We consider the $d=1\mathrm{}$ nonlinear Fokker-Planck-like equation with fractional derivatives $(\partial /\partial t)P(x,t)=D({\partial }^{\gamma }/\partial x^{\gamma })\left[P\right(x,t)\mathrm{}{]}^{\nu }.$ Exact time-dependent solutions are found for $\nu =\left(2\mathrm{}-\gamma \right)/(1+\gamma )(-\infty <\gamma <~2).$ By considering the long-distance asymptotic behavior of these solutions, a connection is established, namely, $q=(\gamma +3)/(\gamma +1)(0<\gamma <~2),$ with the solutions optimizing the nonextensive entropy characterized by index q. Interestingly enough, this relation coincides with the one already known for Lévy-like superdiffusion (i.e., $\nu =1\mathrm{}$ and $0<\mathrm{}\gamma <~2).$ Finally, for $(\gamma ,\nu )=(2,0)\mathrm{}$ we obtain $q=5/3,$ which differs from the value $q=2\mathrm{}$ corresponding to the $\gamma =2\mathrm{}$ solutions available in the literature $(\nu <1\mathrm{}$ porous medium equation), thus exhibiting nonuniform convergence.

• Received 30 March 2000

DOI:https://doi.org/10.1103/PhysRevE.62.2213