Parabolic boundary value problems
The present monograph is devoted to the theory of general parabolic boundary value problems. In the first chapter the authors define the basic notions at the origin of the theory of parabolic problems and give various examples of illustrative and descriptive character. Chapters II to V are devoted to detailed and systematic exposition of the \(L_2\)-theory of parabolic boundary value problems in Hilbert spaces of smooth functions and distributions of arbitrary finite order and with some natural applications of the theory. The parabolic problem is related with the scale of pairs \(\{U_s, F_s\}\) of Hilbert spaces of distributions depending on the real parameter \(s\) so that the operator of the problem realizes an isomorphism between the spaces \(U_s\) and \(F_s\). For sufficiently large \(s\) the solution is a usual classical solution of the boundary value problem. In Chapter VI the authors give a survey of results to Cauchy problems and boundary value problems in the traditional spaces of smooth functions. Special attention is paid to a detailed analysis of examples illustrating and complementing the results formulated. One can read the chapter alone, skipping the previous chapters. Chapter VII is devoted to theorems on stabilization of solutions of parabolic problems. The monograph is very useful both for graduate and post-graduate students, and for scientific researchers whose interests lie in the field of differential equations. For the Russian original (1992) see Zbl 0855.35002.
scientific article; zbMATH DE number 1120673