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Linear second-order partial differential equations of the parabolic type. (English. Russian original) Zbl 0988.35001

J. Math. Sci., New York 108, No. 4, 435-542 (2002); translation from Tr. Semin. Im. I. G. Petrovskogo 21, 9-193 (2001).
The article is a translation from the original Russian version [the authors, Russ. Math. Surv. 17, No. 3, 3-146 (1962); translation from Usp. Mat. Nauk 17, No. 3(105), 3-146 (1962; Zbl 0108.28401)]. It is an outgrow of a special course of lectures given by Professor Olga A. Oleynik in 1960 at the Department of Mechanics and Mathematics of the Moscow State University and presents a brilliant source of the classical theory of linear second-order parabolic equations. Detailed proofs are given of the basic facts of that theory, paying special attention to the various methods of constructing classical and generalized solutions to the boundary-value problems (BVP) and to the Cauchy problem.
Section 1 proposes different forms of the maximum principle for homogeneous and non-homogeneous equations in bounded and unbounded domains. The main boundary value problems are stated, and uniqueness of their solutions is proved. The a priori estimates for solutions of the boundary-value problems are formulated in Section 2 in terms of the corresponding data (initial/boundary condition, coefficients of the equation, boundary of the domain). The authors give detailed proofs of S. N. Bernstein’s interior and up-to-the boundary estimates. Integral \((L^2)\) estimates of solutions are also derived. Section 3 is devoted to the construction of solutions to the first (Dirichlet) and second (conormal) boundary value problems as well as to the Cauchy problem by means of Rothe’s method. The precise dependence of solution’s regularity on the regularity of the data is also investigated. Further study is proposed on the first boundary value problem in domains of noncylindrical type, considering various conditions ensuring existence of barriers. In Section 4 the authors construct the fundamental solution of general form, second-order linear parabolic equation with Hölder continuous coefficients. It is employed later in defining a solution to Cauchy’s problem and also in constructing the Green function for the first BVP. A brief outline is proposed of the method of integral equations in solving initial and boundary value problems. Section 5 introduces the notion of a generalized solution of the first BVP and collects uniqueness theorems and various auxiliary results to be used later. In Sections 6-9, the generalized solution of the first BVP is constructed by means of different methods (finite differences, functional methods, continuation with respect to a parameter, Galerkin’s method). In Section 10 the authors prove existence and uniqueness of generalized solutions to the Cauchy problem in classes of rapidly growing functions, while Section 11 is devoted to the method of difference quotients in the study of regularity of the generalized solution in terms of regularity of the data. Finally, Section 12 proposes some simple results concerning the asymptotic behaviour as \(t\to\infty\) of solutions to the parabolic equations under consideration.

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35K20 Initial-boundary value problems for second-order parabolic equations

Citations:

Zbl 0108.28401
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