On Correctness of a Mixed Problem for the Heat Equation with the Mixed Derivative in the Boundary Condition

Abstract

We consider an initial-boundary value problem for the heat equation with an inhomogeneous initial condition and boundary conditions. One of the boundary conditions contains a mixed derivative. When solving this problem by the method of separation of variables, a spectral problem arises. A system of eigenfunctions of this spectral problem and a biorthogonally conjugate system, are constructed explicitly. Also we obtain an asymptotic formula for the eigenvalues. In this paper we formulate theorems on the properties of the system of eigenfunctions of the spectral problem and a theorem about representing the solution of the initial initial-boundary value problem in the form of a Fourier series in the system of eigenfunctions. Thus, the existence of a solution is shown if the initial condition belongs to the Holder class. However, it has been shown that the solution is not unique. We show that additional condition guarantees the uniqueness of the solution. The unique solution of this problem is also obtained in the article.