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

Abstract

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 $u_t=\frac{n-1}{m}Δu^m$, u>0, in $R^n\times (0,\infty)$, u(x,0)=u_0(x) in $\R^n$. If in addition 0 0, q 2, if $g_{ij}=u^{\frac{4}{n+2}}δ_{ij}$ 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.