Exponential propagation conjecture for fast doubly nonlinear diffusion

Let $m>0$ and $p>1$ satisfy $-p/N<\gamma<0$, with $N\geq 1$, and let $u=u(x,t)$ solve the doubly nonlinear reaction-diffusion problem with the paper's prescribed initial datum. Set

$$\sigma_{\ast}=-\frac{\gamma}{p}f'(0)>0.$$

Fast-diffusion exponential propagation conjecture. For every $\sigma<\sigma_{\ast}$,

$$u(x,t)\to 1\quad\text{uniformly in }\{|x|\leq e^{\sigma t}\}\quad\text{as }t\to\infty,$$

whereas for every $\sigma>\sigma_{\ast}$,

$$u(x,t)\to 0\quad\text{uniformly in }\{|x|\geq e^{\sigma t}\}\quad\text{as }t\to\infty.$$

This predicts exponential spatial expansion in the fast-diffusion regime, in contrast with the linearly propagating traveling-wave behavior of the standard Fisher–KPP model. It is motivated by the corresponding porous-medium result, but the exact propagation behavior for doubly nonlinear diffusion is presented as an open problem.