On the First Initial-Boundary Value Problem for Parabolic Systems in a Bounded Domain with Curvilinear Lateral Boundaries

The first initial-boundary value problem for a second-order Petrovskii parabolic system in a bounded plane domain is considered. The coefficients of the system satisfy the double Dini condition. The lateral boundaries of the domain are defined by continuously differentiable functions. Assuming that the right-hand sides of the Dirichlet boundary conditions are continuously differentiable and the initial function is continuous and bounded together with its first and second derivatives, we establish that the solution of the problem under study belongs to the space of functions that are continuous, together with their higher derivatives, in the closure of the domain. Corresponding estimates are proved. An integral representation of the solution is obtained. If the lateral boundaries of the domain have corners and the boundary functions have piecewise continuous derivatives, we establish that the highest derivatives of the solution are continuous everywhere in the closure of the domain, except at the corner points, and are bounded.