Asymptotic behaviour of the finite blow-up points solutions of the fast diffusion equation

TL;DR

This paper investigates the long-term behavior of solutions with finite blow-up points for the fast diffusion equation, establishing asymptotics, oscillation between harmonic functions, and existence of minimal solutions.

Contribution

It introduces new results on the asymptotic behavior and oscillation of finite blow-up solutions, and proves the existence of minimal solutions for the fast diffusion equation.

Findings

Asymptotic behavior of solutions as time approaches infinity.

Construction of solutions oscillating between harmonic functions.

Existence of minimal solutions with prescribed blow-up behavior.

Abstract

Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $i_0\in\mathbb{Z}^+$, $\Omega\subset\mathbb{R}^n$ be a smooth bounded domain, $a_1,a_2,\dots,a_{i_0}\in\Omega$, $\widehat{\Omega}=\Omega\setminus\{a_1,a_2,\dots,a_{i_0}\}$, $0\le f\in L^{\infty}(\partial\Omega)$ and $0\le u_0\in L_{loc}^p(\widehat{\Omega})$ for some constant $p>\frac{n(1-m)}{2}$ which satisfies $\lambda_i|x-a_i|^{-\gamma_i}\le u_0(x)\le \lambda_i'|x-a_i|^{-\gamma_i'}\,\,\forall 0<|x-a_i|<\delta$, $i=1,\dots, i_0$ where $\delta>0$, $\lambda_i'\ge\lambda_i>0$ and $\frac{2}{1-m}<\gamma_i\le\gamma_i'<\frac{n-2}{m}$ $\forall i=1,2,\dots, i_0$ are constants. We will prove the asymptotic behaviour of the finite blow-up points solution $u$ of $u_t=\Delta u^m$ in $\widehat{\Omega}\times (0,\infty)$, $u(a_i,t)=\infty\,\,\forall i=1,\dots,i_0, t>0$, $u(x,0)=u_0(x)$ in $\widehat{\Omega}$ and $u=f$ on $\partial\Omega\times (0,\infty)$, as $t\to\infty$. We will construct finite blow-up points solution in bounded cylindrical domain with appropriate lateral boundary value such that the finite blow-up points solution oscillates between two given harmonic functions as $t\to\infty$. We will also prove the existence of the minimal solution of $u_t=\Delta u^m$ in $\widehat{\Omega}\times (0,\infty)$, $u(x,0)=u_0(x)$ in $\widehat{\Omega}$, $u(a_i,t)=\infty\quad\forall t>0, i=1,2\dots,i_0$ and $u=\infty$ on $\partial{\Omega}\times (0,\infty)$.