Existence and large time behaviour of finite points blow-up solutions of the fast diffusion equation

TL;DR

This paper establishes the existence, blow-up rates, and long-term behavior of solutions to the fast diffusion equation with singularities at finite points, including convergence to infinity or harmonic functions over time.

Contribution

It provides new results on the existence and asymptotic behavior of singular solutions with blow-up at specified points for the fast diffusion equation.

Findings

Solutions blow up at prescribed points for all positive times.

Solutions either tend to infinity or converge to harmonic functions as time approaches infinity.

The paper characterizes the blow-up rates near singularities and long-term asymptotics.

Abstract

Let $\Omega\subset\R^n$ be a smooth bounded domain and let $a_1,a_2,\dots,a_{i_0}\in\Omega$, $\widehat{\Omega}=\Omega\setminus\{a_1,a_2,\dots,a_{i_0}\}$ and $\widehat{R^n}=\R^n\setminus\{a_1,a_2,\dots,a_{i_0}\}$. We prove the existence of solution $u$ of the fast diffusion equation $u_t=\Delta u^m$, $u>0$, in $\widehat{\Omega}\times (0,\infty)$ ($\widehat{R^n}\times (0,\infty)$ respectively) which satisfies $u(x,t)\to\infty$ as $x\to a_i$ for any $t>0$ and $i=1,\cdots,i_0$, when $0<m<\frac{n-2}{n}$, $n\geq 3$, and the initial value satisfies $0\le u_0\in L^p_{loc}(\2{\Omega}\setminus\{a_1,\cdots,a_{i_0}\})$ ($u_0\in L^p_{loc}(\widehat{R^n})$ respectively) for some constant $p>\frac{n(1-m)}{2}$ and $u_0(x)\ge \lambda_i|x-a_i|^{-\gamma_i}$ for $x\approx a_i$ and some constants $\gamma_i>\frac{2}{1-m},\lambda_i>0$, for all $i=1,2,\dots,i_0$. We also find the blow-up rate of such solutions near the blow-up points $a_1,a_2,\dots,a_{i_0}$, and obtain the asymptotic large time behaviour of such singular solutions. More precisely we prove that if $u_0\ge\mu_0$ on $\widehat{\Omega}$ ($\widehat{R^n}$, respectively) for some constant $\mu_0>0$ and $\gamma_1>\frac{n-2}{m}$, then the singular solution $u$ converges locally uniformly on every compact subset of $\widehat{\Omega}$ (or $\widehat{R^n}$ respectively) to infinity as $t\to\infty$. If $u_0\ge\mu_0$ on $\widehat{\Omega}$ ($\widehat{R^n}$, respectively) for some constant $\mu_0>0$ and satisfies $\lambda_i|x-a_i|^{-\gamma_i}\le u_0(x)\le \lambda_i'|x-a_i|^{-\gamma_i'}$ for $x\approx a_i$ and some constants $\frac{2}{1-m}<\gamma_i\le\gamma_i'<\frac{n-2}{m}$, $\lambda_i>0$, $\lambda_i'>0$, $i=1,2,\dots,i_0$, we prove that $u$ converges in $C^2(K)$ for any compact subset $K$ of $\2{\Omega}\setminus\{a_1,a_2,\dots,a_{i_0}\}$ (or $\widehat{R^n}$ respectively) to a harmonic function as $t\to\infty$.