Corner boundary layer in boundary value problems for singularly perturbed parabolic equations with cubic nonlinearities

For a singularly perturbed parabolic equation
$${{\epsilon }^{2}}\left( {{{a}^{2}}\frac{{{{\partial }^{2}}u}}{{\partial {{x}^{2}}}}-\frac{{\partial u}}{{\partial t}}}\right)=F(u,x,t,\epsilon) $$
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 $u$. A complete asymptotic expansion of the solution at $\epsilon\to0$ is constructed, and its uniformity in a closed rectangle is substantiated.