BOUNDARY VALUE PROBLEMS FOR A PARABOLIC EQUATION DEGENERATING AT THE BOUND OF THE DOMAIN

Abstract

In the work  boundary value problems have been formulated  and investigated for a parabolic equation degenerating at the bound of the domain. The existence, uniqueness and stability  of the solution of the problem have been  proved. The uniqueness of the solution of the problem was proved by the method of energy integrals. At the same time, by applying the method of separation of variables to the considered problem, a spectral problem for an ordinary differential equation has been obtained. Next, the Green's function of the spectral problem was constructed, with the help of which it is equivalently reduced to an the second kind Fredholm integral equation  with a symmetric kernel. The solution of the considered problem has been written as the sum of a Fourier series with respect to the system of eigenfunctions of the spectral problem. An estimate for solution the problem was obtained, from which follows its continuous dependence on the given functions.