Uniqueness of Piecewise Smooth Weak Solutions of Multidimensional Degenerate Parabolic Equations1

We study the degenerate parabolic equation ut +∇ · f = ∇ · (Q∇u) + g, where (x, t) ∈ R ×R+, the flux ~ f , the viscosity coefficient Q and the source term g depend on (x, t, u) and Q is nonnegative definite. Due to the possible degeneracy, weak solutions are considered. In general, these solutions are not uniquely determined by the initial data and, therefore, additional conditions must be imposed in order to guarantee uniqueness. We consider here the subclass of piecewise smooth weak solutions, i.e., continuous solutions which are Csmooth everywhere apart from a closed nowhere dense collection of smooth manifolds. We show that the solution operator is L-stable in this subclass and, consequently, that piecewise smooth weak solutions are uniquely determined by the initial data.