Mathematics – Analysis of PDEs
Scientific paper
2006-09-19
Mathematics
Analysis of PDEs
Scientific paper
We study the extinction behavior of solutions to the fast diffusion equation $u_t = \Delta u^m$ 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.
Daskalopoulos Panagiota
Sesum Natasa
No associations
LandOfFree
On the extinction profile of solutions to fast-diffusion does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On the extinction profile of solutions to fast-diffusion, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the extinction profile of solutions to fast-diffusion will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-466271