Uniqueness for a forward backward diffusion equation

Abstract:

Let $\phi$ be continuous, have at most finitely many local extrema on any bounded interval, be twice continuously differentiable on any closed interval on which there is no local extremum and be strictly decreasing on any closed interval on which it is decreasing. We show that the initial-boundary value problem for ${u_t} = \phi {({u_x})_x}$ with Neumann boundary conditions has at most one smooth solution.