Fast and Superfast Diffusion Processes

For $1\le \alpha \le 2$, the family of fast diffusions, $u_t={\left[u^{-\alpha }{u}_x\right]}_x$, $0<\mathrm{}\alpha <2\mathrm{}$, coexists with superfast processes which, prior to termination within a finite time, assume a time-space separable form. The remarkable properties of $u_t{\left[\ln \mathrm{}\right(u\left)\right]}_{\mathrm{xx}}$ guide the understanding of these processes; two interacting kinks form a superfast shrinking pattern or a persisting motion of two poles. For $\alpha =1\mathrm{}$, the superfast axisymmetric diffusion coexists with a modified fundamental diffusion: a response to a singular core and a ring of $\delta$ sources. In 3D, fast ($0<\mathrm{}\alpha <\frac{2}{3}$) and the separable ($\frac{4}{5}\le \alpha <1\mathrm{}$) superfast processes are distinct. Inhomogeneity of the medium is considered.