Existence of Neumann and singular solutions of the fast diffusion equation

Abstract

Let $\Omega$ be a smooth bounded domain in ${\mathbb{R}}^n$, $n\ge 3$, $0<m\le \frac{n-2}{n}$, $a_1,a_2,...,a_{i_0}\in \Omega$, $δ_0=\min_{1\le i\le i_0}{dist }(a_i,\1Ω)$ 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 $a_1,...,a_{i_0}$.

27 pages