\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 136, pp. 1--16.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2016 Texas State University.} \vspace{7mm}} \begin{document} \title[\hfilneg EJDE-2016/136\hfil Quenching phenomenon] {Quenching phenomenon of singular parabolic problems with $L^1$ initial data } \author[A. N. Dao, J. I. D\'iaz, P. Sauvy \hfil EJDE-2016/136\hfilneg] {Anh Nguyen Dao, Jesus Ildefonso D\'iaz, Paul Sauvy} \address{Anh Nguyen Dao (corresponding author)\newline Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam} \email{daonguyenanh@tdt.edu.vn} \address{Jesus Ildefonso D\'iaz \newline Instituto de Matem\'atica Interdisciplinar, Universidad Complutense de Madrid, 28040 Madrid, Spain} \email{ildefonso.diaz@mat.ucm.es} \address{Paul Sauvy \newline Institut Math\`ematique de Toulouse, Universit\'e Toulouse 1, 31000 Toulouse France} \email{paul.sauvy@ut-capitole.fr} \thanks{Submitted January 15, 2016. Published June 8, 2016.} \subjclass[2010]{35K55, 35K67, 35K65} \keywords{Quenching type parabolic equations; $L^1$-initial datum; free boundary} \begin{abstract} We extend some previous existence results for quenching type parabolic problems involving a negative power of the unknown in the equation to the case of merely integrable initial data. We show that $L^1(\Omega)$ is the suitable framework to obtain the continuous dependence with respect to some norm of the initial datum. This way we answer to the question raised by several authors in the previous literature. We also show the complete quenching phenomena for such a $L^1$-initial datum. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} The main purpose of this paper is to study the existence of nonnegative mild solution and the ``quenching phenomenon'' of the singular parabolic equation \begin{equation} \begin{gathered} \partial_tu-\Delta u+\chi_{\{u>0\}}u^{-\beta}=0 \quad \text{in }\Omega\times(0,T),\\ u=0 \quad \text{on }\partial\Omega\times(0,T),\\ u{(\cdot,0)}=u_0(\cdot) \quad \text{on }\Omega, \end{gathered} \label{1.1} \end{equation} where $\beta\in(0,1)$, $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$, $0\leq u_0$ and $\chi_{\{u>0\}}$ denotes the characteristic function of the set of points $(x,t)$ where $u(x,t)>0$. Parabolic equations involving as zero order term a negative exponent of the unknown are quite common in the literature since 1960. The pioneering paper by Fulks and Maybee \cite{Fulks Maybee} was motivated by the study of the heat conduction in an electric medium but in the modelling the singular term was of a sourcing nature and so in the right hand side of the equation: the differences {between} the behavior of solutions of such model with respect {to} our problem \eqref{1.1} are today well-known. Perhaps, one of the first papers dealing with the equation of \eqref{1.1} was \cite{Ka} in the study of Electric Current Transient in Polarized Ionic Conductors (in fact for $\beta=1$). The literature on this type of problems increased then very quickly and models arising in other contexts were mentioned by different authors, specially when regarding the equation \eqref{1.1} as the limit case of models in chemical catalyst kinetics (Langmuir-Hinshelwood model) or of models in enzyme kinetics (see {\cite{D Pitman,DMO}} for the elliptic case and {\cite{Bandle Brauner,Philips}} for the parabolic equation). See also many references in the survey \cite{Hernandez-Mancebo survey} and the monograph \cite{Gher-Rad}). Obviously, what makes specially interesting equations like \eqref{1.1} is the fact that the solutions may raise to a free boundary defined as $\partial \{(x,t)$: $u(x,t)>0\}$ (see e.g. \cite{D1} and its references). In many contexts the boundary conditions are not zero but, for instance $u=1$ and thus the terminology of quenching problem was used in the literature to denote the appearance of blow-up result on $\partial_tu$ for the first time in which $u=0$ (see, e.g., \cite{Ka,Levine,Philips}). In spite of such a long list of references, most of the theory in the literature deals with bounded (quite often even assumed continuous) initial data. We must add that even so, it is today well-known that the uniqueness of solution fails (see \cite{Winkler-non uniquenes}), except for the case in which there is not a free boundary, see (\cite{Davila-Montenegro survey}). The main purpose of this work is to deal with initial data satisfying merely \begin{equation*} 0\leq u_0\in L^1(\Omega). \end{equation*} Let us introduce the notion of solution we shall use in this paper: \begin{definition} \label{def1} \rm A function $u\in\mathcal{C}([0,T);L^1(\Omega))$ is called a mild solution of \eqref{1.1} if $\chi_{\{u>0\}}u^{-\beta}\in L^1(\Omega\times(0,T))$ and $u$ fulfills \begin{equation} u(\cdot,t)=S(t)u_0(\cdot)-\int_0^tS(t-s)\chi_{\{u>0\}}u^{-\beta} (\cdot,s)ds,\quad\text{in }L^1(\Omega), \label{1.2} \end{equation} where $S(t)$ is the semigroup corresponding to the Laplace operator with homogeneous Dirichlet boundary conditions. \end{definition} We recall that the $L^1(\Omega)$-semigroup $S(t)$ corresponding to the Laplace operator with homogeneous Dirichlet boundary conditions was considered by many authors since the seventies (or even earlier of the past century and that the associated weak solutions $S(t)u_0$ can be characterized by multiplying by suitable test functions (see, e.g., \cite{BaPi,BoGa,BS} and the exposition made in Chapter 4 of \cite{D Pitman}). In particular, we know that any mild solution $u$ belongs to the space $L^s(0,T;W_0^{1,s}(\Omega))$, for any $s\in(1,\frac{N+2}{N+1})$, and satisfies \begin{align*} &\int_{\Omega}u(x,t)\psi(x,t)dx+\int_0^t\int_{\Omega}\nabla u(x,s)\cdot \nabla\psi(x,s)\,dx\,ds \\ &+\int_0^t\int_{\Omega}\chi_{\{u>0\}}u^{-\beta}(x,s)\psi(x,s)\,dx\,ds\\ &=\int_0^t\int_{\Omega}u(x,s)\partial_t\psi(x,s)\,dx\,ds +\int_{\Omega}u_0 (x)\psi(x,0)dx, \end{align*} for any test function \ $\psi\in W^{1,\infty}(0,T;L^1(\Omega))\cap L^{\infty}(0,T;W_0^{1,\infty}(\Omega))$, and for every $t\in(0,T)$. The main results of this article are the following: \begin{theorem}\label{thm1} Let $0\leq u_0\in L^1(\Omega)$. Then, there exists the a maximal nonnegative mild solution $u$ of \eqref{1.1} in $\Omega\times(0,\infty)$, i.e., for any other mild solution $v$ of \eqref{1.1} we have $0\leq v\leq u$ in $\Omega\times[0,\infty)$. \end{theorem} Concerning the quenching phenomenon, we recall that since there is lack of uniqueness of solutions, it seems to be difficult to apply, directly, super and sub-solutions methods to study it. Our approach is to use the energy methods, but with the new fact that our initial datum does not need to be in the natural energy space defined over $L^2(\Omega)$. \begin{theorem}\label{thm2} Let $0\leq u_0\in L^1(\Omega)$. Then, if $v$ is any nonnegative mild solution of \eqref{1.1}, there exists a finite time $T^{\ast}>0$ such that \[ v(x,t)=0, \quad\text{for a.e. } (x,t)\in \Omega\times(T^\ast,\infty). \] Moreover, $T^{\ast}$ only depends on $\| u_0\|_{L^1(\Omega)}$, $N$ and $|\Omega|$. \end{theorem} This article is organized as follows: Section 2 is devoted to the proof of Theorem \ref{thm1}. In Section $3$, we will consider the quenching phenomenon. We also prove the uniqueness result under additional assumption. \section{Proof of Theorem \ref{thm1}} We shall follow a scheme of approximation similar to the one used in \cite{Winkler-non uniquenes}. We start by considering the problem \begin{equation} \begin{gathered} \partial_tu_{\varepsilon}-\Delta u_{\varepsilon}+g_{\varepsilon }(u_{\varepsilon})=0 \quad \text{in }\Omega\times(0,\infty),\\ u_{\varepsilon}=0 \quad \text{on } \partial\Omega\times(0,\infty),\\ u_{\varepsilon}{(\cdot,0)}=u_0(\cdot) \quad \text{on }\Omega \end{gathered} \label{2.2} \end{equation} with \[ g_{\varepsilon}(s)=\begin{cases} 0 & \text{if} s\leq0,\\ \psi_{\varepsilon}(s)s^{-\beta} & \text{if }s>0. \end{cases} \] where $\psi_{\varepsilon}(s)=\psi({\frac{s}{\varepsilon}})$ and $\psi \in{\mathcal{C}^{\infty}(\mathbb{R})}$ is a non-decreasing function on $\mathbb{R}$ such that $\psi(s)=0$ for $s\leq1$, $\psi(s)=1$ for $s\geq2$. The main idea of the proof is to pass to the limit in equation \eqref{2.2} as $\varepsilon\to 0$ to obtain a solution of \eqref{1.1}, which is the maximal solution. First of all, we observe that for any fixed $\varepsilon>0$, $g_{\varepsilon}$ is a global Lipschitz function. Then, we have the following result. \begin{theorem} \label{thm2.0} There exists a unique nonnegative mild solution to problem \eqref{2.2}, $u_{\varepsilon}\in \mathcal{C}([0,+\infty);L^1(\Omega))$; i.e. satisfying that for any $t>0$, \begin{equation} u_{\varepsilon}(t)=S(t)u_0-\int_0^tS(t-s)g_{\varepsilon }(u_{\varepsilon}(s))ds. \label{2.3} \end{equation} Moreover, for any $0<\tauk, \end{cases} \] and its primitive integral \[ S_k(u):=\int_0^{u}T_k(s)ds=\frac{1}{2}|u|^2\chi_{\{|u|0, \\ y(0)=0. \end{gather*} Gronwall's inequality implies $y(t)=0$, and so $v_{+} (t)=0$. This completes the proof. \end{proof} Next, we shall show the existence of solution of \eqref{1.1} by passing to the limit as $\varepsilon\to 0$. \begin{theorem}\label{thm2.1} The sequence $\{u_\varepsilon\}_{\varepsilon>0}$ is nondecreasing, so $u_{\varepsilon}$ converges to a function $u$ in $L^{r}(0,T;W_0^{1,r}(\Omega))$, which is a solution of \eqref{1.1}, for $r\in(1,\frac{N+2}{N+1})$. Furthermore, $u$ is a mild solution of \eqref{1.1}. \end{theorem} \begin{proof} It follows from \eqref{2.3} that for any $t>0$, \begin{equation} 0\leq u_{\varepsilon}(x,t)\leq S(t)u_0(x)\leq Ct^{-\frac{N}{2}}\| u_0\|_{L^1(\Omega)}.\label{2.4} \end{equation} The constant $C$ in \eqref{2.4} merely depends on $N,|\Omega|$, see \cite{BeDa,CazHar}. Then $u_{\varepsilon}$ is bounded locally in time. For any $0<\tauN+1$. Combining \eqref{2.5} and \eqref{2.6} we obtain \begin{equation} \| u_{\varepsilon}\|_{L^{r}(0,T;W_0^{1,r}(\Omega))}\leq C(r,T,\Omega )\| u_0\|_{L^1(\Omega)},\label{2.7} \end{equation} with $r=s\in[1,\frac{N+2}{N+1})$. Thus, for any $r\in(1,\frac{N+2}{N+1})$, $\{\partial_tu_{\varepsilon}\}_{\varepsilon>0}$ is bounded in $L^1(0,T;W^{-1,r'}(\Omega))+L^1\left( \Omega\times(0,T)\right)$ by a constant independent of $\varepsilon$. Then, the sequence $\{u_{\varepsilon}\}_{\varepsilon}$ is relatively compact in $L^1(\Omega\times(0,T))$ (see \cite{Si}) and there is a subsequence of $\{u_{\varepsilon}\}_{\varepsilon}$ (still denoted as $\{u_{\varepsilon}\}_{\varepsilon}$) such that \begin{equation} u_{\varepsilon} \to u,\quad\text{in}\quad L^1(\Omega \times(0,T)).\label{2.8} \end{equation} Next, we claim that \begin{equation} u_{\varepsilon}(x,t)\downarrow u(x,t),\quad\text{for a.e. }(x,t)\in\Omega \times(0,T).\label{2.8a} \end{equation} It is sufficient to show that $\{u_{\varepsilon}\}_{\varepsilon}$ is a non-decreasing sequence. Indeed, for any $\varepsilon>\varepsilon'>0$, we have \[ g_{\varepsilon}(s)\leq g_{\varepsilon'}(s),\quad \forall s\in\mathbb{R}. \] Then \[ \partial_tu_{\varepsilon}-\Delta u_{\varepsilon}+g_{\varepsilon' }(u_{\varepsilon})\geq\partial_tu_{\varepsilon}-\Delta u_{\varepsilon }+g_{\varepsilon}(u_{\varepsilon})=0. \] This implies that $u_{\varepsilon}$ is a super-solution of the equation satisfied by $u_{\varepsilon'}$. Thanks to Lemma \ref{lemmaunique}, we obtain $u_{\varepsilon}(x,t)\geq u_{\varepsilon'}(x,t)$, for a.e. $(x,t)\in\Omega\times(0,T)$, thereby we obtain the claim \eqref{2.8a}. Next, we shall show the convergence of the gradients. Let us first demonstrate that \begin{equation} \nabla u_{\varepsilon}\xrightarrow[\varepsilon\to 0]{}\nabla u_{{}} \quad\text{in } L^1(\Omega\times(0,T)),\label{2.9} \end{equation} For any $\varepsilon,\varepsilon'>0$, we consider function $v_{\varepsilon,\varepsilon'}=u_{\varepsilon}-u_{\varepsilon' }$, and the difference between the equations satisfied by $u_{\varepsilon}$ and $u_{\varepsilon'}$ \begin{equation} \partial_tv_{\varepsilon,\varepsilon'}-\Delta v_{\varepsilon ,\varepsilon'}+g_{\varepsilon}(u_{\varepsilon})-g_{\varepsilon '}(u_{\varepsilon'})=0.\label{2.10} \end{equation} For any $\delta>0$, and any $00$, so we obtain \eqref{2.9}. As a consequence of \eqref{2.9}, there is a sub-sequence of $\{u_\varepsilon\}_{\varepsilon>0}$ such that \begin{equation}\label{2.9a} \nabla u_\varepsilon \to \nabla u, \quad\text{for a.e } (x,t)\in \Omega\times(0,\infty). \end{equation} Let us show now a sharper convergence: for any $r\in(1,\frac{N+2}{N+1})$, \begin{equation} u_{\varepsilon}\to u,\quad\text{in } L^{r}(0,T_0;W_0^{1,r}(\Omega)).\label{2.18} \end{equation} Indeed, conclusion \eqref{2.18} just follows from \eqref{2.7}, \eqref{2.8}, \eqref{2.9} and Vitali's theorem. Next, we show that there is a subsequence of $\{g_{\varepsilon}(u_{\varepsilon})\}_{\varepsilon>0}$ such that \begin{equation} g_{\varepsilon}(u_{\varepsilon}) \to u^{-\beta}\chi_{\{u>0\}},\quad\text{in } L^1(\Omega\times(0,T_0)).\label{2.19} \end{equation} More precisely, we claim that the above subsequence satisfies, from Fatou's lemma, that \begin{equation} \liminf_{\varepsilon\to 0}g_{\varepsilon}(u_{\varepsilon})=u^{-\beta }\chi_{\{u>0\}},\quad\text{in }L^1(\Omega\times(0,T_0)),\label{2.24} \end{equation} this conclusion allows us to obtain \begin{equation} u\in\mathcal{C}([0,T_0];L^1(\Omega)).\label{2.20} \end{equation} Let us skip the proof of \eqref{2.19} (or \eqref{2.24}) at the moment and we will show \eqref{2.20} if \eqref{2.19} holds. For any $00$ such that for any $\tau>0$ fixed, we have \begin{equation}\label{2.23c} |\nabla u(x,t)|\leq Cu^{\frac{1-\beta}{2}}\Big( 1+\big( \tau^{-\frac{N}{2}}\| u_0\|_{L^1(\Omega)}\big) ^{\frac{\beta +1}{2}}\Big) \Big( 1+(t-\tau)^{-\frac{1}{2}}+{d(x)}^{-1}\Big), \end{equation} with $C=C(N,|\Omega|,\beta)>0$, and $d(x)=\inf_{y\in\partial\Omega}\{\| x-y\|\}$, the distance from $x$ to the boundary of the domain $\Omega$. \end{lemma} In fact, we observe that, for any $\tau>0$, ${u_{\varepsilon}(\tau)}\in\mathcal{C}_0(\Omega)$. Then, we apply \cite[Lemma 3.3]{Winkler-non uniquenes} to $u_\varepsilon$, by considering ${u_{\varepsilon}(\tau)}$ as the initial condition instead of $u_\varepsilon(0)$ to obtain \begin{equation} |\nabla u_{\varepsilon}(x,t)|\leq C(\beta)u_{\varepsilon}^{\frac{1-\beta}{2} }\Big( 1+{\| u_{\varepsilon}(\tau)^{\frac{\beta+1}{2}}\| _{L^{\infty}(\Omega)}}\Big) \Big( (t-\tau)^{-\frac{1}{2}}+{d(x)}^{-1}\Big). \label{2.23b} \end{equation} Combining \eqref{2.4} and \eqref{2.23b} deduce that there is a positive constant $C=C(N, \beta, |\Omega|)$ such that \begin{equation} |\nabla u_{\varepsilon}(x,t)| \leq C u_{\varepsilon}^{\frac{1-\beta}{2}} \Big( 1+\big( \tau^{-\frac{N}{2}}\| u_0 \|_{L^1(\Omega)}\big) ^{\frac{\beta+1}{2}}\Big) \Big( 1+(t-\tau)^{-\frac{1}{2}}+{d(x)}^{-1}\big) .\label{2.23bb} \end{equation} Now let $\varepsilon\to 0$ in \eqref{2.23bb}. Then, inequality \eqref{2.23c} follows from \eqref{2.9a}, and the monotonicity of $u_\varepsilon$. This implies \[ \nabla u_{\varepsilon }\to \nabla u, \quad\text{in } L_{ loc}^{q}(\Omega\times(0,+\infty)),\; \forall q\in(1, \infty). \] Now, it remains to show claim \eqref{2.24}. Indeed, using \eqref{2.5} and Fatou's lemma asserts that for any $T_0\in(0,\infty)$, there is a non-negative function $\Phi\in L^1(\Omega\times(0,T_0))$ such that \begin{equation} \liminf_{\varepsilon\to 0}g_{\varepsilon}(u_{\varepsilon})=\Phi ,\quad\text{in }L^1(\Omega\times(0,T_0)).\label{2.25} \end{equation} Furthermore, we observe that \[ g_{\varepsilon}(u_{\varepsilon})(x,t)\geq g_{\varepsilon}(u_{\varepsilon} )\chi_{\{u>0\}}(x,t),\quad\text{for a.e. }(x,t)\in\Omega\times(0,T_0), \] which implies \begin{equation} \liminf_{\varepsilon\to 0}g_{\varepsilon}(u_{\varepsilon})(x,t)\geq u^{-\beta}\chi_{\{u>0\}}(x,t),\quad\text{for a.e. }(x,t)\in\Omega \times(0,T_0).\label{2.26} \end{equation} It follows from the Lebesgue dominated convergence theorem that \begin{equation} u^{-\beta}\chi_{\{u>0\}}\leq\Phi\quad\text{and}\quad u^{-\beta}\chi_{\{u>0\}}\in L^1(\Omega\times(0,T_0)).\label{2.27} \end{equation} For any $\eta>0$ fixed, we use the test function $\psi_{\eta}(u_{\varepsilon})\phi$, $\phi\in\mathcal{C}_c^{\infty}(\Omega\times(0,T_0))$ to the equation satisfied by $u_{\varepsilon}$. Then, integration by parts gives us \begin{align*} &\int_{\operatorname{supp}\phi)}\Big(-\Psi_{\eta}(u_{\varepsilon}) \partial_t\phi+\frac{1}{\eta}|\nabla u_{\varepsilon}|^2\psi{'}\big( \frac{u_{\varepsilon}}{\eta}\big) \phi \\ &+ \nabla u_{\varepsilon} \cdot \nabla\phi \psi_{\eta}(u_{\varepsilon}) + g_{\varepsilon}(u_{\varepsilon})\psi_{\eta }(u_{\varepsilon})\phi\quad\Big) \,dx\,ds=0, \end{align*} where \[ \Psi_{\eta}(u)=\int_0^{u}\psi_{\eta}(s)ds. \] By \eqref{2.8a} and \eqref{2.23c}, we can pass to the limit as $\varepsilon \to 0$ in the above inequality to obtain \begin{equation} \begin{aligned} &\int_{\operatorname{supp}\phi)}\Big(-\Psi_{\eta}(u)\partial_t\phi +\frac{1}{\eta}|\nabla u|^2 \psi'\big(\frac{u}{\eta}\big) \phi\\ &+\nabla u\cdot\nabla \phi\psi_{\eta}(u) +u^{-\beta}\psi_{\eta}(u)\phi\Big)\,dx\,ds=0, \end{aligned}\label{2.28} \end{equation} From \eqref{2.23c}, \eqref{2.27}, and the dominated convergence theorem, it is not difficult to verify that \begin{equation}\label{2.29} \begin{split} &\lim_{\eta\to 0}\int_{\operatorname{supp}\phi)} \Big(-\Psi_{\eta}(u)\partial_t\phi+\nabla u\cdot\nabla\phi \psi_{\eta}(u) +u^{-\beta}\psi_{\eta}(u)\phi\Big) \,dx\,ds\\ &=\int_{\operatorname{supp}\phi)} \Big(-u \partial_t\phi+\nabla u\cdot\nabla\phi+u^{-\beta}\chi _{\{u>0\}}\phi\Big)\,dx\,ds, \end{split} \end{equation} with any term of the left-hand side converges to any term of the right-hand side in order. On the other hand, it follows from \eqref{2.23c} that \begin{align*} \frac{1}{\eta}\int_{\operatorname{supp}\phi)}|\nabla u|^2{\left\vert \psi{'}\big(\frac{u}{\eta}\big) \phi\right\vert }\,dx\,ds & \leq C(\phi)\frac{1}{\eta}\int_{\operatorname{supp}\phi)\cap\{\eta0\}}\phi\right) \,dx\,ds=0.\label{2.31} \end{equation} Note that \eqref{2.31} says that $u$ is a weak solution of \eqref{1.1} in $\Omega\times(0, T_0)$. However, this is not sufficient to conclude that $u$ is a mild solution of \eqref{1.1}. Next, since $u_{\varepsilon}$ is a weak solution of \eqref{2.2}, we have \[ \int_{\operatorname{supp}\phi)} \Big( -u_{\varepsilon}\partial_t\phi+\nabla u_{\varepsilon} \cdot\nabla\phi+g_{\varepsilon}(u_{\varepsilon})\phi\Big) \,dx\,ds=0. \] The passage to the limit as $\varepsilon\to 0$ provides us \begin{equation} \int_{\operatorname{supp}\phi)}\big( -u\partial_t \phi+\nabla u\cdot\nabla\phi\big) \,dx\,ds+\lim_{\varepsilon\to 0}\int_{\operatorname{supp}\phi)}g_{\varepsilon }(u_{\varepsilon})\phi\;\,dx\,ds=0.\label{2.32} \end{equation} By \eqref{2.31} and \eqref{2.32}, we obtain \begin{equation} \lim_{\varepsilon\to 0}\int_0^{\infty}\int_{\Omega}g_{\varepsilon }(u_{\varepsilon})\phi \,dx\,ds =\int_0^{\infty}\int_{\Omega}u^{-\beta} \chi_{\{u>0\}}\phi\;\,dx\,ds.\label{2.33} \end{equation} Thanks to Fatou's lemma, \eqref{2.25} and \eqref{2.33}, we obtain for any non-negative $\phi\in\mathcal{C}_c^{\infty}(\Omega\times(0,\infty))$, \[ \int_0^{\infty}\int_{\Omega}u^{-\beta}\chi_{\{u>0\}}\phi \,dx\,ds\geq\int _0^{\infty}\int_{\Omega}\Phi\phi \,dx\,ds. \] We deduce from the last inequality and \eqref{2.27} that \[ u^{-\beta}\chi_{\{u>0\}}=\Phi,\quad\text{a.e. in }\Omega\times(0,\infty). \] In other words, we obtain the claim \eqref{2.24}. Since $u_\varepsilon$ is a mild solution of equation \eqref{2.2}, we have \begin{equation} u_{\varepsilon}(t)=S(t)u_0-\int_0^tS(t-s)g_{\varepsilon}\big(u_{\varepsilon }(s)\big) ds.\label{2.34} \end{equation} By \eqref{2.24}, we can pass to the limit as $\varepsilon\to 0$ in \eqref{2.34} to obtain \[ u(t)=S(t)u_0-\int_0^tS(t-s)u^{-\beta}\chi_{\{u>0\}}(s) ds, \] or $u$ is a mild solution of equation \eqref{1.1}. Finally, we prove that the solution $u$ constructed above is the maximal solution of \eqref{1.1}. \begin{proposition}\label{pro2.2} Let $v$ be any mild solution of equation \eqref{1.1}. Then, we have \[ v\leq u,\quad\text{in }\Omega\times(0,\infty). \] \end{proposition} First of all, we observe that any mild solution $v$ of \eqref{1.1} satisfies \begin{equation} v\in L^2(\tau,T;W_0^{1,2}(\Omega))\cap L^{\infty}(\Omega\times(\tau ,\infty)),\quad\text{for }0<\tau0$, \[ 0=\partial_tv-\Delta v+v^{-\beta}\chi_{\{v>0\}}\geq\partial_tv-\Delta v++g_{\varepsilon}(v). \] This implies that $v$ is a sub-solution of \eqref{2.2}. Applying Lemma \ref{lemmaunique} to $v$ and $u_{\varepsilon}$ we obtain \[ v\leq u_{\varepsilon},\quad\text{in }\Omega\times(0,\infty). \] Letting $\varepsilon\to 0$, we obtain the desired conclusion. \end{proof} \section{Quenching phenomenon in a finite time} Since $u$ above is the maximal solution, then it is sufficient to show the quenching property for $u$. \begin{theorem} \label{thm3.1} Let $u$ be the maximal solution of equation \eqref{1.1}, see Theorem \ref{thm1}. Then, there exists a finite time $T^{\ast}>0$ such that \[ u(x,t)=0, \quad\forall (x,t)\in \Omega\times(T^\ast, \infty). \] Moreover, $T^{\ast}$ only depends on $\|u_0\|_{L^1(\Omega)},N, \beta$, and $|\Omega|$. \end{theorem} \begin{proof} First of all, we establish the energy equation for $u$ (local in time). By multiplying \eqref{2.2} by $u_{\varepsilon}$, and integrating by parts, we obtain that for any $0<\tau0$ fixed, \eqref{2.4} yields \[ \sup_{t\geq\tau}\| u(t)\|_{L^{\infty}(\Omega)} \leq C(N,|\Omega |)\tau^{-\frac{N}{2}}\| u_0\|_{L^1(\Omega)}:=M_{\tau}. \] Thus, we have that for any $t\geq \tau$, \begin{equation} \int_{\Omega}u^{1-\beta}(t)dx\geq M_{\tau}^{-\beta}\int_{\Omega} u(t)dx. \label{3.8} \end{equation} Combining \eqref{3.7} and \eqref{3.8}, we deduce \begin{align*} M_{\tau}^{-\beta(1-\theta)}\| u(t)\|_{L^2(\Omega)} &\leq C(N)\Big( \int_{\Omega}|\nabla u(t)|^2dx\Big)^{\theta/2} \Big( M_{\tau}^{-\beta}\int_{\Omega}u(t)dx\Big)^{1-\theta} \\ & \leq C(N)\Big( \int_{\Omega}|\nabla u(t)|^2dx\Big)^{\theta/2} \Big( \int_{\Omega}u^{1-\beta}(t)dx\Big) ^{1-\theta} \\ & \leq C(N)\Big( \int_{\Omega}|\nabla u(t)|^2 dx+\int_{\Omega}u^{1-\beta}(t)dx\Big) ^{\frac{\theta}{2}+1-\theta}. \end{align*} Then, \begin{equation} M_{\tau}^{-\frac{2\beta(1-\theta)}{2-\theta}}\Big( \int_{\Omega} |u(t)|^2dx\Big) ^{\gamma}\leq C(N)\Big( \int_{\Omega}|\nabla u(t)|^2dx+\int_{\Omega}u^{1-\beta}(t)dx\Big) , \label{3.9} \end{equation} with $\gamma=\frac{1}{2-\theta}=\frac{N+2}{N+4}$. Hence, from \eqref{3.6} and \eqref{3.9}, we obtain \begin{equation} \frac{d}{dt}w(t)+K(\tau)w^{\gamma}(t)\leq0, \quad \text{for any } t\geq \tau. \label{3.10} \end{equation} where \[ w(t)=\int_{\Omega}|u(t)|^2dx,\quad\text{and} \quad K(\tau)=2C(N)^{-1}M_{\tau}^{-\frac{2\beta(1-\theta)}{2-\theta}}. \] Clearly, if there is a finite time $\tau_0>\tau$, such that $w(\tau_0)=0$, it follows from \eqref{3.10} that \[ w(t)=0, \quad\forall t>\tau_0. \] If not, $w(t)>0$, for $t>\tau$, then solving the ODE \eqref{3.10} yields \begin{equation}\label{3.100} w^{1-\gamma}(t)-w^{1-\gamma}(\tau)\leq -(1-\gamma)K(\tau)(t-\tau),\quad \forall t>\tau. \end{equation} Inequality \eqref{3.100} holds for any $t>\tau$, so it deduces a contradiction as $t$ is large enough. Finally, we shall show that the vanishing time (i.e. the quenching time) of $u(t)$ can be estimated by a constant only depending on $\|u_0\|_{L^1(\Omega)}$ and $N$, $\beta$, $|\Omega|$. In fact, by the basic semigroup estimate (see \cite{CazHar,BeDa}), we have \[ w(\tau)^{1/2}=\| u(\tau)\|_{L^2(\Omega)} \leq C\tau^{-\frac {N}{4}}\| u_0\|_{L^1(\Omega)}. \] Combining this fact, and \eqref{3.100} yields \begin{equation} w^{1-\gamma}(t)+(1-\gamma)K(\tau)(t-\tau)\leq\left(C\tau^{-\frac{N}{4}}\| u_0\|_{L^1(\Omega)}\right)^{2(1-\gamma)},\quad\text{for }t>\tau. \label{3.12} \end{equation} Let $T_{\rm min}$ be a minimum vanishing time of $u(t)$. According to \eqref{3.12}, we have for any $\tau>0$, \[ T_{\rm min}\leq T(\tau)=\tau+C_1 \tau^{-\frac{N}{2}(1-\gamma )}K(\tau)^{-1}\| u_0\|_{L^1(\Omega)}^{2(1-\gamma)}, \] with $C_1=C_1(N, |\Omega|)$. By a computation based on the definition of $K(\tau)$ and $M(\tau)$, we obtain \[ T(\tau)=\tau+C_2\tau^{-\left( \frac{N(1-\gamma)} {2}+N\beta(1-\theta)\gamma\right) }\| u_0\|_{L^1(\Omega)} ^{2\beta(1-\theta)\gamma+2(1-\gamma)}:=\tau+C_2\tau^{-\alpha_1}\| u_0\|_{L^1(\Omega)}^{\alpha_2}. \] However, \[ \min_{\tau>0}\{\tau+C_2\tau^{-\alpha_1}\| u_0\|_{L^1(\Omega )}^{\alpha_2}\}=\tau_0+C_2\tau_0^{-\alpha_1}\| u_0\| _{L^1(\Omega)}^{\alpha_2}, \] with $\tau_0^{\alpha_1+1}=\alpha_1C_2\| u_0\|_{L^1(\Omega )}^{\alpha_2}$. Then the previous equality gives us \[ \min_{\tau>0}\{\tau+C_2\tau^{-\alpha_1}\| u_0\|_{L^1(\Omega )}^{\alpha_2}\}=C_3\| u_0\|_{L^1(\Omega)}^{\frac{\alpha_2 }{\alpha_1+1}}=C_3\| u_0\|_{L^1(\Omega)}^{\frac{2(1+\beta) }{3+\beta}}:=T^{\ast}, \] with $C_3=C_3(N,\gamma,|\Omega|)$. Then, $T_{\rm min}\leq T^{\ast}$, which completes the proof. \end{proof} \begin{remark} \label{rmk10} \rm The quenching property was established in the literature (see e.g., \cite{GiSaSer, Philips}) only for the special case of bounded initial data or $u_0\in L^2(\Omega)$, so the obtained quenching time $T^{\ast}$ always depends on $\| u_0\| _{L^{\infty}(\Omega)}$ or $\| u_0\|_{L^2(\Omega)}$. Thus, our result is sharper in the sense that we merely require that $u_0\in L^1(\Omega)$. \end{remark} Next, we show that the uniqueness result holds for a class of weak solutions satisfying some conditions. Let $\mathcal{A}$ be the set of weak solution of equation \eqref{1.1} such that any solution $v\in \mathcal{A}$, $v(x,t)>0$ in $\Omega\times(0,T_0)$. In other words, the set $\mathcal{A}$ contains the weak solutions such that they have the same quenching time $T_0$ as the maximal solution $u$. \begin{theorem} \label{thm11} Assume $\beta\in(0,1)$. Then \eqref{1.1} has at most one solution in the set $\mathcal{A}$. \end{theorem} \begin{remark} \label{rmk12} \rm To obtain a solution, which stays positive for some time, we refer to \cite[Lemma 1.9]{Davila-Montenegro survey}. \end{remark} \begin{proof}[Proof of Theorem \ref{thm11}] Let $v\in\mathcal{A}$. Thanks to Theorem \ref{thm1}, we obtain \begin{equation}\label{1.1a} v\leq u, \quad \text{in } \Omega\times(0,T^\ast). \end{equation} Since $u$ is a weak solution of \eqref{1.1}, we have that for any $s\in(0,T^\ast)$, \begin{align*} &\int_{\Omega} u(T^\ast)\phi(T^\ast) dx + \int^{T^\ast}_s\int_{\Omega} \nabla u \nabla \phi \,dx\,d\sigma + \int^{T^\ast}_s\int_{\Omega} \chi_{\{u>0\}}u^{-\beta} \phi \,dx\,d\sigma \\ & = \int_{\Omega} u(s)\phi(s) dx, \end{align*} for any test function $ \phi\in L^{\infty}_{loc}((0, \infty); L^{\infty}(\Omega)) \cap L^2_{loc}((0,\infty); H^1_0(\Omega))$. The fact that $u\in\mathcal{A}$ implies \begin{equation}\label{2} \int^{T^\ast}_s\int_{\Omega} \nabla u \nabla \phi \,dx\,d\sigma + \int^{T^\ast}_s\int_{\Omega} u^{-\beta} \phi \,dx\,d\sigma = \int_{\Omega} u(s)\phi(s) dx, \end{equation} By choosing $\phi=v$ as a test function in \eqref{2}, we obtain \begin{equation}\label{3} \int^{T^\ast}_s\int_{\Omega} \nabla u \nabla v \,dx\,d\sigma + \int^{T^\ast}_s\int_{\Omega} u^{-\beta} v \quad \,dx\,d\sigma = \int_{\Omega} u(s) v(s) dx, \end{equation} Similarly, we also get the following equation by changing the roles of $u$ and $v$, \begin{equation}\label{4} \int^{T^\ast}_s\int_{\Omega} \nabla u \nabla v \,dx\,d\sigma + \int^{T^\ast}_s\int_{\Omega} v^{-\beta} u \quad \,dx\,d\sigma = \int_{\Omega} u(s) v(s) dx, \end{equation} Combining \eqref{3} and \eqref{4}, we obtain \[ \int^{T^\ast}_s\int_{\Omega} v^{-\beta} u \quad \,dx\,d\sigma = \int^{T^\ast}_s\int_{\Omega} u^{-\beta} v \quad \,dx\,d\sigma . \] The above equation and \eqref{1.1a} imply $u=v$ in $\Omega\times(s, T^\ast)$. This conclusion holds for any $s>0$, thus we complete the proof. \end{proof} \section{Appendix} \subsection{Proof of Theorem \ref{thm2.0}} Let us regularize the initial condition $u_0$ by considering a nonnegative sequence $\{u_{0,k}\}_k\subset\mathcal{C}_c^{\infty}(\Omega)$ such that $u_{0,k}\to u_0$ in $L^1(\Omega)$ as $k\to \infty$, and consider the problem \begin{equation} \begin{gathered} \partial_tv_k-\Delta v_k+g_{\varepsilon}(v_k)=0, \quad \text{in } \Omega\times(0,T),\\ v_k=0, \quad \text{on }\partial\Omega\times(0,T),\\ v_k(\cdot,0)=u_{0,k}(\cdot) \quad \text{on }\Omega. \end{gathered} \label{4.1} \end{equation} Since $g_{\varepsilon}$ is a global Lipschitz-continuous function, the classical result ensures the existence and {the} uniqueness of a classical solution $v_k$. Moreover, $v_k$ fulfils that for any $t>0$, \begin{equation} v_k(t)=S(t)u_{0,k}-\int_0^tS(t-s)g_{\varepsilon}(v_k(s))ds. \label{4.2} \end{equation} Next, we claim that for any $T>0$, $v_k\geq0$ in $\Omega\times(0,T)$. Indeed, it is sufficient to show that \[ \min_{(x,t)\in\Omega\times(0,T)}v_k(x,t)\geq0. \] We can assume by contradiction that there is a point $(x_0,t_0)\in \Omega\times(0,T)$ such that \[ \min_{\Omega\times(0,T)}v_k(x,t)=v_k(x_0,t_0)<0. \] Let $\overline{v}_k(x,t):=v_k(x,t)+\delta t$, with $\delta>0$ small enough such that $\overline{v}_k(x_0,t_0)=v_k(x_0,t_0)+\delta t_0<0$. This implies that $\overline{v}_k$ attains its minimum at a point inside of $\Omega\times(0,T)$, say $(x_1,t_1)\in\Omega\times(0,T)$, and $\overline{v}_k(x_1,t_1)\leq\overline{v}_k(x_0,t_0)<0$. Then, we have $\partial_t\overline{v}_k(x_1,t_1)=0$ and $\Delta\overline {v}_k(x_1,t_1)\geq0$, so \[ 0=\partial_tv_k(x_1,t_1)-\Delta v_k(x_1,t_1)+g_{\varepsilon }(v_k(x_1,t_1))=(\partial_t\overline{v}_k(x_1,t_1)-\delta )-\Delta\overline{v}_k(x_1,t_1)+0. \] This leads to a contradiction. Thus, we obtain the claim. Next, we proceed as in the proof of Theorem \ref{thm2.1} to obtain $v_k\to u_{\varepsilon}$, in the space $L^{r}(0,T;W_0^{1,r}(\Omega))$, as $k\to{+}\infty$ ({up} to a subsequence if necessary), and that $u_{\varepsilon }\in\mathcal{C}([0,T];L^1(\Omega))$. Then, it suffices to pass to the limit in \eqref{4.2} as $k\to {+}\infty$ in obtain \eqref{2.3}. It remains to show now that $u_{\varepsilon}\in\mathcal{C}_{x,t}^{2+\alpha,1+\frac{\alpha}{2}} (\overline{\Omega}\times[\tau,T])$ for any $0<\tau1$. When $p$ is large enough (such as $p{>}N+2$), we have $v_k\in\mathcal{C}_{x,t}^{\gamma,\frac{\gamma}{2} }(\overline{\Omega}\times[\tau,T])$, for some $\gamma\in(0,1)$. Note that $v_k$ is bounded in $\mathcal{C}_{x,t}^{\gamma,\frac{\gamma}{2}} (\overline{\Omega}\times[\tau,T])$ by a constant independent of $k$. Therefore, Ascoli's theorem implies that, there is a subsequence (still denoted as $\{v_k\}_{{k}}$) such that \[ v_k \to u_{\varepsilon},\quad\text{in } \mathcal{C}_{x,t}^{\gamma,\frac{\gamma}{2}}(\overline{\Omega }\times[\tau,T]). \] On the other hand, $u_{\varepsilon}$ satisfies \[ \partial_tu_{\varepsilon}-\Delta u_{\varepsilon}=-g_{\varepsilon }(u_{\varepsilon}). \] But, since $g_{\varepsilon}$ is Lipschitz-{continuous}, we have that $g_{\varepsilon}(u_{\varepsilon})\in\mathcal{C}_{x,t}^{\gamma,\frac{\gamma} {2}}(\overline{\Omega}\times(\tau,T))$. Then, the conclusion $u_{\varepsilon }\in\mathcal{C}_{x,t}^{2+\gamma,1+\frac{\gamma}{2}}(\overline{\Omega} \times(\tau,T))$ follows from the $\alpha$-Holder regularity of parabolic equations. \subsection*{Proof of claim \eqref{2.35}} Let $v$ be a mild solution of \eqref{1.1} and let us consider the problem \begin{equation} \begin{gathered} \partial_t\overline{v}-\Delta\overline{v}+f=0, \quad \text{in }\Omega\times(0,T),\\ \overline{v}=0, \quad \text{on }\partial\Omega\times(0,T),\\ \overline{v}(\cdot,0)=u_0(\cdot) \quad \text{on }\Omega. \end{gathered} \label{4.3} \end{equation} where $f:=v^{-\beta}\chi_{\{v>0\}}\in L^1(\Omega\times(0,T))$ and $0