The nonlinear diffusion equation $\partial \rho /\partial t=D\tilde{\Delta }{\rho }^{\nu }$ is analyzed here, where $\tilde{\Delta }\equiv {(1/r}^{d-1}\left)\right(\partial /\partial {r)r}^{d-1-\theta }\partial /\partial r,$ and d, $\theta ,$ and $\nu$ are real parameters. This equation unifies the anomalous diffusion equation on fractals $(\nu =1)$ and the spherical anomalous diffusion for porous media $(\theta =0).\mathrm{}$ An exact point-source solution is obtained, enabling us to describe a large class of subdiffusion $[\theta >(1\mathrm{}-\nu \left)d\right],$ “normal” diffusion $[\theta =(1\mathrm{}-\nu \left)d\right]$ and superdiffusion $[\theta <(1\mathrm{}-\nu \left)d\right].$ Furthermore, a thermostatistical basis for this solution is given from the maximum entropic principle applied to the Tsallis entropy.

• Received 9 October 2000

DOI:https://doi.org/10.1103/PhysRevE.63.030101