The Cauchy problem for degenerate parabolic equations with discontinuous drift
Edward D. Conway · Transactions of the American Mathematical Society · 1973
The coefficient of the gradient is allowed to be discontinuous but is assumed to satisfy a “one-sided” Lipschitz condition. This condition insures the pathwise uniqueness of the underlying Markov process which in turn yields the existence of a unique stable generalized solution of the parabolic equation. If the data is Lipschitz continuous, then so is the solution.