Cauchy's problem for a singular parabolic partial differential equation

Abstract

In recent years analytic function theory has been shown to play a basic role in the investigation of existence and uniqueness theorems for solutions to elliptic partial differential equations ([6], [8], [17]). An approach which has proved particularly fruitful is that of integral operators ([l], [12], [17]), from whose use complete families of solutions can be obtained, thus enabling one to construct the Bergman kernel function and solve the Dirichlet and Neumann problems ([2]). For singular elliptic equations serious difficulties arise due to the nonregularities in the kernels of such operators, as well as the failure of Green’s representation to hold in a neighborhood of the singular curve. In such cases recourse is often made to the use of operators whose path of integration is a contour in the complex plane ([7], [15]). In this paper we apply integral operator techniques in conjunction with function theoretic methods to establish an existence, uniqueness, and representation theorem to Cauchy’s problem for the singular parabolic equation