Some non-linear diffusion equations and fractal diffusion

Some scaling solutions of a class of radially symmetric non-linear diffusion equations in an arbitrary dimension d are obtained, (A) for an initial point source with a fixed total amount of material, and (B) for a radial flux of material through a hyper-spherical surface. In this macroscopic model the flux density depends on powers of the concentration and its (radial) gradient. The dimensional dependence of these solutions is analyzed and comparison made with scaling solutions of the corresponding linear equations for fractal diffusion. The non-linear equations contain arbitrary exponents which can be related to an effective fractal dimension of the underlying diffusion process.