On singular diffusion equations with applications to self‐organized criticality

We consider solutions of the singular diffusion equation t , = ( u m −1 u x ) x , m ≦ 0, associated with the flux boundary condition lim x →−∞ ( u m −1 u x ) x = λ > 0. The evolutions defined by this problem depend on both m and λ. We prove existence and stability of traveling wave solutions, parameterized by λ. Each traveling wave is stable in its appropriate evolution. These traveling waves are in L 1 for −1 < m ≦ 0, but have non‐integrable tails for m ≦ −1. We also show that these traveling waves are the same as those for the scalar conservation law u t = −[ f ( u )] x + u xx for a particular nonlinear convection term f ( u ) = f ( u ; m , λ). © 1993 John Wiley & Sons, Inc.