The asymptotic solution of a singularly perturbed parabolic equation with piecewise-smooth boundary condition (Q1842294)

We consider the first boundary-value problem for a singularly perturbed equation of parabolic type \[ \begin{aligned} &u_ t- \varepsilon^ 2 Lu\equiv u_ t- \varepsilon^ 2 (a_{11} u_{xx}+ 2a_{12} u_{xy}+ a_{22} u_{yy})= f(x,y,t),\\ &x,y,t\in \Omega_ T= \{x,y\in \Omega_{XY},\;0<t<T\}, \end{aligned} \tag{1} \] \[ u(x,y,0)=0, \qquad u(x,y,t)= \varphi (x,y,t), \qquad x,y\in \partial \Omega_{XY}, \quad t>0, \tag{2} \] where \(0<\varepsilon \ll 1\) is a small parameter. The asymptotic expansion (AE) of the solution of problem (1)--(2), given smoothness of the functions \(f\) and \(\varphi\) and the boundary \(\Gamma\) of the region \(\Omega_{XY}\) and \(\varphi (x,y, 0)=0\), can easily be constructed by the method of Vishik and Lyusternik, and if \(\Gamma\) contains corner points (correspondingly, the boundary \(\Gamma_ T\) of region \(\Omega_ T\) contains edges) it can be constructed by the method of corner boundary functions arising near edges. If \(\varphi\) is piecewise-smooth on \(\Gamma_ T\), it becomes impossible to construct the AE of the solution of problem as before, because the corner boundary functions have singularities. In this paper a modification of the boundary functions method is proposed, by means of which the complete AE of the solution of problem (1), (2) can be constructed for this case also.