Corner boundary layer in boundary value problems for singularly perturbed parabolic equations with monotonic nonlinearity

A singularly perturbed parabolic equation
$$ \varepsilon^2\left(a^2\frac{\partial^2u}{\partial x^2}-\frac{\partial u}{\partial t}\right)=F(u,x,t,\varepsilon) $$
is considered in a rectangle with boundary conditions of the first kind. The function $F$ at the corner points of the rectangle is assumed to be monotonic with respect to the variable $u$ on the interval from the root of the degenerate equation to the boundary condition. A complete asymptotic expansion of the solution as $\varepsilon\to0$ is constructed, and its uniformity in the closed rectangle is proven.