Degeneration of Boundary Layer at Singular Points

We study the Dirichlet problem in a bounded plane domain for the heat equation with small parameter multiplying the derivative in $t$. The behaviour of solution at characteristic points of the boundary is of special interest. The behaviour is well understood if a characteristic line is tangent to the boundary with contact degree at least $2$. We allow the boundary to not only have contact of degree less than $2$ with a characteristic line but also a cuspidal singularity at a characteristic point. We construct an asymptotic solution of the problem near the characteristic point to describe how the boundary layer degenerates.