Asymptotics of the solution of the first boundary value problem for a singularly perturbed differential equation in partial derivatives of the second order of parabolic type

The article constructs a complete uniform asymptotic expansion in a small parameter of the solution of the first boundary value problem. The first boundary value problem is posed for a singularly perturbed linear inhomogeneous second-order partial differential equation with two independent variables of parabolic type. The problem is investigated on a rectangle. The peculiarities of the problem are the presence of a small parameter in front of the heat conduction operator, the existence of corner boundary layers at the lower corners of the rectangle. It is required to construct a uniform asymptotic expansion of the solution of the first boundary value problem on a rectangle, with any degree of accuracy, as the small parameter tends to zero. The asymptotic expansion of the solution in terms of a small parameter is constructed by the Vishik-Lyusternik method. When solving the problem, we use: methods of integration of ordinary differential equations, the classical method of a small parameter, the Vishik-Lyusternik boundary function method, and the maximum principle. As usual, the problem is solved in two stages: in the first stage, a formal expansion of the solution of the first boundary value problem is constructed; and in the second stage, the remainder of the resulting expansion is estimated and this proves that the resulting expansion is indeed asymptotic over the entire rectangle. In the first stage, a formal asymptotic solution is sought in the form of a sum of six functions (solutions): an external solution defined on the entire rectangle; boundary layer solution in a small neighborhood of the lower side of the rectangle; two lateral boundary layer solutions in a small neighborhood of the lateral sides of the rectangle and two corner boundary layer solutions in the neighborhood of the lower vertices of the rectangle. All these boundary layer solutions exponentially decrease outside the boundary layers.