A solvability conditions of mixed problems for equations of parabolic type with involution

Abstract

In this work the partial differential equations with involutions are considered. The mixed problems for the parabolic type equation, with constant and variable constants, corresponding to the Dirichlet type boundary conditions is investigated. The involution is contained by the second derivative with respect to the variable x, which is the difficult case for investigations. One-dimensional differential operators with involution have an infinite number of positive and negative eigenvalues. This means that the operator on the right-hand side of the equation under study is not semi-bounded. In the case of classical problems, ordinary differential operators usually appear on the right-hand side of the equations, which are semi¬bounded. Therefore, the incorrectness of mixed problems for a parabolic equation with an involution is discussed in this paper. Examples are given. Sufficient conditions for the initial data are found when the problem under study has a unique solution. The representation of the solution in the form of partial sums of the Fourier series in eigenfunctions is found. The density in the space L2 (—1,1) of the set of initial functions is proved everywhere, when the problem has a unique solution.