Existence of Neumann and singular solutions of the fast diffusion equation

TL;DR

This paper investigates the existence and uniqueness of positive solutions with Neumann boundary conditions and singular solutions that blow up at specific points for the fast diffusion equation in bounded domains.

Contribution

It establishes the existence and uniqueness of positive solutions with Neumann boundary conditions and constructs singular solutions that blow up at designated points.

Findings

Existence and uniqueness of positive solutions in punctured domains.

Construction of singular solutions blowing up at specified points.

Results applicable for certain ranges of the parameter m.

Abstract

Let $\Omega$ be a smooth bounded domain in $\R^n$, $n\ge 3$, $0<m\le\frac{n-2}{n}$, $a_1,a_2,..., a_{i_0}\in\Omega$, $\delta_0=\min_{1\le i\le i_0}{dist }(a_i,\1\Omega)$ and let $\Omega_{\delta}=\Omega\setminus\cup_{i=1}^{i_0}B_{\delta}(a_i)$ and $\hat{\Omega}=\Omega\setminus\{a_1\,...,a_{i_0}\}$. For any $0<\delta<\delta_0$ we will prove the existence and uniqueness of positive solution of the Neumann problem for the equation $u_t=\Delta u^m$ in $\Omega_{\delta}\times (0,T)$ for some $T>0$. We will prove the existence of singular solutions of this equation in $\hat{\Omega}\times (0,T)$ for some $T>0$ that blow-up at the points $a_1,..., a_{i_0}$.