  We prove an existence and uniqueness result for Neumann boundary problem of a
parabolic partial differential equation (PDE for short) with a singular
nonlinear divergence term which can only be understood in a weak sense. A
probabilistic approach is applied by studying the backward stochastic
differential equations (BSDEs for short) corresponding to the PDEs, the
solution of which turns out to be a limit of a sequence of BSDEs constructed by
penalization method.
