Nonlinear anomalous diffusion equation and fractal dimension: Exact generalized gaussian solution

Abstract

In this work we incorporate, in a unified way, two anomalous behaviors, the power law and stretched exponential ones, by considering the radial dependence of the $N$-dimensional nonlinear diffusion equation $\partialρ/\partial{t}={\bf \nabla} \cdot (K{\bf \nabla} ρ^ν)-{\bf \nabla}\cdot(μ{\bf F} ρ)-αρ,$ where $K=D r^{-θ}$, $ν$, $θ$, $μ$ and $D$ are real parameters and $α$ is a time-dependent source. This equation unifies the O'Shaugnessy-Procaccia anomalous diffusion equation on fractals ($ν=1$) and the spherical anomalous diffusion for porous media ($θ=0$). An exact spherical symmetric solution of this nonlinear Fokker-Planck equation is obtained, leading to a large class of anomalous behaviors. Stationary solutions for this Fokker-Planck-like equation are also discussed by introducing an effective potential.