On the extinction profile of solutions to fast-diffusion
| dc.creator | Daskalopoulos, Panagiota | |
| dc.creator | Sesum, Natasa | |
| dc.date | 2006-09-19 | |
| dc.date | 2007-05-02 | |
| dc.date.accessioned | 2026-07-07T07:59:00Z | |
| dc.date.available | 2026-07-07T07:59:00Z | |
| dc.description | We study the extinction behavior of solutions to the fast diffusion equation $u_t = Δu^m$ on $\R^N\times (0,T)$, in the range of exponents $m \in (0, \frac{N-2}{N})$, $N > 2$. We show that if the initial data $u_0$ is trapped in between two Barenblatt solutions vanishing at time $T$, then the vanishing behaviour of $u$ at $T$ is given by a Barenblatt solution. We also give an example showing that for such a behavior the bound from above by a Barenblatt solution $B$ (vanishing at $T$) is crucial: we construct a class of solutions $u$ with initial data $u_0 = B (1 + o(1))$, near $ |x| >> 1$, which live longer than $B$ and change behaviour at $T$. The behavior of such solutions is governed by $B(\cdot,t)$ up to $T$, while for $t >T$ the solutions become integrable and exhibit a different vanishing profile. For the Yamabe flow ($m = \frac{N-2}{N+2}$) the above means that these solutions $u$ develop a singularity at time $T$, when the Barenblatt solution disappears, and at $t >T$ they immediately smoothen up and exhibit the vanishing profile of a sphere. In the appendix we show how to remove the assumption on the bound on $u_0$ from below by a Barenblatt. | |
| dc.identifier | https://arxiv.org/abs/math/0609513 | |
| dc.identifier | http://arxiv.org/abs/math/0609513 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/128210 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 35B40 | |
| dc.title | On the extinction profile of solutions to fast-diffusion | |
| dc.type | text |