\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 168, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/168\hfil Existence and non-existence of global solutions] {Existence and non-existence of global solutions for a semilinear heat equation \\ on a general domain} \author[M. Loayza, C. Paix\~ao \hfil EJDE-2014/168\hfilneg] {Miguel Loayza, Crislene S. da Paix\~ao} % in alphabetical order \address{Miguel Loayza \newline Departamento de Matem\'atica, Universidade Federal de Pernambuco - UFPE, 50740-540, Recife, PE, Brazil} \email{miguel@dmat.ufpe.br} \address{Crislene S. da Paix\~ao \newline Departamento de Matem\'atica, Universidade Federal de Pernambuco - UFPE, 50740-540, Recife, PE, Brazil} \email{crisspx@gmail.com} \thanks{Submitted May 27, 2014. Published July 31, 2014.} \subjclass[2000]{35K58, 35B33, 35B44} \keywords{Parabolic equation; blow up; global solution} \begin{abstract} We consider the parabolic problem $u_t-\Delta u=h(t) f(u)$ in $\Omega \times (0,T)$ with a Dirichlet condition on the boundary and $f, h \in C[0,\infty)$. The initial data is assumed in the space $\{ u_0 \in C_0(\Omega); u_0\geq 0\}$, where $\Omega$ is a either bounded or unbounded domain. We find conditions that guarantee the global existence (or the blow up in finite time) of nonnegative solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Let $\Omega\subset \mathbb{R}^N$ be either a bounded or unbounded domain with smooth boundary. Meier \cite{Meier1} considered the blow up phenomenon of the solutions of the parabolic problem \begin{equation}\label{In.me} \begin{gathered} u_t- Lu =h(x,t) f(u) \quad \text{in }\Omega \times (0,T),\\ u=0 \quad \text{on }\partial \Omega \times (0,T),\\ u(0)=u_0\geq 0 \quad \text{in }\Omega, \end{gathered} \end{equation} where \[ L=\sum_{i,j=1}^N a_{ij}(x,t) \frac{\partial^2}{\partial x_i \partial x_j} +\sum_{i=1}^N b_i(x,t) \frac{\partial}{\partial x_i} \] is an uniformly elliptic operator in $\Omega$ with bounded coefficients $a_{ij}=a_{ji}$ and $h$ is a continuous function with $h(\cdot, t)$ bounded. The assumptions on the functions $f$ are the following: \begin{gather}\label{Cd.f} f \in C^1[0,\infty); \quad f(s)>0 \text{ for }s>0; \quad f(0)\geq 0; \quad f'\geq 0 ; \\ \label{Cd.fa} G(w)=\int_w^\infty \frac{d\sigma}{f(\sigma)}< \infty \quad\text{if }w>0. \end{gather} When $h(x,t)=h(t)$ we have the following result which follows from \cite[Theorem 2]{Meier1}. In this article, we denote by $(S(t))_{t\geq 0}$ the heat semigroup with the homogeneous Dirichlet condition on the boundary. \begin{theorem}[\cite{Meier1}] \label{thm1.1} Assume that $f$ satisfies conditions \eqref{Cd.f} and \eqref{Cd.fa} and $h(x,\cdot)=h(\cdot)\in C[0,\infty) $. \begin{itemize} \item[(i)] Let $f$ be convex with $f(0)=0$. Then the solution $u$ of \eqref{In.me} blows up in finite time, if there exists $\tau>0$ such that \begin{equation}\label{C3.Mei} G(\| S(\tau)u_0\|_\infty) \leq \int_0^\tau h(\sigma)d\sigma. \end{equation} \item[(ii)] Let $f(0)>0$. If there exists $\tau>0$ such that \begin{equation}\label{Mei.dos} G(0)\leq \| S(\tau)u_0\|_\infty \int_0^\tau \frac{h(\sigma)}{\| S(t)u_0\|_\infty}d\sigma, \end{equation} then the solution of \eqref{In.me} blows up in finite time. \end{itemize} \end{theorem} Meier \cite{Meier2} also considered the semilinear parabolic equation \begin{equation}\label{In.uno} \begin{gathered} u_t-\Delta u= h(t)u^p \quad \text{in }\Omega \times (0, T),\\ u=0 \quad \text{in }\partial \Omega \times (0,T),\\ u(0)=u_0\geq 0 \quad \text{in }\Omega, \end{gathered} \end{equation} where $h \in C[0,\infty)$, $p>1$ and $u_0 \in L^\infty(\Omega)$. He studied the existence of the Fujita critical exponent $p^*$ of \eqref{In.uno}, that is, a number such that if $1p^*$, then there exists a nontrivial global solution of problem \eqref{In.uno}. Determining the value of the Fujita critical for problem \eqref{In.uno} and its extensions has been objective of research of many authors, see for instance \cite{DLevine, Fujita, Levine, Meier3, Meier2, Meier1, W1, W2}. Below we list some values of $p^*$, which depend of the domain $\Omega$ and the function $h$. For instance, \begin{itemize} \item[(i)] If $\Omega=\mathbb{R}^N$ and $h=1$, then Fujita's result in \cite{Fujita} means that $p^*=1+2/N$; \item[(ii)] If $\Omega=R^N_k=\{x; x_i>0, i=1,...,k\}$ and $h(t) \sim t^{q}$ for $t$ large( i.e. there exist constants $c_0, c_1>0$ such that $c_0 t^q \leq h(t)\leq c_1 t^{q}$ for $t$ large) and $q>-1$, then $p^*=1+2(q+1)/(N+k)$, see \cite{Meier1}; \item[(iii)] If $\Omega$ bounded and $h(t) \sim e^{\beta t}$ for $t$ large, $\beta>0$, then $p^*=1+\beta/\lambda_1$, where $\lambda_1$ is the first Dirichlet eigenvalue of the Laplacian in $\Omega$, see \cite{Meier2}. \end{itemize} The results above can be obtained from the following general theorem, using only of the asymptotic behavior of the solution $u(t)=S(t)u_0$, $t\geq 0$, of the linear problem $u_t-\Delta u=0$, in $\Omega \times (0,\infty)$ and the function $h$. \begin{theorem}[\cite{Meier2}] \label{thm1.2} Let $p>1$, $h\in C [0,\infty)$. (i) If there exists $u_0\in L^\infty(\Omega)$, $u_0 \geq 0$ such that \begin{equation}\label{C1.Mei} \int_0^\infty h(\sigma )\| S(\sigma)u_0\|^{p-1}_\infty d\sigma <\infty, \end{equation} then there exists a global solution of \eqref{In.uno} with $\lim_{t \to \infty}\| u(t)\|_\infty=0$. (ii) If \begin{equation}\label{C2.Mei} \limsup_{t \to \infty}\|S(t)u_0\|^{p-1}\int_0^t h(\sigma)d\sigma=\infty \end{equation} for all $u_0 \in L^\infty(\Omega), u_0\geq 0$, then every nontrivial nonnegative solution of \eqref{In.uno} blows up in finite time. \label{Th.Meier1} \end{theorem} Condition \eqref{C1.Mei}, was used by Weissler \cite{W1}, when $h=1$ and $\Omega=\mathbb{R}^N$, to find a non negative global solution of \eqref{In.uno}. This is clear since we can choose $a_0$ so that $\overline u(t)=a(t) S(t)u_0$, where $$ a(t)=\Big[a_0^{-(p-1)}-(p-1)\int_0^t h(\sigma)\|S(\sigma)u_0\|_\infty^{p-1}d\sigma \Big]^{-1/(p-1)}, $$ is a supersolution of \eqref{In.uno} defined for all $t\geq 0$. In this work we are interested in the parabolic problem \begin{equation}\label{In.dos} \begin{gathered} u_t-\Delta u= h(t)f(u) \quad \text{in }\Omega \times (0, T),\\ u=0 \quad \text{on }\partial \Omega \times (0,T),\\ u(0)=u_0\geq 0 \quad \text{in }\Omega, \end{gathered} \end{equation} where $h\in C[0,\infty)$, $f \in C[0,\infty)$ is a locally Lipschitz function and $u_0 \in C_0(\Omega)$. Firstly, we are interested in finding conditions that guarantee the global existence of solutions of problem \eqref{In.dos}. In particular, we would like obtain a similar condition to Theorem 1.1(i). In second place, we are interested in the blow up in finite time of nonnegative solutions of \eqref{In.dos} assuming only $f$ locally Lipschitz, that is, without condition \eqref{Cd.f}. It is well known that if $f$ is locally Lipschitz, $f(0)=0$ and $u_0 \in C_0(\Omega)$, $u_0 \geq 0$, problem \eqref{In.dos} has a unique nonnegative solution $u \in C([0, T_{\rm max}), C_0(\Omega))$ defined in the maximal interval $[0,T_{\rm max})$ and verifying the equation \begin{equation}\label{Re.uno} u(t)=S(t)u_0 + \int_0^t S(t-\sigma)h(\sigma)f(u(\sigma))d\sigma, \end{equation} for all $t \in [0,T_{\rm max})$. Moreover, we have the blow up alternative: either $T_{\rm max}=\infty$(global solution) or $T_{\rm max}<\infty$ and $\lim_{t\to T_{\rm max}}\|u(t)\|_\infty=\infty$ (blow up solution). Throughout this work a nonnegative function $u \in C([0,T), C_0(\Omega))$ is said to be a solution of \eqref{In.dos} in a interval $[0, T)$ if satisfies equation \eqref{Re.uno}. Our first result is about the existence of a global solution of problem \eqref{In.dos}. \begin{theorem}\label{Th.dos} Assume that $f$ is locally Lipschitz and $f(0)=0$. Suppose that there exists $a>0$ such that the functions $f$ and $g:(0,\infty) \to [0,\infty)$, defined by $g(s)=f(s)/s$, are nondecreasing in $(0,a]$. If $v_0\in C_0(\Omega)$, $v_0\geq 0, v_0 \neq 0, \|v_0\|_\infty\leq a $ verifies \begin{equation}\label{In.cua} \int_0^\infty h(\sigma) g(\|S(\sigma)v_0\|_\infty) d\sigma <1, \end{equation} then there exists $u_0^* \in C_0(\Omega)$, $0\leq u_0^* \leq v_0$ such that for any $u_0 \in C_0(\Omega)$ $0\leq u_0\leq u_0^*, u_0 \neq 0$ the solution of \eqref{In.dos} is a global solution. Moreover, there exists a constant $\gamma>0$ so that $u(t)\leq \gamma \cdot S(t)u_0$ for all $t\geq 0$. In particular, $\lim_{t \to \infty}\|u(t)\|_\infty=0$. \end{theorem} \begin{remark} \rm (i) In Theorem \ref{Th.dos} we assume that $g$ is nondecreasing in some interval $(0,a]$. This condition is verified, for instance, if $f$ is a convex function. An analogous condition on $g$ was used also in \cite[Theorem 7]{Meier2}, but there it is assumed that $f(0)=f'(0)=0$ and $\Omega=\mathbb{R}^N_k$. (ii) If $f(t)=t^p$ for all $t\geq 0$ and $p>1$, we have that $G(w)=w^{1-p}/(p-1)$ and $g(s)=s^{p-1}$. Thus, condition \eqref{In.cua} reduces to condition \eqref{C1.Mei}. \end{remark} Our second result is the following. \begin{theorem} \label{Th.uno} Let $f$ be a locally Lipschitz function, $f(0)=0$, $f(s)>0$ for all $s>0$ and $G$ given by \eqref{Cd.fa}. Assume that the following conditions are satisfied: \begin{itemize} \item[(i)] The function $f$ is nondecreasing and verifies the following property \begin{equation}\label{Cf} f(S(t)v_0) \leq S(t) f(v_0), \end{equation} for all $v_0 \in C_0(\Omega), v_0\geq 0$ and $t>0$. \item[(ii)] There exist $\tau>0$ and $u_0\in C_0(\Omega)$, $u_0\geq 0, u_0 \neq 0$ such that \begin{equation}\label{In.tre} G(\|S(\tau )u_0\|_\infty) \leq \int_0^{\tau }h(\sigma)d\sigma. \end{equation} \end{itemize} Then the solution of problem \eqref{In.dos} blows up in finite time $T_{\rm max}\leq \tau$. \end{theorem} \begin{remark} \label{Rmk.uno} \rm Regarding Theorem \ref{Th.uno} we have the following comments: \begin{itemize} \item[(i)] By the positivity of the heat semigroup, we have that $S(t)v_0\geq 0$ if $v_0 \geq 0$. Hence, the left side of \eqref{Cf} is well defined. \item[(ii)] If $f$ is a convex function and $\Omega=\mathbb{R}^N$, then \eqref{Cf} holds. It is clear, by Jensen's inequality since $S(t)u_0 =k_t \star u_0$, where $k_t$ is a heat kernel. \item[(iii)] If $f$ is twice differentiable and convex, then \eqref{Cf} holds. Indeed, if $w(t) = f(S(t)v_0)$, then $w_t-\Delta w=-f''(S(t)v_0) |\nabla S(t)v_0|^2 \leq 0$. We then conclude using the maximum principle. \end{itemize} \end{remark} Theorem \ref{Th.dos} is proved using a monotone sequence method, see \cite{P2,W1}. Our arguments for proving Theorem \ref{Th.uno} are different to the arguments in Meier. Precisely, Meier uses the subsolutions method for problem \eqref{In.me}, whereas we use the formulation \eqref{Re.uno} to get an ordinary differential inequality, see inequality \eqref{Pr.odi}. We now apply our results to the heat equation with logarithmic nonlinearity \begin{equation}\label{Ej.dos} \begin{gathered} u_t-\Delta u= h(t) (1+u)[\ln (1+u)]^{q} \quad \text{in }\mathbb{R}^N \times (0, T),\\ u(0)=u_0\geq 0 \quad\text{in }\mathbb{R}^N, \end{gathered} \end{equation} where $q>1$ and $h:[0,\infty)\to [0,\infty)$ is a continuous function. Problem \eqref{Ej.dos} was introduced in \cite{73}, is a particular case of more general quasilinear models with common properties of convergence to Hamilton-Jacobi equations studied in \cite{76}, where the asymptotic of global in time solutions were established. For the mathematical theory of blow-up, see \cite{89} and the references therein. We have the following result. \begin{theorem}\label{Th.Ej2} Assume that $q>1$, $h:[0,\infty) \to [0,\infty)$ is a continuous function such that $h(t)\sim t^{r}$ for $t$ large enough and $r>-1$. (i) If $11+\frac{2}{N}(r+1)$, there exists $u_0 \in C_0(\mathbb{R}^N)$, $u_0\neq 0, u_0\geq 0$ so that the solution of \eqref{Ej.dos} is a global solution. \end{theorem} We also apply our results to the exponential reaction model \begin{equation}\label{Ej.uno} \begin{gathered} u_t-\Delta u = h(t)[\exp (\alpha u)-1]\quad\text{in }\Omega \times (0, T),\\ u=0 \quad \text{on }\partial \Omega \times (0,T),\\ u(0)=u_0\geq 0 \quad\text{in }\Omega, \end{gathered} \end{equation} with $\alpha>0$, $h\in C [0,\infty)$ and $\Omega$ a bounded domain with smooth boundary. These problems are important in combustion theory \cite{172} under the name of solid-fuel model (Frank-Kamenetsky equation). \begin{theorem}\label{Th.Ej1} Let $\alpha>0$ and $h\in C[0,\infty)$. \begin{itemize} \item[(i)] If there exists $\tau>0$ such that $\int_0^\tau h(\sigma)d\sigma\geq 1/\alpha$, then there exists $u_0 \in C_0(\Omega), u_0\geq 0$ so that the solution of problem \eqref{Ej.uno} blows up in finite time. \item[(ii)] If $\int_0^\infty h(\sigma)d\sigma<1/\alpha$, then there exists $u_0 \in C_0(\Omega), u_0\geq 0$ such that the solution of problem \eqref{Ej.uno} is global. \end{itemize} \end{theorem} \section{Proof of the main results} \begin{lemma}\label{Lem.com} Assume $h, f:[0,\infty) \to [0, \infty)$ with $h$ continuous, $f$ locally Lipschitz and nondecreasing. Let $u , v \in C([0,T], C_0(\Omega)) $ be solutions of problem \eqref{In.dos}(in the sense of \eqref{Re.uno}) with $u(0)=u_0\geq 0$ and $v(0)=v_0 \geq 0$. If $u_0\leq v_0$, then $u(t)\leq v(t)$ for all $t \in [0,T]$. \end{lemma} \begin{proof} Let $M=\max\{\|u(t)\|_\infty, \|v(t)\|_\infty; t \in [0,T]\}$. Since $u_0 \leq v_0$ we have \begin{equation}\label{Comp.uno} u(t)-v(t)\leq \int_0^t S(t-\sigma) h(\sigma)[f(u(\sigma))-f(v(\sigma))]d\sigma. \end{equation} On the other hand, since $u\leq u^+$, $f$ is nondecreasing and locally Lipschitz, we have $$ [f(u)-f(v)]\leq [f(u)-f(v)]^{+}\leq L_M (u-v)^+, $$ where $L_M$ is the Lipschitz constant in $[0,M]$. Thus, it follows from inequality \eqref{Comp.uno} that $$ \|[u(t)-v(t)]^{+}\|_\infty\leq L_M\int_0^t h(\sigma)\|[u(\sigma)-v(\sigma)]^{+}\|_\infty. $$ The conclusion follows from Gronwall's inequality. \end{proof} \begin{proof}[Proof of Theorem \ref{Th.uno}] We adopt the argument used in the proof of\cite[Lemma 15.6]{QS}. Assume that $u$ is a global solution and let $00$, then $$ \frac{d}{dt}(\Psi(\psi(t)))=-\frac{\psi'(t)}{f(\psi(t))}\leq - h(t). $$ Thus, $$ \int_0^s h(\sigma)d\sigma \leq \Psi(\psi(0))- \Psi(\psi(s)) = \int_{\psi(0)}^{\psi(s)} \frac{d\sigma}{f(\sigma)} < \int_{S(s)u_0}^\infty \frac{d\sigma}{f(\sigma)}=G(S(s)u_0) $$ for every $s>0$. This fact, contradicts inequality \eqref{In.tre}. \end{proof} \begin{proof}[Proof of Theorem \ref{Th.dos}] We use the monotone sequence argument (see \cite{P2,W1}). Since $\int_0^\infty h(\sigma)g(\|S(\sigma)v_0\|_\infty)d\sigma<1$, there exists $\beta>0$ such that \begin{equation}\label{Pr.dos} \int_0^\infty h(\sigma)g(\|S(\sigma)v_0\|_\infty)<\frac{\beta}{\beta+1}<1. \end{equation} Set \begin{equation}\label{Rev.uno} 0<\lambda< \frac{1}{\beta+1}\min\big\{1, \frac{a}{\|v_0\|_\infty}\big\}. \end{equation} From Lemma \ref{Lem.com}, it suffices to show that the corresponding solution $u$ of \eqref{In.dos} with $u(0)=u_0^*=\lambda v_0$ is global. We define a sequence $(u^n)_{n\geq 1}$ by $u^0=S(t)u_0^*$ and \begin{equation}\label{Pr.tre} u^n(t)=S(t)u_0^*+\int_0^t S(t-\sigma)h(\sigma)f(u^{n-1}(\sigma))d\sigma, \end{equation} for $n \in \mathbb{N}$ and all $t\geq 0$. Now, we claim that \begin{equation}\label{Ind} u^n(t)\leq (1+\beta) S(t)u_0^*, \end{equation} for all $t \geq 0$. We argue by induction on $n$. It is clear that \eqref{Ind} holds for $n=0$. Assume now that inequality \eqref{Ind} holds. It follows from \eqref{Rev.uno} and \eqref{Ind} that \begin{equation}\label{He.uno} \|u^n(t)\|_\infty \leq \lambda(1+\beta)\|v_0\|_\infty 0$ for all $s>0$. By Remark \ref{Rmk.uno}(iii), condition \eqref{Cf} is verified. Set $G(w)=\int_w^\infty \frac{ds}{(s+1) [\ln(1+s)]^q}=\frac{[\ln(1+w)]^{1-q}}{q-1}$. From here, \begin{equation}\label{Ej.cua} [G(\|S(t)u_0\|_\infty)]^{-1}\int_0^t h(\sigma)d\sigma = (q-1) [\ln (1+\|S(t)u_0\|_\infty) ]^{q-1}\int_0^t h(\sigma)d\sigma. \end{equation} To verify condition \eqref{In.tre}, we use the following result, which follows directly from L'H\^opital's rule: \begin{equation}\label{Ej.cin} \lim_{t \to \infty} \frac{\ln (1+c_0t^{-\beta})}{t^{-\alpha}} =\begin{cases} (c_0\beta)/\alpha &\text{if } \alpha=\beta,\\ 0 &\text{if } \beta>\alpha,\\ \infty &\text{if } \beta<\alpha, \end{cases} \end{equation} for $\alpha, \beta, c_0>0$. From \cite{LeeNi}(Lemma 2.12), we know that $\| S(t)u_0\|_\infty \geq c_0 t^{-N/2}$ for $t$ large and $u_0 \in C_0(\mathbb{R}^N), u_0\geq 0, u_0\neq 0$. Therefore, it follows from \eqref{Ej.cua} and \eqref{Ej.cin} that if $h(t)\geq c_1 t^{r}, r>-1$, for $t$ large enough then there exists a constant $c>0$ so that \begin{align*} [G(\|S(t)u_0\|_\infty)]^{-1}\int_0^t h(\sigma)d\sigma &\geq c [\ln (1+ c_0 t^{-\frac{N}{2}})]^{q-1}t^{r+1}\\ &\geq c (c_0t^{-\frac{N}{2}})^{q-1}t^{r+1}>1, \end{align*} if $q< 1+\frac{2}{N}(r+1)$. Hence, condition \eqref{In.tre} is verified and the conclusion follows of Theorem \ref{Th.uno}. We now analyze global existence using Theorem \ref{Th.dos}. It is clear that $f$ and $g(s)=f(s)/s$, where $f$ is given by \eqref{Ej.sei} are nondecreasing functions. Let $\psi \in C_0(\mathbb{R}^N)$ with $\|\psi\|_\infty=1$. From \cite{LeeNi}(Lemma 2.12) there exists $c_1, t_0>0$ such that \begin{equation}\label{Ej.onc} \|S(t)\psi \|_\infty \leq c_1 t^{-N/2}, \end{equation} for all $t \geq t_0$. Let $\epsilon>0$ so that $1+r-\frac{N}{2}(q-1)+\epsilon q<0$. From \eqref{Ej.cin} there exists $t_1>0$ such \begin{equation}\label{Ej.cat} \ln (1+ c_1 t^{-N/2})\leq t^{N/2 -\epsilon}, \end{equation} for all $t \geq t_1$. Let $t_2>0$ such that \begin{equation}\label{Fe.uno} h(t)\leq c_2t^{r}, \end{equation} for all $t\geq t_2$ and fix $t_3>\max\{1,t_0, t_1, t_2\}$ satisfying \begin{equation}\label{Ej.doc} c_4 t_3^{1+r-\frac{N}{2}(q-1)+\epsilon q}<\frac{1}{2}, \end{equation} where $c_4=c_3 c_2/[N(q-1)/2-r-1-\epsilon q ]>0$ and $c_3=(1+1/c_1)$. Consider $v_0=\mu \psi$ with $0<\mu\leq 1$ and \begin{equation} \label{Ej.trec} c_5(t_3) g(\mu) < \frac{1}{2}, \end{equation} where $c_5(t_3)=\int_0^{t_3} h(\sigma)d\sigma$. This fact is possible because $\lim_{\mu \to 0^{+}} g(\mu)=0$. It follows of \eqref{Ej.onc} that $\|S(t)v_0\|_\infty\leq c_1 \mu t^{-N/2}\leq c_1 t^{-N/2}$ for all $t\geq t_0$. Thus, $g(\|S(t)v_0\|_\infty)\leq g(c_1t^{-N/2})$ for all $t\geq t_0$. Hence, by \eqref{Ej.cat} - \eqref{Ej.trec} we have \begin{align*} &\int_0^\infty h(\sigma) g(\|S(\sigma)v_0\|_\infty) d\sigma\\ &\leq g(\|v_0\|_\infty)\int_0^{t_3}h(\sigma)d\sigma + \int_{t_3}^{\infty } h(\sigma) g(c_1 \sigma^{-N/2})d\sigma\\ &\leq g(\mu)\int_0^{t_3}h(\sigma)d\sigma + \int_{t_3}^{\infty } h(\sigma) (1+ \frac{1}{c_1 \sigma^{-N/2}}) [\ln (1+c_1 \sigma^{-N/2})]^q d\sigma\\ &< \frac{1}{2}+c_3 \int_{t_3}^\infty h(\sigma)\sigma^{N/2}[\ln (1+c_1 \sigma^{-N/2})]^qd\sigma\\ &\leq \frac{1}{2}+c_3 c_2\int_{t_3}^\infty \sigma^{r}\sigma^{N/2}\sigma^{-(N/2-\epsilon)q}d\sigma\\ &\leq \frac{1}{2}+c_4 {t_3}^{1+r -\frac{N}{2}(q-1) +\epsilon q}<1. \end{align*} Therefore, estimate \eqref{In.cua} is satisfied. \end{proof} \begin{remark} \rm We can see from \eqref{Ej.cua} (fixing $t$), that if $u_0=\lambda \psi$ with $\psi \in C_0(\mathbb{R}^N), \psi\geq 0, \psi \neq 0$, then condition \eqref{Cf} is satisfied when $\lambda>0$ is large. In other words, if initial data is large enough, then the corresponding solution of problem \eqref{Ej.dos} blows up in finite time. \end{remark} \begin{proof}[Proof of Theorem \ref{Th.Ej1}] (i) Note that \[ G(w)= \int_w^{\infty} \frac{d\sigma}{\exp(\alpha \sigma)-1} =-\frac{1}{\alpha} \ln [1-\exp(-\alpha w)]. \] Let $w_0>0$ such that $\ln (1-\exp(-\alpha w_0))=-1$. Set $u_0=\lambda \varphi_1$, where $\lambda\geq w_0e^{\lambda_1 \tau}$ and $\varphi_1$ is the first eigenfunction associated to first eigenvalue $\lambda_1$ of the Laplacian with Dirichlet condition on the boundary $\partial \Omega$. We suppose that $\|\varphi_1\|_\infty=1$. Hence, $\| S(\tau)u_0\|_\infty=\lambda e^{-\lambda_1 \tau}\geq w_0$. Thus, $G(\|S(\tau)u_0\|_\infty) \leq G(w_0)\leq \int_0^\tau h(\sigma)d\sigma$. From Theorem \ref{Th.uno}, the result follows. (ii) We use Theorem \ref{Th.dos}. Let $g(s)= \frac{e^{\alpha s}-1}{s}$ for all $s>0$ and let $\epsilon >0$ so that $\int_0^\infty h(\sigma)d\sigma<1/(\alpha +\epsilon)$. Since $\lim_{s\to 0^{+}} g(s)=\alpha $, there exist $s_0>0$ such that $g(s)<\alpha+\epsilon$ for all $0