Uniqueness and time oscillating behaviour of finite points blow-up solutions of the fast diffusion equation

TL;DR

This paper proves the uniqueness of finite points blow-up solutions for the fast diffusion equation and constructs initial data leading to oscillating solutions between infinity and a positive constant over time.

Contribution

It extends previous results by establishing uniqueness and demonstrating oscillatory behavior of solutions in bounded domains and the entire space.

Findings

Proved uniqueness of finite points blow-up solutions.

Constructed initial data causing oscillations between infinity and a positive constant.

Extended results to both bounded domains and al^n imes (0,\u221e).

Abstract

Let $n\ge 3$ and $0<m<\frac{n-2}{n}$. We will extend the results of J.L. Vazquez and M. Winkler and prove the uniqueness of finite points blow-up solutions of the fast diffusion equation $u_t=\Delta u^m$ in both bounded domains and $\mathbb{R}^n\times (0,\infty)$. We will also construct initial data such that the corresponding solution of the fast diffusion equation in bounded domain oscillate between infinity and some positive constant as $t\to\infty$.