Jm. Goard et al., THE INTEGRABLE NONLINEAR DEGENERATE DIFFUSION EQUATION U(T)=[F(U)U(X)(-1)](X) AND ITS RELATIVES, Zeitschrift fur angewandte Mathematik und Physik, 47(6), 1996, pp. 926-942
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.