  We solve a class of BSDE with a power function $f(y) = y^q$, $q > 1$, driving
its drift and with the terminal boundary condition $ \xi = \infty \cdot
\mathbf{1}_{B(m,r)^c}$ (for which $q > 2$ is assumed) or $ \xi = \infty \cdot
\mathbf{1}_{B(m,r)}$, where $B(m,r)$ is the ball in the path space $C([0,T])$
of the underlying Brownian motion centered at the constant function $m$ and
radius $r$. The solution involves the derivation and solution of a related heat
equation in which $f$ serves as a reaction term and which is accompanied by
singular and discontinuous Dirichlet boundary conditions. Although the solution
of the heat equation is discontinuous at the corners of the domain the BSDE has
continuous sample paths with the prescribed terminal value.
