Corner boundary layer in boundary value problems with nonlinearities having stationary points

For a singularly perturbed parabolic equation
$$ \varepsilon^2\biggl(a^2\frac{\partial^2u}{\partial x^2}-\frac{\partial u}{\partial t}\biggr)=F(u,x,t,\epsilon) $$
in a rectangle, a problem with boundary conditions of the first kind is considered. It is assumed that, at the corner points of the rectangle, the function $F$ is cubic in the variable $u$. The zero of the derivative of $F$ and the boundary value of the problem at each corner point of the rectangle lie on one side of the solution of the degenerate equation. A complete asymptotic expansion of the solution at $\varepsilon\to0$ is constructed, and its uniformity in the closed rectangle is substantiated.