A Hyperbolic PDE with Parabolic Behavior

In this paper we present a hyperbolic partial differential equation (PDE) in one space and one time dimension. This equation arose in a study of numerical schemes for simulating evolving river topographies. The solution of this PDE, whose initial data are specified along a characteristic, is very similar to that of the canonical diffusion equation. This interesting example provides insight into the solution of hyperbolic PDEs when data is specified in this pathological way as well as illustrating some connections between the parabolic and hyperbolic classes of evolution equations.