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Linear and nonlinear equations of Sobolev type. (Линейные и нелинейные уравнения соболевского типа.) (Russian) Zbl 1179.35007
Matematika, Prikladnaya Matematika. Moskva: Fizmatlit (ISBN 978-5-9221-0779-2/hbk). 734 p. (2007).
This innovative text/reference contains a detailed study of linear, semilinear and nonlinear partial differential equations not solvable with respect to the highest-order derivative which have recently been called “Sobolev type equations”. The main goal of this monograph is the investigation of 99 new linear and nonlinear Sobolev type equations arising in mathematical models of quasistationary electromagnetic waves in anisotropic non-metal media with dispersion, mathematical models of cold plasma in outer magnetic fields and more.
The contents of this treatise is divided into three parts. The first part named “Linear equations” contains two chapters: “Model linear Sobolev type equation of high order” and “Two-dimensional Sobolev type equations and new classes of generalized analytic functions”. In the second part entitled “Nonlinear equations” are contained five chapters: “Model nonlinear equations of pseudoparabolic type”, “Blowup of solutions to a class strongly nonlinear pseudoparabolic equations with sources and nonlinear dissipation”, “Blowup of solutions to strongly nonlinear wave pseudoparabolic equations”, “Blowup of solutions to strongly nonlinear wave dissipative pseudoparabolic equations with sources”, “Certain particular problems for nonlinear equations of pseudoparabolic type”. The last part is devoted to a method of calculation for solving initial-boundary problems for partial differential equations not solvable with respect to the highest-order time derivative.

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35K99 Parabolic equations and parabolic systems
35L99 Hyperbolic equations and hyperbolic systems
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