Existence of Neumann and singular solutions of the fast diffusion equation

Abstract

Let $Ω$ be a smooth bounded domain in $\R^n$, $n\ge 3$, $0<m\le\frac{n-2}{n}$, $a_1,a_2,..., a_{i_0}\inΩ$, $δ_0=\min_{1\le i\le i_0}{dist }(a_i,\1Ω)$ and let $Ω_δ=Ω\setminus\cup_{i=1}^{i_0}B_δ(a_i)$ and $\hatΩ=Ω\setminus\{a_1\,...,a_{i_0}\}$. For any $0<δ<δ_0$ we will prove the existence and uniqueness of positive solution of the Neumann problem for the equation $u_t=Δu^m$ in $Ω_δ\times (0,T)$ for some $T>0$. We will prove the existence of singular solutions of this equation in $\hatΩ\times (0,T)$ for some $T>0$ that blow-up at the points $a_1,..., a_{i_0}$.

27 pages