Corner Boundary Layer in Boundary Value Problems for Singularly Perturbed Parabolic Equations with Monotonic Nonlinearity

A singularly perturbed parabolic equation

$${\varepsilon ^2}\left( {{{\text{a}}^2}\frac{{{\partial ^2}u}}{{\partial {x^2}}} - \frac{{\partial u}}{{\partial t}}} \right) = F\left( {u,x,t,\varepsilon } \right)$$

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 ε → 0 is constructed, and its uniformity in the closed rectangle is proven.