Angular boundary layer in boundary value problems for singularly perturbed parabolic equations with quadratic nonlinearity

A singularly perturbed parabolic equation $\varepsilon^2\left(a^2\frac{\partial^2u}{\partial x^2}-\frac{\partial u}{\partial t}\right)=F(u,x,t,\varepsilon)$ with the boundary conditions of the first kind is considered in a rectangle. The function $F$ at the angular points is assumed to be quadratic. The full asymptotic approximation of the solution as $\varepsilon\to 0$ is constructed, and its uniformity in the closed rectangle is substantiated.