Existence and asymptotic behaviour of solutions of the very fast diffusion equation

Details Existence and asymptotic behaviour of solutions of the very fast diffusion equation Existence and asymptotic behaviour of solutions of the very fast diffusion equation

2011-09-16

arxiv.org/abs/1109.3618v1

Mathematics

Analysis of PDEs

19 pages

Scientific paper

Let n>2, $0\max(1,(1-m)n/2), and $0\le u_0\in L_{loc}^p(R^n)$ satisfy $\liminf_{R\to\infty}R^{-n+\frac{2}{1-m}}\int_{|x|\le R}u_0\,dx=\infty$. We prove the existence of unique global classical solution of $u_t=\frac{n-1}{m}\Delta u^m$, u>0, in $R^n\times (0,\infty)$, u(x,0)=u_0(x) in $\R^n$. If in addition 00, q2, if $g_{ij}=u^{\frac{4}{n+2}}\delta_{ij}$ is a metric on $R^n$ that evolves by the Yamabe flow $\partial g_{ij}/\partial t=-Rg_{ij}$ with u(x,0)=u_0(x) in $R^n$ where $R$ is the scalar curvature, then u(x,t) is a global solution of the above fast diffusion equation.

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