\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 10, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/10\hfil Nonlinear convection] {Nonlinear convection in reaction-diffusion equations under dynamical boundary conditions} \author[G. Pincet Mailly, J.-F. Rault \hfil EJDE-2013/10\hfilneg] {Ga\"elle Pincet Mailly, Jean-Fran\c cois Rault} % in alphabetical order \address{Ga\"elle Pincet Mailly \newline LMPA Joseph Liouville FR 2956 CNRS, Universit\'e Lille Nord de France\\ 50 rue F. Buisson, B. P. 699, F-62228 Calais Cedex, France} \email{mailly@lmpa.univ-littoral.fr} \address{Jean-Fran\c cois Rault \newline LMPA Joseph Liouville FR 2956 CNRS, Universit\'e Lille Nord de France\\ 50 rue F. Buisson, B. P. 699, F-62228 Calais Cedex, France} \email{jfrault@lmpa.univ-littoral.fr} \thanks{Submitted July 10, 2012. Published January 9, 2013.} \subjclass[2000]{35K55, 35B44} \keywords{Nonlinear parabolic problem; dynamical boundary conditions; \hfill\break\indent lower and upper-solution; blow-up; global solution} \begin{abstract} We study the blow-up phenomena for positive solutions of nonlinear reaction-diffusion equations including a nonlinear convection term $\partial_t u = \Delta u - g(u) \cdot \nabla u + f(u)$ in a bounded domain of $\mathbb{R}^N$ under the dissipative dynamical boundary conditions $\sigma \partial_t u + \partial_\nu u =0$. Some conditions on $g$ and $f$ are discussed to state if the positive solutions blow up in finite time or not. Moreover, for certain classes of nonlinearities, an upper-bound for the blow-up time can be derived and the blow-up rate can be determined. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} We consider the nonlinear parabolic problem \begin{equation}\label{CRD} \begin{gathered} \partial_t u = \Delta u - g(u) \cdot \nabla u + f(u) \quad\text{in } \Omega \text{ for } t>0, \\ \sigma \partial_t u + \partial_\nu u =0 \quad\text{on } \partial \Omega \text{ for } t>0, \\ u(\cdot,0) = u_0 \geq 0 \quad \text{in } \overline{\Omega}, \end{gathered} \end{equation} where $g:\mathbb{R} \to \mathbb{R}^N$, $f:\mathbb{R} \to \mathbb{R}$, $\Omega$ is a bounded domain of $\mathbb{R^N}$ with $\mathcal{C}^2$-boundary $\partial \Omega$. We denote by $\nu:\partial \Omega \to \mathbb{R}^N$ the outer unit normal vector field, and by $\partial_\nu $ the outer normal derivative. These equations arise in different areas, especially in population growth, chemical reactions and heat conduction. For instance, in the case of a heat transfer in a medium $\Omega$, the first equation $\partial_t u = \Delta u - g(u) \cdot \nabla u + f(u)$ is a heat equation including a nonlinear convection term $g(u)\cdot \nabla u$ and a nonlinear source $f$. On the boundary $\partial \Omega$, if $\sigma$ is positive, the dynamical boundary conditions describe the fact that a heat wave with the propagation speed $\frac{1}{\sigma}$ is sent into the region into an infinitesimal layer near the boundary due to the heat flux across the boundary (see \cite{CE} and \cite{Goldstein}). There are various results in the literature about the theory of blow-up for semilinear parabolic equations, in particular for reaction-diffusion equations, see e.g. \cite{Vazquez,F,FML,K}. In this work, we discuss a problem involving a nonlinear convection term. Whereas a Burgers' equation has been studied in \cite{BMR} in the one-dimensional case, we now consider a more general convection term and we set in a regular domain of $\mathbb{R}^N$. After recalling some qualitative properties in Section \ref{S1}, we construct a global upper-solution for Problem \eqref{CRD} in Section \ref{SGE} and we deduce some conditions on $f$ and $g$ guaranteeing global existence of the solutions (Theorem \ref{GE_up}). In Section \ref{SBU}, we investigate two methods to ensure the blow-up of solutions of Problem \eqref{CRD}. The first one is an eigenfunction method valid for the model problem \begin{equation}\label{pbm2} \begin{gathered} \partial_t u = \Delta u - g(u) \cdot \nabla u + u^p \quad\text{in } \overline{\Omega} \text{ for } t>0, \\ \sigma \partial_t u + \partial_\nu u =0 \quad \text{on } \partial \Omega \text{ for } t>0, \\ u(\cdot,0) = u_0 \quad \text{in } \overline{\Omega}, \end{gathered} \end{equation} with $p>1$ (Theorem \ref{BU_up}). We also derive some upper bounds for the blow-up time. The second method, devoted to the problem \begin{equation}\label{pbm3} \begin{gathered} \partial_t u = \Delta u - g(u) \cdot \nabla u + e^{pu} \quad\text{in } \overline{\Omega} \text{ for } t>0, \\ \sigma \partial_t u + \partial_\nu u =0 \quad \text{on } \partial \Omega \text{ for } t>0, \\ u(\cdot,0) = u_0 \quad \text{in } \overline{\Omega}, \end{gathered} \end{equation} with $p>0$, requires a self-similar lower-solution which blows up in finite time (Theorem \ref{BUthm3}). We prove the blow-up of solutions of Problem \eqref{pbm3}. Finally, in Section \ref{SGO}, we determine the blow-up rate of the solutions of Problem \eqref{pbm2} in the $L^\infty$-norm when approaching the blow-up time (Theorem \ref{thmgo1}). Throughout, we shall assume the dissipativity condition \begin{equation}\label{sigma0} \sigma \geq 0 \quad \text{on } \partial \Omega \times (0,\infty). \end{equation} To study classical solutions, we always assume that the parameters in the equations of Problem \eqref{CRD} are smooth \begin{gather}\label{sigma1} \sigma \in \mathcal{C}^1_b(\partial \Omega \times (0,\infty) ) , \\ \label{reactionf} f \in \mathcal{C}^1(\mathbb{R} ), \quad f(s) >0 \quad \text{for } s> 0 , \\ \label{convectiong} g \in \mathcal{C}^1(\mathbb{R},\mathbb{R}^N ) . \end{gather} The initial data is continuous, non-trivial and non-negative in $\overline{\Omega}$ \begin{equation}\label{idata} u_0 \in \mathcal{C}(\overline{\Omega}), \quad u_0 \not \equiv 0, \quad u_0 \geq 0 . \end{equation} Let $T = T(\sigma, u_0)$ denote the maximal existence time of the unique maximal classical solution of Problem \eqref{CRD}, \[ u_\sigma \in \mathcal{C}(\overline{\Omega} \times [0,T) ) \cap \mathcal{C}^{2,1}(\overline{\Omega} \times (0,T) ) \] with the coefficient $\sigma$ in the boundary conditions and the initial data $u_0$. As for the well-posedness and the local existence of the solutions of Problem \eqref{CRD}, we refer to \cite{vBDC}, \cite{CE} and \cite{E}. From \cite{CE}, since the convection term depends linearly on the gradiant $\nabla u$ of the solution, the maximal existence time $T$ is the blow-up time of the solution with respect to the $L^\infty$-norm: \[ T = \inf \big\{ \ s>0 : \lim_{t \nearrow s} \sup_{\overline{\Omega}} |u(x,t)| = \infty \big\}\,. \] \section{Qualitative properties}\label{S1} The aim of this section is to compare the solutions for different parameters $\sigma$ and initial data $u_0$ and to summarize some positivity results on the classical solutions of Problem \eqref{CRD}. Using the maximum principle from \cite{vBDC}, we extend some results obtained in \cite{vBPM2} in the case of reaction-diffusion to our problem with convection. \begin{theorem}\label{ut>d} Assume hypotheses \eqref{sigma0}--\eqref{idata}. Suppose that $\sigma$ does not depend on time \begin{equation}\label{dts0} \sigma \in \mathcal{C}^1(\partial \Omega). \end{equation} Then the solution $u$ of Problem \eqref{CRD} satisfies \begin{gather*} u > 0 \quad\text{in } \overline{\Omega} \times (0,T(\sigma,u_0) ),\\ \partial_t u \geq 0 \quad\text{in } \overline{\Omega} \times [0,T(\sigma,u_0)),\\ \partial_t u > 0 \quad\text{in } \overline{\Omega} \times (0,T(\sigma,u_0) ). \end{gather*} Moreover, for all $\xi \in(0,T(\sigma,u_0))$, there exists $d>0$ such that \[ \partial_t u > d \quad\text{in } \overline{\Omega} \times [\xi,T(\sigma,u_0) ). \] \end{theorem} \begin{proof} Let $\tau \in (0,T(\sigma,u_0))$. Since $u$ is $\mathcal{C}^{2,1}(\overline{\Omega} \times [0,\tau])$ and because $f$ and $g$ are smooth (\eqref{reactionf} and \eqref{convectiong}), we can define these constants \[ C = \sup_{\overline{\Omega} \times [0,\tau]} g(u),\quad M = \sup_{\overline{\Omega} \times [0,\tau]} g'(u) \cdot \nabla u - f'(u) . \] First, the positivity principle \cite[Corollary 2.4]{vBDC} applied to Problem \eqref{CRD} implies $u \geq 0$ in $\overline{\Omega} \times [0,\tau]$ since $f\geq 0$ by condition \eqref{reactionf}. Thus we obtain \begin{gather*} \partial_t u \geq \Delta u - g(u) \cdot \nabla u \geq \Delta u - C |\nabla u| \quad \text{in } \Omega \text{ for } t>0, \\ \sigma \partial_t u + \partial_\nu u =0 \quad \text{on } \partial \Omega \text{ for } t>0, \\ u(\cdot,0) = u_0 \quad \text{in } \overline{\Omega}. \end{gather*} The strong maximum principle from \cite{vBDC} implies \[ m:=\min_{\overline{\Omega} \times [0,\tau]} u = \min_{\overline{\Omega} } u_0 \, , \] and if this minimum $m$ is attained in $\overline{\Omega} \times (0,\tau]$, $u\equiv m$ in $\overline{\Omega} \times [0,\tau]$. Since $f>0$ in $(0,\infty)$, the first equation in Problem \eqref{CRD} leads to $m=0$, and we obtain $u_0\equiv 0$, a contradiction with equation \eqref{idata}. Hence $u>m\geq 0$ in $\overline{\Omega} \times (0,\tau]$. Then, since the coefficients in the equations of Problem \eqref{CRD} are sufficiently smooth, classical regularity results in \cite{LSU} imply that $u \in \mathcal{C}^{2,2}(\overline{\Omega} \times [0,\tau])$. Thus $y=\partial_t u \in \mathcal{C}^{2,1}(\overline{\Omega} \times [0,\tau])$ and satisfies \begin{gather*} \partial_t y = \Delta y - g(u) \cdot \nabla y - (g'(u) \cdot \nabla u )y + f'(u)y \quad \text{in } \Omega \text{ for } t>0, \\ \sigma \partial_t y + \partial_\nu y =0 \quad \text{on } \partial \Omega \text{ for } t>0. \end{gather*} By continuity, condition \eqref{idata} implies $y(\cdot,0) \geq 0$ in $\overline{\Omega}$. Again, Corollary 2.4 from \cite{vBDC} implies $y\geq 0$ in $\overline{\Omega} \times [0,\tau]$. To apply properly the strong maximum principle, we have to introduce $w= y e^{Mt}\geq0$. By definition of $C$ and $M$, we obtain \begin{gather*} \partial_t w \geq \Delta w - g(u) \cdot \nabla w \geq \Delta w - C |\nabla w| \quad\text{in } \Omega \text{ for } t>0, \\ \sigma \partial_t w + \partial_\nu w \geq 0 \quad\text{on } \partial \Omega \text{ for } t>0. \end{gather*} Again, the strong maximum principle from \cite{vBDC} implies \[ \tilde{m}:=\min_{\overline{\Omega} \times [0,\tau]} w = \min_{\overline{\Omega} } w(\cdot,0) \,, \] and if this minimum $\tilde{m}$ is attained in $\overline{\Omega} \times (0,\tau]$, $w\equiv \tilde{m}$ in $\overline{\Omega} \times [0,\tau]$. In particular, if $\tilde{m}=0$, we have $\partial_t u \equiv 0$ in $\overline{\Omega} \times [0,\tau]$, thus $u(\cdot,t)=u_0$ for all $t \in [0,\tau]$. Hence $u$ attains its minimum in $\overline{\Omega} \times (0,\tau]$, which is impossible according to the first part of the proof. Thus $w$ and $\partial_t u$ are positive in $\overline{\Omega} \times (0,\tau]$. Finally, let $\xi \in (0,\tau)$. Because $y$ is continuous and thanks to the previous point, there exists $d>0$ such that $y(\cdot,\xi)>d$ in $\overline{\Omega}$. As $y$ satisfies \begin{gather*} \partial_t y = \Delta y - g(u) \cdot \nabla y - \Big( g'(u) \cdot \nabla u + f'(u) \Big) y\quad \text{in } \Omega \times [\xi,\tau], \\ \sigma \partial_t y + \partial_\nu y =0 \quad \text{on } \partial \Omega \times [\xi,\tau], \end{gather*} the weak maximum principle from \cite{vBDC} implies \[ \min_{\overline{\Omega} \times [\xi,\tau]} y = \min_{\overline{\Omega} } y(\cdot,\xi) \,. \] Hence $y>d$ in $\overline{\Omega} \times [\xi,\tau]$. Note that $d$ depends only on $\xi$, not on $\tau$. Without this step, we only have $y \geq \tilde{m} e^{-M\tau}$ which may vanish as $\tau \to T(\sigma,u_0)$. \end{proof} Let $0 \leq \sigma_1 \leq \sigma_2$ be two coefficients satisfying condition \eqref{sigma1}, $v_0 \leq u_0$ be two initial data fulfilling hypothesis \eqref{idata} and $w_0$ a function in $\mathcal{C}_0(\overline{\Omega})$ with $0 \leq w_0 \leq v_0$. Denote by $u_{\sigma_1}$, $u_{\sigma_2}$, $v$ and $w$ the maximal solutions of the following four problems \begin{gather*} \partial_t u_{\sigma_1} = \Delta u_{\sigma_1} - g(u_{\sigma_1}) \cdot \nabla u_{\sigma_1} + f(u_{\sigma_1}) \quad \text{in } \Omega \text{ for } t>0, \\ \sigma_1 \partial_t u_{\sigma_1} + \partial_\nu u_{\sigma_1} =0 \quad \text{ on } \partial \Omega \text{ for } t>0, \\ u_{\sigma_1}(\cdot,0) = u_0 \quad \text{ in } \overline{\Omega}; \end{gather*} \begin{gather*} \partial_t u_{\sigma_2} = \Delta u_{\sigma_2} - g(u_{\sigma_2}) \cdot \nabla u_{\sigma_2} + f(u_{\sigma_2}) \quad \text{in } \Omega \text{ for } t>0, \\ \sigma_2 \partial_t u_{\sigma_2} + \partial_\nu u_{\sigma_2} =0 \quad \text{on } \partial \Omega \text{ for } t>0, \\ u_{\sigma_2}(\cdot,0) = u_0 \quad \text{in } \overline{\Omega}; \end{gather*} \begin{gather*} \partial_t v = \Delta v - g(v) \cdot \nabla v + f(v) \quad\text{ in } \Omega \text{ for } t>0, \\ \sigma_2 \partial_t v + \partial_\nu v =0 \quad \text{ on } \partial \Omega \text{ for } t>0, \\ v(\cdot,0) = v_0 \quad \text{ in } \overline{\Omega}; \end{gather*} and \begin{gather*} \partial_t w = \Delta w - g(w) \cdot \nabla w + f(w) \quad \text{ in } \Omega \text{ for } t>0, \\ w =0 \quad \text{on } \partial \Omega \text{ for } t>0, \\ w(\cdot,0) = w_0 \quad \text{in } \overline{\Omega}. \end{gather*} Let $T(\sigma_1,u_0)$, $T(\sigma_2,u_0)$, $T(\sigma_2,v_0)$ and $T(w_0)$ be their respective maximal existence times. For the reader convenience, we recall some results stemming from the comparison principle \cite{vBDC}. \begin{theorem}[\cite{vBPM}]\label{compare_thm} Under the aforementioned hypotheses, we have \begin{gather*} T(\sigma_2,u_0) \leq T(\sigma_2,v_0) \leq T(w_0),\\ 0\leq w \leq v \leq u_{\sigma_2} \quad\text{in } \overline{\Omega} \times [0,T(\sigma_2,u_0) ) \,. \end{gather*} In addition, if $u_0 \in \mathcal{C}^2(\overline{\Omega})$ with \begin{equation}\label{idata2} \Delta u_0 - g(u_0)\cdot \nabla u_0 + f(u_0) \geq 0 \text{ in } \Omega , \end{equation} we have \begin{gather*} T(\sigma_1,u_0) \leq T(\sigma_2,u_0),\\ u_{\sigma_2} \leq u_{\sigma_1} \quad \text{in } \overline{\Omega} \times [0,T(\sigma_1,u_0) ) \,. \end{gather*} \end{theorem} An important fact comes from the last statement of Theorem \ref{ut>d}. For any positive solution $u$ of Problem \eqref{CRD}, the maximum principle implies that for any $s \in (0 , T(\sigma,u_0)) $, there exists $c>0$ such that $u (\cdot,s) \geq c$ in $\overline{\Omega}$. Then, consider the solution $\tilde{u}$ of \eqref{CRD} with the constant initial data $c$ and $\tilde{\sigma} = \sup \sigma$ in the boundary conditions. Theorem \ref{compare_thm} implies $\tilde{u} \leq u$. Since $c$ satisfies \eqref{idata2}, Theorem \ref{ut>d} leads to $\partial_t \tilde{u} > d>0$. Thus, $\tilde{u}$ can be big enough after a long time (maybe it blows up). So does $u$, even if $u_0$ does not satisfy condition \eqref{idata2}. \section{Existence of global solutions}\label{SGE} In this section, we give some conditions on the function $g$ in the convection term, which ensure the existence of global solutions to Problem \eqref{CRD} for various reaction terms $f$. We use the comparison method from \cite{vBDC}. Thus, we just need to find an appropriate upper-solution of Problem \eqref{CRD} which does not blow up. This is our first lemma. \begin{lemma}\label{supersol} Let $\alpha>0$ and $K>0$ be two real numbers and let $\eta \in \mathcal{C}^1([0,\infty))$ with $\eta' \geq \alpha^2$. For any integer $1 \leq j \leq N$, the function $U$ defined in $\Omega \times [0,\infty)$ by \[ U(x,t) = K \exp \big( \alpha x_j + \eta(t) \big) , \] satisfies \begin{gather*} \partial_t U \geq \Delta U - g(U) \cdot \nabla U + f(U) \quad\text{in } \Omega \text{ for } t>0, \\ \sigma \partial_t U + \partial_\nu U \geq 0 \quad\text{on } \partial \Omega \text{ for } t>0, \\ U(\cdot,0) > 0 \quad\text{in } \overline{\Omega}, \end{gather*} if \begin{equation}\label{Conv>RD} \alpha g_j (\omega) \geq \frac{f(\omega)}{\omega} \quad \text{for all } \omega \geq 0 \end{equation} and if \begin{equation}\label{sigmamin} \sigma (x,t) \geq \frac{\alpha}{\eta'(t)} \quad \text{for all } t>0. \end{equation} \end{lemma} \begin{proof} A simple computation of the derivatives of $U$ leads us to \[ \partial_t U - \Delta U + g(U) \cdot \nabla U = \big( \eta' - \alpha^2 \big) U + \alpha g_j(U) U \quad\text{in } \Omega \text{ for } t>0. \] Since we assume $\eta' \geq \alpha^2$, hypothesis \eqref{Conv>RD} implies \[ \partial_t U - \Delta U + g(U) \cdot \nabla U -f(U) \geq 0 \quad\text{in } \Omega \times (0,\infty). \] Furthermore, on the boundary $\partial \Omega$, for $t>0$, we have \begin{equation}\label{U_bOm} \begin{aligned} \sigma \partial_t U +\partial_\nu U & = \big( \sigma \eta'(t)+ \alpha \nu_j(x) \big) U \\ &\geq \big( \sigma \eta'(t)- \alpha \big) U \geq 0 , \end{aligned} \end{equation} by hypothesis \eqref{sigmamin} since $\nu$ is normalized, and clearly $U(x,0) =K \exp \big( \alpha x_j + \eta(0) \big) >0$ in $\overline{\Omega}$. \end{proof} \begin{remark}\rm In the case of the Dirichlet boundary conditions, we can use this upper-solution with the special choice $\eta \equiv 0$ (see \cite{QS}). However, for the dynamical boundary conditions, we must use a positive time-dependent $\eta$ because our solutions are not bounded, see Theorem \ref{ut>d}. \end{remark} Now we can state the following theorems for a nonlinear reaction term $f$ growing as a power of $u$ (Problem \eqref{pbm2}), or as an exponential function (Problem \eqref{pbm3}). \begin{theorem}\label{GE_up} Let $\sigma$ be a coefficient fulfilling conditions \eqref{sigma0}, \eqref{sigma1} and such that there exists $\delta >0$ with $$ \inf_{\partial \Omega} \sigma \geq \delta \sup_{\partial \Omega} \sigma \quad \text{for } t>0 \quad\text{and } \quad \Big(\sup_{x\in \partial \Omega} \sigma(x,\cdot)\Big)^{-1} \in L^1_{\rm loc}(\mathbb{R}^+) . $$ Assume $u_0$ satisfies condition \eqref{idata}. If there exists an integer $1 \leq j \leq N$ such that \begin{equation}\label{Conv>RD2} \liminf_{\omega \to \infty} \frac{g_j(\omega)}{\omega^{p-1}} > 0, \end{equation} then the solution of Problem \eqref{pbm2} is a global solution. \end{theorem} \begin{proof} In view of Theorem \ref{ut>d} and \eqref{Conv>RD2}, we can suppose that $u_0$ is sufficiently big such that there exists $C>0$ with \[ g_j(u) \geq C u^{p-1} \text{ in } \Omega \text{ for } t>0. \] For \[ \eta (t) = C\delta^{-1} \int_0^t \Big( \sup_{x\in \partial \Omega} \sigma(x,s)\Big)^{-1} \,ds + C^2 t. \] we have $\eta' \geq C^2$ and \eqref{sigmamin} is satisfied. Let $K$ be a positive number such that \[ K \geq u_0 (x) e^{-C x_j -\eta(0)} \quad \text{for all } x \in \overline{\Omega}. \] Then by hypotheses \eqref{sigma1}, \eqref{idata} and \eqref{sigmamin}, the function $U$ defined in Lemma \ref{supersol} is an upper-solution of \eqref{pbm2} since $U(\cdot,0) \geq u_0$ in $\overline{\Omega}$. Using the comparison principle from \cite{vBDC}, the unique solution $u$ of Problem \eqref{CRD} satisfies $$ 0 \leq u(x,t) \leq U(x,t) \quad \text{for all } x \in \overline{\Omega} \text{ and } t>0 \,, $$ thus $u$ does not blow up. \end{proof} This theorem holds in particular for a nonlinearity $g$ in the form \[ g(u)=( \alpha_1 u^{q_1} , \dots, \alpha_i u^{q_i}, \dots, \alpha_N u^{q_N} ) \] with at least one integer $j$ such that $\alpha_j >0$ and $q_j \geq p-1$. A similar result can be derived for Problem \eqref{pbm3}. \begin{theorem} Under the aforementioned assumptions, the solution of Problem \eqref{pbm3} is a global solution if the convection term $g(u)\cdot \nabla u$ has (at least) one component $g_j$ satisfying $g_j(u)= \alpha_j e^{q_j u}$ with $\alpha_j >0$ and $q_j>p$. \end{theorem} \begin{proof} Thanks to $q_j>p$, condition \eqref{Conv>RD} is fulfilled because $\alpha_j e^{q_j u} \geq \alpha_j e^{p u}/u$ for $u$ sufficiently big. \end{proof} \begin{remark} \rm Condition \eqref{Conv>RD2} is optimal for Problem \eqref{pbm2}, see Theorems \ref{GE_up} and \ref{BU_up}. But it can be improved in some special cases, for example, if the reaction term is $f(u)=u \ln u$. Lemma \ref{supersol} implies that all solutions of Problem \eqref{CRD} are global if one component $g_j$ of $g$ satisfies $g_j(u) \geq \alpha_j \ln u$. In fact, in that case, every positive solution of \eqref{CRD} is global, without any assumption on the convection term $g$, since $\int_c ^\infty 1/f(y) \, dy = \infty$ for $c>0$, see \cite[Theorem 3.2]{CE}. \end{remark} Condition \eqref{sigmamin} on $\sigma$ allows us to consider fast decaying functions $\sigma$, but, to ensure global existence, it is essential that $\sigma$ does not vanish on the whole $\partial \Omega$. Indeed let us prove the following blow-up result related to the Neumann boundary conditions, for $\sigma \equiv 0$ on $\partial \Omega$. \begin{theorem} Assume that $\sigma \equiv 0$, $u_0$ fulfills hypothesis \eqref{idata} and $f $ is positive in $(0,\infty)$ such that \begin{equation}\label{NeumanBU} \int_c^\infty \frac{1}{f(y)} \,dy < \infty \quad \text{for some } c>0 \,. \end{equation} Then every positive solution of \eqref{CRD} blows up in finite time. \end{theorem} \begin{proof} Let $u$ be a non-trivial positive solution of \begin{equation}\label{NeuCRD} \begin{gathered} \partial_t u = \Delta u - g(u) \cdot \nabla u + f(u) \quad\text{in } \Omega \text{ for } t>0, \\ \partial_\nu u =0 \quad\text{on } \partial \Omega \text{ for } t>0, \\ u(\cdot,0) = u_0 \quad\text{in } \overline{\Omega}. \end{gathered} \end{equation} Using the maximum principle from \cite{vBDC}, we have $u(\cdot,\xi)>0$ in $\overline{\Omega}$ for $\xi>0$. Hence, without loss of generality, we suppose $u_0 > c$ in $\overline{\Omega}$. Now, consider the maximal solution $z$ of the ODE $\dot{z} = f(z)$ with the initial data $ z(0)= \inf \{ u_0(x): x\in \overline{\Omega} \} $. Condition \eqref{NeumanBU} implies that its maximal existence time $T_z$ is finite: \[ T_z=\int_{z(0)} ^\infty \frac{1}{f(y)} \,dy < \infty . \] Since $\nabla z= 0$, $z$ is a lower solution of Problem \eqref{NeuCRD}. Using the comparison principle from \cite{vBDC}, we obtain $z(t) \leq u(\cdot,t)$ in $\overline{\Omega}$ for $t>0$. Thus, $u$ must blow up in finite time with $00$ and $q0. \end{equation} By applying the eigenfunction method (see \cite{vBPM,F,K}), we obtain some conditions on the initial data $u_0$ which guarantee the finite time blow-up and we derive some upper bounds for the blow-up times. This is a general technique which can be applied to the following problem, where the boundary behaviour of the solutions is not involved: \begin{equation}\label{pbm2positive} \begin{gathered} \partial_t u = \Delta u - g(u) \cdot \nabla u + u^p \quad\text{in } \overline{\Omega} \text{ for } t>0, \\ u \geq 0 \quad\text{on } \partial \Omega \text{ for } t>0, \\ u(\cdot,0) = u_0 \quad\text{in } \overline{\Omega}. \end{gathered} \end{equation} Henceforth, we denote by $\lambda$ the first eigenvalue of $-\Delta$ in $H_0^1(\Omega)$ and by $\varphi$ an eigenfunction associated to $\lambda$ satisfying \begin{equation}\label{phi} \varphi \in H_0^1(\Omega) ,\quad 0<\varphi \leq 1 \quad \text{in } \Omega. \end{equation} \begin{theorem}\label{thmbu1} Let $\alpha>0$, $1 (2|\Omega|^{p-1}C)^{1/p} \end{equation} with \begin{gather*} C= (p-1)|\Omega|\Big(\frac{4\lambda}{p-q}\Big)^\frac{1}{p-1} + \Big(\frac{4q}{p-q}\Big)^\frac{q}{p-q}\alpha^m \int_\Omega |\nabla \varphi|^m \,dx \,, \end{gather*} then the maximal classical solutions $u$ of Problem \eqref{pbm2positive} blow up in finite time $T$ satisfying \begin{equation}\label{Tborne} T\leq \frac{2 \int_\Omega u_0\varphi^m\,dx}{(p-1)\Big(|\Omega|^{1-p} \Big(\int_\Omega u_0\varphi^m\,dx\Big)^p-2C \Big)}=:\tilde T. \end{equation} \end{theorem} \begin{proof} Define \[ M(t)=\int_\Omega u(x,t)\varphi(x)^m\,dx. \] Thus, \[ \dot{M}(t)=\int_\Omega \Delta u\varphi^m\,dx-\int_\Omega g(u)\cdot\nabla u\, \varphi^m\,dx +\int_\Omega u^p\varphi^m\,dx. \] First, we prove that \begin{equation}\label{eqdelta2} \int_\Omega \Delta u\varphi^m\,dx\geq -m\lambda|\Omega|^{\frac{p-1}{p}}\Big(\int_\Omega u^p\varphi^m\,dx\Big)^{1/p}. \end{equation} Observe that the behaviour of $\varphi$ and $\partial_\nu\varphi$ on $\partial\Omega$ imply \begin{equation}\label{phinu} \int_{\partial \Omega} \partial_\nu u \varphi^m \,ds = 0\quad\text{and} \quad \int_{\partial \Omega} u\partial_\nu(\varphi^m) \,ds \leq 0 , \end{equation} since $u\geq 0$ on $\partial\Omega$ for $t>0$. As in \cite{QS}, Equation \eqref{phinu} and Green's formula yield \begin{equation}\label{modif_coro} \int_\Omega \Delta u\varphi^m\,dx\geq -m\lambda \int_\Omega u\varphi^m\,dx. \end{equation} Since $\varphi \leq 1$, $\int_\Omega u\varphi^m\,dx\leq\int_\Omega u\varphi^{\frac{m}{p}}\,dx$ and by H\"older's inequality, \eqref{eqdelta2} holds.\\ Now, we show that \begin{equation}\label{nabla} -\int_\Omega g(u)\cdot\nabla u\,\varphi^m\,dx \geq -m\alpha \Big(\int_\Omega |\nabla\varphi|^m \,dx\Big)^{1/m} \Big(\int_\Omega u^p\varphi^m\,dx\Big)^{q/p}. \end{equation} By Green's formula and by definition of $G$ and $\varphi$, we have \[ -\int_\Omega g(u)\cdot\nabla u\,\varphi^m\,dx =-\int_\Omega \operatorname{div}(G(u))\varphi^m\,dx = m\int_\Omega (G(u)\cdot \nabla\varphi) \varphi^{m-1} \,dx \,. \] Equation \eqref{G0$ and $ \varepsilon>0$, \[ a= \frac{\varepsilon a}{\varepsilon} \leq \frac{\varepsilon^r a^r}{r} + \frac{1}{s\varepsilon^s}\] for $r,s>1$ with $r^{-1} + s^{-1} =1$. It yields \[ C_1 \Big( \int_\Omega u^p\,\varphi^m\,dx\Big)^{1/p} \leq\frac{1}{4}\int_\Omega u^p\,\varphi^m\,dx +\underbrace{\frac{p-1}{p \varepsilon_1^\frac{p}{p-1}}}_{:=C_3}, \] and in the same way we have \[ C_2 \Big( \int_\Omega u^p\,\varphi^m\,dx\Big)^{q/p} \leq \frac{1}{4} \int_\Omega u^p\,\varphi^m\,dx+C_4, \] with \[ C_4=\frac{1}{m \varepsilon_2^m}\,. \] Then \[ \dot{M}(t)\geq\frac{1}{2}\int_\Omega u^p\,\varphi^m\,dx-C \] with $C=C_3+C_4>0$. By \eqref{phi} and H\"older's inequality, we obtain that \[ \dot{M}(t)\geq\frac{1}{2}|\Omega|^{1-p}M^p-C. \] Since $M$ is increasing with respect to $t$, by \eqref{eqphiu0} we have \[ \dot{M}(t)\geq\Big( \frac{1}{2}|\Omega|^{1-p} -CM(0)^{-p} \Big) M^p , \] and we can conclude that $u$ can not exist globally. To derive an upper bound for the blow-up time, we integrate the previous differential inequality between 0 and $t>0$. We obtain \[ M(t)\geq \Big(M(0)^{1-p}-(p-1)\Big( \frac{1}{2}|\Omega|^{1-p} -CM(0)^{-p} \Big) t \Big)^{\frac{-1}{p-1}}. \] Hence $M$ blows up before $\tilde{T}= M(0)^{1-p}(p-1)^{-1} \big( \frac{1}{2}|\Omega|^{1-p} -CM(0)^{-p} \big)^{-1}$, so does $u$. Thus, $T\leq \tilde T$. \end{proof} Note that Condition \eqref{eqphiu0} on the initial data is only necessary to derive an upper bound for the maximal existence time. Thanks to Theorem \ref{ut>d}, we obtain the following result. \begin{theorem}\label{BU_up} Let $qd} permits to ensure that there exist $t_0>0$ and $C>0$ such that $u(\cdot,t_0)$ is big enough to satisfy Equation \eqref{eqphiu0} and $G(u) \leq C u^q$ in $\Omega$ for $t>t_0$. Thus applying Theorem \ref{thmbu1} to $v(x,t)=u(x,t+t_0)$, we prove that $v$ blows up in a finite time $T_v$ satisfying \eqref{Tborne}. Hence, $u$ blows up in a finite time $T_u=t_0 +T_v$.\qed \end{proof} Now, we prove the blow-up of positive solutions of Problem \eqref{pbm3}. \begin{theorem}\label{BUthm3} Assume $\sigma$ and $u_0$ satisfy conditions \eqref{sigma0}, \eqref{sigma1}, \eqref{convectiong} and \eqref{idata}. If \[ \limsup_{\omega \to \infty} \frac{|g(\omega)|}{e^{q\omega}} < \infty, \] then all the positive solutions of Problem \eqref{pbm3} blow up in finite time. \end{theorem} \begin{proof} Let $u$ be a positive solution of Problem \eqref{pbm3} and define $v=e^{\gamma u}$ with $\gamma \in (q,p)$ and $\gamma>1/2$. As in the previous proof, we suppose that $u$ is sufficiently big such that for some $C>0$ \begin{equation}\label{HBU2} |g(u)| \leq C e^{qu} \quad \text{in } \Omega \text{ for } t>0. \end{equation} Computing the derivatives of $v$, we obtain \[ \partial_ t v = \Delta v - \frac{1}{v} |\nabla v|^2 - g(u)\cdot \nabla v + \gamma v^{\frac{p+\gamma}{\gamma}} \quad \text{in } \Omega \text{ for } t>0. \] Using condition \eqref{HBU2}, we obtain \[ \partial_ t v \geq\Delta v - \frac{1}{ v} |\nabla v|^2 - C v^{q/\gamma}|\nabla v| + \gamma v^{\frac{p+\gamma}{\gamma}} \quad \text{ in } \Omega \text{ for } t>0. \] Young's inequality \[ C v^{q/\gamma}|\nabla v| \leq \frac{C^2}{2} |\nabla v | ^2 + \frac{1}{2} v ^{2q/\gamma}, \] leads to \[ \partial_ t v \geq \Delta v - \frac{2+C^2}{2} |\nabla v|^2 + \gamma v^{\frac{p+\gamma}{\gamma}} - \frac{1}{2}v^{\frac{2q}{\gamma}} \quad \text{in } \Omega \text{ for } t>0, \] since $v\geq 1$. Morevover, we have \[ \gamma v^{\frac{p+\gamma}{\gamma}} - \frac{1}{2}v^{\frac{2q}{\gamma}} \geq (\gamma -\frac{1}{2}) v^\frac{p+\gamma}{\gamma} \] by definition of $\gamma$. Thus, we obtain \begin{equation}\label{pbmSW} \begin{gathered} \partial_t v \geq \Delta v - \mu |\nabla v|^2 + \kappa v^\frac{p+\gamma}{\gamma} \quad\text{in } \overline{\Omega} \text{ for } t>0, \\ v \geq 0 \quad\text{on } \partial \Omega \text{ for } t>0, \\ v(\cdot,0) > 0 \quad\text{in } \overline{\Omega} , \end{gathered} \end{equation} with $\mu =(2+C^2)/2$ and $\kappa=\gamma-1/2$. Without loss of generality (see Theorem \ref{ut>d}), we can suppose that $v(\cdot,0) \geq V(\cdot,0)$ in $\overline{\Omega}$, where \[ V(x,t) = (1-\varepsilon t)^\frac{-1}{p-1} W\Big( \frac{ |x| }{(1-\varepsilon t)^m}\Big), \] with $0\frac{1}{m(p-1)}$ and $\varepsilon < \frac{2\kappa (p-1)}{2+A}$. According to Souplet \& Weissler \cite{SW}, $V$ is a blowing-up sub-solution for Problem \eqref{pbmSW}. By the comparison principle from \cite{vBDC}, $v \geq V$ and $u$ blows up in finite time. \end{proof} \begin{remark}\rm In this section, we point out the accelerating effect of the dynamical boundary conditions, in comparison with the Dirichlet boundary conditions. Indeed, we prove that, even if the initial data $u_0$ is small, the solutions of Problem \eqref{pbm2} blow up in finite time. But, if we replace the dynamical boundary conditions by the Dirichlet boundary conditions in the second equation of Problem \eqref{pbm2}, it is well known that the solutions are global and decay to $0$ if the initial data are small enough, see for instance references \cite{Straughan} and \cite{Weissler}. \end{remark} \section{Growth Order}\label{SGO} In this section, we are interested in the blow-up rate for Problem \eqref{pbm2} when approaching the blow-up time $T$. For the convection term, we assume that \begin{equation}\label{eqg2} g(u)=(g_1(u),\dots,g_n(u)) \quad \text{with } g_i(u)=u^q \; \forall i=1,\dots,n, \ 1q+1$, valid for any non-negative initial data $u_0\in \mathcal{C}(\overline{\Omega})$. \begin{lemma} Let $p>q+1$, and assume hypotheses \eqref{sigma0}--\eqref{idata}. Then the classical maximal solution $u$ of \eqref{pbm2} satisfies \[ \|u(\cdot,t)\|_\infty\geq (p-1)^{\frac{-1}{p-1}}(T-t)^{\frac{-1}{p-1}} \] for $02q+1>3$, when the time $t$ approaches the blow-up time $T$. \begin{theorem}\label{thmgo1} Suppose conditions \eqref{sigma0}, \eqref{idata}, \eqref{dts0} and \eqref{eqg2} are fulfilled. For \begin{equation}\label{p>p*} p>2q+1\,, \end{equation} there exists a positive constant $C$ such that the classical maximal solution $u$ of \eqref{pbm2} satisfies \[ \|u(\cdot,t)\|_\infty \leq \frac{C}{(T - t)^{1/p-1}} \quad for\,\,t\in [0,T). \] \end{theorem} \begin{proof} Let $\beta>1$ such that \begin{equation}\label{eqpq} p(p-1)(p-2q-1) = \frac{Nq^2}{\beta} >0 , \end{equation} and choose $M>1$ such that \[ M \geq \frac{Nq}{2(2q+1)}\beta^{\frac{2q}{p-2q-1}}. \] First, for $\xi\in\,(0,T)$, we shall prove that there exists $\delta>0$ such that \[ \partial_tu\geq\delta e^{-Mt}(u^p+\beta u^{2q+1}) \] in $\overline{\Omega}\times[\xi,T)$. Introduce \[ J=\partial_tu-\delta d(t)k(u) \] with $d(t)=e^{-Mt}$ and $k(u)=u^p+\beta u^{2q+1}$. Note that classical regularity results from \cite{LSU} yield $J\in\ \mathcal{C}^{2,1}\left(\overline{\Omega}\times [\xi,T)\right)$. We recall that Theorem \ref{ut>d} implies that there exists $c>0$ such that $\partial_t u\geq c>0$ in $\overline{\Omega}\times[\xi,T)$. Thus, we can choose $\delta>0$ sufficiently small such that \[ J(\cdot,\xi)\geq 0 \quad \text{in } \overline{\Omega}. \] The function $J$ satisfies the boundary condition \begin{align*} \sigma \partial_t J + \partial_\nu J &=\partial_t ( \sigma \partial_t u + \partial_\nu u) -\delta dk'(u)( \sigma \partial_t u + \partial_\nu u) -\sigma \delta d'k(u) \\ &=\sigma \delta Me^{-Mt}k(u)\geq 0. \end{align*} Furthermore, $J$ satisfies \[ \partial_t J-\Delta J+g(u)\cdot\nabla J-(pu^{p-1}- g'(u)\cdot\nabla u)J =\delta dH(u)\text{ in } \overline{\Omega}\times [\xi,T), \] where \[ H(u):=pu^{p-1}k(u)-k'(u)u^p+k^{''}(u)|\nabla u|^2-\frac{d'}{d}k(u)-k(u) g'(u)\cdot\nabla u. \] To prove that $H(u)\geq 0$, we shall show that \begin{equation}\label{eqH} \begin{aligned} q\sqrt{N}u^{q-1}|\nabla u|(u^p+ \beta u^{2q+1}) &\leq M(u^p+ \beta u^{2q+1})+ \beta(p-2q-1)u^{p+2q} \\ &\quad +(p(p-1)u^{p-2}+2q(2q+1)\beta u^{2q-1})|\nabla u|^2. \end{aligned} \end{equation} Inequality \eqref{eqH} is trivial in the case where $M\geq q\sqrt{N}u^{q-1}|\nabla u|$. Now, suppose that $M< q\sqrt{N}u^{q-1}|\nabla u|$. When $q\sqrt{N}u^{q+1}\leq 2q(2q+1)|\nabla u|$, we have $q\sqrt{N}u^{q-1}u^p|\nabla u|\leq p(p-1)u^{p-2}|\nabla u|^2$ and $q\sqrt{N}u^{3q}|\nabla u|\leq 2q(2q+1)u^{2q-1}|\nabla u|^2$ since $p>3$ then \eqref{eqH} follows. In the case where $q\sqrt{N}u^{q+1}> 2q(2q+1)|\nabla u|$, since \[ u>\Big(\frac{2(2q+1)}{Nq}M\Big)^{1/2q}\geq\beta ^{\frac{1}{p-2q-1}}, \] we obtain \begin{equation}\label{eq2546} u^p+ \beta u^{2q+1}\leq 2 u^p. \end{equation} Moreover, \eqref{eqpq} yields \begin{align*} 2 \sqrt{N}\,qu^{q+1}|\nabla u| & = 2\sqrt{\beta p(p-1)(p-2q-1)}\,u^{q+1}|\nabla u|\\ & \leq \Big(\sqrt{\beta (p-2q-1)}\,u^{q+1}-\sqrt{p(p-1)}\,|\nabla u|\Big)^2\\ &\quad + 2\sqrt{\beta p(p-1)(p-2q-1)}\,u^{q+1}|\nabla u|\\ & \leq \beta (p-2q-1)u^{2(q+1)}+p(p-1)|\nabla u|^2. \end{align*} Thus, multiplying by $u^{p-2}$ leads to \[ 2 \sqrt{N}qu^{q-1}|\nabla u|u^p\leq \beta(p-2q-1)u^{p+2q} +p(p-1)u^{p-2}|\nabla u|^2, \] and by \eqref{eq2546}, the inequality \eqref{eqH} holds. Finally, we can conclude by the comparison principle from \cite{vBDC} that $J\geq 0$ in $\overline{\Omega}\times [\xi,T)$, in particular, $\partial_t u \geq \varepsilon u^p$ with $\varepsilon>0$. Now, we shall derive the upper blow-up rate estimate of $\|u(\cdot,t)\|_\infty$ for $t\in\,[\xi,T)$. For each $x \in \Omega$, the integral \[ \int_t^\tau \frac{\partial_t u(x,s)}{u^p(x,s) }\,ds = \int_{u(x,t)}^{u(x,\tau)} \frac{1}{\eta^p}\,d\eta \] converges as $\tau \to T$. Integrating the inequality $\partial_t u \geq \varepsilon u^p$ leads to \[ \varepsilon (\tau -t) \leq \frac{u(x,\tau)^{1-p} - u(x,t)^{1-p}}{1-p} \leq \frac{u(x,t)^{1-p}}{p-1} \,. \] Letting $\tau \to T$ implies $u(x,t) \leq \Big( \varepsilon (p-1)(T-t) \Big)^\frac{-1}{p-1}$ and we can conclude as in the proof of Theorem 2.3 from \cite{BMR}. \end{proof} \subsection*{Acknowledgments} The authors would like to thank Dr. Mabel Cuesta for her helpful discussions and valuable suggestions. \begin{thebibliography}{00} \bibitem{BL} C. Bandle, H. Levine; \emph{Fujita type results for convective-like reaction diffusion equations in exterior domains.} ZAMP \textbf{40} (1989), 665--676. \bibitem{vBDC} J. von Below, C. 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