The First Initial Boundary Value Problem for Parabolic Systems in a Semi-Bounded Domain With Curvilinear Lateral Boundary

The first initial boundary value problem for a second-order parabolic system in a semi-bounded domain on the plane 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 the closed interval. When the right-hand side of the boundary condition of the first kind is continuously differentiable and the initial function is continuous and bounded together with its first and second derivatives, it is established that the solution of the problem is continuous and bounded in the closure of the domain together with its higher order derivatives. The 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, it is proved that, despite the lateral boundary and the boundary function being non-smooth, the higher order derivatives of the solution are continuous everywhere in the closure of the domain, except the corner points, and are bounded.