One-Dimensional Super-Fast Diffusion: Persistence Versus Extinction Revisited—Extinction at Spatial Infinity

Abstract

We consider positive classical solutions of

$$\begin{aligned} v_t=(v^{m-1}v_x)_x, \qquad x\in {\mathbb {R}}, \ t>0, \qquad (\star ) \end{aligned}$$

in the super-fast diffusion range $m<-1$. Our main interest is in smooth positive initial data $v_0=v(\cdot ,0)$ which decay as $x\rightarrow +\infty$, but which are possibly unbounded as $x\rightarrow -\infty$, having in mind monotonically decreasing data as prototypes. It is firstly proved that if $v_0$ decays sufficiently fast only in one direction by satisfying

$$\begin{aligned} v_0(x) \le cx^{-\beta } \qquad \text{ for } \text{ all } ~x>0 \quad \hbox { with some }\quad \beta >\frac{2}{1-m} \end{aligned}$$

and some $c>0$, then the so-called proper solution of ($\star$) vanishes identically in ${\mathbb {R}}\times (0,\infty )$, and accordingly no positive classical solution exists in any time interval in this case. Complemented by some sufficient criteria for solutions to remain positive either locally or globally in time, this condition for instantaneous extinction is shown to be optimal at least with respect to algebraic decay of the initial data. This partially extends some known nonexistence results for ($\star$) (Daskalopoulos and Del Pino in Arch Rat Mech Anal 137(4):363–380, 1997) in that it does not require any knowledge on the behavior of $v_0(x)$ for $x<0$. Next focusing on the phenomenon of extinction in finite time, we show that in this respect a mass influx from $x=-\infty$ can interact with mass loss at $x=+\infty$ in a nontrivial manner. Namely, we shall detect examples of monotone initial data, with critical decay as $x\rightarrow +\infty$ and exponential growth as $x\rightarrow -\infty$, that lead to solutions of ($\star$) which become extinct at a finite positive time, but which have empty extinction sets. This is in sharp contrast to known extinction mechanisms which are such that the corresponding extinction sets coincide with all of ${\mathbb {R}}$.