Stanislav I. Pohozaev – författare

Blow-up for Higher-Order Parabolic, Hyperbolic, Dispersion and Schrödinger Equations shows how four types of higher-order nonlinear evolution partial differential equations (PDEs) have many commonalities through their special quasilinear degenerate representations. The authors present a unified approach to deal with these quasilinear PDEs.The book first studies the particular self-similar singularity solutions (patterns) of the equations. This approach allows four different classes of nonlinear PDEs to be treated simultaneously to establish their striking common features. The book describes many properties of the equations and examines traditional questions of existence/nonexistence, uniqueness/nonuniqueness, global asymptotics, regularizations, shock-wave theory, and various blow-up singularities.Preparing readers for more advanced mathematical PDE analysis, the book demonstrates that quasilinear degenerate higher-order PDEs, even exotic and awkward ones, are not as daunting as they first appear. It also illustrates the deep features shared by several types of nonlinear PDEs and encourages readers to develop further this unifying PDE approach from other viewpoints.

Many physical processes in fields such as mechanics, thermodynamics, electricity, magnetism or optics are described by means of partial differential equations. In particular, the methods of conformal mappings, Fourier analysis and Green`s functions are considered, as well as the perturbation method and integral transformation method, among others.

This handbook is addressed to students of technology institutf's where a course on mathematical physics of relatively reduced volume is offered, as well as to engineers and scientists. The aim of the handbook is to treat (demonstrate) the basic methods for solving the simplest problems of classical mathematical physics. The most basic among the methods considered hrre i8 the superposition method. It allows one, based on particular linearly indepmdent HolutionH (solution "atoms"), to obtain the solution of a given problem. To that end the "Hupply" of solution atoms must be complete. This method is a development of the well-known method of particular solutions from the theory of ordinar~' differelltial equations. In contrast to the case of ordinary differential equations, where the number of linearly independent 80lutions is always finite, for a linear partial differrntial equation a complete "supply" of solution atoms is always infinite. This infinite set of Holutions may be discrete (for example, for regular boundary vahlP problems in a bounded domain), or form a continuum (for example, in the case of problems in the whole space). In the first case the superposition method reduces to tlH' construction of a series in the indicated solution atoms with unknown coefficipnts, while in the second case the series is replaced by an integral with respect to the corm:iponding parameters (variables). This first step leads us to the general solution of the associated hOlllogeneous equation under the assumption that the set of solution atoms i;; "complete.

Semilinear elliptic equations play an important role in many areas of mathematics and its applications to physics and other sciences. This book presents a wealth of modern methods to solve such equations, including the systematic use of the Pohozaev identities for the description of sharp estimates for radial solutions and the fibring method. Existence results for equations with supercritical growth and non-zero right-hand sides are given.Readers of this exposition will be advanced students and researchers in mathematics, physics and other sciences who want to learn about specific methods to tackle problems involving semilinear elliptic equations.