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

Abstract

A singularly perturbed parabolic equation

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

is considered in a rectangle with the boundary conditions of the first kind. At the corner points of the rectangle, the monotonicity of the function $F$ with respect to the variable $u$ in the interval from the root of the degenerate equation to the boundary value is not required. The asymptotic approximation of the solution is constructed under the assumption that the principal term of the corner part exists. A complete asymptotic expansion of the solution as $\varepsilon \to 0$ is constructed, and its uniformity in a closed rectangle is proved.