---
abstract: 'Let $\Omega$ be a smooth bounded domain in ${{\mathbb{R}}}^n$, $n\ge 3$, $0<m\le\frac{n-2}{n}$, $a_1,a_2,\dots, a_{i_0}\in\Omega$, $\delta_0=\min_{1\le i\le i_0}\mbox{dist }(a_i,{\partial}\Omega)$ and let $\Omega_{\delta}=\Omega\setminus\cup_{i=1}^{i_0}B_{\delta}(a_i)$ and $\hat{\Omega}=\Omega\setminus\{a_1\,\dots,a_{i_0}\}$. 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 $a_1,\dots, a_{i_0}$.'
address:
- |
    Kin Ming Hui: Institute of Mathematics, Academia Sinica,\
    Taipei, 10617, Taiwan, R.O.C.
- 'Sunghoon Kim: Department of Mathematics, School of Natural Sciences, The Catholic University of Korea, 43 Jibong-ro, Wonmi-gu, Bucheon-si, Gyeonggi-do, 420-743, Republic of Korea'
author:
- Kin Ming Hui
- Sunghoon Kim
title: Existence of Neumann and singular solutions of the fast diffusion equation
---

0.2truein

Introduction
============

Recently there is a lot of research on the equation $$\label{fast-diffusion-eqn}
u_t=\Delta u^m$$ by M. Bonforte, E. Chasseigne, M. Fila, G. Grillo, J.L. Vazquez, M. Winkler, E. Yanagida @BGV1, @BGV2, @BV1, @BV2, @BV3, @CV, @FVWY, P. Daskalopoulos, M.Del Pino and N. Sesum @DPS, @DS1, @DS2, S.Y. Hsu \[Hs2-3\], K.M. Hui \[Hu2-3\], M.Del Pino and M. Sáez @PS, L.A. Peletier and H. Zhang @PZ, etc. This equation arises in many physical models. When $m>1$, it is called the porous medium which models the diffusion of gases through porous media @A. When $m=1$, is the heat equation. When $0<m<1$, it is usually called the fast diffusion equation. When $m=\frac{n-2}{n+2}$, appears in the study of Yamabe flow on ${{\mathbb{R}}}^n$. In fact the metric $g_{ij}=u^{\frac{4}{n+2}}dx^2$, $u>0$, is a solution of the Yamabe flow @DS2, @PS, $$\frac{{\partial}g_{ij}}{{\partial}t}=-Rg_{ij}$$ in ${{\mathbb{R}}}^n$, $n\ge 3$, if and only if $u$ is a solution of $$u_t=\frac{n-1}{m}\Delta u^m$$ with $m=\frac{n-2}{n+2}$ where $R$ is the scalar curvature of $g_{ij}$. We refer the readers to the book @V3 by J.L. Vazquez for the basics of and the books @DK, @V2, by P. Daskalopoulos, C.E. Kenig and J.L. Vazquez for the most recent results of . We also refer to the paper @BV3, by M. Bonforte and J. L. Vazquez for the non local version of .

As observed by L. Peletier @P and J.L Vazquez @V1 there is a big difference on the behaviour of solutions of for $(n-2)/n<m<1$, $n\ge 3$, and for $0<m\le (n-2)/n$, $n\ge 3$. For example there is a $L^1-L^{\infty}$ regularizing effect for the solutions of $$\label{Cauchy-problem}
\left\{\begin{aligned}
u_t=&\Delta u^m, u\ge 0,\quad\mbox{ in }{{\mathbb{R}}}^n\times (0,T)\\
u(x,0)=&u_0\qquad\qquad\quad\mbox{ in }{{\mathbb{R}}}^n
\end{aligned}\right.$$ with $0\le u_0\in L^1_{loc}({{\mathbb{R}}}^n)$ for any $(n-2)/n<m<1$ @HP, @DaK. However there is no such $L^1-L^{\infty}$ regularizing effect for solutions of when $0<m\le (n-2)/n$, $n\ge 3$, @V2. When $\frac{n-2}{n}<m<1$, existence and uniqueness of global weak solution of for any $0\le u_0\in L^1_{loc}({{\mathbb{R}}}^n)$ has been proved by M.A. Herrero and M. Pierre in @HP. When $0<m\le (n-2)/n$ and $n\ge 3$, existence of positive smooth solutions of for any $0\le u_0\in L_{loc}^p({{\mathbb{R}}}^n)$, $p>\max(1,(1-m)n/2)$, satisfying the condition, $$\label{initial-value-average-lower-bd-0}
\liminf_{R\to\infty}\frac{1}{R^{n-\frac{2}{1-m}}}\int_{|x|\le R}u_0\,dx
\ge C_1T^{\frac{1}{1-m}}$$ for some constant $C_1>0$ is proved by S.Y. Hsu in @Hs3.

In this paper we will study the existence of singular solutions of . Study of singular solutions of nonlinear elliptic equations were also obtained by H. Brezis and L. Veron @BrV, B. Gidas and J. Spruck @GS, etc. In order to study the singular solutions of we will first prove the existence of positive smooth solution of the Neumann problem for in smooth bounded domains with a finite numbers of holes when $0<m\le (n-2)/n$, $n\ge 3$. When $n\ge 3$ and $m=(n-2)/n$, we will prove the existence of singular solutions of in a smooth bounded domain that blow-up at a finite number of points in the domain. More precisely let $\Omega$ be a smooth bounded domain in ${{\mathbb{R}}}^n$, $n\ge 1$, $a_1, a_2, \dots, a_{i_0}\in\Omega$, $\delta_0=\min_{1\le i,j\le i_0}\left(\mbox{dist }(a_i,{\partial}\Omega),|a_i-a_j|\right)/2$, $\Omega_{\delta}=\Omega\setminus\cup_{i=1}^{i_0}B_{\delta}(a_i)$, $\hat{\Omega}=\Omega\setminus\{a_1,\dots,a_{i_0}\}$ and $\hat{{{\mathbb{R}}}}={{\mathbb{R}}}\setminus\{a_1,\dots,a_{i_0}\}$. We will prove the following three main theorems.

\[Neumann-holes-problem-existence-thm\] Let $n\ge 3$, $0<m\le\frac{n-2}{n}$, $0<\delta<\delta_0$, $0\le u_0\in L^p(\Omega_{\delta})$ for some constant $p>\frac{n(1-m)}{2}$, $0\le f\in L_{loc}^{\infty}({\partial}\Omega\times [0,\infty))$ and $0\le g_i\in L_{loc}^{\infty}({\partial}B_{\delta}(a_i)\times [0,\infty))$ for all $i=1,\cdots,i_0$. Suppose either $u_0\not\equiv 0$ on $\Omega_{\delta}$ or $$\label{f-g-L1-positive-condition}
\int_0^t\int_{\partial\Omega}f\,d\sigma ds
+\sum_{i=1}^{i_0}\int_0^t\int_{\partial B_{\delta}(a_i)}g_i\,d\sigma ds>0\quad\forall t>0.$$ Then there exists a unique solution $u$ for the equation $$\label{Neumann-holes-eqn}
\begin{cases}
\begin{aligned}
u_t=&{\triangle}u^m\qquad\,\,\,\mbox{ in }\Omega_{\delta}\times(0,\infty)\\
\frac{\partial u^m}{\partial\nu}&=f\qquad\,\,\,\mbox{ on }\partial\Omega\times(0,\infty)\\
\frac{\partial u^m}{\partial\nu}&=g_i\qquad\,\,\mbox{ on $(\bigcup_{i=1}^{i_0}\partial B_{\delta}(a_i))\times(0,\infty)$}
\quad\forall i=1,\dots,i_0\\
u(x,0)&=u_0(x)\quad\,\mbox{in }\Omega_{\delta}
\end{aligned}
\end{cases}$$ that satisfies $$\label{mass-formulua-holes}
\int_{\Omega_{\delta}}u(x,t)\,dx=\int_{\Omega_{\delta}}u_0\,dx+\int_0^t\int_{\partial\Omega}f\,d\sigma ds
+\sum_{i=1}^{i_0}\int_0^t\int_{\partial B_{\delta}(a_i)}g_i\,d\sigma ds
\quad \forall t>0$$ where $\frac{\partial}{\partial\nu}$ is the derivative on $\partial\Omega_{\delta}$ with respect to the unit outward normal of the domain $\Omega_{\delta}$ Moreover if $f\equiv 0$ on ${\partial}\Omega\times (0,\infty)$ and $g_i$, $i=1,\dots, i_0$, are nonnegative monotone decreasing functions of $t>0$, then $$\label{Aronson-Benilan-ineqn}
u_t\le\frac{u}{(1-m)t}$$ in $\Omega_{\delta}\times(0,\infty)$.

\[singular-soln-existence-thm\] Let $n\ge 3$, $0<m\le\frac{n-2}{n}$, $p>\frac{n(1-m)}{2}$, $0\le f\in L_{loc}^{\infty}({\partial}\Omega\times [0,\infty))$. Let $0\le u_0\in L_{loc}^p(\hat{\Omega})$ be such that $$\label{u0-blow-up-condition2}
\frac{C_1}{|x-a_i|^qe^{\frac{1}{\delta_1^2-|x-a_i|^2}}}\le u_0(x)\le\frac{C_2}{|x-a_i|^q}\qquad \forall 0<|x-a_i|\le \delta_1, i=1,\cdots,i_0$$ for some constants $C_1>0$, $C_2>0$, $q\ge\max\left(\frac{n}{2m},\frac{n-2}{m}\right)$ and $0<\delta_1<\min\left(\frac{(1-m)q}{4+(1-m)q},\delta_0\right)$. Then there exists a solution $u$ of $$\label{singular-Neumann-problem}
\left\{\begin{aligned}
u_t=&\Delta u^m\quad\mbox{ in }\hat{\Omega}\times(0,\infty)\\
\frac{\partial u^m}{\partial\nu}=&f\qquad\, \mbox{ on }\partial\Omega\times(0,\infty)\\
u(x,0)=&u_0(x)\quad\mbox{ in }\hat{\Omega}
\end{aligned}\right.$$ such that $$\label{singular-soln-upper-lower-bd}
u(x,t)\ge\frac{C_1}{|x-a_i|^qe^{\frac{1}{\delta_1^2-|x-a_i|^2}}}\quad\forall 0<|x-a_i|<\delta_1,t>0, i=1,\cdots,i_0$$ and $$\label{singular-soln-upper-lower-bd2}
u(x,t)\le\frac{C_T}{|x|^q}\qquad\qquad\forall 0<|x-a_i|\le\frac{\delta_1}{2},0<t\le T, i=1,\cdots,i_0$$ hold for some constant $C_T>0$ where ${\partial}/{\partial}\nu$ is the derivative with respect to the unit outward normal on ${\partial}\Omega$.

\[singular-soln-existence-thm2\] Let $n\ge 3$, $0<m\le\frac{n-2}{n}$, $p>\frac{n(1-m)}{2}$. Let $0\le u_0\in L_{loc}^p(\hat{{{\mathbb{R}}}})$ be such that holds for some constants $C_1>0$, $C_2>0$, $q\ge\max\left(\frac{n}{2m},\frac{n-2}{m}\right)$ and $$0<\delta_1<\min\left(\frac{(1-m)q}{4+(1-m)q},\frac{1}{2}\min_{1\le i,j\le i_0}|a_i-a_j|\right).$$ Then there exists a solution $u$ of $$\label{singular-Neumann-problem2}
\left\{\begin{aligned}
u_t=&\Delta u^m\quad\,\,\mbox{ in }\hat{{{\mathbb{R}}}}\times(0,\infty)\\
u(x,0)=&u_0(x)\quad\mbox{ in }\hat{{{\mathbb{R}}}}
\end{aligned}\right.$$ such that and hold for some constant $C_T>0$.

The plan of the paper is as follows. In section two we will prove some a priori estimates for $C^{2,1}$ solution of . In section three we will prove Theorem \[Neumann-holes-problem-existence-thm\]. In section four we will prove Theorem \[singular-soln-existence-thm\] and Theorem \[singular-soln-existence-thm2\].

We start with some notations and definitions that will be used in this paper. Let $\Omega_1\subset{{\mathbb{R}}}^n$ be a smooth bounded domain and let $\Sigma_1$, $\Sigma_2$ be relatively open subsets of $\partial\Omega_1$ such that $\partial\Omega_1=\Sigma_1\cup\Sigma_2$ and if $n\geq 2$, then $\overline{\Sigma}_1\cap\overline{\Sigma}_2$ is a $C^2$ manifold of dimension $n-2$. For any $0<m<1$, $0\le u_0\in L^1(\Omega_1)$, $f\in L_{loc}^1(\Sigma_1\times [0,\infty))$ and $g\in L_{loc}^1(\Sigma_2\times [0,\infty))$, we say that $u$ is a very weak solution (subsolution, supersolution respectively) of $$\label{Dirichlet-Neumann-problem}
\left\{\begin{aligned}
u_t=&{\triangle}u^m\quad\mbox{in }\Omega_1\times(0,T)\\
\frac{\partial u^m}{\partial\nu}=&f\qquad\mbox{ on }\Sigma_1\times(0,T)\\
u=&g\qquad\mbox{ on }\Sigma_2\times(0,T)\\
u(x,0)=&u_0(x)\quad\mbox{in }\Omega_1
\end{aligned}\right.$$ if $0\le u\in C([0,T);L^1(\Omega_1))$ satisfies ($\ge$, $\le$ respectively) $$\begin{aligned}
\label{weak-soln-defn}
&\int_{t_1}^{t_2}\int_{\Omega_1}(u\eta_t+u^m\Delta\eta)\,dxdt+\int_{t_1}^{t_2}\int_{\Sigma_1}f\eta\,d\sigma dt\notag\\
&\qquad\qquad\qquad\qquad =\int_{t_1}^{t_2}\int_{\Sigma_2}g^m\frac{{\partial}\eta}{{\partial}\nu}\,d\sigma dt+\int_{\Omega_1}u(x,t_2)\eta (x,t_2)\,dx-\int_{\Omega_1}u(x,t_1)\eta (x,t_1)\,dx\end{aligned}$$ for any $0<t_1<t_2<T$, and $\eta\in C^2({\overline}{\Omega}_1\times (0,T))$ satisfying $\eta =0$ on $\Sigma_2\times (0,T)$, and ${\partial}\eta/{\partial}\nu =0$ on $\Sigma_1\times (0,T)$ and $u$ has initial value $u_0$. We say that $u$ is a solution (subsolution, supersolution respectively) of if $u\in L^{\infty}_{loc}(\overline{\Omega}_1\times(0,T)) $ is positive in $\Omega_1\times (0,T)$ and satisfies in $\Omega_1\times (0,T)$ ($\le$, $\ge$ respectively) in the classical sense with $$\label{u-L1-converge-u0-as-t-goto-0}
\|u(\cdot,t)-u_0\|_{L^1(\Omega_{\delta})}\to 0\quad\mbox { as }t\to 0$$ and also satisfies ($\ge$, $\le$ respectively) for any $0<t_1<t_2<T$, and $\eta\in C^2({\overline}{\Omega}_1\times (0,T))$ satisfying $\eta =0$ on $\Sigma_2\times (0,T)$, and ${\partial}\eta/{\partial}\nu =0$ on $\Sigma_1\times (0,T)$.

We say that $u$ is a solution (subsolution, supersolution respectively) of if $u\in L^{\infty}_{loc}((\overline{\Omega}\setminus\{a_1,\dots,a_{i_0}\})\times(0,T))\cap C^{2,1}((\overline{\Omega}\setminus\{a_1,\dots,a_{i_0}\})\times(0,T))$ is positive in $\hat{\Omega}\times (0,T)$ and satisfies in $\hat{\Omega}\times (0,T)$ ($\le$, $\ge$ respectively) in the classical sense with $$\label{u-L1-converge-u0-as-t-goto-0-Omega-hat}
\lim_{t\to 0}\int_{\hat{\Omega}}u(x,t)\eta (x)\,dx=\int_{\hat{\Omega}}u_0\eta\,dx\quad\forall\eta\in C_0^{\infty}(\hat{\Omega})$$ and also satisfies $$\label{weak-singular-soln-defn}
\int_{t_1}^{t_2}\int_{\hat{\Omega}}(u\eta_t+u^m\Delta\eta)\,dxdt+\int_{t_1}^{t_2}\int_{{\partial}\Omega}f\eta\,d\sigma dt
=\int_{\Omega_1}u(x,t_2)\eta (x,t_2)\,dx-\int_{\Omega_1}u(x,t_1)\eta (x,t_1)\,dx$$ ($\ge$, $\le$ respectively) for any $0<t_1<t_2<T$, and $\eta\in C_0^2((\overline{\Omega}\setminus\{a_1,\dots,a_{i_0}\})\times(0,T))$ satisfying ${\partial}\eta/{\partial}\nu =0$ on $\partial\Omega\times (0,T)$.

We say that $u$ is a solution (subsolution, supersolution respectively) of if $u\in L^{\infty}_{loc}(\hat{{{\mathbb{R}}}}\times(0,T))\cap C^{2,1}(\hat{{{\mathbb{R}}}}\times(0,T))$ is positive in $\hat{{{\mathbb{R}}}}\times (0,T)$ and satisfies in $\hat{{{\mathbb{R}}}}\times (0,T)$ ($\le$, $\ge$ respectively) in the classical sense with $$\label{u-L1-converge-u0-as-t-goto-0-Omega-hat*}
\lim_{t\to 0}\int_{\hat{\Omega}}u(x,t)\eta (x)\,dx=\int_{\hat{\Omega}}u_0\eta\,dx\quad\forall\eta\in C_0^{\infty}(\hat{{{\mathbb{R}}}}).$$

For any $x_0\in{{\mathbb{R}}}^n$, $x_0'\in{{\mathbb{R}}}^{n-1}$, $R>0$, we let $B_R(x_0)=\{x\in{{\mathbb{R}}}^n:|x-x_0|<R\}$, $B'_R(x_0')=\{x'\in{{\mathbb{R}}}^{n-1}:|x-x_0'|<R\}$, $\hat{B}_R(x_0)=B_R(x_0)\setminus\{x_0\}$, $B_R=B_R(0)$, $\hat{B}_R=\hat{B}_R(0)$ and $B_R'=B_R'(0)$. For any $a\in{{\mathbb{R}}}$, we let $a_+=\max (a,0)$ and $a_-=\max (-a,0)$. For any set $A\in{{\mathbb{R}}}^n$, we let $\chi_A$ be the characteristic function of the set $A$.

A priori estimates
==================

In this section we will prove some a priori estimates for the solutions of . We will also prove a $L^p-L^{\infty}$ estimates for the solutions of for some constant $p>1$. These $L^p-L^{\infty}$ estimates will be used in section three to give uniform upper bound for the approximating $C^{2,1}$ solutions of which appear in the construction of solution of . Note that similar $L^{\infty}$ estimates are also obtained in @BV2, @CD, @CV, @D, @DK,@DGV1, @DGV2, @DK, @DKV and @HP. We first observe that by an argument similar to the proof of Lemma 2.3 of @DaK we have the following result.

\[cf. Lemma 1.1 of @Hs2\]\[comparsion-lemma1\] Let $\Omega_1\subset{{\mathbb{R}}}^n$ be a smooth bounded domain and let $\Sigma_1$, $\Sigma_2$ be relatively open subsets of $\partial\Omega_1$ such that $\partial\Omega_1=\Sigma_1\cup\Sigma_2$ and if $n\ge 2$ then $\overline{\Sigma}_1\cap\overline{\Sigma}_2$ is a $C^2$ manifold of dimension $n-2$. Let $0\leq u_{0,1}$, $u_{0,2}\in L^{1}(\Omega)$, $f_1$, $f_2\in L^1(\Sigma_1\times(0,T))$ and $g_1$, $g_2\in C(\Sigma_2\times(0,T))$ be such that $0\le g_1\leq g_2$ on $\Sigma_2\times(0,T)$. Suppose $u_1$, $u_2$, are subsolution and supersolution of in $\Omega_1\times(0,T)$ with $f=f_1$, $f_2$, $g=g_1$, $g_2$ and $u_0=u_{0,1}$, $u_{0,2}$, respectively. Then $$\int_{\Omega_1}(u_1-u_2)_+(x,t)\,dx\leq \int_{\Omega_1}(u_{0,1}-u_{0,2})_+(x,t)\,dx+\int_0^t\int_{\Sigma_1}(f_1-f_2)_+\,d\sigma ds
\qquad \forall 0\leq t<T.$$

By the same argument as the proof of Lemma 3.4 of @Hu1 we have the following result.

\[cf. Lemma 3.4 of @Hu1\]\[positive-angle-at-bdary-lemma\] Suppose $\Omega\subset{{\mathbb{R}}}^n$ is a smooth bounded convex domain. For any $x\in{\partial}\Omega$, $x_0\in\Omega$, let $n(x)$ be the unit outward normal vector at $x$ with respect to $\Omega$ and let $n_1(x)$ be the unit vector along the line segment $\overrightarrow{x_0x}$ from $x_0$ to $x$. If $\theta (x)$ is the angle between $n(x)$ and $n_1(x)$, then there exists a constant $0<c_0\le 1$ such that $$0<c_0\le\text{cos }\theta (x)\le 1\quad\forall x\in{\partial}\Omega.$$

Now, we are going to prove some estimates for the solutions of .

\[soln-bded-holes-lemma\] Let $n\ge 1$, $0<m<1$, $T>0$, $q\ge\frac{2}{1-m}$, $0<\delta_1<\min\left(\frac{(1-m)q}{4+(1-m)q},\delta_0\right)$, and $0<\delta\le\delta_2<\left(\frac{(1-m)q}{4+(1-m)q}\right)\delta_1$. Let $0\leq u_0\in 
L^1(\overline{\Omega}_{\delta})$ such that $$\label{u0-blow-up-condition}
u_0(x)\le\frac{C_2}{|x-a_i|^{q}} \qquad \forall \delta\le |x-a_i|<\delta_1, i=1,\cdots,i_0$$ holds for some constant $C_2>0$. Let $f\in L^1(\partial\Omega\times(0,T))$ and $g_i\in L^1(\partial B_{\delta}(a_i)\times(0,T))$, $i=1,2,\dots,i_0$, be such that $\sup f<\infty$, $\sup g_i<\infty$, for all $i=1,2,\dots,i_0$. Let $u$ be a solution of $$\label{Neumann-problem-holes}
\left\{\begin{aligned}
u_t=&{\triangle}u^m\qquad \mbox{ in }\Omega_{\delta}\times(0,T)\\
\frac{\partial u^m}{\partial\nu}=&f\qquad\quad\, \mbox{ on }\partial\Omega\times(0,T)\\
\frac{\partial u^m}{\partial\nu}=&\frac{g_i}{\delta^{mq+1}}\quad\mbox{ on }B_{\delta}(a_i)\times(0,T)\qquad\forall i=1,\dots,i_0\\
u(x,0)=&u_0(x)\qquad\mbox{in }\Omega_{\delta}.
\end{aligned}\right.$$ Then there exists a constant $A_1>0$ such that $$\label{u-upper-bd-phi-tidle}
u(x,t)\leq \phi_{A_1}(x-a_i,t) \qquad \forall\delta\le|x-a_i|<\delta_1,\,\,0\leq t<T, i=1,\cdots,i_0$$ holds for all $0<\delta\leq \delta_2$ where $$\label{phi-A-defn}
\phi_{A_1}(x,t)=\frac{A_1(1+t)^{\frac{1}{1-m}}}{|x|^{q}(\delta_1-|x|)^{\frac{2}{1-m}}}.$$ If $\Omega$ is a smooth convex domain and $$\label{u0-L-infty-bd-condition}
\|u_0\|_{L^{\infty}(\overline{\Omega}_{\delta})}\le M_0$$ holds for some constant $M_0>0$, then there exists a constant $M_1>0$ depending on $M_0$ such that $$\label{u-upper-bd1}
u(x,t)\leq M_1 \quad \forall (x,t)\in\overline{\Omega}_{\delta_2}\times[0,T)$$ holds for any $0<\delta\leq\delta_2$.

We will use a modification of the proof of Lemma 1.3 of @Hs2 to prove the lemma. Without loss of generality it suffices to prove for $i=i_0=1$. Let $$\label{eq-condition-of-A-1-by-max-1}
A_1=\max\left\{C_2,\left(\frac{m(mq^2+2q+2n+4)}{1-m}\right)^{\frac{1}{1-m}},\,\,\left(\frac{2(\sup g_1)_+}{mq}\right)^{\frac{1}{m}}\right\}$$ and let $\delta'\in(\delta_2,\delta_1)$. For any $0<t_1<t_2<T$, let $$\label{M-defn}
M=\max_{\substack{\delta'\leq |x-a_1|\leq\delta_1\\t_1\leq t\leq t_2}}\frac{m|\nabla u|}{u^{1-m}}.$$ Since $$\frac{\partial \phi_{A_1}^{m}}{\partial r}(x-a_1,t)=\frac{mA_1^m(1+t)^{\frac{m}{1-m}}}{r^{mq}(\delta_1-r)^{\frac{2m}{1-m}}}\left(-\frac{q}{r}+\frac{2}{(1-m)(\delta_1-r)}\right)\to\infty$$ uniformly on $t\in [0,\infty]$ as $r=|x-a_1|\to\delta_1^-$, there exists a constant $\delta''\in(\delta',\delta_1)$ such that $$\label{eq-condition-of-tilde-phi-on-boundary-1}
\frac{\partial \phi_{A_1}^{m}}{\partial r}(x-a_1,t)>M\quad\forall\delta''\leq |x-a_1|\leq \delta_1,t\geq 0.$$ Since $q/\delta\ge 4/[(1-m)(\delta_1-\delta)]$, by direct computation, $$\begin{aligned}
\label{eq-condition-of-tilde-phi-on-boundary-2}
\frac{\partial \phi_{A_1}^{m}}{\partial \nu}(x-a_1,t)&=\frac{mA_1^m(1+t)^{\frac{m}{1-m}}}{{\delta}^{mq}(\delta_1-\delta)^{\frac{2m}{1-m}}}\left[\frac{q}{\delta}-\frac{2}{(1-m)(\delta_1-\delta)}\right]\notag\\
&\geq \frac{mA_1^mq}{2{\delta}^{mq+1}(\delta_1-\delta)^{\frac{2m}{1-m}}}\notag\\
&\geq \frac{mA_1^mq}{2{\delta}^{mq+1}}\geq \frac{g_1}{{\delta}^{mq+1}} \qquad \quad \mbox{on }\partial B_{\delta}(a_1)\times(0,\infty).\end{aligned}$$ By , $$\label{eq-condition-of-tilde-phi-in-interior-1}
\begin{aligned}
{\triangle}\phi^m_{A_1}
=&A_1^m(1+t)^{\frac{m}{1-m}}\left[\frac{mq(mq-n+2)}{|x|^{mq+2}(\delta_1-|x|)^{\frac{2m}{1-m}}}-\frac{2m(2mq-n+1)}{(1-m)|x|^{mq+1}(\delta_1-|x|)^{\frac{1+m}{1-m}}}+\frac{2m(1+m)}{(1-m)^2|x|^{mq}(\delta_1-|x|)^{\frac{2}{1-m}}}\right]\\
\le&\frac{mA_1^m(mq^2+2q+2n+4)(1+t)^{\frac{m}{1-m}}}{(1-m)^2|x|^{mq+2}(\delta_1-|x|)^{\frac{2}{1-m}}}\\
\le&\phi_{A_1,t}\qquad\qquad\qquad\qquad\qquad\qquad \forall 0<|x|<\delta_1, t\geq 0.
\end{aligned}$$ Hence by , and for any $\delta_1'\in (\delta'',\delta_1)$, $\phi_{A_1}(x-a_1,t)$ is a supersolution of $$\label{w-eqn}
\begin{cases}
w_t={\triangle}w^m\qquad\quad \mbox{ in }(B_{\delta_1'}(a_1){\backslash}B_{\delta}(a_1))\times(0,T)\\
\frac{\partial w^m}{\partial\nu}=M \qquad \quad\,\, \mbox{ on }\partial B_{\delta_1'}(a_1)\times(0,T)\\
\frac{\partial w^m}{\partial\nu}=\frac{g_1}{\delta^{mq+1}}\qquad\,\,\mbox{ on }\partial B_{\delta}(a_1)\times(0,T)\\
w(x,0)=u_0(x)\quad\mbox{ in }B_{\delta_1'}(a_1){\backslash}B_{\delta}(a_1)
\end{cases}$$ where ${\partial}/{\partial}\nu$ is the derivative with respect to the unit outward normal at the boundary of the domain $B_{\delta_1'}(a_1){\backslash}B_{\delta}(a_1)$. Since $u$ is a subsolution of , by Lemma \[comparsion-lemma1\], $\forall 0<t_1<t_2<T$, $\delta_1'\in (\delta'',\delta_1)$, $$\begin{aligned}
&\int_{B_{\delta_1'}(a_1){\backslash}B_{\delta}(a_1)}(u(x,t)-\phi_{A_1}(x-a_1,t))_+\,dx\leq \int_{B_{\delta_1'}(a_1){\backslash}B_{\delta}(a_1)}(u(x,t_1)-\phi_{A_1}(x-a_1,t_1))_+\,dx\quad\forall t_1\le t\le t_2\\
\Rightarrow\quad&\int_{B_{\delta_1}(a_1){\backslash}B_{\delta}(a_1)}(u(x,t)-\phi_{A_1}(x-a_1,t))_+\,dx\leq \int_{B_{\delta_1}(a_1){\backslash}B_{\delta}(a_1)}(u(x,t_1)-\phi_{A_1}(x-a_1,t_1))_+\,dx\quad\mbox{ as }
\delta_1'\to\delta_1\\
\Rightarrow\quad&\int_{B_{\delta_1}(a_1){\backslash}B_{\delta}(a_1)}(u(x,t)-\phi_{A_1}(x-a_1,t))_+\,dx\le 0\qquad\forall 0<t<T\qquad\qquad\qquad\mbox{ as }t_1\to 0, 
t_2\to T\end{aligned}$$ and follows.\
Suppose now $\Omega$ is a smooth convex domain and holds for some constant $M_0>0$. Let $n(x)$, $n_1(x)$, $\theta(x)$ and $c_0$ be as in Lemma \[positive-angle-at-bdary-lemma\] with $x_0=a_1$. By there exists a constant $M>0$ such that $$u(x,t)\le M\qquad \forall (x,t)\in\left(\bigcup_{i=1}^{i_0} \partial B_{\delta_2}(a_i)\right)\times(0,T)$$ holds for all $0<\delta\le\delta_2$. Let $$w(x,t)=A_2(1+t)^{\frac{1}{1-m}}e^{\frac{|x-a_1|}{\delta_2}}$$ where $$A_2=\max\left\{\left(\frac{m(1-m)(m+n-1)}{\delta_2^2}\right)^{\frac{1}{1-m}},\left(\frac{\delta_2}{mc_0}(sup f)_+\right)^{\frac{1}{m}},\,\,M,\,\,\|u_0\|_{L^{\infty}(\Omega_{\delta_2})}\right\}.$$ Then $$w(x,0)\ge u_0(x)\quad\mbox{ in }\Omega_{\delta_2}$$ and $$w(x,t)\ge u(x,t)\quad\mbox{on }\left(\bigcup_{i=1}^{i_0}\partial B_{\delta_2}(a_i)\right)\times(0,T).$$ By direct computation, $$\begin{aligned}
{\triangle}w^m=&\frac{mA_2^m}{\delta_2}\left[\frac{m}{\delta_2}+\frac{n-1}{|x-a_1|}\right](1+t)^{\frac{m}{1-m}}e^{\frac{m|x-a_1|}{\delta_2}}
\le\frac{mA_2^m(m+n-1)}{\delta_2^2}(1+t)^{\frac{m}{1-m}}e^{\frac{m|x-a_1|}{\delta_2}}\\
\le&\frac{A_2}{1-m}(1+t)^{\frac{m}{1-m}}e^{\frac{|x-a_1|}{\delta_2}}=w_t, \qquad \mbox{in }\Omega_{\delta_2}\times(0,T).
\end{aligned}$$ Moreover by Lemma \[positive-angle-at-bdary-lemma\], $$\frac{\partial w^m}{\partial\nu}(x,t)=\frac{\partial w^m}{\partial n_1}(x,t)\cdot\cos\theta(x)
=\frac{mA_2^{m}}{\delta_2}\cos\theta(x)(1+t)^{\frac{m}{1-m}}e^{\frac{m|x-a_1|}{\delta_2}}\ge\frac{mA_2^mc_0}{\delta_2}
\ge f(x,t)$$ for all $x\in\partial\Omega$, $0<t<T$. Hence by Lemma \[comparsion-lemma1\], $$u(x,t)\leq w(x,t)\leq A_2(1+T)e^{M_2}\quad\mbox{ in }\overline{\Omega}_{\delta_2}\times[0,T)$$ where $M_2=\max_{x\in\overline{\Omega}_{\delta_2}}\left(\frac{|x-a_1|}{\delta_2}\right)$ and follows.

\[comparison-lem2\] Let $n\ge 1$, $0<m<1$, $0<\delta<\delta_1<\min (\delta_0, m/2)$, $0\le f\in L_{loc}^{\infty}({\partial}\Omega\times [0,\infty))$ and $g_i\in L_{loc}^{\infty}([0,\infty))$, $i=1,\cdots, i_0$, be such that $$\label{gi-lower-bd-condition}
\inf_{[0,\infty)} g_i(t)>m(q+4\delta_1^{-2})\quad\forall i=1,\cdots, i_0.$$ Let $0\le u_0\in L^1(\Omega_{\delta})$ be such that $$\label{u0-lower-blow-up-condition}
u_0(x)\ge\frac{C_1}{|x-a_i|^qe^{\frac{1}{\delta_1^2-|x-a_i|^2}}}\quad \forall \delta\le |x-a_i|<\delta_1, i=1,\cdots,i_0$$ for some constants $C_1>0$ and $q\ge\max\left(\frac{n}{2m},\frac{n-2}{m}\right)$ and let $u$ be a solution of . Then $$\label{singular-soln-lower-bd}
u(x,t)\ge\frac{C_1}{|x-a_i|^qe^{\frac{1}{\delta_1^2-|x-a_i|^2}}}\quad\forall \delta\le |x|<\delta_1,t>0, i=1,\cdots,i_0$$ holds for any $0<\delta\le\delta_1/2$.

Without loss of generality it suffices to prove the lemma when $i_0=1$ and $a_1=0$. Let $$\phi(x)=|x|^{-q}\psi(x), \qquad\psi(x)=e^{-\frac{1}{\delta_1^2-|x|^2}}.$$ By direct computation, $$\Delta |x|^{-mq}=mq(mq+2-n)|x|^{-mq-2}\ge 0,$$ $$\Delta \psi^m=\frac{2m\psi^m}{(\delta_1^2-|x|^2)^4}\left(2m|x|^2-4|x|^2(\delta_1^2-|x|^2)-n(\delta_1^2-|x|^2)^2\right)$$ Hence $$\begin{aligned}
\label{phi^m-subharmonic}
\Delta\phi^m=&\psi^m\Delta |x|^{-mq}+2\nabla\psi^m\cdot\nabla |x|^{-mq}+|x|^{-mq}\Delta\psi^m\notag\\
\ge&\frac{4m^2q|x|^{-mq}\psi^m}{(\delta_1^2-|x|^2)^2}+|x|^{-mq}\Delta\psi^m\notag\\
\ge&\frac{2m|x|^{-mq}\psi^m}{(\delta_1^2-|x|^2)^4}\left((2m-4\delta_1^2)|x|^2+(2mq-n)(\delta_1^2-|x|^2)^2\right)\notag\\
\ge&0\quad\mbox{ in }B_{\delta_1}.\end{aligned}$$ By , $$\label{phi-derivative-near-zero}
\left.\frac{{\partial}\phi^m}{{\partial}\nu}\right|_{|x|=\delta}=\frac{m\psi(\delta)^m}{\delta^{qm+1}}\left(q+\frac{2}{(\delta_1^2-\delta^2)^2}\right)
\le\frac{g_i}{\delta^{qm+1}}\quad\forall0<\delta<\frac{\delta_1}{2}.$$ By and for any $0<\delta<\frac{\delta_1}{2}$, $\phi$ is a subsolution of $$\label{dirichlet-neumann-problem}
\left\{\begin{aligned}
u_t=&{\triangle}u^m\qquad \mbox{ in }B_{\delta_1}\times(0,\infty)\\
\frac{\partial u^m}{\partial\nu}=&\frac{g_i}{\delta^{mq+1}}\quad\mbox{ on }B_{\delta}\times(0,\infty)\\
u=&0\qquad\quad\, \mbox{ on }\partial B_{\delta_1}\times(0,\infty)\\
u(x,0)=&u_0(x)\qquad\mbox{in }B_{\delta_1}\setminus B_{\delta}.
\end{aligned}\right.$$ Since $u$ is a supersolution of , by Lemma \[comparsion-lemma1\] follows.

We will now prove a $L^p-L^{\infty}$ estimates for the solution of . Since the proof is similar to the proof in section 1 of @Hu1, we will only sketch the argument here. For any smooth bounded domain $\Omega\subset{{\mathbb{R}}}^n$, let $x_0\in\partial\Omega$. When $n\ge 2$, by rotation and translation of the coordinate axis we may assume that $x_0$ is at the origin and the tangent plane to $\partial\Omega$ at $x_0$ is ${{\mathbb{R}}}^{n-1}\times\{0\}$ and there exists a constant $0<R_0\le\delta_0/2$ and a smooth function $$\phi_1:{{\mathbb{R}}}^{n-1}\to {{\mathbb{R}}}$$ with $\phi_1\in C_0^{\infty}({{\mathbb{R}}}^{n-1})$, $\phi_1(0)=0$, $\nabla\phi_1(0)=0$, and $\sup_{x'\in {{\mathbb{R}}}^{n-1}}|\nabla\phi_1(x')|\le\frac{1}{10}$ such that $$\begin{cases}
\begin{aligned}
\Omega\cap B_{R_0}=&\left\{(x',x_n):x'\in{{\mathbb{R}}}^{N-1}, x_n>\phi_1(x')\right\}\cap B_{R_0}\\
\partial\Omega\cap B_{R_0}=&\left\{(x',x_n):x'\in{{\mathbb{R}}}^{N-1}, x_n=\phi_1(x')\right\}\cap B_{R_0}.
\end{aligned}
\end{cases}$$ Then $(\cup_{i=1}^{i_0}B_{\delta_0/2}(a_i))\cap B_{R_0}(x_0)=\phi$. For any $x=(x',x_n)\in \Omega\cap B_{R_0}$ with $x'\in{{\mathbb{R}}}^{n-1}$, $x_n\in{{\mathbb{R}}}$, let $$\label{domain-transformation}
\psi_1(x)=(x',x_n-\phi_1(x'))$$ and $\Omega'_1=\psi_1(\Omega\cap B_{R_0})$. Then $\psi_1$ is a differeomorphism between $\Omega\cap B_{R_0}$ and $\Omega'_1$. Let $$\tilde{u}(y,t)=u(y',y_n+\phi_1(y'),t) \qquad \forall (y',y_n)\in\Omega'_1.$$ For $0<r<R_0/5$, let $$D_r'=\left\{\begin{aligned}
&\{(y',y_n)\in{{\mathbb{R}}}^{n-1}\times{{\mathbb{R}}}:|y'|<r, 0<y_n<r\}\quad\mbox{ if }n\ge 2\\
&\{y\in{{\mathbb{R}}}:|y-x_0|<r\}\qquad\qquad\qquad\qquad\qquad\mbox{if }n=1
\end{aligned}\right.$$ and $D_r=\psi_1^{-1}(D_r')$ where $\psi_1$ is given by if $n\ge 2$ and $\psi_1$ is the identity map if $n=1$. For $0<r<\rho<R_0/5$ and $0<t_1<t_2<t_0<T$, $t_2<t<T$, we let $S(t)=\psi_1^{-1}(D_r')\times(t_2,t]$, $R(t)=\psi_1^{-1}(D_{\rho}')\times(t_1,t]$, $S=S(t_0)$ and $R=R(t_0)$.

\[weak-w12-soln-lem\] Let $0<\delta<\delta_0$, $f\in L^1(\partial\Omega\times(0,T))$, and $g_i\in L^1(\partial B_{\delta}(a_i)\times(0,T))$, $i=1,2,\dots,i_0$. Suppose $u\in C^{2,1}({\overline}{\Omega}_{\delta}\times (0,T))$ is a solution of in $\Omega_{\delta}\times (0,T)$. For any $a>0$, let $v=\max (u,a)$. Then $$\begin{aligned}
\label{weak-w12-soln}
&\int_{\Omega_{\delta}}v(x,t_2)\phi (x,t_2)\,dx+\int_{t_1}^{t_2}\int_{\Omega}\nabla v^m\cdot\nabla\phi \,dx\,dt\notag\\
\le&\int_{\Omega_{\delta}}v(x,t_1)\phi (x,t_1)\,dx+\int_{t_1}^{t_2}\int_{\Omega}v\phi_t\,dx\,dt+\int_{t_1}^{t_2}\int_{{\partial}\Omega}f\phi\,d\sigma\,dt+\int_{t_1}^{t_2}\int_{\cup_{i=0}^{i_0}{\partial}B_{\delta}(a_i)}g_i\phi\,d\sigma\,dt\notag\\\end{aligned}$$ holds for any $0<t_1<t_2<T$ and $\phi\in C({\overline}{\Omega}_{\delta}\times (0,T))$ such that the weak derivatives $\phi_t,\nabla\phi$, exist and belong to $L^2(\Omega_{\delta}\times (0,T))$ and $$\label{weak-w12-soln2}
\int_{\Omega_{\delta}}v_t\phi\,dx+\int_{\Omega}\nabla v^m\cdot\nabla \phi\,dx
\le\int_{{\partial}\Omega}f\phi\,d\sigma+\int_{\cup_{i=0}^{i_0}{\partial}B_{\delta}(a_i)}g_i\phi\,d\sigma$$ holds for any $0<t<T$ and $\phi\in C({\overline}{\Omega}_{\delta}\times (0,T))$ such that for any $0<t'<T$ the weak derivatives $\nabla\phi (x,t')$ exists and belong to $L^2(\Omega_{\delta})$.

By approximation it suffices to show that holds for any $\phi\in C^{\infty}({\overline}{\Omega}_{\delta}\times (0,T))$. Since the proof of the lemma is similar to the proof of Lemma 1.1 of @Hu1 we will only sketch the argument here. We choose a sequence of functions $\{p_k\}_{k=1}^{\infty}\subset C^{\infty}({{\mathbb{R}}})$, $0\le p_k\le 1$, $p_k(s)=0$ for all $s\le a+(2k)^{-1}$, $p_k(s)=1$ for all $s\ge a+k^{-1}$, $0\le p_k'(s)\le Ck$ for some constant $C>0$, and $p_k(s)\to\chi_{(a,\infty)}(s)$ as $k\to\infty$. Multiplying by $p_k(u)\phi$ and integrating by parts, $$\begin{aligned}
\label{weak-soln-k-approx-eqn}
&\int_{\Omega}q_k(u(x,t_2))\phi(x,t_2)\,dx+\int_{t_1}^{t_2}\int_{\Omega}\nabla u^m\cdot\nabla \phi\,dx\,dt\notag\\
\le&\int_{\Omega}q_k(u(x,t_2))\phi(x,t_2)\,dx+\int_{t_1}^{t_2}\int_{\Omega}\nabla u^m\cdot\nabla (p_k(u)\phi)\,dx\,dt\notag\\
\le&\int_{\Omega}q_k(u(x,t_1))\phi(x,t_1)\,dx+\int_{t_1}^{t_2}\int_{\Omega}q_k(u)\phi_t\,dx\,dt+\int_0^T\int_{{\partial}\Omega}f\phi\,d\sigma\,dt+\int_{t_1}^{t_2}\int_{\cup_{i=1}^{i_0}{\partial}B_{\delta}(a_i)}g_i\phi\,d\sigma\,dt\end{aligned}$$ where $$q_k(s)=\int_a^sp_k(\hat{s})\,d\hat{s}$$ Since $q_k(u)\to v-a$ as $k\to\infty$, letting $k\to\infty$ in we get . By a similar argument we get and the lemma follows.

\[lem-the-first-lemma-for-moser-iteration-i3\] Let $0<\delta<\delta_0$, $f\in L^1(\partial\Omega\times(0,T))$, and $g_i\in L^1(\partial B_{\delta}(a_i)\times(0,T))$, $i=1,2,\dots,i_0$, be such that $f\leq M$ for some constant $M>0$. Suppose $u\in C^{2,1}({\overline}{\Omega}_{\delta}\times (0,T))$ is a solution of in $\Omega_{\delta}\times (0,T)$. Let $v=\max (u,1)$. Then there exists a constant $C>0$ such that $$\label{step1-ineqn}
\sup_{t_2\le t\le t_0}\int_{D_r}v^{\alpha+1}(x,t)\,dx+\iint_{S}|\nabla v^{\frac{\alpha+m}{2}}|^2\,dx\,dt\leq C\{(\rho-r)^{-2}+(t_2-t_1)^{-1}\}(1+\alpha^2)\iint_{R}v^{\alpha+1}\,dx\,dt$$ holds for any $\alpha>0$, $0<\rho<R_0/5$, and $0<m<1$.

Since the proof is similar to that of Lemma 1.1 of @Hu1, we will only sketch the argument here. We first choose $\eta\in C^{\infty}({{\mathbb{R}}}^{n+1})$, $0\le\eta\le 1$, such that $\eta(x,s)=0$ for all $(x,s)\not\in R$, $\eta(x,s)=1$ for all $(x,s)\in S$, and $|\eta_t|\le C(t_2-t_1)^{-1}$, $|\nabla\eta|\le C(\rho-r)^{-1}$, for some constant $C>0$. By Lemma \[weak-w12-soln-lem\] the function $v=\max(u,1)$ satisfies . Since the function $v^{\alpha}\eta^2\in C({\overline}{\Omega}_{\delta}\times (0,T))$ and its first order weak derivatives belong to $L^2({\Omega}_{\delta}\times (0,T))$, putting $\phi=v^{\alpha}\eta^2$ in and simplifying $$\label{eq-aligned-the-first-step-for-energy-inequality-34t}
\begin{aligned}
&\frac{1}{\alpha+1}\int_{D_r}v^{\alpha+1}(x,t)\,dx+\iint_{R(t)}\nabla (v^{m})\cdot
\nabla(v^{\alpha}\eta^2)\,dxds\\
\le&\frac{2}{\alpha+1}\iint_{R(t)}v^{\alpha+1}\eta\eta_t\,dxds+M\iint_{(\partial D_{\rho}\cap\partial\Omega)\times(t_1,t)}v^{\alpha}\eta^2\,d\sigma ds.
\end{aligned}$$ Then $$\begin{aligned}
&\frac{1}{\alpha+1}\int_{D_r}v^{\alpha+1}(x,t)\,dx
+\frac{4\alpha m}{(\alpha+m)^2}\iint_{R(t)}\eta^2|\nabla v^{\frac{\alpha+m}{2}}|^2\,dxds\\
\le &\frac{2}{\alpha+1}\iint_{R(t)}v^{\alpha+1}\eta\eta_t\,dxds+2m\iint_{R(t)}v^{\alpha+m-1}\eta\left|\nabla v\right|\left|\nabla\eta\right|\,dxds
+M\iint_{(\partial D_{\rho}\cap\partial\Omega)\times(t_1,t_0)}v^{\alpha}\eta^2\,d\sigma ds\\
\le&\frac{2}{\alpha+1}\iint_{R(t)}v^{\alpha+1}\eta\eta_t\,dxds+\frac{2m\alpha}{(\alpha+m)^2}
\iint_{R(t)}\eta^2|\nabla v^{\frac{\alpha+m}{2}}|^2\,dxds
+\frac{2m}{\alpha}\iint_{R(t)}v^{\alpha+m}|\nabla\eta|^2\,dxds\\
&\qquad  +M\iint_{(\partial D_{\rho}\cap\partial\Omega)\times(t_1,t)}v^{\alpha}\eta^2\,d\sigma ds.
\end{aligned}$$ Hence $$\label{ineqn-1}
\frac{\alpha}{\alpha+1}\int_{D_r}v^{\alpha+1}(x,t)\,dx
+\frac{2m\alpha^2}{(\alpha+m)^2}\iint_{R(t)}\eta^2|\nabla v^{\frac{\alpha+m}{2}}|^2\,dxds\\
\le 2\iint_{R(t)}v^{\alpha+1}(|\eta_t|+|\nabla\eta|^2)\,dxds
+ MI$$ where $$I=\alpha\iint_{(\partial D_{\rho}\cap\partial\Omega)\times(t_1,t)}v^{\alpha}\eta^2
\,d\sigma ds.$$ By an argument similar to the proof of Lemma 1.1 of @Hu1, $$\label{bd-estimate}
\begin{aligned}
I&\le\alpha\int_{t_1}^t\int_{\Omega}|\partial_{x_n}(v^{\alpha}\eta^2)|\,dxdt\\
&\le\alpha^2\iint_{R(t)}v^{\alpha-1}\eta^2|\nabla v|\,dxds
+2\alpha\iint_{R(t)}v^{\alpha}\eta|\nabla\eta|\,dxds\\
&\le\alpha^2\iint_{R(t)}v^{\alpha+m-1}\eta^2|\nabla v|\,dxds
+2\alpha\iint_{R(t)}v^{\alpha}\eta|\nabla\eta|\,dxds\\
&\le\frac{2\alpha^2}{\alpha+m}\iint_{R(t)}\eta^2v^{\frac{\alpha+m}{2}}|\nabla v^{\frac{\alpha+m}{2}}|\,dxds
+2\alpha\iint_{R(t)}v^{\alpha}\eta|\nabla\eta|\,dxds\\
&\le\frac{m\alpha^2}{(\alpha+m)^2(M+1)}\iint_{R}\eta^2\left|\nabla v^{\frac{\alpha+m}{2}}\right|^2\,dxds
+\alpha^2m^{-1}(M+1)\iint_{R} v^{\alpha+m}\eta^2\,dxds+2\alpha\iint_{R(t)}v^{\alpha}\eta|\nabla\eta|\,dxds.
\end{aligned}$$ By and we get and the lemma follows.

By Lemma \[lem-the-first-lemma-for-moser-iteration-i3\], an argument similar to the proof in Section 2 of @Hs3, and a compactness argument we have the following result.

\[Lp-L-infty-thm\] Let $n\geq 3$, $0<m\leq \frac{n-2}{n}$, $p>\frac{n(1-m)}{2}$, $0<\delta<\delta_0$, and let $f\in L^1(\partial\Omega\times(0,T))$ be such that $f\leq M$ for some constant $M>0$. Suppose $u\in C^{2,1}({\overline}{\Omega}_{\delta}\times (0,T))$ is a solution of in $\Omega_{\delta}\times (0,T)$. Then for any $0<t_1<t_2<T$, $r_1\in (0,\delta_0/2)$, there exists constants $\theta>0$ and $C>0$ depending on $M$, $m$, and $n$ such that $$\|u\|_{L^{\infty}(E_{r_1}\times(t_2,T))}\leq C\left\{1+\int_{t_1}^T\int_{E_{2r_1}}u^p\,dxdt\right\}^{\frac{\theta}{p}}$$ where $E_{r_1}=\{x\in\Omega : \textbf{dist}(x,\partial\Omega)<r_1\}$.

Similarly we have the following result.

\[Lp-L-infty-thm-1\] Let $n\ge 3$, $0<m\le \frac{n-2}{n}$, $p>\frac{n(1-m)}{2}$, $0<\delta<\delta_0$, and let $f\in L^1(\partial\Omega\times(0,T))$ and $g_i\in L^1(\partial B_{\delta}(a_i)\times (0,T))$ be such that $f\leq M$ and $g_i\leq M$ for all $i=1,\dots,i_0$ and some constant $M>0$. Suppose $u\in C^{2,1}({\overline}{\Omega}_{\delta}\times (0,T))$ is a solution of in $\Omega_{\delta}\times (0,T)$. Then for any $0<t_1<t_2<T$, there exist constants $\theta>0$ and $C>0$ depending on $M$, $\delta$, $m$, and $n$ such that $$\|u\|_{L^{\infty}(\Omega_{\delta}\times(t_2,T])}\leq C\left(1+\iint_{\Omega_{\delta}\times(t_1,T]}u^p\,dxdt\right)^{\frac{\theta}{p}}.$$

\[Lp-L-infty-thm-2\] Let $n\ge 3$, $0<m\le \frac{n-2}{n}$, $p>\frac{n(1-m)}{2}$, $0<\delta<\delta_0$, and let $f\in L^1(\partial\Omega\times(0,T))$ and $g_i\in L^1(\partial B_{\delta}(a_i)\times (0,T))$ be such that $f\leq M$ and $g_i\leq M$ for all $i=1,\dots,i_0$ and some constant $M>0$. Suppose $0\le u_0\in L^p(\Omega_{\delta})$ and $u\in C^{2,1}({\overline}{\Omega}_{\delta}\times (0,T))$ is a solution of in $\Omega_{\delta}\times (0,T)$. Then for any $0<t_1<T$ there exist constants $\theta>0$ and $C>0$ depending on $M$, $\delta$, $m$, and $n$ such that $$\|u\|_{L^{\infty}(\Omega_{\delta}\times(t_1,T])}\leq C\left(1+\int_{\Omega_{\delta}}u_0^p\,dx\right)^{\frac{\theta}{p}}.$$

Let $0<a\leq 1$ and $v=\max(u,a)$. As before by Lemma \[weak-w12-soln-lem\] the function $v$ satisfies . Since the function $v^{p-1}\in C({\overline}{\Omega}_{\delta}\times (0,T))$ and its first order weak derivatives belong to $L^2({\Omega}_{\delta}\times (0,T))$, by , $$\begin{aligned}
\label{v-integral-time-derivative}
\frac{{\partial}}{{\partial}t}\left(\int_{\Omega_{\delta}}v^p\,dx\right)
=&p\int_{\Omega_{\delta}}v^{p-1}v_t\,dx\notag\\
\le&-p\int_{\Omega_{\delta}}\nabla v^{p-1}\cdot\nabla v^m\,dx
+p\int_{{\partial}\Omega_{\delta}\cup(\cup_{i=1}^{i_0}{\partial}B_{\delta}(a_i))}v^{p-1}\frac{{\partial}v^m}{{\partial}\nu}\,d\sigma\notag\\
\le &-p(p-1)m\int_{\Omega_{\delta}}v^{p+m-3}|\nabla v|^2\,dx
+Mp\int_{{\partial}\Omega_{\delta}\cup(\cup_{i=1}^{i_0}{\partial}B_{\delta}(a_i))}v^{p-1}\,d\sigma.\end{aligned}$$ By an argument similar to the proof of Lemma 1.1 of @Hu1 there exists a constant $C_1>0$ such that, $$\begin{aligned}
\label{v-bdary-integral-ineqn}
\int_{{\partial}\Omega_{\delta}\cup(\cup_{i=1}^{i_0}{\partial}B_{\delta}(a_i))}v^{p-1}\,d\sigma
\le&C_1\left(\int_{\Omega_{\delta}}|\nabla v^{p-1}|\,dx+\int_{\Omega_{\delta}}v^{p-1}\,dx\right)\notag\\
=&(p-1)C_1\int_{\Omega_{\delta}}v^{p-2}|\nabla v|\,dx+C_1\int_{\Omega_{\delta}}v^{p-1}\,dx\notag\\
\le&\frac{m(p-1)}{2M}\int_{\Omega_{\delta}}v^{p+m-3}|\nabla v|^2\,dx
+\frac{(p-1)MC_1^2}{2m}\int_{\Omega_{\delta}}v^{p-m-1}\,dx+C_1\int_{\Omega_{\delta}}v^{p-1}\,dx.\end{aligned}$$ By and , $$\begin{aligned}
\frac{{\partial}}{{\partial}t}\left(\int_{\Omega_{\delta}}v^p\,dx\right)
\le&C_2\left(\int_{\Omega_{\delta}}v^{p-m-1}\,dx+\int_{\Omega_{\delta}}v^{p-1}\,dx\right)\\
\le&\frac{2C_2}{a^m}\int_{\Omega_{\delta}}v^{p-1}\,dx\\
\le&\frac{2C_2|\Omega|^{1/p}}{a^m}\left(\int_{\Omega_{\delta}}v^p\,dx\right)^{\frac{p-1}{p}}\end{aligned}$$ for some constant $C_2>0$. Hence $$\label{u^p-integral-bd}
\left(\int_{\Omega_{\delta}}u^p(x,t)\,dx\right)^{\frac{1}{p}}\le\left(\int_{\Omega_{\delta}}v^p(x,t)\,dx\right)^{\frac{1}{p}}\le\left(\int_{\Omega_{\delta}}(u_0+a)^p\,dx\right)^{\frac{1}{p}}+2C_2p^{-1}a^{-m}|\Omega|^{1/p}t\quad\forall 0<t<T.$$ By and Proposition \[Lp-L-infty-thm-1\], Proposition \[Lp-L-infty-thm-2\] follows.

\[u-Lp-loc-bd-u0-Lp-loc-lemma\] Let $n\ge 3$, $0<m\le \frac{n-2}{n}$, $p>\frac{n(1-m)}{2}$, $0<\delta<\delta_2<\delta_1<\delta_0$, and $g_i\in L^1(\partial B_{\delta}(a_i)\times (0,T))$ for all $i=1,\dots,i_0$ and let $f\in L^1(\partial\Omega\times(0,T))$ be such that $f\leq M$ for some constant $M>0$. Suppose $0\le u_0\in L^p(\Omega_{\delta})$ and $u\in C^{2,1}({\overline}{\Omega}_{\delta}\times (0,T))$ is a solution of in $\Omega_{\delta}\times (0,T)$. Then there exists a constant $C>0$ depending on $M$, $\delta_1$, $\delta_2$, $m$, and $n$ such that $$\left(\int_{E_{\delta_0-\delta_1}}u^p(x,t)\,dx\right)^{\frac{1-m}{p}}
\le\left(\int_{E_{\delta_0-\delta_3}}(u_0+1)^p\,dx\right)^{\frac{1-m}{p}}+Ct\quad\forall 0<t<T$$ holds for any $0<\delta\le\delta_2$ where $\delta_3=(\delta_1+\delta_2)/2$ and $E_r=\{x\in\Omega:\mbox{dist}\, (x,{\partial}\Omega)<r\}$ for any $0<r<\delta_0$.

We choose $\phi\in C^{\infty}({\overline}{\Omega})$, $0\le\phi\le 1$, such that $\phi (x)=1$ for all $x\in E_{\delta_0-\delta_1}$, $\phi (x)=0$ for all $x\not\in E_{\delta_0-\delta_3}$. Let $\eta=\phi^{\alpha}$ for some constant $\alpha>0$ to be chosen later and let $v=\max (u,1)$. By Lemma \[weak-w12-soln-lem\] the function $v$ satisfies . Since the function $v^{p-1}\eta^2\in C({\overline}{\Omega}_{\delta}\times (0,T))$ and its first order weak derivatives belong to $L^2({\Omega}_{\delta}\times (0,T))$, by , $$\begin{aligned}
\label{v-eta-integral-time-derivative}
\frac{{\partial}}{{\partial}t}\left(\int_{\Omega_{\delta}}v^p\eta^2\,dx\right)
=&p\int_{\Omega_{\delta}}v^{p-1}\eta^2v_t\,dx\notag\\
\le&-p\int_{\Omega_{\delta}}\nabla (v^{p-1}\eta^2)\cdot\nabla v^m\,dx
+p\int_{{\partial}\Omega}v^{p-1}\eta^2\frac{{\partial}v^m}{{\partial}\nu}\,d\sigma\notag\\
\le&-p(p-1)m\int_{\Omega_{\delta}}v^{p+m-3}\eta^2|\nabla v|^2\,dx
+2pm\int_{\Omega_{\delta}}v^{p+m-2}\eta|\nabla\eta||\nabla v|\,dx\notag\\
&\qquad +pM\int_{{\partial}\Omega}v^{p-1}\eta^2\,d\sigma.\end{aligned}$$ Now $$\label{v-Lq-1}
2\int_{\Omega_{\delta}}v^{p+m-2}\eta|\nabla\eta||\nabla v|\,dx
\le\frac{(p-1)}{4}\int_{\Omega_{\delta}}v^{p+m-3}\eta^2|\nabla v|^2\,dx
+\frac{4}{p-1}\int_{\Omega_{\delta}}v^{p+m-1}|\nabla\eta|^2\,dx.$$ By an argument similar to the proof of Lemma 1.1 of @Hu1 there exists a constant $C_1>0$ such that, $$\begin{aligned}
\label{v-eta-bdary-integral-ineqn}
&\int_{{\partial}\Omega}v^{p-1}\eta^2\,d\sigma\notag\\
&\qquad \le C_1\int_{\Omega_{\delta}}|\nabla (v^{p-1}\eta^2)|\,dx \notag\\
&\qquad  \le (p-1)C_1\int_{\Omega_{\delta}}v^{p-2}\eta^2|\nabla v|\,dx+2C_1\int_{\Omega_{\delta}}v^{p-1}\eta|\nabla\eta|\,dx\notag\\
&\qquad \le\frac{m(p-1)}{2M}\int_{\Omega_{\delta}}v^{p+m-3}\eta^2|\nabla v|^2\,dx
+\frac{(p-1)MC_1^2}{2m}\int_{\Omega_{\delta}}v^{p-m-1}\eta^2\,dx
+2C_1\int_{\Omega_{\delta}}v^{p-1}\eta |\nabla\eta|\,dx.\end{aligned}$$ By , and , $$\begin{aligned}
\label{v^q-time-derivative-ineqn}
&\frac{{\partial}}{{\partial}t}\left(\int_{\Omega_{\delta}}v^p\eta^2\,dx\right)\notag\\
&\qquad \le C_2\left(\int_{\Omega_{\delta}}v^{p-m-1}\eta^2\,dx+\int_{\Omega_{\delta}}v^{p-1}\eta |\nabla\eta|\,dx
+\int_{\Omega_{\delta}}v^{p+m-1}|\nabla\eta|^2\,dx\right)\notag\\
&\qquad \le C_2\left[\int_{\Omega_{\delta}}v^{p-1}\eta^2\,dx
+\int_{\Omega_{\delta}}(v^p\eta^2)^{\frac{p-1}{p}} |\nabla\eta|\eta^{-1+\frac{2}{p}}\,dx
+\int_{\Omega_{\delta}}(v^p\eta^2)^{\frac{p+m-1}{p}}(|\nabla\eta|\eta^{-1+\frac{1-m}{p}})^2\,dx\right]\end{aligned}$$ for some constant $C_2>0$. Now $$\label{eta-expression-bd1}
|\nabla\eta|\eta^{-1+\frac{2}{p}}=\alpha\phi^{\frac{2\alpha}{p}-1}|\nabla\phi|\in L^{\infty}\quad\mbox{ if }\alpha>\frac{p}{2}$$ and $$\label{eta-expression-bd2}
|\nabla\eta|\eta^{-1+\frac{1-m}{p}}=\alpha\phi^{\frac{(1-m)}{p}\alpha-1}|\nabla\phi|\in L^{\infty}\quad\mbox{ if }
\alpha>\frac{p}{1-m}.$$ We now choose $\alpha>\max \left(\frac{p}{2},\frac{p}{1-m}\right)$. Since $$\int_{\Omega_{\delta}}v^p\eta^2\,dx\ge\int_{\Omega}\eta^2\,dx>0,$$ by , and , $$\begin{aligned}
\label{v^q-time-derivative-ineqn2}
\frac{{\partial}}{{\partial}t}\left(\int_{\Omega_{\delta}}v^p\eta^2\,dx\right)
\le&C_3\left[\int_{\Omega_{\delta}}v^{p-1}\eta^2\,dx
+\int_{\Omega_{\delta}}(v^p\eta^2)^{\frac{p-1}{p}} \,dx
+\int_{\Omega_{\delta}}(v^p\eta^2)^{\frac{p+m-1}{p}}\,dx\right]\notag\\
\le&C_4\left[\left(\int_{\Omega_{\delta}}v^p\eta^2\,dx\right)^{\frac{p-1}{p}}+\left(\int_{\Omega_{\delta}}v^p\eta^2\,dx\right)^{\frac{p+m-1}{p}}\right]\notag\\
\le&C_5\left(\int_{\Omega_{\delta}}v^p\eta^2\,dx\right)^{\frac{p+m-1}{p}}\end{aligned}$$ for some constants $C_3>0$, $C_4>0$, $C_5>0$. Integrating , $$\left(\int_{\Omega_{\delta}}u^p(x,t)\eta^2(x)\,dx\right)^{\frac{1-m}{p}}\le\left(\int_{\Omega_{\delta}}v^p(x,t)\eta^2(x)\,dx\right)^{\frac{1-m}{p}}\le\left(\int_{\Omega_{\delta}}(u_0+1)^p\eta^2\,dx\right)^{\frac{1-m}{p}}+C_6t\quad\forall 0<t<T$$ for some constant $C_6>0$ and the lemma follows.

By a similar argument we also have the following result.

\[Lp-L-infty-thm-3\] Let $n\ge 3$, $0<m\le \frac{n-2}{n}$, $p>\frac{n(1-m)}{2}$, $0<\delta<\delta_0$. Let $0\le u_0\in L^p(\Omega_{\delta})$, $g_i\in L^1(\partial B_{\delta}(a_i)\times (0,T))$ for all $i=1,2,\dots,i_0$, $f\in L^1(\partial\Omega\times(0,T))$ be such that $f\leq M$ for some constant $M>0$. Suppose $u\in C^{2,1}({\overline}{\Omega}_{\delta}\times (0,T))$ is a solution of in $\Omega_{\delta}\times (0,T)$. Then for any $0<t_1<T$, $\delta<\delta_1<\delta_2\le\delta_0$, there exist constants $\theta>0$ and $C>0$ depending on $M$, $m$, $n$, $t_1$, $\delta_1$, and $\delta_2$ such that $$\|u\|_{L^{\infty}(\Omega_{\delta_2}\times(t_1,T])}\leq C\left\{1+\int_{\Omega_{\delta_1}}u_0^p\,dx\right\}^{\frac{\theta}{p}}.$$ and for any $0<t_1<T$, $R_2>R_1>0$ such that $B_{R_2}(x_1)\subset\Omega_{\delta}$, there exist constants $\theta>0$ and $C>0$ depending on $m$, $n$, $t_1$, $R_1$ and $R_2$, but independent of $M$ such that $$\|u\|_{L^{\infty}(B_{R_1}(x_1)\times(t_1,T])}\le C\left\{1+\int_{B_{R_2}(x_1)}u_0^p\,dx\right\}^{\frac{\theta}{p}}.$$

Existence of solutions for the Neumann problem
==============================================

In this section we will prove the existence of solutions of the Neumann problem . We first observe that by an argument similar to the proof of Proposition A.1 of @BV1 we have the following lemma.

\[Alexsandrov-reflection-lemma\] Let $n\ge 1$, $0<m<1$, and $0\le v_0(x)\in L^1(B_{5R}(x_0))$, $v_0\not\equiv 0$, such that $\text{supp }v_0\subset B_{R}(x_0)\subset{{\mathbb{R}}}^n$. Let $v$ be a weak solution of $$\label{Dirichlet-problem}
\left\{\begin{aligned}
v_t=&\Delta v^m\qquad\mbox{ in }B_{5R}(x_0)\times(0,T_{v_0})\\
v(x,t)=&0\qquad\quad\,\mbox{ on }\partial B_{5R}(x_0)\times(0,T_{v_0})\\
v(x,0)=&v_0(x)\quad\,\,\mbox{ in }B_{5R}(x_0)
\end{aligned}\right.$$ where $T_{v_0}>0$ is the extinction time of $v$, then $$v(x_1,t)\ge v(x_2,t)\quad\forall |x_1|\le R, 4R\le |x_2|\le 5R.$$

By Lemma \[Alexsandrov-reflection-lemma\] and an argument similar to the proof in section 1 of @BV2 we have the following result.

\[Dirichlet-soln-positive-lemma\](cf. (1.18) and (1.27) of @BV2) Let $n$, $m$, and $v_0$, $v$, and $T_{v_0}$, be as in Lemma \[Alexsandrov-reflection-lemma\].Then there exist constants $C_1>0$, $C_2>0$, such that $$T_{v_0}\ge C_1R^{2-n(1-m)}\left(\int_{B_R(x_0)}v_0\,dx\right)^{1-m}$$ and $$v^m(x,t)\ge C_2R^{2-n}\|v_0\|_{L^1(B_{R}(x_0))}T_{v_0}^{-\frac{1}{1-m}}t^{\frac{m}{1-m}}\quad \forall |x|\le R, t\in (0,t_{\ast}]$$ where $$t_{\ast}=\frac{C_1}{2}R^{2-n(1-m)}\left(\int_{B_R(x_0)}v_0\,dx\right)^{1-m}.$$

\[positivity-lemma\] Let $n\ge 3$, $0<m\leq\frac{n-2}{n}$ and $T>t_0>0$. Suppose $0\le u\in C(\Omega\times(t_0,T])$ satisfies in $\mathcal{D}(\Omega\times (t_0,T))$ and $$\label{u-integral-positive}
\int_{\Omega}u(x,T)\,dx>0.$$ Then $$u(x,T)>0 \qquad \forall x\in\Omega.$$

Let $$D(T)=\left\{x\in\Omega :u(x,T)>0\right\}.$$ Since by there exists a point $x(T)\in \Omega$ such that $u(x(T),T)>0$, $D(T)\ne\phi$. Suppose that $D(T)\neq\Omega$. Then there exist a point $x_0\in\Omega\cap{\partial}D(T)$ such that $$\label{u=0-at-x0}
u(x_0,T)=0.$$ We choose a constant $R>0$ such that $B_{5R}(x_0)\subset\Omega$. We claim that $$\label{eq-properties-from-assumption-of-u-epsilon-at-T}
\int_{B_{R/2}(x_0)}u(x,T)\,dx>0.$$ Suppose not. Then $$\int_{B_{R/2}(x_0)}u(x,T)\,dx=0\quad\Rightarrow\quad u(x,T)=0\quad\forall |x-x_0|\le\frac{R}{2}$$ which contradicts the fact that $x_0\in\partial D(T)$. Hence holds. Let $\psi\in C_0^{\infty}(B_{5R}(x_0))$, $0\le\psi\le 1$, be such that $\psi(x)=1$ for all $x\in B_{R/2}(x_0)$ and $\psi(x)=0$ for all $x\in B_{5R}(x_0){\backslash}B_R(x_0)$. By the proof of Lemma 3.1 of @HP there exist constants $\alpha>1$ and $C_R>0$ such that $$\begin{aligned}
\label{eq-the-difference-between-1-m-norm-and-T-s234}
&\left|\frac{{\partial}}{{\partial}t}\left(\int_{B_{5R}(x_0)}u\psi^{\alpha}\,dx\right)\right|\le C_R\left(\int_{B_{5R}(x_0)}u\psi^{\alpha}\,dx\right)^m\quad\forall 0<t<T\notag\\
\Rightarrow\quad&\left|\left(\int_{B_{5R}(x_0)}u(x,T)\psi^{\alpha}(x)\,dx\right)^{1-m}-\left(\int_{B_{5R}(x_0)}u(x,s)\psi^{\alpha}(x)\,dx\right)^{1-m}\right|\le (1-m)C_R(T-s) \quad \forall T>s>0.\end{aligned}$$ Let $${\varepsilon}_1=\frac{1-2^{m-1}}{(1-m)C_{R_1}}\left(\int_{B_{5R}(x_0)}u(x,T)\psi^{\alpha}(x)\,dx\right)^{1-m}.$$ By for any $0<T-s\leq{\varepsilon}_1$ $$\begin{aligned}
\label{eq-aligned-uniform-positivity-of-mass-of-u-near-T-0--}
\int_{B_R(x_0)}u(x,s)\,dx\ge&\int_{B_{5R}(x_0)}u(x,s)\psi^{\alpha}(x)\,dx\notag\\
\ge&\left[\left(\int_{B_{5R}(x_0)}u(x,T)\psi^{\alpha}(x)\,dx\right)^{1-m}-(1-m)C_R(T-s)\right]^{\frac{1}{1-m}}\notag\\
\ge&\frac{1}{2}\int_{B_{5R}(x_0)}u(x,T)\psi^{\alpha}(x)\,dx\notag\\
\ge&\frac{1}{2}\int_{B_{R/2}(x_0)}u(x,T)\,dx.\end{aligned}$$ Let $v_{0,\tau}(x)=u(x,T-\tau)\chi_{B_R(x_0)}$. By the discussion on P.537 of @BV2 there exists a unique weak minimal solution $v^{\tau}$ of with initial value $v_{0,\tau}$. Let $T_{v_{0,\tau}}$ be the extinction time of $v^{\tau}$. Then by , and Lemma \[Dirichlet-soln-positive-lemma\], $$T_{v_{0,\tau}}\geq c_0>0\quad\forall 0<\tau\le{\varepsilon}_1.$$ where $$c_0=C_1R^{2-n(1-m)}\left(\frac{1}{2}\int_{B_{R/2}(x_0)}u(x,T)\,dx\right)^{1-m} \qquad \mbox{and} \qquad  \mbox{$C_1>0$ is as in Lemma \ref{Dirichlet-soln-positive-lemma}.}$$ Let $\tau_1=\min (c_0,{\varepsilon}_1)/2$. By Lemma \[Dirichlet-soln-positive-lemma\] there exists a constant $c_1>0$ such that $$\label{min-soln-bded-below}
v^{\tau_1}(x,\tau_1)\ge c_1\quad\forall |x|\le R.$$ Since $v^{\tau_1}$ is the unique weak minimal solution of with initial value $v_{0,\tau_1}$, by the maximum principle, $$\label{u-lower-bd}
u^m(x,T)\ge (v^{\tau_1})^m(x,\tau_1)>0\quad\forall |x|\le R.$$ By and we get $u^m(x_0,T)\ge c_1>0$. This contradicts . Hence $D(T)=\Omega$ and the lemma follows.

\[Aronson-Benilan-ineqn-lemma\] Let $n\ge 1$, $0<m<1$, $0<\delta<\delta_0$, $0<u_0\in C^2({\overline}{\Omega}_{\delta})$. Let $f\in C^{\infty}({\partial}\Omega\times [0,T))$ and $g_i\in C^{\infty}({\partial}B_{\delta}(a_i)\times [0,T))$ for all $i=1,\cdots,i_0$ be such that $f$, $g_i$, are nonnegative monotone decreasing functions of $t\in (0,T)$, and $f\le M$, $g_i\le M$, for all $i=1,\cdots,i_0$ and some constant $M>0$. Suppose $u\in C^{2,1}({\overline}{\Omega}_{\delta}\times [0,T))$ is a positive solution of . Then $u$ satisfies in $\Omega_{\delta}\times (0,T)$.

Let $$q=\frac{u_t}{u}, \qquad {\varepsilon}=\min_{\overline{\Omega}_{\delta}}u_0, \qquad a=\frac{{\varepsilon}}{(1-m)\left(\|{\triangle}u_0^m\|_{L^{\infty}(\Omega_{\delta})}+1\right)}$$ and $$\overline{q}=q-\frac{1}{(1-m)(a+t)}.$$ By direct computation, $$\label{q-eqn}
q_t=mu^{m-1}{\triangle}q+2mu^{m-2}\nabla u\cdot\nabla q+(m-1)q^2 \qquad \mbox{in $\Omega_{\delta}\times(0,T)$}.$$ Then ${\overline}{q}$ satisfies $$\label{q-bar-eqn}
\begin{cases}
\begin{aligned}
&\overline{q}_t=mu^{m-1}{\triangle}\overline{q}+2mu^{m-2}\nabla u\cdot\nabla\overline{q}-(1-m)\overline{q}\left(\overline{q}+\frac{2}{(1-m)(a+t)}\right)\qquad \mbox{in $\Omega_{\delta}\times(0,T)$}\\
&\overline{q}(x,0)\leq 0 \qquad \qquad \qquad \qquad \mbox{on $\Omega_{\delta}$}.
\end{aligned}
\end{cases}$$ Since $$\begin{aligned}
&f_t=\frac{\partial}{\partial t}\left(\frac{\partial u^m}{\partial \nu}\right)=mu^{m-1}u_{\nu t}-m(1-m)u^{m-2}u_{\nu}u_t\quad
\quad\forall (x,t)\in\partial\Omega\times(0,T)\notag\\
&\qquad \Rightarrow\quad u_{\nu t}=\frac{f_tu^{1-m}}{m}+(1-m)\frac{u_{\nu}u_t}{u}\qquad\qquad\qquad\,
\forall (x,t)\in\partial\Omega\times(0,T),\end{aligned}$$ we have $$\begin{aligned}
\label{eq-of-overline-q-sub-nu-on-partial-Omega-with-f-t-and-u-and-a}
\frac{\partial q}{\partial\nu}&=\frac{u_{\nu t}}{u}-\frac{u_{\nu}u_t}{u^2}=\frac{f_t}{mu^{m}}-\frac{qf}{u^m}\qquad\quad\,\mbox{ on }\partial\Omega\times(0,T)\notag\\
&=\frac{f_t}{mu^{m}}-\frac{{\overline}{q}f}{u^m}-\frac{f}{(1-m)(a+t)u^m}\quad \mbox{on $\partial\Omega\times(0,T)$}.\end{aligned}$$ Similarly $$\frac{\partial q}{\partial\nu}=\frac{g_{i,t}}{mu^{m}}-\frac{{\overline}{q}g_i}{u^m}-\frac{g_i}{(1-m)(a+t)u^m}\qquad \mbox{ on }\partial B_{\delta}(a_i)\times(0,T)\quad\forall i=1,\cdots,i_0.$$ Let $0<T_1<T$ and suppose $\overline{q}$ attains a positive maximum on $\overline{\Omega}_{\delta}\times(0,T_1]$ at $(x_0,t_0)$ for some $x_0\in{\overline}{\Omega}_{\delta}$, $0\le t_0\le T_1$. Suppose $x_0\in\Omega_{\delta}$ and $t_0>0$. Then $$\label{q-bar-ineqns-at-max}
\overline{q}_t\geq 0, \qquad \nabla\overline{q}=0 \qquad \mbox{and} \qquad {\triangle}\overline{q}\leq 0  \qquad \mbox{at $(x_0,t_0)$}.$$ Hence by and , $$\begin{aligned}
0\le\overline{q}_t=&mu^{m-1}{\triangle}\overline{q}+2mu^{m-2}\nabla u\cdot\nabla\overline{q}-(1-m)\overline{q}\left(\overline{q}+\frac{2}{(1+m)(a+t)}\right)\notag\\
\le&-(1-m)\overline{q}\left(\overline{q}+\frac{2}{(1+m)(a+t)}\right)<0\qquad \qquad \mbox{at $(x_0,t_0)$}.\notag\end{aligned}$$ Contradiction arises. Hence either $x_0\in\partial\Omega_{\delta}$ or $t_0=0$. Suppose $x_0\in\partial\Omega_{\delta}$ and $t_0>0$. Without loss of generality we may assume that $x_0\in\partial\Omega$. By the strong maximum principle, $$\label{q-bar-bdary-derivative-+ve}
\frac{{\partial}{\overline}{q}}{{\partial}\nu}(x_0,t_0)>0.$$ Then by and , $$\begin{aligned}
&0<\frac{f_t}{mu^{m}}-\frac{{\overline}{q}f}{u^m}-\frac{f}{(1-m)(a+t)u^m}\le-\frac{{\overline}{q}f}{u^m}\quad\mbox{ at }(x_0,t_0)\\
&\Rightarrow\quad {\overline}{q}f<0\quad\mbox{ at }(x_0,t_0)\end{aligned}$$ Since ${\overline}{q}(x_0,t_0)>0$ and $f(x_0,t_0)\ge 0$, contradiction arises. Hence $t_0=0$. Since $\overline{q}(x,0)\leq 0$ on $\Omega_{\delta}$, $$\begin{aligned}
&\overline{q}(x,t)\le 0\quad\forall x\in\Omega_{\delta}, 0<t<T_1\notag\\
&\Rightarrow\quad\overline{q}(x,t)\le 0 \quad\forall x\in\Omega_{\delta}, 0<t<T\quad\mbox{ as }T_1\to T\\
&\Rightarrow\quad u_t\le\frac{u}{(1-m)(a+t)}\le\frac{u}{(1-m)t}\quad\mbox{ in }\Omega_{\delta}\times (0,T)\end{aligned}$$ and the lemma follows.

We are now ready for the proof of Theorem \[Neumann-holes-problem-existence-thm\].

\[Proof of Theorem \[Neumann-holes-problem-existence-thm\]\] By Lemma \[comparsion-lemma1\] the solution of is unique. Hence it remains to prove the existence of solution of . We will use a modification of the proof of Lemma 2.1 of @Hs2 to prove this theorem. We divide the proof into three cases.\
**Case 1**: $0<u_0\in C^{\infty}(\overline{\Omega}_{\delta})$, $f\in C^{\infty}({\partial}\Omega\times [0,\infty))$, $g_i\in C^{\infty}({\partial}B_{\delta}(a_i)\times [0,\infty))$, such that $g_{i}(x,t)=\alpha_i$ for all $(x,t)\in\partial B_{\delta}(a_i)\times[0,\delta')$, $i=1,\cdots,i_0$, and $f(x,t)=0$ for all $(x,t)\in\partial \Omega\times[0,\delta')$ for some constants $\delta'>0$ and $\alpha_1,\cdots,\alpha_{i_0}\in{{\mathbb{R}}}^+$, respectively.

Let ${\varepsilon}_0=\inf_{\overline{\Omega}_{\delta}}u_0$. Then ${\varepsilon}_0>0$. We first choose a sequence of functions $\{\phi_j\}_{j=1}^{\infty}\subset C^{\infty}(\Omega)$, $0\le\phi_j\le 1$ for all $j\in\mathbb{Z}^+$, such that $$\phi_j(x)=\begin{cases}
1\qquad \mbox{if $\textbf{dist}(x,\partial\Omega)\leq \frac{\delta_0-\delta}{3(j+1)}$}\\
0\qquad \mbox{if $\textbf{dist}(x,\partial\Omega)> \frac{\delta_0-\delta}{3j}$}
\end{cases}$$ for all $j\in\mathbb{Z}^+$ and a sequence of functions $\{\psi_j\}_{j=1}^{\infty}\subset C_0^{\infty}({{\mathbb{R}}}^N)$, $0\le\psi_j\le 1$ for all $j\in\mathbb{Z}^+$, such that $$\psi_j(x)=\begin{cases}
1\qquad\mbox{ if }|x|\le\delta+\frac{\delta_0-\delta}{3(j+1)}\\
0\qquad\mbox{ if }|x|\ge\delta+\frac{\delta_0-\delta}{3j}
\end{cases}$$ for all $j\in\mathbb{Z}^+$. Then $0\le\phi_{j+1}\le\phi_{j}\leq 1$ and $0\le\psi_{j+1}\le\psi_{j}\le 1$ for any $j\in{{\mathbb{Z}}}^+$. For any $j\in\mathbb{Z}^+$ and $x\in\Omega$ let $$u_{0,j}(x)=u_{0}(x)\left(1-\phi_j(x)-\sum_{i=1}^{i_0}\psi_j(x-a_i)\right)+\delta^{\frac{2}{m}}\sum_{i=1}^{i_0}\frac{\alpha_i^{\frac{1}{m}}}{|x-a_i|^{\frac{1}{m}}}\cdot\psi_{j}(x-a_i)+{\varepsilon}_0\phi_j$$ and $${\varepsilon}_1=\min\left\{{\varepsilon}_0,\left(\frac{\delta^2\alpha}{\textbf{diam}\,\Omega}\right)^{\frac{1}{m}}\right\}, \qquad \mbox{where $\alpha=\min\left\{\alpha_1,\cdots,\alpha_{i_0}\right\}$}.$$ Then $$\label{u0j-upper-lower-bd}
{\varepsilon}_1\le u_{0,j} \le u_{0}+C_1 \quad\mbox{ in }\Omega_{\delta}\quad \forall j\in\mathbb{Z}^+\qquad\mbox{and}\qquad\|u_{0,j}-u_0\|_{L^{1}(\Omega_{\delta})}\to 0\quad\mbox{ as }j\to\infty$$ for some constant $C_1>0$ depending on $m$, $\alpha_1,\cdots,\alpha_{i_0}$, $\delta$ and $$\begin{cases}
\begin{aligned}
\frac{\partial u_{0,j}^m}{\partial\nu}=&0\quad\,\mbox{ on }\partial\Omega\quad\forall j\in{{\mathbb{Z}}}^+\\
\frac{\partial u_{0,j}^m}{\partial\nu}=&\alpha_i\quad\mbox{ on }\partial B_{\delta}(a_i)\quad\forall
j\in{{\mathbb{Z}}}^+,i=1,2,\cdots,i_0.
\end{aligned}
\end{cases}$$ We will now use an argument similar to the proof of Theorem 3.5 of @Hu1 to prove the existence of a solution of in $\Omega_{\delta}\times (0,\infty)$ with initial value $u_{0,j}$. Let $n_1={\varepsilon}_1^m$ and $n_2=(\|u_0\|_{L^{\infty}}+C_1)^m$. We choose a monotone decreasing function $H\in C^{\infty}({{\mathbb{R}}})$, $H>0$, such that $H(s)=ms^{1-\frac{1}{m}}$ for $n_1/2\le s\le 2n_2$, $H(s)=m(n_1/4)^{1-\frac{1}{m}}$ for $s\le n_1/4$, $H(s)=m(4n_1/4)^{1-\frac{1}{m}}$ for $s\ge 4n_2$. Then by standard theory for non-degerenate parabolic equation @LSU such that the problem $$\left\{\begin{aligned}
v_t=&H(v)\Delta v\quad\,\,\,\mbox{ in }\Omega_{\delta}\times (0,\infty)\\
\frac{{\partial}v}{{\partial}\nu}=&f\qquad\qquad\mbox{ on }{\partial}\Omega\times (0,\infty)\\
\frac{{\partial}v}{{\partial}\nu}=&g_i\qquad\qquad\mbox{ on }{\partial}B_{\delta}(a_i)\times (0,\infty)\quad\forall i=1,\cdots,i_0\\
v(x,0)=&u_{0,j}(x,0)^m\quad\mbox{ in }\Omega_{\delta}
\end{aligned}\right.$$ has a classical solution $v_j\in C^{2,1}(\overline{\Omega}_{\delta}\times (0,\infty))$. By the maximum principle for non-degerenate parabolic equation @LSU, $n_1\le v_j\le n_2$ on $\overline{\Omega}_{\delta}\times (0,\infty)$. Hence $H(v_j)=mv_j^{1-\frac{1}{m}}$ and thus the function $u_j=v_j^{\frac{1}{m}}\in C^{2,1}(\overline{\Omega}_{\delta}\times (0,\infty))$ is a solution of in $\Omega_{\delta}\times (0,\infty)$ with initial value $u_{0,j}$ such that $$\label{uj-lower-bd}
u_j\ge{\varepsilon}_1\quad\mbox{ in }\overline{\Omega}_{\delta}\times[0,\infty).$$

Since $$\frac{{\partial}}{\partial t}\left(\int_{\Omega_{\delta}}u_j\,dx\right)=\int_{\Omega_{\delta}}{\triangle}u_j^m\,dx
=\int_{\partial\Omega}f\,d\sigma+\sum_{i=1}^{i_0}\int_{\partial B_{\delta}(a_i)}g_i\,d\sigma\quad \forall t>0, j\in{{\mathbb{Z}}}^+,$$ integrating over $t$ we have $$\label{eq-integration-form-from-main-construction-equation-1-with-j-1}
\int_{\Omega_{\delta}}u_j(x,t)\,dx=\int_{\Omega_{\delta}}u_{0,j}\,dx+\int_0^t\int_{\partial\Omega}fd\sigma ds+\sum_{i=1}^{i_0}\int_0^t\int_{\partial B_{\delta}(a_i)}g_i\,d\sigma ds \qquad t>0.$$

We will now show that $u_j$ converges to a solution $u$ of as $j\to \infty$. Let $t_2>t_1>0$. Then by and Proposition \[Lp-L-infty-thm-2\] there exists a constant $C_2>0$ such that $$\label{uj-uniform-upper-bd}
u_j\le C_2 \qquad \forall x\in\overline{\Omega}_{\delta},t_1\le t\le t_2,j\in{{\mathbb{Z}}}^+.$$ Hence by and the equation for the sequence $\{u_j\}_{j=1}^{\infty}$ is uniformly parabolic on $\overline{\Omega}_{\delta}\times [t_1,t_2]$ for any $t_2>t_1>0$. Thus by the Schauder estimates @LSU (Theorem 3.1 and Theorem 5.4 in chapter 5 of @LSU), for any $t_2>t_1>0$, $$\sup_{\overline{\Omega}_{\delta}\times[t_1,t_2]}(|\nabla u_j|+|{\partial}^2_{x_kx_l}u_j|+|u_{j,t}|)\le C_3\quad\forall k,l=1,\cdots,n, j\in{{\mathbb{Z}}}^+$$ and $$\sup_{\substack{y,y'\in\overline{\Omega}_{\delta}\\s,s'\in [t_1,t_2]}}\frac{|{\partial}^2_{x_kx_l}u_j(y,s)-{\partial}^2_{x_kx_l}u_j(y',s')|}{|y-y'|^{\alpha}}+\sup_{\substack{y,y'\in\overline{\Omega}_{\delta}\\s,s'\in [t_1,t_2]}}\frac{|u_{j,t}(y,s)-u_{j,t}(y',s')|}{|s-s'|^{\frac{\alpha}{2}}}\le C_4\quad\forall k,l=1,\cdots,n, j\in{{\mathbb{Z}}}^+$$ for some constants $C_3>0$, $C_4>0$, $0<\alpha<1$. Hence the sequence $\{u_{j}\}_{j=1}^{\infty}$ is uniformly Hölder continuous in $C^{2,1}(\overline{\Omega}_{\delta}\times[t_1,t_2])$ for any $t_2>t_1>0$. By the Ascoli Theorem and a diagonalization argument the sequence $\{u_{j}\}_{j=1}^{\infty}$ has a subsequence which we may assume without loss of generality to be the sequence itself that converges uniformly on every compact subset of $\overline{\Omega}_{\delta}\times (0,\infty)$ to a solution $u\in C^{2,1}(\overline{\Omega}_{\delta}\times(0,\infty))$ of in $\overline{\Omega}_{\delta}\times(0,\infty)$ which satisfies $$\frac{{\partial}u^m}{{\partial}\nu}=f\quad\mbox{ on }{\partial}\Omega\times (0,\infty)\qquad\mbox{ and }\qquad\frac{{\partial}u^m}{{\partial}\nu}=g_i\quad\mbox{ on }{\partial}B_{\delta}(a_i)\times (0,\infty)\quad\forall i=1,\dots,i_0$$ as $j\to\infty$. Letting $j\to\infty$ in , $u$ satisfies .

It remains to show that $u$ has initial value $u_0$. For any $\psi\in C^{\infty}_0(\Omega_{\delta})$, by , $$\begin{aligned}
&\left|\int_{\Omega_{\delta}}u_j(x,t)\psi(x)\,dx-\int_{\Omega_{\delta}}u_{0,j}(x)\psi(x)\,dx\right|\\
&\qquad \qquad =\left|\int_0^t\int_{\Omega_{\delta}}u_{j,t}(x,s)\psi(x)\,dxds\right|=\left|\int_0^t\int_{\Omega_{\delta}}u_{j}^m(x,s){\triangle}\psi(x)\,dxds\right|\\
&\qquad \qquad \le\left|\Omega\right|^{1-m}\left\|{\triangle}\psi\right\|_{L^{\infty}}\int_{0}^{t}\left(\int_{\Omega_{\delta}}u_{j}(x,s)\,dx)\right)^{m}\,ds\\
&\qquad \qquad \le\left|\Omega\right|^{1-m}\left\|{\triangle}\psi\right\|_{L^{\infty}}\left(\int_{\Omega_{\delta}}u_{0,j}(x)\,dx+\int_0^t\int_{\partial\Omega}f\,d\sigma ds+\sum_{i=1}^{i_0}\int_0^t\int_{\partial B_{\delta}(a_i)}g_i\,d\sigma ds\right)^m t\quad\forall t>0.
\end{aligned}$$ Letting $j\to\infty$, $$\begin{aligned}
\label{u-u0-L1-bd}
&\left|\int_{\Omega_{\delta}}u(x,t)\psi(x)\,dx-\int_{\Omega_{\delta}}u_0(x)\psi(x)\,dx\right|\notag\\
&\qquad \qquad \le\left|\Omega\right|^{1-m}\left\|{\triangle}\psi\right\|_{L^{\infty}}\left(\int_{\Omega_{\delta}}u_0(x)\,dx+\int_0^t\int_{\partial\Omega}f\,d\sigma ds+\sum_{i=1}^{i_0}\int_0^t\int_{\partial B_{\delta}(a_i)}g_i\,d\sigma ds\right)^m t
\quad\forall t>0.\end{aligned}$$ Letting $t\to 0$ in , $$\label{u-converge-u0-weakly}
\lim_{t\to 0}\int_{\Omega_{\delta}}u(x,t)\psi(x)\,dx=\int_{\Omega_{\delta}}u_0(x)\psi(x)\,dx\qquad\forall\psi\in C^{\infty}_0(\Omega).$$ By , any sequence $\{t_k\}_{k=1}^{\infty}$ converging to $0$ as $k\to\infty$ will have a convergent subsequence $\{t_{k_l}\}_{l=1}^{\infty}$ such that $u(x,t_{k_l})$ converges to $u_0(x)$ a.e. $x\in\Omega_{\delta}$ as $l\to\infty$. By the Lebesgue Dominated Convergence Theorem $$\lim_{l\to\infty}\int_{\Omega_{\delta}}|u(x,t_{k_l})-u_0(x)|\,dx=0$$ Since the sequence $\{t_k\}_{k=1}^{\infty}$ is arbitrary, $u$ satisfies .

**Case 2:** $0\leq u_0\in L^{\infty}\left(\overline{\Omega}_{\delta}\right)$.

We choose a sequence of functions $\{u_{0,j}\}_{j=1}^{\infty}\subset C^{\infty}(\overline{\Omega}_{\delta})$ such that $b_j:=\min_{\Omega_{\delta}}u_{0,j}>0$ on $\overline{\Omega}_{\delta}$ for all $j\in{{\mathbb{Z}}}^+$ and $$\label{eq-cases-aligned-approximations-to-u-by-u-j-08}
\begin{cases}
\begin{aligned}
&\|u_{0,j}-u_0\|_{L^1(\Omega_{\delta})}\to 0 \qquad  \qquad \mbox{as $j\to\infty$}\\
&\qquad u_{0,j}(x)\to u_0(x) \qquad \qquad \mbox{a.e. $x\in\Omega$} \quad \mbox{ as }j\to\infty\\
&\|u_{0,j}\|_{L^{\infty}(\Omega_{\delta})}\leq \|u_0\|_{L^{\infty}(\Omega_{\delta})}+\frac{1}{j}\quad\forall j\in\mathbb{Z}^+.
\end{aligned}
\end{cases}$$ For each $i=1,\cdots,i_0$, we choose a sequence of positive functions $\{g_{i,j}\}_{j=1}^{\infty}\subset C^{\infty}(\partial B_{\delta}(a_i)\times[0,\infty))$ satisfying $$\label{eq-cases-aligned-construction-of-g-i-j-converging-to-g-i-08}
\begin{cases}
\begin{aligned}
g_{i,j}&\to g_i \qquad \qquad\qquad\qquad\qquad\mbox{in $L^1_{loc}(\partial B_{\delta}(a_i)\times[0,\infty))$} \qquad \mbox{as $j\to\infty$}\qquad \forall i=1,\cdots, i_0\\
g_{i,j}(x,t)&\to g_{i}(x,t) \qquad\qquad\qquad\qquad\mbox{a.e. $(x,t)\in\partial B_{\delta}(a_i)\times[0,\infty)$} \quad \mbox{as $j\to\infty$}\quad\,\,\, \forall i=1,\cdots, i_0\\
g_{i,j}(x,t)&\le \|g_i\|_{L^{\infty}({\partial}B_{\delta}(a_i)\times [0,T])}+1\qquad\mbox{ on }\partial B_{\delta}(a_i)\times[0,T)\qquad\qquad\qquad\forall j\in\mathbb{Z}^+,\, i=1,\cdots, i_0, T>0\\
g_{i,j}&=\alpha_{i,j}\qquad \qquad\qquad\qquad\quad \mbox{ on }\partial B_{\delta}(a_i)\times  [0,1/j)\qquad\qquad\qquad\qquad\forall j\in\mathbb{Z}^+,\, i=1,\cdots, i_0
\end{aligned}
\end{cases}$$ for some positive constants $\alpha_{i,j}$ and choose a sequence of nonnegative functions $\{f_j\}_{j=1}^{\infty}\subset C^{\infty}\left(\partial\Omega\times [0,\infty)\right)$ satisfying $$\label{eq-cases-aligned-construction-of-f-i-converging-to-g-i-08}
\begin{cases}
\begin{aligned}
f_j&\to f \qquad \qquad\qquad \quad\,\mbox{ in }L^1_{loc}(\partial\Omega\times[0,\infty)) \quad \mbox{as $j\to\infty$}\\
f_j(x,t)&\to f(x,t) \qquad\qquad\quad\,\, \,\mbox{a.e. $(x,t)\in\partial\Omega\times[0,\infty)$} \quad \mbox{as $j\to\infty$}\\
f_j(x,t)&\le\|f\|_{L^{\infty}({\partial}\Omega\times [0,T])}+1\quad\mbox{on }\partial \Omega_{\delta}\times[0,T)\quad
\forall j\in\mathbb{Z}^+, T>0\\
f_j&= 0 \qquad\qquad \qquad\quad \,\,\,\mbox{ on }\partial\Omega\times [0,1/j)\quad \forall j\in\mathbb{Z}^+.
\end{aligned}
\end{cases}$$ If $f\equiv 0$ on ${\partial}\Omega\times (0,\infty)$ and $g_i$ are nonnegative monotone decreasing function of $t\in (0,\infty)$ for $i=1,\dots,i_0$, then we choose $f_j\equiv 0$ on ${\partial}\Omega\times (0,\infty)$ for all $j\in{{\mathbb{Z}}}^+$ and the functions $g_{i,j}$ such that they are positive monotone decreasing functions of $t\in (0,\infty)$ for $i=1,\dots,i_0$ and $j\in{{\mathbb{Z}}}^+$. Then by case 1 for any $j\in{{\mathbb{Z}}}^+$ there exists a solution $u_j\in C^{2,1}({\overline}{\Omega}_{\delta}\times(0,\infty))$ of in $\Omega_{\delta}\times(0,\infty)$ with initial value $u_{0,j}$ that satisfies $$\label{eq-integration-form-from-main-construction-equation-1-with-j-2}
\int_{\Omega_{\delta}}u_j(x,t)\,dx=\int_{\Omega_{\delta}}u_{0,j}\,dx+\iint_{\partial\Omega\times(0,t)}f_j\,d\sigma ds
+\sum_{i=1}^{i_0}\iint_{\partial B_{\delta}(a_i)\times(0,t)}g_{i,j}\,d\sigma ds \qquad \forall t>0,j\in{{\mathbb{Z}}}^+.$$ Let $t_2>t_1>0$. Then by and Proposition \[Lp-L-infty-thm-2\], there exists a constant $C>0$ such that holds. Since the constant function $b_j$ is a subsolution of in $\Omega_{\delta}\times(0,\infty)$ with $u_0=b_j$. By Lemma \[comparsion-lemma1\], $$\label{uj-uniform-lower-bd}
u_j(x,t)\ge b_j>0\quad\forall x\in\Omega_{\delta}, t\ge 0.$$ Then by and the equation is uniformly parabolic for the sequence $\{u_j\}_{j=1}^{\infty}$ on every compact subset of ${\partial}{\Omega}_{\delta}\times (0,\infty)$. By the Schauder estimates @LSU (Theorem 3.1 and Theorem 5.4 in chapter 5 of @LSU), for any compact subset $K$ of $\Omega_{\delta}\times (0,\infty)$, $$\label{1st-2nd-derivatives-infty-bd}
\sup_K(|\nabla u_j|+|{\partial}^2_{x_kx_l}u_j|+|u_{j,t}|)\le C_3\quad\forall k,l=1,\cdots,n, j\in{{\mathbb{Z}}}^+$$ and $$\label{2nd-derivatives-holder}
\sup_{(y,s), (y',s')\in  K}\frac{|{\partial}^2_{x_kx_l}u_j(y,s)-{\partial}^2_{x_kx_l}u_j(y',s')|}{|y-y'|^{\alpha}}+\sup_{(y,s), (y',s')\in  K}\frac{|u_{j,t}(y,s)-u_{j,t}(y',s')|}{|s-s'|^{\frac{\alpha}{2}}}\le C_4\quad\forall k,l=1,\cdots,n, j\in{{\mathbb{Z}}}^+$$ for some constants $C_3>0$, $C_4>0$, $0<\alpha<1$.

Hence by the Schauder estimates @LSU the sequence $\{u_j\}_{j=1}^{\infty}$ is equi-Hölder continuous in $C^{2,1}(K)$ for any compact subset $K$ of $\Omega_{\delta}\times (0,T)$. Hence by the Ascoli Theorem and a diagonalization argument the sequence $\{u_j\}_{j=1}^{\infty}$ has a subsequence which we may assume without loss of generality to be the sequence itself that converges uniformly in $C^{2,1}(K)$ for any compact subset $K\subset\Omega_{\delta}\times(0,\infty)$ as $j\to\infty$ to some function $u\in C^{2,1}(\Omega_{\delta}\times(0,\infty))$ that satisfies $$\begin{aligned}
\label{weak-soln-eqn}
&\int_{t_1}^{t_2}\int_{\Omega_{\delta}}(u\eta_t+u^m\Delta\eta)\,dxdt+\int_{t_1}^{t_2}\int_{{\partial}\Omega}f\eta\,d\sigma dt
+\sum_{i=1}^{i_0}\int_{t_1}^{t_2}\int_{{\partial}B_{\delta}(a_i)}g_i\eta\,d\sigma dt\notag\\
&\qquad \qquad =\int_{\Omega_{\delta}}u(x,t_2)\eta (x)\,dx-\int_{\Omega_{\delta}}u(x,t_1)\eta (x)\,dx\end{aligned}$$ for any $t_2>t_1>0$, and $\eta\in C^2({\overline}{\Omega}_{\delta}\times (0,\infty))$ satisfying ${\partial}\eta/{\partial}\nu =0$ on ${\partial}\Omega_{\delta}\times (0,\infty)$. Letting $j\to\infty$ in , by , , and the Lebesgue Dominated Convergence Theorem, we get that $u$ satisfies . By Lemma \[positivity-lemma\], $$\label{eq-strictly-positivity-of-limit-function-u-through-whole-domain-324}
u(x,t)>0\quad\forall\, (x,t)\in\Omega_{\delta}\times(0,\infty).$$ By an argument same as case 1, $u$ satisfies . Hence $u$ is a solution of .\
**Case 3:** $0\leq u_0\in L^p(\Omega_{\delta})$.\
For any $j\in{{\mathbb{Z}}}^+$, let $u_{0,j}(x)=\min\left(u_0(x),j\right)$. By case 2 there exists a solution $u_j$ of in $\Omega_{\delta}\times(0,T_j)$ with initial value $u_{0,j}$ that satisfies . Since $\|u_{0,j}\|_{L^{p}(\Omega_{\delta})}\le\|u_0\|_{L^{p}(\Omega_{\delta})}$, by Proposition \[Lp-L-infty-thm-2\] for any $t_2>t_1>0$ there exists a constant $C$ such that holds. Since $u_{0,j}$ increases and converges to $u_0$ as $j\to\infty$, by Lemma \[comparsion-lemma1\], $$\label{eq-increasing-sequence-of-u-sub-j-on-L-1-space03}
u_j(x,t)\le u_{j+1}(x,t)\quad\forall x\in\Omega_{\delta},0<t<T_j,j\in{{\mathbb{Z}}}^+.$$ Since $u_j>0$ in $\Omega_{\delta}\times (0,\infty)$, by and the equation for the sequence $\{u_j\}_{j=1}^{\infty}$ is uniformly parabolic on ${\overline}{\Omega}_{\delta}\times [t_1,t_2]$ for any $t_2>t_1>0$. By the Schauder estimates @LSU (Theorem 3.1 and Theorem 5.4 in chapter 5 of @LSU), for any compact subset $K$ of $\Omega_{\delta}\times (0,\infty)$, and hold for some constants $C_3>0$, $C_4>0$, $0<\alpha<1$.

Hence the sequence $\{u_j\}_{j=1}^{\infty}$ is uniformly equi-Hölder continuous in $C^{2,1}(K)$ for any compact set $K\subset\Omega_{\delta}\times (0,\infty)$. Hence by , the Ascoli theorem, and a diagonalization argument and the sequence $\{u_j\}_{j=1}^{\infty}$ increases and converges in $C^{2,1}(K)$ for any compact set $K\subset\Omega_{\delta}\times (0,\infty)$ to a solution $u\in C^{2,1}(\Omega_{\delta}\times (0,\infty))$ of in $\Omega_{\delta}\times (0,T)$. Putting $u=u_j$ and $u_0=u_{0,j}$ in , , , and letting $j\to\infty$, by we get that $u$ satisfies , and $u\in L^{\infty}({\partial}{\Omega}_{\delta}\times (0,\infty))$. It remains to show that $u$ has initial value $u_0$.\
Let $t_k\to 0$ as $k\to\infty$. By the same argument as in the Case 1, the sequence $\left\{u(x,t_k)\right\}_{k=1}^{\infty}$ has a subsequence which we may assume without loss of generality to the sequence itself such that $u(x,t_k)\to u_0$ weakly in $L^1(\Omega_{\delta})$ and a.e. $x\in\Omega_{\delta}$ as $k\to\infty$. By the proof of Proposition \[Lp-L-infty-thm-2\], holds. Hence $u(x,t_{k})$ converges weakly in $L^p(\Omega_{\delta})$ to some function $v_0$ as $k\to\infty$. Then there exists a subsequence of $\{t_{k}\}_{k=1}^{\infty}$ which we may assume without loss of generality to be the sequence $\{t_{k}\}_{k=1}^{\infty}$ such that $u(x,t_{k})$ converges to $v_0(x)$ a.e. $x\in\Omega_{\delta}$ as $k\to\infty$. Hence $v(x)=u_0(x)$ a.e. $x\in\Omega_{\delta}$. Thus $$\label{u0<u-lp-norm-at-t=0}
\int_{\Omega_{\delta}}u_0^p\,dx\le\liminf_{k\to\infty}\int_{\Omega_{\delta}}u^p(x,t_{k})\,dx.$$ Letting first $t\to 0$ and then $a\to 0$ in , $$\label{u0>u-lp-norm-at-t=0}
\limsup_{t\to 0}\int_{\Omega_{\delta}}u^p(x,t)\,dx\le\int_{\Omega_{\delta}}u_0^p\,dx.$$ By and , $$\label{u=u0-lp-norm-at-t=0}
\lim_{l\to 0}\int_{\Omega_{\delta}}u^p(x,t_{k})\,dx=\int_{\Omega_{\delta}}u_0^p\,dx.$$ Now consider the function $$w_k(x)=2^p(u^p(x,t_{k})+u_0^p(x))-|u(x,t_{k})-u_0(x)|^p.$$ Note that $w_k(x)\ge 0$ on $\Omega_{\delta}$ and $w_k(x)\to 2^{p+1}u_0^p(x)$ a.e. $x\in\Omega_{\delta}$ as $k\to\infty$. Hence by the Fatou Lemma and , $$\begin{aligned}
2^{p+1}\int_{\Omega_{\delta}}u_0^p\,dx\le&\liminf_{k\to\infty}\int_{\Omega_{\delta}}2^p(u^p(x,t_{k})+u_0^p(x))-|u(x,t_{k})-u_0(x)|^p\,dx\\
=&2^{p+1}\int_{\Omega_{\delta}}u_0^p\,dx-\limsup_{k\to\infty}\int_{\Omega_{\delta}}|u(x,t_{k})-u_0(x)|^p\,dx\\
\Rightarrow\qquad\lim_{l\to\infty}\int_{\Omega_{\delta}}|u(x,t_{k})-&u_0(x)|^p\,dx=0.\end{aligned}$$ Since the sequence $\left\{t_k\right\}_{k=1}^{\infty}$ is arbitrary, $u$ satisfies . Hence, $u$ is a solution of in $\Omega_{\delta}\times (0,\infty)$.\
If $f\equiv 0$ on ${\partial}\Omega\times (0,\infty)$ and $g_i$, $i=1,\dots, i_0$, are positive monotone decreasing functions of $t>0$, then by the choice of the approximating functions for $f$ and $g_i$ in the construction of solutions of cases 1,2,3 above and Lemma \[positivity-lemma\] $u$ satisfies in $\Omega_{\delta}\times(0,\infty)$ and the theorem follows.

By the same argument as the proof of Theorem \[Neumann-holes-problem-existence-thm\] and Lemma \[comparsion-lemma1\] we have the following two results.

\[Neumann-problem-bded-domain-thm\] Let $n\ge 3$, $0<m\le\frac{n-2}{n}$, $0\le u_0\in L^p(\Omega)$ for some constant $p>\frac{n(1-m)}{2}$, and $0\le f\in L_{loc}^{\infty}(\partial\Omega\times [0,\infty))$. Suppose either $u_0\not\equiv 0$ on $\Omega_{\delta}$ or $$\int_0^t\int_{\partial\Omega}f\,d\sigma ds>0\quad \forall t>0$$ holds. Then there exists a unique solution $u$ for the Neumann problem, $$\begin{cases}
\begin{aligned}
u_t=&{\triangle}u^m \quad\mbox{in $\Omega\times(0,\infty)$}\\
\frac{\partial u^m}{\partial\nu}=&f \qquad \mbox{ on }\partial\Omega\times(0,\infty)\\
u(x,0)=&u_0(x)\quad\mbox{in }\Omega
\end{aligned}
\end{cases}$$ which satisfies $$\int_{\Omega}u(x,t)\,dx=\int_{\Omega}u_0\,dx+\int_0^t\int_{\partial\Omega}f\,d\sigma ds\quad \forall t>0.$$ Moreover if $f\equiv 0$ on ${\partial}\Omega\times (0,\infty)$, then $u$ satisfies in $\Omega_{\delta}\times(0,T)$.

\[existence-radially-symmetric-soln-Neumann-with-1-hole-cor\] Let $n\ge 3$, $0<\delta<R$, $0<m\leq\frac{n-2}{n}$, $p>\frac{n(1-m)}{2}$ and let $f,g\in L_{loc}^{\infty}([0,\infty))$ be two nonnegative functions. Suppose $0\le u_0\in L^p(B_R{\backslash}B_{\delta})$ is a radially symmetric function such that either $u_0\not\equiv 0$ on $\Omega_{\delta}$ or $$\int_0^t\int_{\partial B_R}f\,d\sigma ds+\int_0^t\int_{\partial B_{\delta}}g\,d\sigma ds>0\quad \forall t>0$$ holds. Then there exists a unique solution of $$\left\{\begin{aligned}
u_t=&{\triangle}u^m\quad\mbox{in }(B_R{\backslash}B_{\delta})\times(0,\infty)\\
\frac{\partial u^m}{\partial\nu}=&f\qquad\mbox{ on }\partial B_R\times(0,\infty)\\
\frac{\partial u^m}{\partial\nu}=&g\qquad\mbox{ on }\partial B_{\delta}\times(0,\infty)\\
u(x,0)=&u_0(x)\quad\mbox{in }B_R
\end{aligned}\right.$$ in $(B_R{\backslash}B_{\delta})\times(0,\infty)$ which is radially symmetric and satisfies with $\Omega=B_{R}$, $i_0=1$ and $a_1=0$.

Existence of singular solutions
===============================

In this section we will prove the existence of singular solutions of in $\hat{\Omega}\times (0,T)$.

\[Proof of Theorem \[singular-soln-existence-thm\]\]

Let $\alpha=2m(q+4\delta_1^{-2})$ and let $$0<{\varepsilon}_j<\min\left(\frac{\delta_1}{2},\frac{(1-m)^2q^2}{(4+(1-m)q)^2},\frac{(1-m)q\delta_0}{4+(1-m)q}\right)\quad\forall j\in{{\mathbb{Z}}}^+.$$ be a sequence decreasing to zero as $j\to\infty$. By Theorem \[Neumann-holes-problem-existence-thm\] for any $j\in{{\mathbb{Z}}}^+$ there exists a solution $u_j$ of $$\label{Neumann-problem-in-punctured-ball}
\left\{\begin{aligned}
u_t=&\Delta u^m\qquad\mbox{in }\Omega_{\delta}\times(0,\infty)\\
\frac{\partial u^m}{\partial\nu}=&f\qquad\quad\mbox{ on }\partial\Omega\times(0,\infty)\\
\frac{\partial u^m}{\partial\nu}=&\frac{\alpha}{{\varepsilon}_j^{qm+1}}\quad\,\,\mbox{ on }\partial B_{{\varepsilon}_j}(a_i)\times(0,\infty)\quad\forall
i=1,\cdots, i_0\\
u(x,0)=&u_0(x)\qquad\mbox{in }B_R
\end{aligned}\right.$$ By Lemma \[soln-bded-holes-lemma\] there exists a constant $0<\delta_2<\delta_1$ such that for any $T>0$ there exists a constant $A_1>0$ such that $$\label{uj-upper-bd-phi-tidle}
u_j(x,t)\le\phi_{A_1}(x-a_i,t) \qquad \forall{\varepsilon}_j\le |x-a_i|<\delta_1,\,\,0\leq t<T, i=1,\cdots,i_0,{\varepsilon}_j<\delta_2$$ where $\phi_{A_1}$ is given by . By Lemma \[comparison-lem2\], $$\label{uj-lower-bd-at-blow-up-pt}
u_j(x,t)\ge\frac{C_1}{|x-a_i|^qe^{\frac{1}{\delta_1^2-|x-a_i|^2}}}\quad\forall {\varepsilon}_j\le |x-a_i|<\delta_1,t>0, {\varepsilon}_j\le\frac{\delta_1}{2}, i=1,\cdots,i_0$$ holds for any $0<\delta\le\delta_1/2$. By Proposition \[Lp-L-infty-thm-3\] for any $t_2>t_1>0$ and $\delta'<\delta_1$ there exists a constant $M_{\delta', t_1,t_2}>0$ such that $$\label{uj-upper-bd5}
u_j(x,t)\le M_{\delta', t_1,t_2}\quad\forall x\in\Omega_{\delta'}, t_1\le t\le t_2, {\varepsilon}_j<\frac{\delta'}{2}.$$ By and Theorem 1.1 of @S the sequence $\{u_j\}_{{\varepsilon}_j<\frac{\delta'}{2}}$ is equi-Hölder continuous on $\Omega_{\delta'}\times [t_1,t_2]$ for any $0<\delta'<\delta_1$ and $t_2>t_1>0$. By the Ascoli Theorem and a diagonalization argument the sequence $\{u_{j}\}_{j=1}^{\infty}$ has a subsequence which we may assume without loss of generality to be the sequence itself that converges uniformly on every compact subset of $\hat{\Omega}\times(0,\infty)$ to some continuous function $u$ that satisfies in $\mathcal{D}(\hat{\Omega}\times (0,\infty))$ as $j\to\infty$. Letting $j\to\infty$ in , , and , we get , $$\label{u-upper-bd-phi-tidle2}
u(x,t)\le\phi_{A_1}(x-a_i,t) \qquad \forall 0< |x-a_i|<\delta_1,\,\,0\leq t<T, i=1,\cdots,i_0$$ and $$\label{u-upper-bd5}
u(x,t)\le M_{\delta', t_1,t_2}\quad\forall x\in\Omega_{\delta'}, t_1\le t\le t_2.$$ By , follows. By and Lemma \[positivity-lemma\], $$u>0\quad\mbox{ in }\hat{\Omega}\times (0,\infty).$$ Hence for any $t_2>t_1>0$ and $0<\delta'<\delta_0$, there exists a constant $M'>0$ such that $$\label{u-lower-bd5}
u\ge M'\quad\mbox{ in }\Omega_{\delta'}\times (t_1,t_2).$$ By and the equation for $u$ is uniformly parabolic on $\Omega_{\delta'}\times (t_1,t_2)$ for any $t_2>t_1>0$ and $0<\delta'<\delta_0$. Hence by the Schauder estimates @LSU (Theorem 3.1 and Theorem 5.4 in chapter 5 of @LSU), $u\in C^{2+\beta,1+(\beta/2)}(\hat{\Omega}\times (0,\infty))$ for some constant $0<\beta<1$ is a classical solution of in $\hat{\Omega}\times (0,\infty)$. Putting $u=u_j$ in and letting $j\to\infty$, we get that $u$ satisfies for any $0<t_1<t_2<T$ and $\eta\in C_0^2((\overline{\Omega}\setminus\{a_1,\dots,a_{i_0}\})\times(0,T))$ satisfying ${\partial}\eta/{\partial}\nu =0$ on $\partial\Omega\times (0,T)$.

Let $\eta\in C_0^{\infty}(\hat{\Omega})$ and $K=\mbox{supp}\,\eta$. Then ${\varepsilon}=\mbox{dist}(K,{\partial}\Omega\cup\{0\})>0$. Let $\psi\in C_0^{\infty}(\hat{\Omega})$, $0\le\psi\le 1$, be such that $\psi (x)=1$ for all $x\in K$ and $\psi (x)=0$ for all $\mbox{dist}(x,K)\ge{\varepsilon}/2$. Let $K_1=\{x\in\Omega:\mbox{dist}(x,K)\le{\varepsilon}/2\}$. By the proof of Lemma 3.1 of @HP there exist constants $\alpha>1$ and $C>0$ such that $$\begin{aligned}
\label{uj-L1-uniform-bd}
&\left|\frac{{\partial}}{{\partial}t}\left(\int_{\Omega_{{\varepsilon}_j}}u_j\psi^{\alpha}\,dx\right)\right|\le C\left(\int_{\Omega_{{\varepsilon}_j}}u_j\psi^{\alpha}\,dx\right)^m\quad\forall t>0,{\varepsilon}_j<{\varepsilon}/2\notag\\ 
\Rightarrow\quad&\left(\int_{\Omega_{{\varepsilon}_j}}u_j(x,t)\psi^{\alpha}(x)\,dx\right)^{1-m}\le\left(\int_{\Omega_{{\varepsilon}_j}}u_0\psi^{\alpha}\,dx\right)^{1-m}+(1-m)Ct\qquad \forall t>0,{\varepsilon}_j<{\varepsilon}/2\notag\\
\Rightarrow\quad&\int_Ku_j(x,t)\,dx\le c_{K_1}\qquad \forall 0<t<1,{\varepsilon}_j<{\varepsilon}/2.\end{aligned}$$ where $$c_{K_1}=\left(\left(\int_{K_1}u_0\,dx\right)^{1-m}+(1-m)C\right)^{\frac{1}{1-m}}$$ Hence by , $$\begin{aligned}
\label{uj-u0-L1-difference}
\left|\int_{\Omega_{{\varepsilon}_j}}u_j\eta\,dx-\int_{\Omega_{{\varepsilon}_j}}u_0\eta\,dx\right|\le&\int_0^t\int_{\Omega_{{\varepsilon}_j}}u_j^m|\Delta\eta|\,dx\,dt\notag\\
\le&\|\Delta\eta\|_{L^{\infty}}|K|^{1-m}t\left(\int_Ku_j(x,t)\,dx\right)^m\notag\\
\le&\|\Delta\eta\|_{L^{\infty}}|K|^{1-m}c_{K_1}^mt\quad\forall 0<t<1,{\varepsilon}_j<{\varepsilon}/2.\end{aligned}$$ Letting $j\to\infty$ in , $$\label{u-u0-L1-difference}
\left|\int_{\hat{\Omega}}u\eta\,dx-\int_{\hat{\Omega}}u_0\eta\,dx\right|\le\|\Delta\eta\|_{L^{\infty}}|K|^{1-m}c_{K_1}^mt\quad\forall 0<t<1.$$ Letting $t\to 0$ in we get and Theorem \[singular-soln-existence-thm\] follows.

We are now ready for the proof of Theorem \[singular-soln-existence-thm2\].

\[Proof of Theorem \[singular-soln-existence-thm2\]\] Let $R_0>3\max_{1\le i\le i_0}|a_i|$. Then for any integer $j>R_0$ by Theorem \[singular-soln-existence-thm\] there exists a solution $u_j$ of $$\left\{\begin{aligned}
u_t=&\Delta u^m\quad\mbox{ in }\hat{B}_j\times(0,\infty)\\
\frac{\partial u^m}{\partial\nu}=&0\qquad\, \mbox{ on }\partial B_j\times(0,\infty)\\
u(x,0)=&u_0(x)\quad\mbox{ in }\hat{B}_j
\end{aligned}\right.$$ which satisfies $$\label{uj-uniform-lower-blow-up-rate}
u_j(x,t)\ge\frac{C_1}{|x-a_i|^qe^{\frac{1}{\delta_1^2-|x-a_i|^2}}}\quad\forall 0<|x-a_i|<\delta_1,t>0, i=1,\cdots,i_0$$ and for any $T>0$ there exists a constant $A_1>0$ such that $$\label{uj-uniform-upper-blow-up-rate}
u_j(x,t)\le\phi_{A_1}(x-a_i,t) \qquad \forall 0< |x-a_i|<\delta_1,\,\,0\leq t<T, i=1,\cdots,i_0,$$ where $\phi_{A_1}$ is given by . By , and the same argument as the proof of Theorem \[singular-soln-existence-thm\] the sequence $\{u_j\}_{j>R_0}$ has a subsequence that converges to a $C^{2,1}$ solution $u$ of that satisfies and for some constant $C_T>0$. This completes the proof of Theorem \[singular-soln-existence-thm2\].

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