THE INTEGRABLE NONLINEAR DEGENERATE DIFFUSION EQUATION U(T)=[F(U)U(X)(-1)](X) AND ITS RELATIVES

The nonlinear diffusion equation u(t) = [f(u)g(u(x))](x) arises in rec ent models of turbulent transport and of stress dissipation in rock bl asting. A Lie point symmetry analysis produces many similarity reducti ons of exponential and power-law forms, and reveals that for all choic es of f the the equation is always integrable when g(ur) = 1/u(x). We identify the functions f(u) which guarantee equivalence to the linear heat equation. For all other choices of f, the linear canonical form l eads to a self-adjoint differential equation by separation of variable s x and t. We construct a number of explicit solutions with simple bou ndary conditions, which illustrate behavior in the vicinity of the deg enerate region with u(x) = infinity. If zero flux and constant concent ration are maintained on free boundaries, then steep concentration gra dients may evolve from smooth initial conditions. For other boundary c onditions, unlike the examples of strong degeneracy, smoothing will oc cur at initial step discontinuities.