ON SINGULAR DIFFUSION-EQUATIONS WITH APPLICATIONS TO SELF-ORGANIZED CRITICALITY

We consider solutions of the singular diffusion equation u(t) = (u(m-1 )u(x))x, m less-than-or equal-to 0, associated with the flux boundary condition lim(x -->-infinity) u(m-1)u(x) = lambda > 0. The evolutions defined by this problem depend on both m and lambda. We prove existenc e and stability of traveling wave solutions, parameterized by lambda. Each traveling wave is stable in its appropriate evolution. These trav eling waves are in L1 for -1 < m less-than-or-equal-to 0, but have non -integrable tails for m less-than-or-equal-to -1. We also show that th ese 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, lambda). (C) 1993 John Wiley & Sons, Inc.