in the presence of free charge sources within a self-consistent electric field. The monograph features an extensive reference list (440 items), providing a comprehensive overview of the current state of mathematical modeling for various critical problems in physics. Notably, it includes many references to papers previously published only in Russian journals, serving as a valuable resource for exploring Russian literature.
