On Solving Initial Boundary Value Problems for a Parabolic Equation in Dini Classes

Abstract

The method of potentials is one of the classical methods for solving initial boundary value problems for parabolic equations and systems. The method is based on studying the smoothness of parabolic potentials in various functional spaces and studying the solvability of the corresponding integral equations and systems to which the problems are reduced. This method allows one to find a constructive form of solutions to initial-boundary value problems and directly investigate the smoothness of solutions. This article continues a series of works on constructing the solvability theory of parabolic initial boundary value problems in domains with nonsmooth lateral boundaries in the time variable. The article considers the first and second initial boundary value problems with zero initial condition for a uniformly parabolic equation that is one-dimensional in the spatial variable. It is assumed that the equation coefficients are bounded and uniformly continuous with a modulus of continuity satisfying twice the Dini condition. The specific feature of the study is that the right side of the equation can grow to infinity in a certain way when approaching the initial data assignment line. The solvability (in the classical sense) of these problems is established in a semi-bounded curvilinear domain in a plane with a nonsmooth lateral boundary from the Dini-Hölder class, admitting, in particular, so-called "beaks". The solution is constructed as a sum of the plane parabolic potential and the single layer potential, the kernel of which is the fundamental solution of the equation. The smoothness of the solution of these problems in the Dini-Hölder classes is investigated, and the corresponding correctness estimates are obtained. These estimates characterize the behavior of solutions and their first-order spatial derivative in the closure of the domain. The results obtained can be used to study heat and mass transfer processes. For the proof, the smoothness in the Dini-Hölder space of a plane parabolic potential with an unlimited density of the Dini class in a curvilinear domain with a nonsmooth lateral boundary is investigated. The obtained result can be used in solving other initial boundary value problems for an inhomogeneous parabolic equation.