Regular Boundary Value Problems for the Heat Equation with Scalar Parameters

摘要

This paper belongs to the general theory of well-posed initial-boundary value problems for parabolic equations. The classical construction of a boundary value problem is as follows: an equation and a boundary condition are given. It is necessary to investigate the solvability of this problem and properties of the solution if it exists (in the sense of belonging to some space). Beginning with the papers of J. von Neumann and M.I. Vishik (1951), there exists another more general approach: an equation and a space are given, right-hand parts of the equation and boundary conditions, and a solution must belong to this space. It is necessary to describe all the boundary conditions, for which the problem is correctly solvable in this space. Further development of this theory was given by M. Otelbaev, who constructed a complete theory for ordinary differential operators and for symmetric semibounded operators in a Banach space. In this paper we find regular solution of the regular boundary problem for the heat equation with scalar parameter.