Extinction profile of the logarithmic diffusion equation

Details Extinction profile of the logarithmic diffusion equation Extinction profile of the logarithmic diffusion equation

2010-12-09

arxiv.org/abs/1012.1915v2

Mathematics

Analysis of PDEs

some typos and errors are corrected, 29 pages

Scientific paper

Let $u$ be the solution of $u_t=\Delta\log u$ in $\R^N\times (0,T)$, N=3 or $N\ge 5$, with initial value $u_0$ satisfying $B_{k_1}(x,0)\le u_0\le B_{k_2}(x,0)$ for some constants $k_1>k_2>0$ where $B_k(x,t) =2(N-2)(T-t)_+^{N/(N-2)}/(k+(T-t)_+^{2/(N-2)}|x|^2)$ is the Barenblatt solution for the equation. We prove that the rescaled function $\4{u}(x,s)=(T-t)^{-N/(N-2)}u(x/(T-t)^{-1/(N-2)},t)$, $s=-\log (T-t)$, converges uniformly on $\R^N$ to the rescaled Barenblatt solution $\4{B}_{k_0}(x)=2(N-2)/(k_0+|x|^2)$ for some $k_0>0$ as $s\to\infty$. We also obtain convergence of the rescaled solution $\4{u}(x,s)$ as $s\to\infty$ when the initial data satisfies $0\le u_0(x)\le B_{k_0}(x,0)$ in $\R^N$ and $|u_0(x)-B_{k_0}(x,0)|\le f(|x|)\in L^1(\R^N)$ for some constant $k_0>0$ and some radially symmetric function $f$.

Affiliated with

Also associated with

No associations

Rating

Extinction profile of the logarithmic diffusion equation 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 Extinction profile of the logarithmic diffusion equation, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Extinction profile of the logarithmic diffusion equation will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-76473