ON THE FIRST INITIAL BOUNDARY VALUE PROBLEM FOR PARABOLIC SYSTEMS IN A BOUNDED DOMAIN WITH CURVED LATERAL BOUNDARIES

The first initial boundary value problem for a second-order Petrovsky parabolic system with coefficients satisfying the double Dini condition in a bounded domain on the plane is considered. The lateral boundaries of the region are defined by continuously differentiable functions. It is established that if the right-hand parts of the boundary conditions of the first kind are continuously differentiable, and the initial function is continuous and bounded together with its first and second derivatives, so the solution to the problem belongs to the space of functions that are continuous and bounded together with their higher derivatives in the closure of the domain. The corresponding estimates have been proved. An integral representation of the solution is obtained. If the lateral boundaries of the domain admit the presence of angles, and the boundary functions have piecewise continuous derivatives, then in this case it is established that the higher derivatives of the solution are continuous everywhere in the closure of the domain, with the exception of corner points, and thus bounded.

parabolic systems, initial boundary value problems, non-smooth lateral boundaries of a domain, boundary integral equations, Dini condition