First Initial-Boundary Value Problem for Parabolic Systems in a Semibounded Domain with Curvilinear Lateral Boundary

The first initial-boundary value problem for a second-order parabolic system in a semibounded plane domain is considered. The coefficients of the system satisfy the double Dini condition. The function defining the lateral boundary of the domain is continuously differentiable on an interval. Assuming that the right-hand side of the Dirichlet boundary condition is 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 is continuous and bounded in the closure of the domain together with its higher derivatives. Corresponding estimates are proved. An integral representation of the solution is given. If the lateral boundary of the domain has corners and the boundary function has a piecewise continuous derivative, then it is proved that, despite the nonsmoothness of the lateral boundary and the boundary function, the highest derivatives of the solution are continuous everywhere in the closure of the domain, except the corner points, and are bounded.