Corner Boundary Layer in Boundary Value Problems for Singularly Perturbed Parabolic Equations with Nonmonotonic Nonlinearities

Abstract

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. At the corner points of the rectangle, the function \(F\) is assumed to be quadratic and nonmonotonic with respect to the variable \(u\) on the interval from the root of the degenerate equation to the boundary value. The main attention is paid to constructing the main term of the corner part of the asymptotics of the solution as \(\epsilon \to 0\).