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 $\partial \rho /\partial t=\nabla \cdot (K\nabla {\rho }^{\nu })-\nabla \cdot (\mu \mathbf{F}\rho )-\alpha \rho ,$ where $K=D{r}^{-\theta }$, $\nu$, $\theta$, $\mu$ and $D$ are real parameters and $\alpha$ is a time-dependent source. This equation unifies the O'Shaugnessy-Procaccia anomalous diffusion equation on fractals ($\nu =1$) and the spherical anomalous diffusion for porous media ($\theta =0$). 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.

Latex, 6 pages. To appear in Phys. Rev. E

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