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

Abstract

Let n>2, $0<m\le (n-2)/n$, p>\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 0<m<(n-2)/n and $u_0(x)\approx A|x|^{-q}$ as $|x|\to\infty$ for some constants A>0, q<n/p, we prove that there exist constants $\alpha$, $\beta$, such that the function $v(x,t)=t^{\alpha}u(t^{\beta}x,t)$ converges uniformly on every compact subset of $R^n$ to the self-similar solution $\psi(x,1)$ of the equation with $\psi(x,0)=A|x|^{-q}$ as $t\to\infty$. Note that when m=(n-2)/(n+2), n>2, 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.