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

TL;DR

This paper proves the existence, uniqueness, and asymptotic behavior of solutions to a very fast diffusion equation, including convergence to self-similar solutions under specific initial conditions.

Contribution

It establishes the global existence and asymptotic convergence of solutions to the fast diffusion equation with particular initial data.

Findings

Existence of unique global classical solutions for specified initial conditions.

Convergence of rescaled solutions to self-similar profiles as time approaches infinity.

Connection of the equation to the Yamabe flow in geometric analysis.

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.