\documentclass[twoside]{article} \usepackage{amssymb,amsmath} % font used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Blow-up for $p$-Laplacian parabolic equations \hfil EJDE--2003/20} {EJDE--2003/20\hfil Yuxiang Li \& Chunhong Xie \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2003}(2003), No. 20, pp. 1--12. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Blow-up for $p$-Laplacian parabolic equations % \thanks{ {\em Mathematics Subject Classifications:} 35K20, 35K55, 35K57, 35K65. \hfil\break\indent {\em Key words:} $p$-Laplacian parabolic equations, blow-up, global existence, first eigenvalue. \hfil\break\indent \copyright 2003 Southwest Texas State University. \hfil\break\indent Submitted October 20, 2002. Published February 28, 2003.} } \date{} % \author{Yuxiang Li \& Chunhong Xie} \maketitle \begin{abstract} In this article we give a complete picture of the blow-up criteria for weak solutions of the Dirichlet problem $u_t=\nabla(|\nabla u|^{p-2}\nabla u)+\lambda |u|^{q-2}u,\quad \mbox{in } \Omega_T,$ where $p>1$. In particular, for $p>2$, $q=p$ is the blow-up critical exponent and we show that the sharp blow-up condition involves the first eigenvalue of the problem $-\nabla(|\nabla \psi|^{p-2}\nabla \psi)=\lambda |\psi|^{p-2}\psi,\quad\mbox{in } \Omega;\quad \psi|_{\partial\Omega}=0.$ \end{abstract} \newtheorem{thm}{Theorem}[section] \newtheorem{lem}[thm]{Lemma} \newtheorem{defn}[thm]{Definition} \newtheorem{rem}[thm]{Remark} \numberwithin{equation}{section} \section{Introduction} In this paper we study the Dirichlet problem \begin{eqnarray}\label{e:main} \begin{gathered} u_t=\nabla(|\nabla u|^{p-2}\nabla u)+\lambda |u|^{q-2}u,\quad \mbox{in }\Omega_T,\\ u=0, \quad\mbox{on } S_T,\\ u(x,0)=u_0(x),\quad\mbox{in } \Omega, \end{gathered} \end{eqnarray} $u_0(x)\in C_0(\overline{\Omega})$, where $p>1$, $q>2$, $\lambda>0$ and $\Omega\subset \mathbb{R}^N$ is an open bounded domain with smooth boundary $\partial\Omega$. When $p=2$, the blow-up properties of the semilinear heat equation (\ref{e:main}) hasve been investigated by many researchers; see the recent survey paper \cite{GV}. For $p\neq 2$, the main interest in the past twenty years lies in the regularities of weak solutions of the quasilinear parabolic equations; see the monograph \cite{D} and the references therein. When $\Omega=\mathbb{R}^N$, the Fujita exponents have been calculated; see \cite{G2, G3, G4, GL} and also the survey papers \cite{DL, L1}. To the best of our knowledge, when $\Omega$ is a bounded domain, the blow-up conditions are not fully established, especially, in the case $q=p>2$. In \cite{T}, the author showed that $q=p$ is the critical case, that is, if $qp$, there are both nonnegative, nontrivial global weak solutions and solutions which blow up in finite time. The blow-up result for $q>p$ is also proved in \cite{LP}. Furthermore, in \cite{Zh} the author proved that in the critical case $q=p>2$, if the Lebesgue measure of $\Omega$ is sufficiently small, (\ref{e:main}) has a global solution and if $\Omega$ is a sufficiently large ball, it has no global solution. In this paper we shall give a complete picture of the blow-up criteria for (\ref{e:main}). In particular, in the critical case $q=p>2$, we will prove that if $\lambda>\lambda_1$, there are no nontrivial global weak solutions, and if $\lambda\leq\lambda_1$, all weak solutions are global, where $\lambda_1$ is the first eigenvalue of the nonlinear eigenvalue problem $$\label{e:eigenvalue problem} -\nabla(|\nabla \psi|^{p-2}\nabla \psi)=\lambda |\psi|^{p-2}\psi,\quad\mbox{in }\Omega;\quad \psi|_{\partial\Omega}=0.$$ The following lemma concerns the properties of the first eigenvalue $\lambda_1$ and the first eigenfunction $\psi(x)$. \begin{lem}\label{lem:eigenvalue problem} There exists a positive constant $\lambda_1(\Omega)$ with the following properties: \begin{enumerate} \item[(a)] For any $\lambda<\lambda_1(\Omega)$, the eigenvalue problem {\rm (\ref{e:eigenvalue problem})} has only the trivial solution $\psi\equiv 0$. \item[(b)] There exists a positive solution $\psi\in W_0^{1,p}(\Omega)\cap C(\overline{\Omega})$ of {\rm (\ref{e:eigenvalue problem})} if and only if $\lambda=\lambda_1(\Omega)$. \item[(c)] The collection consisting of all solutions of {\rm (\ref{e:eigenvalue problem})} with $\lambda=\lambda_1(\Omega)$ is 1-dimensional vector space. \item[(d)] If $\Omega_j$, $j=1, 2$ are bounded domain with smooth boundary satisfying $\Omega_1\Subset\Omega_2$, then $\lambda_1(\Omega_1)>\lambda_1(\Omega_2)$. \item[(e)] Let $\{\Omega_n\}$ be a sequence of bounded domains with smooth boundaries such that $\Omega_n\Subset\Omega_{n+1}$ and $\bigcup_{n=1}^{\infty}\Omega_n=\Omega$, then $\lim_{n\rightarrow\infty} \lambda_1(\Omega_n)=\lambda_1(\Omega)$. \end{enumerate} \end{lem} \paragraph{Proof} (a)-(d) follow from \cite[Lemma 2.1, 2.2]{FM}. The continuity of $\psi(x)$ is asserted in \cite[Corollary 4.2]{Tr}. We now prove (e). It follows from (d) that $\lambda_1(\Omega_n)$ is strictly decreasing and so it tends to some nonnegative constant $\lambda_1^*(\Omega)$ as $n\rightarrow\infty$. Denote by $\psi_n(x)$ the positive solution of (\ref{e:eigenvalue problem}) on $\Omega_n$ with $\lambda=\lambda_1(\Omega_n)$ such that $\int_{\Omega_n}\psi_ndx=1$. By (c), $\psi_n$ is unique. By the similar method in the proof of \cite[Theorem 2.1]{FM}, one can obtain from $\{\psi_n\}$ a positive solution $\psi^*$ of (\ref{e:eigenvalue problem}) with $\lambda=\lambda_1^*(\Omega)$. Then by (b), we have $\lambda_1^*(\Omega)=\lambda_1(\Omega)$. \hfill$\diamondsuit$\smallskip We note that the blow-up conditions for (\ref{e:main}) are similar to that of the porous media equations; see \cite{G1, LS, PS, Sa}. Also our results clearly illustrate the observation that larger domains are more unstable than smaller domains; see \cite{L1}. To prove that $q=p$ is the critical case, we shall use the method of comparison with suitable blowing-up self-similar sub-solutions introduced by Souplet and Weissler \cite{SW}. This method enables us to treat the singular case $12$. Recently, the self-similar sub-solution method is proven to be useful in proof of blow-up theorems in the semilinear and porous media equations with gradient terms and nonlocal problems; see also \cite{AMST, R, Sou}. This paper shows that this method can apply to the quasilinear problems with gradient diffusion. In the discussion of the critical case, we use a technique of comparison combined with the so-called concavity" method, which is a different treatment with respect to the eigenfunction method for the porous media equations. This paper is organized as follows: In the next section we consider comparison principles of the weak solutions of (\ref{e:main}). In section 3 we first discuss the critical case $q=p>2$. The last section is devoted to the proof of the blow-up results for (\ref{e:main}) with large initial values. \section{Weak solutions and comparison principles} Following the book \cite{D}, we give the definition of the weak solutions of (\ref{e:main}). \begin{defn} \rm A weak sub(super)-solution of the Dirichlet problem (\ref{e:main}) is a measurable function $u(x,t)$ satisfying $u\in C(0,T; L^2(\Omega))\cap L^p(0,T; W_0^{1,p}(\Omega))\cap L^{\infty}(\Omega_T),\ u_t\in L^2(\Omega_T)$ and for all $t\in (0,T]$ \label{defn:weak solutions} \begin{aligned} &\int_{\Omega}u\varphi(x,t)dx+\int_0^t\int_{\Omega}\{-u\varphi_t+ |\nabla u|^{p-2}\nabla u\cdot\nabla\varphi\}dx\,d\tau\\ &\leq(\geq)\int_{\Omega}u_0\varphi(x,0)dx+\lambda\int_0^t \int_{\Omega}|u|^{q-2}u\varphi \,dx\,d\tau\nonumber \end{aligned} for all bounded test functions $\varphi\in W^{1,p}(0,T; L^2(\Omega))\cap L^p(0,T; W_0^{1,p}(\Omega))\cap L^{\infty}(\Omega_T),\quad \varphi\geq 0.$ A function $u$ that is both a sub-solution and a super-solution is a weak solution of the Dirichlet problem (\ref{e:main}). \end{defn} It would be technically convenient to have a formulation of weak solutions that involves $u_t$. The following notion of weak sub(super)-solutions in terms of Steklov averages involves the discrete time derivative of $u$ and is equivalent to (\ref{defn:weak solutions}), $$\label{defn:weak solutions Steklov} \int_{\Omega\times\{t\}}\{u_{h,t}\varphi + [|\nabla u|^{p-2}\nabla u]_h\cdot\nabla\varphi-\lambda[|u|^{q-2}u]_h\varphi \}dx\leq(\geq)0,$$ for all $0T-h. \end{cases} \] The equivalence of (\ref{defn:weak solutions}) and (\ref{defn:weak solutions Steklov}) can be proven by the simple properties of Steklov averages. \begin{lem}[{\cite[Lemma I.3.2]{D}}]\label{lem:Steklov averages} Let$v\in L^{q,r}(\Omega_T)$. Then let$h\rightarrow 0$,$v_h$converges to$v$in$L^{q,r}(\Omega_{T-\varepsilon})$for every$\varepsilon\in (0,T)$. If$v\in C(0,T; L^q(\Omega))$, then as$h\rightarrow 0$,$v_h(\cdot,t)$converges to$v(\cdot,t)$in$L^q(\Omega)$for every$t\in (0,T-\varepsilon)$,$\forall\varepsilon\in (0,T)$. \end{lem} The H\"{o}lder continuity of the above weak solution has been studied by many researchers in the past twenty years; see \cite{D}. The following lemma is a special case. \begin{lem}\label{lem:Holder continuity} For$p>1$, let$u$be a bounded weak solution of the Dirichlet problem {\rm (\ref{e:main})}. If$u_0\in C_0(\overline{\Omega})$, then$u\in C(\overline{\Omega_T})$. Moreover, let$T^*<\infty$be the maximal existence time of$u$, then$\limsup_{t\rightarrow T^*}\|u(\cdot,t)\|_{\infty}=\infty$. \end{lem} The existence of the local weak solutions of the Dirichlet problem (\ref{e:main}) can be proven by Galerkin approximations using the a priori estimates presented in the book \cite[Theorem III.1.2 and Theorem IV.1.2]{D}. For details for$p>2$, we refer to \cite[Theorem 2.1]{Zh}. To establish the comparison principle, we begin with a simple lemma that provides the necessary algebraic inequalities. \begin{lem}\label{lem:algebraic inequalities} For all$\eta, \eta'\in \mathbb{R}^N$, there holds $(|\eta|^{p-2}\eta-|\eta'|^{p-2}\eta')\cdot(\eta-\eta') \geq \begin{cases} c_2(|\eta|+|\eta'|)^{p-2}|\eta-\eta'|^2, & \mbox{if } p>1,\\ c_1|\eta-\eta'|^p, &\mbox{if } p>2, \end{cases}$ where$c_1$and$c_2$are positive constants depending only on$p$. \end{lem} For the detailed proof of this lemma, we refer to \cite[Lemma 2.1]{Da}. \begin{thm}\label{thm:comparison} Let$u,v\in C(\overline{\Omega_T})$be weak sub- and super-solutions of {\rm (\ref{e:main})} respectively and$u(x,0)\leq v(x,0)$, then$u\leq v$in$\overline{\Omega_T}$. \end{thm} \paragraph{Proof} We write (\ref{defn:weak solutions Steklov}) for$u, v$against the testing function $[(u-v)_h]_+(x,t) =\Big[\frac{1}{h}\int_{t}^{t+h}(u-v)(x,\tau)d\tau\Big]_+,$ with$h\in(0,T)$and$t\in[0,T-h)$. Differencing the two inequalities for$u$,$v$and integrating over$(0,t)gives \begin{align*} &\int_{\Omega}[(u-v)_h]_+^2(x,t)dx+2\int_0^t\!\int_{\Omega} [|\nabla u|^{p-2}\nabla u-|\nabla v|^{p-2}\nabla v]_h\cdot \nabla[(u-v)_h]_+dxd\tau\\ &\leq\int_{\Omega}[(u-v)_h]_+(x,0)dx +2\lambda\int_0^t\!\int_{\Omega}[|u|^{q-2}u-|v|^{q-2}v]_h[(u-v)_h]_+dxd\tau. \end{align*} Ash\rightarrow 0$the first term on the right tends to zero since$(u-v)_+\in C(\overline{\Omega_T})$. Applying Lemma~\ref{lem:Steklov averages} and Lemma~\ref{lem:algebraic inequalities}, we arrive at $\int_{\Omega}(u-v)_+^2(x,t)dx\leq c_3\int_0^t\int_{\Omega}(u-v)_+^2dxd\tau.$ The Gronwall's Lemma gives the desired result. \hfill$\diamondsuit$In the following we consider the positivity of the weak solutions of the problem $$\label{e:noreaction} \begin{gathered} v_t=\nabla(|\nabla v|^{p-2}\nabla v),\quad\mbox{in } \Omega\times \mathbb{R}_+,\\ v=0, \quad \mbox{on } \partial\Omega\times \mathbb{R}_+,\\ v(x,0)=v_0(x)\geq 0, \quad\mbox{in } \Omega, \end{gathered}$$ where$p>2$. Let \begin{multline*} u_S(x-x_0,t-t_0)=A_{p,N}[\tau+(t-t_0)]^{-N/[(p-2)N+p]}\\ \times\Big\{\Big[a^{p/p-1}- \big(\frac{|x-x_0|}{[\tau+(t-t_0)]^{1/[(p-2)N+p]}} \big)^{p/(p-1)}\Big]_+\Big\}^{(p-1)/(p-2)}, \end{multline*} where $A_{p,N}=\big(\frac{p-2}{p}\big)^{(p-1)/(p-2)} \big\{\frac{1}{(p-2)N+p}\big\}^{1/(p-2)},$$\tau>0$,$a>0$are arbitrary constants. According to \cite[p. 84 ]{SGKM},$u_S(x-x_0,t-t_0)$satisfies the first equation of (\ref{e:noreaction}). Without loss of generality, we assume that$v_0(x)>0$in a ball$B(x_0, \delta_1)$. Let$\overline{x}\in\Omega$be another point. In the following we prove that there exists a finite time$\overline{t}$and a neighborhood$V_{\overline{x}}$such that$v(x,\overline{t})>0$in$V_{\overline{x}}$. Since$\Omega$is connected, there exists a continuous curve$\Gamma:\gamma(s)\subset\Omega$,$0\leq s\leq 1$, such that$\gamma(0)=x_0$and$\gamma(1)=\overline{x}$. Denote$\delta_2=\mathrm{dist}(\Gamma,\partial\Omega)$and$\delta=\min\{\delta_1,\delta_2\}$. Let$x_1=\Gamma\cap\partial B(x_0, \delta/2)$,$\cdots$,$x_k=\Gamma\cap\partial B(x_{k-1}, \delta/2)$,$\cdots$, such that$x_k\neq x_{k-2}$. It is clear that$\overline{x}\in B(x_n, \delta/2)$for some$n$. Since$\overline{B(x_1,\delta/4)}\subset B(x_0,\delta)$, then$v_0(x)>0$in$\overline{B(x_1,\delta/4)}$. Choose suitable$\tau$and$a$such that$\mathop{\rm supp}u_S\subset B(x_1,\delta/4)$and$\|u_S\|_{\infty}\leq \min_{x\in B(x_1,\delta/4)}v_0(x)$, then$u_S(x-x_1,t)$is a weak sub-solution of (\ref{e:noreaction}) in$B(x_1,\delta)$. The comparison principle implies that there exists$\tau_1>0$such that$v(x,\tau_1)>0$in$B(x_1,\delta)$. Thus$v(x,\tau_1)>0$in$B(x_2,\delta/2)$since$B(x_2,\delta/2)\subset B(x_1,\delta)$. Repeating the above procedure, by finite steps, there exists a finite time$\overline{t}$such that$v(x,\overline{t})>0$in$B(x_n,\delta/2)$. The proof is completed. Thus we have the following lemma. \begin{lem}\label{thm:positivity} Assume that$v_0\in C_0(\overline{\Omega})$is nontrivial. Denote$\Omega_{\rho}=\{x\in\Omega: \mathrm{dist} (x,\partial\Omega) >\rho\}$. Let$v$be the weak solution of {\rm (\ref{e:noreaction})}. Then there exists a finite time$t_{\rho}>0$such that$v(x,t_{\rho})>0$in$\Omega_{\rho}$. \end{lem} \paragraph{Proof} It follows from the above proof that for any$x\in\Omega$, there exist$t_x>0$and a neighborhood$V_x\subset\Omega$such that$v(x,t_x)>0$in$V_x$. Since$\bigcup_{x\in\Omega}V_x\supset\overline{\Omega_{\rho}}$, by the finite covering theorem,$\overline{\Omega_{\rho}}\subset\bigcup_{i=1}^n V_{x_i}$. Put$t_{\rho}=\max\{t_{x_1},\cdots,t_{x_n}\}$. This lemma is proved. \hfill$\diamondsuit$\section{The critical case$q=p>2$} Since in \cite{T, Zh}, the authors have been established that$q=p>2$is the critical case of (\ref{e:main}), we first consider what happens if$q=p$. Zhao showed in \cite{Zh} that if the Lebesgue measure of$\Omega$is sufficiently small, (\ref{e:main}) has a global solution and if$\Omega$is a sufficiently large ball, it has no global solution. In this section we shall prove that if$q=p>2$, the crucial role is played by the first eigenvalue$\lambda_1$of the eigenvalue problem (\ref{e:eigenvalue problem}), as in the porous media equations. First we consider the global existence case$\lambda\leq\lambda_1$. \begin{thm}\label{thm:small domains} Assume that$u_0\in C_0(\overline{\Omega})$and$q=p>2$. If $$\label{e:small domains} \lambda<\lambda_1,$$ then the unique weak solution of {\rm (\ref{e:main})} is globally bounded. \end{thm} \paragraph{Proof} Since$\lambda<\lambda_1$, by Lemma~\ref{lem:eigenvalue problem}, there exists$\Omega_{\varepsilon}\Supset\Omega$such that$\lambda<\lambda_{1,\varepsilon}<\lambda_1$. Let$\psi_{\varepsilon}(x)$be the first eigenfunction with$\sup_{x\in\Omega}\psi_{\varepsilon}(x)=1$of the eigenvalue problem (\ref{e:eigenvalue problem}) with$\Omega=\Omega_{\varepsilon}$. Choose$K$to be so large that$u_0(x)\leq K\psi_{\varepsilon}(x)\equiv v(x)$. For all$00$large. \end{rem} \begin{rem} \rm Theorem~\ref{thm:small domains} and Remark~\ref{rem:lambda lambda1} hold for mixed sign solutions as well. To see this, just use$-K\psi_{\varepsilon}$in Theorem~\ref{thm:small domains} and$-K\psi$in Remark~\ref{rem:lambda lambda1} as weak subsolutions of (\ref{e:main}). \end{rem} Now we consider the blow-up case$\lambda>\lambda_1$. In \cite[Theorem 4.1]{Zh}, using the so-called concavity" method, the author showed that if$u_0\in W_0^{1,p}(\Omega)\cap L^{\infty}(\Omega)$and $$\label{e:Eu} \mathcal{E}(u_0)=\frac{1}{p}\int_{\Omega}|\nabla u_0|^pdx- \frac{\lambda}{p}\int_{\Omega}|u_0|^pdx<0,$$ then there exists$T^*<\infty$such that $$\label{e:blow up} \lim_{t\rightarrow T^*}\parallel u(\cdot,t)\parallel_{L^{\infty}(\Omega)}=\infty.$$ See also \cite{L2}. The result is crucial in the proof of the blow-up case$\lambda>\lambda_1$. The following lemma reproves the result using another version of the concavity" argument. \begin{lem}\label{lem:blow up} Assume that$u_0\in W_0^{1,p}(\Omega)\cap C_0(\overline{\Omega})$satisfies {\rm (\ref{e:Eu})}, then {\rm (\ref{e:blow up})} holds. \end{lem} \paragraph{Proof} Unlike in the usual concavity" argument, we put $\mathcal{H}(t)=\frac{1}{2}\int_{\Omega}u^2dx.$ Taking$u$and$u_t$as testing functions in the weak formulation of (\ref{e:main}), modulo a Steklov average, gives $$\label{e:first derivative} \begin{gathered} \frac{d}{dt}\mathcal{H}(t)=-p\mathcal{E}(u),\quad \mbox{in } \mathcal{D}'(\mathbb{R}_+),\\ -\frac{d}{dt}\mathcal{E}(u)=\int_{\Omega}(u_t)^2dx,\quad \mbox{in } \mathcal{D}'(\mathbb{R}_+). \end{gathered}$$ Differentiating (\ref{e:first derivative}), we have $\frac{d^2}{dt^2}\mathcal{H}(t)=-p\frac{d}{dt}\mathcal{E}(u), \quad \mbox{in } \mathcal{D}'(\mathbb{R}_+).$ Note that $\frac{d}{dt}\mathcal{H}(t)=\int_{\Omega}uu_tdx,\quad \mbox{in } \mathcal{D}'(\mathbb{R}_+).$ Then using the H\"{o}lder inequality, we have $\frac{p}{2}\Big[\frac{d}{dt}\mathcal{H}(t)\Big]^2 =\frac{p}{2}\Big[\int_{\Omega}uu_tdx\Big]^2 \leq \frac{p}{2}\int_{\Omega}u^2dx\int_{\Omega}(u_t)^2dx =\mathcal{H}(t)\frac{d^2}{dt^2}\mathcal{H}(t),$ in$\mathcal{D}'(\mathbb{R}_+)$, which implies $\frac{d^2}{dt^2}\mathcal{H}^{1-\frac{p}{2}}(t)\leq 0, \quad\mbox{in } \mathcal{D}'(\mathbb{R}_+).$ It follows that$T^*<\infty$. Indeed, otherwise, taking into account (\ref{e:Eu}) and the continuity of$\mathcal{H}(t)$, there exists$T<\infty$such that$\lim_{t\rightarrow T}\mathcal{H}(t)=\infty$: a contradiction. The proof is completed. \hfill$\diamondsuit$The following theorem follows from the above lemma. \begin{thm}\label{thm:large domains} For$q=p>2$, the unique weak solution of the Dirichlet problem {\rm (\ref{e:main})} with nontrivial, nonnegative$u_0\in C_0(\overline{\Omega})$blows up in finite time provided that $$\label{e:large domains} \lambda>\lambda_1.$$ \end{thm} \paragraph{Proof} Let$\psi(x)>0$be the first eigenfunction of the eigenvalue problem (\ref{e:eigenvalue problem}) with$\max_{x\in\Omega}\psi(x)=1$. Then we have, for any$k>0$, $\mathcal{E}(k\psi)=\frac{1}{p}\int_{\Omega}|\nabla (k\psi)|^pdx- \frac{\lambda}{p}\int_{\Omega}(k\psi)^pdx =k^p\frac{\lambda_1-\lambda}{p}\int_{\Omega}\psi^pdx<0.$ Therefore, by Lemma~\ref{lem:blow up}, the solution of (\ref{e:main}) with the initial datum$k\psi(x)$blows up in finite time. Given any nontrivial initial datum$u_0(x)\geq 0$, denote by$T^*$the maximal existence time of the weak solution of (\ref{e:main}). Suppose by contradiction that$T^*=\infty$. Combining (\ref{e:large domains}) with Lemma~\ref{lem:eigenvalue problem}, there exists$\Omega_{\rho}\Subset\Omega$such that$\lambda>\lambda_{1,\rho}>\lambda_1$. By Lemma~\ref{thm:positivity} and the comparison principle, there exists$t_{\rho}>0$such that $$\label{e:positivity} u(x,t_{\rho})>0, \quad x\in \overline{\Omega_{\rho}}.$$ Consider the problem (\ref{e:main}) in$\Omega_{\rho}$with the initial datum$k\psi_{\rho}$, where$\psi_{\rho}$is the first eigenfunction of (\ref{e:eigenvalue problem}) in$\Omega_{\rho}$with$\max\psi_{\rho}=1$. We know that the weak solution$u_{\rho}(x,t)$blows up in finite time for any$k>0$. Choose$k$so small that$u(x,t_{\rho})\geq k\psi_{\rho}$in$\Omega_{\rho}$, then a contradiction follows from the comparison principle. The theorem is proved. \hfill$\diamondsuit$\section{Global nonexistence for large initial values} In \cite{Zh}, the author used the so-called concavity" method to prove that if$q>p>2$, the unique weak solution of (\ref{e:main}) blows up in finite time if$\mathcal{E}(u_0)<0$. In this section we use the method of comparison with suitable blowing-up self-similar sub-solution to give a uniform treatment for all$p>1$. In the following theorem we construct a suitable blowing-up self-similar subsolution. \begin{thm}\label{thm:large initial values} Assume that$q>p>1$and$q>2$. Given a nonnegative, nontrivial initial datum$u_0\in C_0(\overline{\Omega})$, there exists$\mu_0>0$(depending only upon$u_0$) such that for all$\mu>\mu_0$, the weak solution$u(x,t)$of the Dirichlet problem {\rm (\ref{e:main})} with initial data$\mu u_0$blows up in a finite time$T^*$. Moreover, there is some$C(u_0)>0$such that $$\label{e:estimate of T^*} T^*(\mu u_0)\leq \frac{C(u_0)}{\mu^{p-1}},\quad \mu\rightarrow\infty.$$ \end{thm} \paragraph{Proof} We seek an unbounded self-similar sub-solution of (\ref{e:main}) on$[t_0,1/\varepsilon)\times \mathbb{R}^N$,$00$and$t_0$to be determined. First note that$\forall t\in [t_0,1/\varepsilon)$, $$\label{e:Supp} {\rm supp}(v(\cdot,t))\subset \overline{B}(0,R(1-\varepsilon t_0)^m),$$ with$R=(A^{\sigma-1}(\sigma+A))^{1/\sigma}$. We compute (by setting$y=|x|/(1-\varepsilon t)^mfor convenience), \begin{align*} Pv &= v_t-\nabla(|\nabla v|^{p-2}\nabla v)-\lambda |v|^{q-2}v\\ &= \frac{\varepsilon(kV(y)+myV'(y))}{(1-\varepsilon t)^{k+1}} -\frac{(|V'(y)|^{p-2}V'(y))'+(N-1)|V'(y)|^{p-2}V'(y)/y} {(1-\varepsilon t)^{(k+m)(p-1)+m}}\\ &\quad-\frac{\lambda}{(1-\varepsilon t)^{k(q-1)}}V^{q-1}(y). \end{align*} It is easy to verify that \begin{gather} 1\leq V(y)\leq 1+\frac{A}{\sigma},\quad -1\leq V'(y)\leq 0,\quad \mbox{for } 0\leq y\leq A, \nonumber\\ 0\leq V(y)\leq 1,\quad -\frac{R^{\sigma-1}}{A^{\sigma-1}}\leq V'(y)\leq -1,\quad \mbox{for } A\leq y\leq R, \label{e:p-Laplace V} \\ (|V'(y)|^{p-2}V'(y))'+(N-1)|V'(y)|^{p-2}V'(y)/y =-\frac{N}{A}\chi_{\{y\frac{k}{m},\quad 0<\varepsilon<\frac{\lambda}{k(1+A/\sigma)}. \end{gather*} Fort_0\leq t<1/\varepsilon$with$t_0$sufficiently close to$1/\varepsilon$, we have, in the case$0\leq y\leq A$, $Pv(x,t)\leq \frac{\varepsilon k(1+A/\sigma)-\lambda}{(1-\varepsilon t)^{k+1}} +\frac{N/A}{(1-\varepsilon t)^{(k+m)(p-1)+m}}\leq 0.$ In the case$A\leq yR$. Since$v(x,t)$is continuous and piecewise$C^2$and due to the sign of the singular measure in (\ref{e:p-Laplace V}) , then$v(x,t)$is a local weak sub-solution of the Dirichlet problem (\ref{e:main}). Now by translation, one can assume without loss of generality that$0\in \Omega$and$u_0(0)=\max_{x\in \Omega}u_0(x)$. It follows from the continuity of$u_0$that $u_0(x)\geq C,\quad \mbox{for all } x\in B(0,\rho),$ for some ball$B(0,\rho)\Subset\Omega$and some constant$C>0$. Taking$t_0$still closer to$1/\varepsilon$if necessary, one can assume that$B(0,R(1-\varepsilon t_0)^m)\subset B(0,\rho)$. Therefore, $$\label{e:large mu} \mu u_0(x)\geq \mu C \geq \frac{V(0)}{(1-\varepsilon t_0)^k}\geq v(x,t_0),\quad x\in \Omega,$$ for all$\mu>\mu_0=V(0)/C(1-\varepsilon t_0)^k$. By the Theorem~\ref{thm:comparison}, it follows that $u(x,t)\geq v(x,t+t_0),\quad x\in \Omega,\; 0V(0)/C(1-\varepsilon t_0)^k, by the previous calculation, whenever t_0\leq T <1/\varepsilon such that \mu\geq V(0)/C(1-\varepsilon T)^k, we have T^*(\mu u_0)\leq 1/\varepsilon-T. Then \[ T^*(\mu u_0)\leq \frac{1}{\varepsilon}\left(\frac{1+A/\sigma}{\mu C}\right)^{q-2},\quad \mbox{for all } \mu\geq \frac{V(0)}{C(1-\varepsilon t_0)^{1/(q-2)}}.$ The proof is completed. \hfill$\diamondsuit$Under the conditions of the above theorem, the solutions of (\ref{e:main}) exist globally for small initial data. \begin{thm} Assume that$q>p>1$and$q>2$. There exists$\eta>0$such that the solution of {\rm (\ref{e:main})} exists globally if$\|u_0\|_{\infty}<\eta$. \end{thm} \paragraph{Proof} Let$\Omega_{\varepsilon}\Supset\Omega$be a bounded domain and$\psi_{\varepsilon}$be the first eigenfunction of (\ref{e:eigenvalue problem}) on$\Omega_{\varepsilon}$with$\sup_{x\in \Omega}\psi_{\varepsilon}(x)=1$. Denote$\delta=\inf_{x\in \Omega}\psi_{\varepsilon}(x)$. Choose$k^{q-p}=\lambda_1/\lambda$and$\eta=k\delta$. A direct computation yields that$k\psi_{\varepsilon}(x)$and$-k\psi_{\varepsilon}(x)$is a weak super- and sub-solution of (\ref{e:main}) respectively. This theorem follows the comparison principle. \hfill$\diamondsuit$\begin{thm} Assume that$2