Dezin problem analog for a parabolic-hyperbolic type equation with periodicity condition

Abstract

In this paper we consider an inhomogeneous second-order parabolic-hyperbolic mixed type equation, represented as one-dimensional heat equation in the parabolic part and the one-dimensional wave equation in the hyperbolic part. For the equation, an analog of the Dezin problem is investigated, which means to find a solution to the equation that satisfies inner-boundary condition, relating the value of the desired function on the equation type change line to the value of the normal derivative on the hyperbolicity region boundary, and inhomogeneous periodicity nonlocal boundary conditions. A substitution is given that allows us to reduce the problem to an equivalent one and, without losing generality, restrict ourselves to investigate the problem with homogeneous conditions for an inhomogeneous equation.The solution is constructed as the Fourier series on the orthonormal system of eigenfunctions of the corresponding one-dimensional spectral problem. A criterion for the solution uniqueness to the problem is established.In case when the uniqueness criterion is violated, an example of a nontrivial solution to a homogeneous problem is given, and a necessary and sufficient condition for the existence of a solution to an inhomogeneous problem is obtained.In justifying the solution existence, the problem of small denominators in the sum of the series with respect to the ratio of the rectangle sides in the hyperbolic part of the domain. An estimate of the denominator separation from zero under certain conditions with respect to the problem parameters is obtained. This estimate allows us to substantiate the uniform convergence of the series and their derivatives up to the second-order inclusive under certain conditions for given functions.