\documentclass[twoside]{article} \usepackage{amssymb} % font used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Parabolic equations with gradient nonlinearities \hfil EJDE--2001/20} {EJDE--2001/20\hfil Philippe Souplet \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2001}(2001), No. 20, pp. 1--19. \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} \\ % Recent results and open problems on parabolic equations with gradient nonlinearities % \thanks{ {\em Mathematics Subject Classifications:} 35K55, 35B35, 35B40, 35B33, 35J60. \hfil\break\indent {\em Key words:} nonlinear parabolic equations, gradient term, finite time blowup, \hfil\break\indent global existence. \hfil\break\indent \copyright 2001 Southwest Texas State University. \hfil\break\indent Submitted February 19, 2001. Published March 26, 2001.} } \date{} % \author{ Philippe Souplet } \maketitle \begin{abstract} We survey recent results and present a number of open problems concerning the large-time behavior of solutions of semilinear parabolic equations with gradient nonlinearities. We focus on the model equation with a dissipative gradient term $$u_t-\Delta u=u^p-b|\nabla u|^q,$$ where $p$, $q>1$, $b>0$, with homogeneous Dirichlet boundary conditions. Numerous papers were devoted to this equation in the last ten years, and we compare the results with those known for the case of the pure power reaction-diffusion equation ($b=0$). In presence of the dissipative gradient term a number of new phenomena appear which do not occur when $b=0$. The questions treated concern: sufficient conditions for blowup, behavior of blowing up solutions, global existence and stability, unbounded global solutions, critical exponents, and stationary states. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \renewcommand{\theequation}{\thesection.\arabic{equation}} \catcode`@=11 \@addtoreset{equation}{section} \catcode`@=12 \section{Introduction} The large-time behavior of solutions of nonlinear reaction-diffusion equations has received considerable interest since the $60$'s. A model case of such equation is \begin{equation} u_t-\Delta u=|u|^{p-1}u. \label{RD} \end{equation} Various sufficient conditions for blowup and global existence were provided and qualitative properties were investigated, such as: nature of the blowup set, rate and profile of blowup, maximum existence time and continuation after blowup, boundedness of global solutions and convergence to a stationary state. We refer for these to the books and survey articles \cite{BE1, SGKM, L2, Vo, St, DL}. More recently, a number of works have addressed the same type of questions for semilinear parabolic equations where the nonlinearity also depends on the spatial derivatives of $u$. A rough and partial classification of such equation can be made according to two criteria. The first one is the nature the gradient dependence of the nonlinearity, namely, through a convective term, like $a\cdot\nabla(u^q)$, or through a term of Hamilton-Jacobi type $b|\nabla u|^q$. The second criterion is the presence (or not) of a reaction term, like $u^p$. Typical equations resulting from the combination of these criteria are \begin{eqnarray} &u_t-\Delta u=a.\nabla (u^q),& \label{CD}\\ &u_t-\Delta u=u^p+a.\nabla (u^q), &\label{CRD}\\ &u_t-\Delta u= b|\nabla u|^q, &\label{VHJ}\\ &u_t-\Delta u=u^p- b|\nabla u|^q. &\label{CW} \end{eqnarray} (Here $u^p\equiv |u|^{p-1}u$, $a\in \mathbb{R}^N$, $b\in \mathbb{R}$.) Each of these equations has been rather well studied in the past ten years. However, reviewing all of them would be somehow too dispersive, and we prefer to focus on one particular equation, which already provides a rich variety of aspects. The purpose of this article is thus to survey the existing literature on the equation (CW). We refer the interested reader to \cite{EVZ} for (\ref{CD}), \cite{AE} for (\ref{CRD}), \cite{BK} for (\ref{VHJ}), and to the references in these papers. Outside of this classification, let us also mention the equation $$u_t-u_{xx}=f(u)|u_x|^{q-1}u_x,$$ which exhibits interesting phenomena (related to derivative blowup -- see e.g. \cite{AF, S5}). We will consider the associated initial-boundary value problem of Dirichlet type: \begin{eqnarray} &u_t-\Delta u=|u|^{p-1}u-b|\nabla u|^q, \quad t>0,\ x\in\Omega,&\nonumber \\ &u(t,x)=0,\quad t>0,\ x\in\partial\Omega,\label{P} \\ &u(0,x)=\phi(x) \geq 0,\quad x\in\Omega.& \nonumber \end{eqnarray} In what follows, we assume that $p>1$, $q\geq 1$, and $\Omega$ is a domain of $\mathbb{R}^N$, bounded or unbounded, sufficiently regular (say, uniformly regular of class $C^2$). Also, {\it unless otherwise stated, we assume $b>0$}. (A few results will however concern the case $b<0$.) It is known that (\ref{P}) admits a unique, maximal in time, classical solution $u\geq 0$, for all $\phi\geq 0$ sufficiently regular, e.g., $\phi\in C^1\bigl(\overline{\Omega}\bigr)$ with $\phi\big|_{\partial\Omega}=0$ if $\Omega$ is bounded, or $\phi\in W^{1,s}_0(\Omega)$ with $s>N\max(p,q)$ if $\Omega$ is unbounded. This regularity of $\phi$ will be assumed throughout the paper, unless otherwise stated. We denote by $T^*=T^*(\phi)$ the maximum existence time of $u$, and we say that $u$ blows up in finite time if $T^*(\phi)<\infty$. When $\phi\geq 0$ and $b>0$, it is known \cite{Q2, SW2} that gradient blowup cannot occur for (\ref{P}), that is: $T^*(\phi)<\infty$ implies $\limsup_{t\to T^*}\|u(t)\|_\infty=\infty$. Since we only consider nonnegative solutions of (\ref{P}), it is clear that the gradient term here represents a dissipation when $b>0$. In fact, the dynamics of this equation can be partially understood as a competition between the reaction term $u^p$, which may cause blowup as in the equation (\ref{RD}), and the gradient term, which fights against blowup. The solutions will exhibit different large-time behaviors, according to the issue of this competition. Similar mechanisms of competition have been studied in the case of nonlinear wave equations of the type $$u_{tt}-\Delta u=|u|^{p-1}u- |u_t|^{q-1}u_t\,,$$ where $p>1$, $q\geq 1$ (see \cite{GT}). Equation (\ref{P}) was first introduced in \cite{CW} in order to investigate the possible effect of a damping gradient term on global existence or nonexistence. On the other hand, a model in population dynamics was proposed in \cite{S2}, where (\ref{P}) describes the evolution of the population density of a biological species, under the effect of certain natural mechanisms. In particular, the dissipative gradient term represents the action of a predator which destroys the individuals during their displacements (it is assumed that the preys are not vulnerable at rest). A further discussion of this model can be found in \cite{AMST}, where the related degenerate equation $$u_t-\Delta(u^m)=u^p-|\nabla(u^\alpha)|^q $$ with $m>1$, $\alpha>0$ was studied. As it will turn out, the large-time behavior of the solution of problem (\ref{P}) will generally depend on all the values of the parameters, on the initial data, and on the domain $\Omega$. However, of particular importance will be the fact that $p>q$ or $q\geq p$. These cases are respectively reviewed in $\S$ 2 and 3. Finally, $\S$ 4 is devoted to stationary solutions of (\ref{P}). Throughout the paper, we will indicate a number of open problems related to the results we will review. \section{The case $p>q$} \subsection{Existence of blowup: the general result} The following result \cite{SW1} states that finite-time blowup occurs for large data whenever $p>q$. \begin{theorem} \label{thm2.1} Assume $p>q$, $\Omega\subset \mathbb{R}^N$ (bounded or unbounded) and $\psi\not\equiv 0$ ($\psi\geq 0$). Then there exists $\lambda_0=\lambda_0(\psi)>0$ such that for all $\lambda>\lambda_0$, the solution of (\ref{P}) with initial data $\phi=\lambda\psi$ blows-up in finite time. \end{theorem} We will see in $\S$ 3 that this result is optimal, in the sense that blowup {\it never} occurs if $q\geq p$, at least in bounded domains. The basic idea of the proof is to compare $u$ with a subsolution that blows up in finite time. In fact, one constructs a {\it self-similar} subsolution, whose profile is compactly supported. Interestingly, it is possible to find blowing-up self-similar subsolutions, whether or not (\ref{P}) has the invariance properties normally associated with self-similar solutions. The similarity exponents depend on $p$ and $q$, and can be chosen within a certain range of values. The result of Theorem \ref{thm2.1} actually extends to more general nonlinearities $F(u,$ $\nabla u)$ and also to some degenerate problems. We mention that the conclusion of Theorem 2.1 was obtained earlier, by completely different methods, in \cite{KP} in the special case $q=2$, and in \cite{Q1} in the special case $N=1$, $b$ small. \subsection{Other conditions for blowup} Besides the preceding general blowup result, various blowup conditions of more specific type are known, often under the restriction $q\leq 2p/(p+1)$. Some of them concern non-decreasing solutions. A sufficient condition on the initial data for having $u_t\geq 0$ is $\Delta \phi+\phi^p-b|\nabla\phi|^q \geq 0$ (see \cite{CW, ST}). The following theorem \cite{CW, AW} establishes blowup under an additional assumption of negative initial energy, in the spirit of the results of \cite{L1} and \cite{B} for equation (\ref{RD}). \begin{theorem} \label{thm2.2} Assume $q\leq 2p/(p+1)$ and $\Omega\subset\mathbb{R}^N$ (bounded or unbounded). Assume that $\phi$ (sufficiently regular) satisfies $$E(\phi)={1\over 2}\|\nabla\phi\|_2^2-{1\over p+1}\|\phi\|_{p+1}^{p+1}<0 $$ and is such that $u_t\geq 0$. Moreover, suppose that $-E(\phi)/\|\phi\|_2^2$ is large enough if $q< 2p/(p+1)$, or that $b$ is sufficiently small if $q=2p/(p+1)$. Then $T^*<\infty$. \end{theorem} In some situations, the energy assumption can be relaxed, leading to blowup of all nontrivial non-decreasing solutions \cite{S1, S2}. \begin{theorem} \label{thm2.3} Assume $q=2p/(p+1)$, $\Omega=\mathbb{R}^N$, $(N-2)p0$ (so that the result is not covered by Theorem 2.2). For equation (\ref{RD}) in $\Omega=\mathbb{R}^N$ a classical result, essentially due to Fujita (see \cite{Fu, L2}), asserts that no nonnegative nontrivial global solutions exist for $p\leq 1+2/N$, whereas both blowing-up and global positive solutions do exist if $p>1+2/N$. The value $p_c=1+2/N$ is thus said to be the Fujita critical exponent of the problem. \paragraph{Open problem 1.} Is there a Fujita critical exponent for equation (\ref{P}) in $\mathbb{R}^N$ when $q=2p/(p+1)$ and $b$ is small? \smallskip Partial facts are known about this problem. First, if $p>1+2/N$, for any $b>0$ (and any $q$ actually), there always exist positive global solutions. This follows from a straightforward comparison argument with the global solutions of the case $b=0$. When $q=2p/(p+1)$ and $b$ is large, both blowing-up and stationary positive solutions do exist. Therefore no Fujita-like result can hold in this case. On the contrary, when $q=2p/(p+1)$, $p\leq 1+2/N$ and $b$ is small, the existence of positive global solutions is unknown (at least it is known that no positive stationary solutions exist). On account of the similarity of scaling properties between equations (\ref{RD}) and (\ref{P}) when $q=2p/(p+1)$, the authors of \cite{AW} conjectured the nonexistence of positive global solutions. In one space dimension on a bounded interval, when $q\leq 2p/(p+1)$, with $b$ small if $q=2p/(p+1)$, it is known \cite{CW} that (\ref{P}) admits a unique positive stationary solution $v$. In this case, a very simple blowup condition, which does not require the monotonicity of $u$, was obtained in \cite{Fi}. \begin{theorem} \label{thm2.4} Assume $\Omega=(a,b)$, $-\infty0$ such that if $\phi$ satisfies \begin{equation} \liminf_{|x|\to \infty,\ x\in \Omega'} |x|^{2/(p-1)}\phi(x)>C, \label{decaycondition} \end{equation} then $T^*<\infty$. \end{theorem} It can be proved that the decay condition (\ref{decaycondition}) is optimal: there exist global solutions for initial data which decay like $\varepsilon |x|^{-2/(p-1)}$ when $\varepsilon>0$ is small. Recently, a similar optimal result was obtained in \cite{R} for a very general class of ``smaller" unbounded domains, of paraboloid type. The corresponding decay condition on the initial data is related in a precise way to the growth of the domain at infinity. \paragraph{Open problem 2.} Does the result of Theorem 2.5 remain valid when $1\leq q<2p/(p+1)$~? \smallskip Let us remark that all the results in \S 2.2 involve the limiting value $q=2p/(p+1)$. The origin of this number can be easily understood from scaling considerations. Indeed, for $q=2p/(p+1)$, the equation (\ref{P}) exhibits the same scale invariance as the equation (\ref{RD}). Namely, if $u$ solves (\ref{P}), say, in $\mathbb{R}^N$, then so does $u_\alpha(t,x)\equiv \alpha^{2/(p-1)} u(\alpha^2 t, \alpha x)$. This property will play an important role in $\S$ 2.3 (self-similar solutions), and in \S 2.4 and \S 3. \subsection{Description of blowup} Several results on the blowup behavior of non-global solutions of (\ref{P}) have been recently obtained, although still relatively little is known in comparison with the most studied case of (\ref{RD}). The estimates of the blowup rates were proved in \cite{CF, ST, CFQ, FS} in the case $q< 2p/(p+1)$. We summarize the results in the following theorem. \begin{theorem} \label{thm2.6} Assume $q<2p/(p+1)$ and let $u\geq 0$ be a solution of (\ref{P}), such that $T<\infty$. The estimate \begin{equation} C_1(T-t)^{1/(p-1)} \leq \|u(t)\|_\infty \leq C_2(T-t)^{1/(p-1)},\quad\hbox{as $t\to T$} \label{upper} \end{equation} holds in each of the following cases: \begin{description} \item{(i)} \cite{CF} $\Omega=\mathbb{R}^N$, $p\leq 1+2/N$; \item{(ii)} \cite{ST} $\Omega=\mathbb{R}^N$ or $\Omega=B_R$, $u$ radially symmetric, $u_r\leq 0$, $u_t\geq 0$, $p<(N+2)/(N-2)_+$. Moreover this remains valid for $q=2p/(p+1)$ and $b$ small; \item{(iii)} \cite{CFQ} $\Omega$ convex bounded and ($u_t\geq 0$ or $p\leq 1+2/N$); \item{(iv)} \cite{FS} $\Omega$ arbitrary, $p\leq 1+2/(N+1)$. \end{description} \end{theorem} This theorem shows that for $q<2p/(p+1)$ (or $=$), the blowup rate is the same as for (\ref{RD}). Recall that for (\ref{RD}), the upper bound in (upper) holds for all subcritical $p$, i.e. $p<(N+2)/(N-2)_+$, (see \cite{W, FM, GK}, and also \cite{MZ} for further recent results), whereas it may fail for large supercritical $p$ (see \cite{HV}). Also, the lower bound in (\ref{upper}) holds for (\ref{RD}) for all $p>1$ (see, e.g., \cite{FM}). There are basically four different techniques to prove the upper blowup estimate in (\ref{upper}) for (\ref{RD}) (the lower bound is much easier). Three of them use some re-scaling arguments, either of elliptic or parabolic type, which means that one re-scales, respectively, only space or both space and time variables, so that the limiting equation obtained is either elliptic or parabolic. The technique of \cite{W}, which relies on elliptic re-scaling (for monotone symmetric solutions) was used (and improved) in \cite{ST}. That of \cite{GK}, relying on elliptic re-scaling and energy methods, does not seem applicable here, because the equation (\ref{P}) has no variational structure. The technique in \cite{FM}, relying on maximum principle arguments, was successfully adapted in \cite{CFQ}. The method of \cite{H}, which relies on parabolic re-scaling and Fujita-type theorems (and was designed for problems with nonlinear boundary conditions), was used in \cite{CF, FS}. Concerning the blowup set and profile of solutions of (\ref{P}), the following very interesting result was proved in \cite{CFQ}. \begin{theorem} \label{thm2.7} Assume that $\Omega$ is a ball, $u$ is radially symmetric and $u_r\leq 0$, $r=|x|$. Then $0$ is the only blowup point and \begin{equation} u(t,r)\leq C_\alpha r^{-\alpha} \quad\hbox{for all $\alpha>\alpha_0$}, \label{profile} \end{equation} where $$ \alpha_0=\left\{ \begin{array}{ll} 2/(p-1), &\mbox{ if } q<2p/(p+1),\\[3pt] q/(p-q), &\mbox{ if } q\geq 2p/(p+1). \end{array}\right. $$ Furthermore, this estimate is optimal in the sense that, if in addition $N=1$ and $u_t\geq 0$, then (\ref{profile}) holds for no $\alpha<\alpha_0$. \end{theorem} The proof relies in particular on nontrivial modifications of the maximum principle arguments of \cite{FM}. Recall that for (\ref{RD}), under the assumptions of Theorem \ref{thm2.7}, (\ref{profile}) holds for all $\alpha>2/(p-1)$ (see \cite{FM}). Actually, the final profile is given by \begin{equation} u(T,r) \sim C(\log r)^{1/(p-1)}r^{-2/(p-1)},\quad\hbox{as $r\to 0$} \label{profileRD} \end{equation} (for radially symmetric decreasing solutions, this is known in $\mathbb{R}^N$ or on a bounded interval -- see \cite{Ve}). Also, observe that $q/(p-q)>2/(p-1)$ for $q>2p/(p+1)$. Theorem \ref{thm2.7} thus indicates that the blowup profile of solutions of (\ref{P}) is basically similar to that in (\ref{RD}) as long as $q<2p/(p+1)$, whereas for $q$ greater than this critical value, the gradient term induces an important effect on the profile, which becomes more singular. Under the assumptions of case (ii) of Theorem \ref{thm2.6}, the following information on the blowup profile is also obtained in \cite{ST}: there exists a constant $C>0$ (independent of $u$) such that $${u(t,|y|\sqrt{T-t})\over u(t,0)} \geq 1-C|y|$$ for $t$ close to $T$. However, this estimate is only of interest for $|y|$ small. As for the blowup set of non-global solutions, it is proved in \cite{CFQ} that when $q<2p/(p+1)$ and $\Omega$ is convex and bounded, the blowup set of any solution of (\ref{P}) is a compact subset of $\Omega$. In some special cases, a further insight into the description of blowup can be gained by studying the existence of backward self-similar solutions, that is, solutions of the form \begin{equation} u(t,x)=(T-t)^{-1/(p-1)} W(x/(T-t)^m),\quad -\infty1$, such that for all $p$ with $10$ such that the corresponding function $W$ satisfies $\lim_{ |x |\to\infty} |x|^{2/(p-1)}W(x)=C$. In particular, $u$ blows up at the single point $x=0$, and it holds $$u(T,x)= C|x|^{-2/(p-1)},\quad\hbox{for all } x\neq 0\,.$$ \end{theorem} It is to be noted that no nontrivial, backward, self-similar solutions exist for $b=0$ and $p$ subcritical. Also the blowup profile above is different from all the profiles known for (\ref{RD}). Namely, it is slightly less singular, by a logarithmic factor, than the corresponding profile for (\ref{RD}) (see formula (\ref{profileRD}) above). Comparison of Theorems 2.7 and 2.8 yields the interesting and a bit surprising observation that the gradient term can have different effects on the blowup profile: when the perturbation is mild ($q=2p/(p+1)$ in Theorem \ref{thm2.8}), slightly less singular profile; when the perturbation is strong ($2p/(p+1)2$ (see \cite{La, KP, GV1, GV2}). The authors of \cite{KP} interpret the above result in the following way. While the term $u^p$ alone would force the solution to develop a spike at the maximum point, hence causing single point blowup, the gradient term tends to push up the steeper parts of the profile $u(t,.)$. This enhances regional or even global blowup, the influence of the gradient term becoming more important as the value of $p$ decreases. Concerning self-similar profiles, in the case $b<0$, $q=2$, for radial solutions in $\mathbb{R}^N$ it is proved in \cite{GV1, GV2} that blowup solutions behave asymptotically like a self-similar solution $w$ of the following Hamilton-Jacobi equation without diffusion: $$ w_t= |\nabla w|^2+w^p,$$ with $w$ having the form (\ref{autosimilaire}), for $m=(2-p)/2(p-1)$. Note that this kind of self-similar behavior is quite different from that in Theorem 2.8 above (or from those known for $b=0$ and $p$ super-critical); indeed, $m$ describes the range $(-\infty,1/2)$ for $p\in (1,\infty)$. Let us mention that for the related equation with exponential source \begin{equation} u_t-\Delta u=e^u-|\nabla u|^2, \label{expeqn} \end{equation} some results on blowup sets and profiles where obtained in \cite{BE2}. The analysis therein is strongly based on the observation that the transformation $v=1-e^{-u}$ changes (\ref{expeqn}) into the linear equation $v_t-\Delta v=1$. \paragraph{Open problem 3.} The value of $p_0$ in Theorem \ref{thm2.8} is not explicitly known (because the proof involves a limiting argument). Can one specify the allowable values of $p$, or even extend the result to all $p>1$, and also to all $b>0$? On the other hand, is the self-similar solution unique for each value of the parameters? Is the self-similar profile of Theorem 2.8 representative of all blowup behaviors when $q=2p/(p+1)$, or do there exist different profiles? \paragraph{Open problem 4.} What is the blowup rate when $2p/(p+1)q=2$. (Note that $2p/(p+1)\to 2$ as $p\to \infty$.) \begin{theorem} \label{thm2.11} Assume $\Omega=(0,L)$, $0q=2$. For some $b=b_0(L)>0$, there exist (infinitely many) $\phi$ such that $u_t\geq 0$, $T^*=\infty$, and $\lim_{t\to \infty}\|u(t)\|_\infty=\infty$. \end{theorem} More precisely, it is proved in \cite{D} that $u(t)$ approaches the (unique) {\it singular} stationary solution $v_s$ as $t\to \infty$, whenever $\phi$ lies between the maximal regular stationary solution and $v_s$. Further sharp stability/instability results for equilibria of (\ref{P}) are given in \cite{D} for $q=2$ and $N=1$. \paragraph{Open problem 5.} What can be said about boundedness of global solutions for $2p/(p+1)0;\ \Omega \hbox{ contains a ball of radius } r\bigl\}\,=\sup_{x\in\Omega}\mathop{\rm dist}(x,\partial\Omega). $$ The following result \cite{SW2, S3} gives a characterization in terms of $\rho(\Omega)$ of the domains $\Omega$ in which all solutions of (\ref{P}) are global and bounded for $q\geq p$. \begin{theorem} \label{thm3.1} Assume $q\geq p$. \begin{description} \item{(i)} If $\rho(\Omega)<\infty$, then for all $\phi$, the solution $u$ of (\ref{P}) is global and bounded. \item{(ii)} If $\rho(\Omega)=\infty$, then there exists $\phi$ such that the solution $u$ of (\ref{P}) is unbounded (with either $T^*<\infty$ and $\limsup_{t\to T^*}\|u(t)\|_{\infty}=\infty,$ or $T^*=\infty$ and $\lim_{t\to \infty}\|u(t)\|_{\infty}$ $=\infty$). \end{description} \end{theorem} (See paragraph after Theorem \ref{thm3.6} below for some ideas on the proof.) One important property of the inradius, is that its finiteness is also equivalent to the validity of the {\it Poincar\'e inequality} in $W^{1,k}_0(\Omega)$, $1\leq k<\infty$: \begin{equation} \|v\|_{k}\leq C_k(\Omega)\|\nabla v\|_{k},\quad \forall v\in W^{1,k}_0(\Omega). \label{P_k} \end{equation} (The equivalence is true under mild regularity assumptions on $\Omega$, for instance if $\Omega$ satisfies a uniform exterior cone condition -- see \cite{S3} and the references therein for details.) As an illustration, we have $\rho(\Omega)<\infty$ if $\Omega$ is contained in a strip, and $\rho(\Omega)=\infty$ if $\Omega$ contains a cone. A typical example of "largest" possible domains satisfying $\rho(\Omega)<\infty$ is the complement of a periodic net of balls $$\Omega=\mathbb{R}^N\setminus\bigcup\limits_{z\in {\bf Z}^N} \overline B(Rz,\epsilon), \quad 0<\epsilonb_0(\Omega)>0$ large enough and $\phi$ is any initial data; \item{(b)} $b>0$ and $\|\phi\|_k$ is sufficiently small (independent of $b$). \end{description} By the way, let us mention that the stability of the $0$ solution for equation (\ref{RD}) in unbounded domains is also strongly related to $\rho(\Omega)$ (see \cite{S3, S4}). Theorem 3.1 (ii) does not conclude whether blowup occurs in finite of infinite time. Some cases of global unbounded solutions -- i.e. $\|u(t)\|_\infty\to\infty$ as $t\to\infty$ -- will be described in \S 3.3. One of the more interesting questions on equation (\ref{P}) then remains the following: \paragraph{Open problem 6.} Can {\it finite time} blowup occur when $q\geq p$ ? This is unknown even for $\Omega=\mathbb{R}^N$ (note that the existence of a blowing-up solution in some domain $\Omega$ would imply the same conclusion in $\mathbb{R}^N$ by comparison). \smallskip However, the following result \cite{SW2} shows that in any domain, finite time blowup cannot occur if $q\geq p$ and $\phi$ is compactly supported. \begin{theorem} \label{thm3.2} Assume $q\geq p$ and $\Omega\subset \mathbb{R}^N$ (bounded or unbounded). If $\phi$ is compactly supported in $\mathbb{R}^N$, then $T^*=\infty$. \end{theorem} Actually, the conclusion of Theorem \ref{thm3.2} remains valid whenever $\phi$ decays exponentially in at least one direction \cite{SW2}. \subsection{Critical blowup exponents} As a consequence of Theorems \ref{thm2.1} and \ref{thm3.1}, it follows that the critical blowup exponent for problem (\ref{P}) is given by $q=p$, whenever $\rho(\Omega)<\infty$. For bounded domains, this was conjectured in \cite{Q1}, where the conjecture was verified in the case when $\Omega$ is a bounded interval and $b$ is small. \begin{corollary} \label{coro3.3} Assume $\rho(\Omega)<\infty$. \begin{description} \item{(i)} If $p>q$, then there exists $\phi$ such that $u$ blows up in finite time. \item{(ii)} If $q\geq p$, then for all $\phi$, $u$ is global and bounded. \end{description} \end{corollary} If one restricts to {\it compactly supported} initial data, it follows from Theorems \ref{thm2.1} and \ref{thm3.2} that the critical blowup exponent is still given by $q=p$ for {\it any} domain, including $\mathbb{R}^N$. \begin{corollary} \label{coro3.4} Assume $\Omega\subset \mathbb{R}^N$ (bounded or unbounded). \begin{description} \item{(i)} If $p>q$, then there exists $\phi$, compactly supported, such that $u$ blows up in finite time. \item{(ii)} If $q\geq p$, then for all $\phi$ compactly supported, $u$ is global (possibly unbounded). \end{description} \end{corollary} \subsection{Unbounded global solutions} Under additional assumptions on $\Omega$, one can prove that some unbounded {\bf global} solutions do actually exist \cite{SW2}. \begin{theorem} \label{thm3.5} Assume that $q\geq p$ and that $\Omega$ contains a cone. Then there exists $\phi$, compactly supported, such that the solution $u$ of (\ref{P}) satisfies $T^*=\infty$ and $$\lim_{t\to \infty}\|u(t)\|_\infty=\infty.$$ \end{theorem} If $\Omega=\mathbb{R}^N$, one further obtains solutions which blow up {\it everywhere} in infinite time \cite{SW2}. \begin{theorem} \label{thm3.6} Assume $q\geq p$ and $\Omega=\mathbb{R}^N$. Then there exists $\phi$, compactly supported, such that the solution $u$ of (\ref{P}) satisfies $T^*=\infty$ and $$\forall x\in \mathbb{R}^N,\ \lim_{t\to \infty}u(t,x)=\infty.$$ \end{theorem} Note that the conclusions of Theorems \ref{thm3.5} and \ref{thm3.6} remain true for large sets of initial data, namely for any compactly supported initial data lying above $\phi$ (this follows from Theorem \ref{thm3.2} and the comparison principle). The proofs of Theorems \ref{thm3.5} and \ref{thm3.6} rely on the construction of ordered, global, unbounded sub- and supersolutions. The main difficulty in constructing the subsolution comes from the gradient term, whose power is larger than that of the source term. The idea is to build a radial expanding wave, whose maximum at the origin grows up to $\infty$ as $t\to\infty$, while its gradient remains uniformly bounded. As for supersolutions, a pair of them is constructed under the form of traveling waves, propagating in two opposite directions. These supersolutions prevent $u$ from blowing up in finite time. The subsolutions above are also an essential ingredient for proving the existence of unbounded global solutions when $\rho(\Omega)=\infty$ (see Theorem \ref{thm3.1} (ii)). More precisely, one superposes a sequence of expanding wave subsolutions, whose supports eventually fill a collection of balls of arbitrary large radii, included in $\Omega$. \paragraph{Open problem 7.} Does there exist unbounded global solutions whenever $\rho(\Omega)=\infty$ and $q\geq p$~? \paragraph{Open problem 8.} What is the precise grow-up rate of $\|u(t)\|_\infty$ for unbounded global solutions of (\ref{P}) ? For the solutions constructed in the proof of Theorem 3.6, we only have the rough estimate $C_1t\leq \|u(t)\|_\infty\leq C_2e^{C_3 t}$, as $t\to \infty$. \smallskip Global blowup, as described in Theorem \ref{thm3.6}, can occur only for $\Omega=\mathbb{R}^N$. Indeed, define the blowup set of $u$ as $$ E=\bigl\{x_0\in\overline{\Omega}\cup\{\infty\};\,\exists x_n\to x_0,\ \exists t_n\to T^*,\ u(t_n,x_n)\to\infty\bigr\}. $$ The blowup set then satisfies the following alternative \cite{SW2}. \begin{theorem} \label{thm3.7} Assume $q\geq p$ and $\Omega\subset\mathbb{R}^N$ (unbounded). Assume that $\phi$ is such that $u$ is unbounded, with either $T^*<\infty$ or $T^*=\infty$. \begin{description} \item{(i)} If $\Omega\not=\mathbb{R}^N$, then $E=\{\infty\}$. \halign{#\cr} \item{(ii)} If $\Omega=\mathbb{R}^N$, then either $E=\mathbb{R}^N\cup\{\infty\}$ or $E=\{\infty\}$. \end{description} \end{theorem} \paragraph{Open problem 9.} Does there exist $\phi$ such that $E=\{\infty\}$ when $q\geq p$ and $\Omega=\mathbb{R}^N$~? Theorem 3.6 provides some $\phi$ such that $\Omega=\mathbb{R}^N$ and $E=\mathbb{R}^N\cup\{\infty\}$. \smallskip Finally, we have the analogue of Theorem \ref{thm2.5} when $q\geq p$, except that it is not known whether $T^*=\infty$ or $T^*<\infty$ \cite{SW2}. \begin{proposition} \label{prop3.8} Assume that $q\geq p$ and that $\Omega$ contains a cone $\Omega'$. There exists a constant $C=C(\Omega')>0$ such that if $\phi$ satisfies $$\liminf_{|x|\to \infty,\ x\in \Omega'} |x|^{2/(p-1)}\phi(x)>C,$$ then the solution $u$ of (\ref{P}) is unbounded (with $T^*\leq\infty$). \end{proposition} \section{Stationary states} The stationary states of (\ref{P}) were thoroughly investigated in \cite{CW, AW, C, FQ, D, Vo, SZ, PSZ}. We conclude this survey by a brief account of results on (positive classical) stationary solutions of (\ref{P}), i.e. solutions of the elliptic problem \begin{eqnarray} &\Delta u+u^{p}-b|\nabla u|^q=0,\quad x\in\Omega & \label{stationary}\\ &u(x)=0,\quad x\in\partial\Omega\,. &\nonumber \end{eqnarray} The best results available concern the case when $\Omega=\mathbb{R}^N$ or $\Omega$ is a ball $B_R$. By the results of \cite{GNN}, any positive solution to (\ref{stationary}) on $\mathbb{R}^N$ or on a ball must be radial. Searching solutions of (\ref{stationary}) thus leads to an ODE. Let $p_S=(N+2)/(N-2)$, with $p_S=\infty$ if $N\leq 2$. For the elliptic problem associated with (\ref{RD}) ((\ref{stationary}) with $b=0$), which is classically known as Lane-Emden's equation, it is well-known that positive solutions exist on a ball (resp. on $\mathbb{R}^N$) if and only if $pp_S$: existence (for all $q>1$) \cite{SZ}; \item{(ii)} If $p=p_S$: existence if and only if $q2p/(p+1)$ \cite{SZ}; \item(iii3) nonexistence if $p< N/(N-2)_+$ and $q=2p/(p+1)$ with $b$ small \cite{CW, FQ, Vo}; \item(iii4) nonexistence if $N\geq 3$, $N/(N-2)\overline{q}$, for some (explicitly determined) $\overline{q}\in (2p/(p+1),p)$ \cite{SZ}. \end{description}\end{description} Moreover, there is numerical evidence that solutions exist for some values of $q$ between $2p/(p+1)$ and $\overline{q}$ \cite{SYZ}. Next we turn to the case when $\Omega$ is a ball $B_R$ in $\mathbb{R}^N$. Contrary to the case $\Omega=\mathbb{R}^N$, the super-critical range $p>p_S$ is hardly explored. We thus classify the results in terms of the value of $q$ as a function of $p$. \begin{description} \item{(i)} If $11$: existence if and only if $b\leq b_0$, for some $b_0=b_0(p,N)>0$ \cite{Q2, Vo}; \end{description} Some partial results are known when $\Omega$ is an arbitrary bounded domain with smooth boundary (these results are obtained via topological degree theory). \begin{description} \item{(i)} If $p1$: existence if and only if $b\leq b_0$, for some $b_0=b_0(p,N)$ \cite{Q2, Vo}; \end{description} Last, we mention that some results on the number of stationary states can be found in \cite{CW, C, D, Vo, PSZ, SZ}. If we analyze the results above, we find several ``critical'' values of the parameters with respect to the existence of positive stationary solutions. The value $p=p_S$ is critical in the case of the whole space, as it is for the equation without gradient term. Concerning $q$, there are at least two critical values $q=2p/(p+1)$ and $q=p$. There might possibly exist a third critical value $\overline{q}\in (2p/(p+1),p)$, in which case $N/(N-2)$ would also be critical for $p$ when $N\geq 3$. (Incidentally, when $q=2p/(p+1)$, it happens that $p\geq N/(N-2)_+$ is a necessary and sufficient condition for the existence of singular stationary solutions of the form $C|x|^{-r}$ for all $b>0$.) Moreover, the size of $b$ can also be determinant when $q\geq 2p/(p+1)$. In comparison with these properties, it is interesting to recall from $\S$ 3.2 that $q=p$ is the only critical blowup exponent for the evolution problem (at least in bounded domains), and that the values of $p>1$ and $b\,(>0)$ do not play much role in global existence or nonexistence. \begin{thebibliography}{00} {\frenchspacing \bibitem{AMST} F. {Andreu}, J. M. {Mazon}, F. {Simondon} and J. {Toledo}, {\it Global existence for a degenerate nonlinear diffusion problem with nonlinear gradient term and source}\/, Math. Ann., 314 ({1999}), 703--728. \bibitem{AE} J. {Aguirre} and M. {Escobedo}, {\it On the blow up of solutions for a convective reaction diffusion equation}\/, Proc. Roy. Soc. Edinburgh., {123A}, 3 ({1993}), 433--460. \bibitem{AW} L. {Alfonsi} and F. B. {Weissler}, {\it Blowup in $\mathbb{R}^N$ for a parabolic equation with a damping nonlinear gradient term\rm, in: Diffusion Equations and Their Equilibrium States, III}\/, N.~G.~Lloyd {\it et al.} (eds.), Birkh\"auser, Boston-Basel-Berlin, 1992, 1--10. \bibitem{AF} S. {Angenent} and J. {Fila}, {\it Interior gradient blowup in a semilinear parabolic equation}\/, Differential Integral Equations, 9 (1996), 865--877. \bibitem{B}J. {Ball}, {\it Remarks on blowup and nonexistence theorems for nonlinear evolution equations}\/, Quarterly J. Math. Oxford Ser. 28 (1977), 473--486. \bibitem{BE1} J. {Bebernes} and D. {Eberly}, Mathematical Problems from Combustion Theory, Applied Mathematical Sciences, Springer-Verlag, 1989. \bibitem{BE2} J. {Bebernes} and D. {Eberly}, {\it Characterization of blow-up for a semilinear heat equation with a convection term}\/, Quart. J. Mech. Appl. Mech., 42 (1989) 447--456. \bibitem{BK} M. {Ben-Artzi} and H. {Koch}, {\it Decay of mass for a semilinear parabolic equation}\/, Commun. Partial Differ. Equations, 24 (1999) 869--881. \bibitem{C} M. {Chipot}, {\it On a class of nonlinear elliptic equations}\/, Proc. of the Banach Center, 27 (1992), 75--80. \bibitem{CW} M. {Chipot} and F. B. {Weissler}, {\it Some blow up results for a nonlinear para\-bolic problem with a gradient term}\/, SIAM J. Math. Anal., 20 (1989), 886--907. \bibitem{CF} M. {Chlebik} and M. {Fila}, {\it From critical exponents to blowup rates for parabolic problems}\/, Rend. Mat. Appl., Ser. VII, 19 (1999), 449--470. \bibitem{CFQ} M. {Chlebik}, M. {Fila} and P. {Quittner}, {\it Blowup of positive solutions of a semilinear parabolic equation with a gradient term}\/, Dyn. Contin. Discrete Impulsive Syst., to appear.\bibitem{D} K. {Deng}, {\it Stabilizat ion of solutions of a nonlinear parabolic equation with a gradient term}\/, Math. Z., 216 (1994), 147--155. \bibitem{DL} K. {Deng} and H. A. {Levine}, {\it The role of critical exponents in blowup theorems: the sequel}\/, J. Math. Anal. Appl., 243 (2000), 85--126. \bibitem{EVZ} M. {Escobedo}, J. L. {Vazquez} and E. {Zuazua}, {\it A diffusion-convection equation in several space dimensions}\/, Indiana Univ. Math. J., 42 (1993), 1413--1440. \bibitem{Fi} M. {Fila}, {\it Remarks on blow up for a nonlinear parabolic equation with a gradient term}\/, Proc. Amer. Math. Soc., 111 (1991), 795--801. \bibitem{FQ} M. {Fila} and P. {Quittner}, {\it Radial positive solutions for a semilinear elliptic equation with a gradient term}\/, Adv. Math. Sci. Appl., 2 (1993), 39--45. \bibitem{FS} M. {Fila} and Ph. {Souplet}, {\it The blowup rate for semilinear parabolic problems on general domains}\/, NoDEA Nonlinear Differential Equations Appl., to appear. \bibitem{FM} A. {Friedman} and J. B. {McLeod}, {\it Blowup of positive solutions of semilinear heat equations}\/, Indiana Univ. Math. J., {34} (1985), 425--447. \bibitem{Fu} H. {Fujita}, {\it On the blowing-up of solutions of the Cauchy problem for $u_t=\Delta u+u^{1+\alpha}$}\/, J. Fac. Sci. Univ. Tokyo, Sect. I, 13 (1966), 109--124. \bibitem{GV1} V. A. {Galaktionov} and J. L. {Vazquez}, {\it Regional blowup in a semilinear heat equation with convergence to a Hamilton-Jacobi equation}\/, SIAM J. Math. Anal., 24 (1993), 1254--1276. \bibitem{GV2} V. A. {Galaktionov} and J. L. {Vazquez}, {\it Blowup for quasilinear heat equations described by means of nonlinear Hamilton-Jacobi equations}\/, J. Differ. Equations, {127} (1996), 1--40. \bibitem{GT} V. {Georgiev} and G. {Todorova}, {\it Existence of a solution of the wave equation with nonlinear damping and source terms}, J. Differ. Equations, 109 (1994), 295--308. \bibitem{GNN} B. {Gidas}, W.-M. {Ni}, L. {Nirenberg}, {\it Symmetry and related properties via the maximum principle}, Commun. Math. Phys., 68 (1979), 209--243. \bibitem{GK} Y. {Giga} and R. V. {Kohn}, {\it Characterizing blowup using similarity variables}\/, Indiana Univ. Math. J., {36} (1987), 1--40. \bibitem{HV} M. A. {Herrero} and J. J. L. {Velazquez}, {\it Explosion de solutions des \'equations paraboliques semi\-lin\'e\-aires supercritiques}\/, C. R. Acad. Sc. Paris, 319, S\'erie I (1994), 141--145. \bibitem{H} B. {Hu}, {\it Remarks on the blowup estimate for solution of the heat equation with a nonlinear boundary condition}\/, Differential Integral Equations, 9 (1996), 891--901. \bibitem{K} S. {Kaplan}, {\it On the growth of solutions of quasilinear parabolic equations}\/, Commun. Pure Appl. Math., 16 (1963), 327--343. \bibitem{KP} B. {Kawohl} and L. {Peletier}, {\it Remarks on blowup and dead cores for nonlinear para\-bolic equations}\/, Math. Z., 202 (1989), 207--217. \bibitem{La} A. A. {Lacey}, {\it Global blowup of a nonlinear heat equation}\/, Proc. Royal Soc. Edinburgh, 104 (1986), 161--167. \bibitem{LN} T. {Lee} and W. {Ni}, {\it Global existence, large time behaviour and life span of solutions of a semilinear parabolic Cauchy problem}\/, Trans. Amer. Math. Soc., 333 (1992), 365--378. \bibitem{L1} H. A. {Levine}, {\it Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_{t}=-Au+F(u)$}\/, Arch. Rational Mech. Anal., 51 (1973), 371--386. \bibitem{L2} H. A. {Levine}, {\it The role of critical exponents in blowup theorems}\/, SIAM Review, 32 (1990), 262--288. \bibitem{MZ} F. {Merle} and H. {Zaag}, {\it Optimal estimates for blowup rate and behavior for nonlinear heat equations}\/, Commun. Pure Appl. Math., 51 (1998), 139--196. \bibitem{PSZ} L. A. {Peletier}, J. {Serrin} and H. {Zou}, {\it Ground states of a quasilinear equation}\/, Differential Integral Equations, 7 (1994), 1063--1082. \bibitem{Q1} P. {Quittner}, {\it Blowup for semilinear parabolic equations with a gradient term}\/, Math. Methods Appl. Sci., 14 (1991), 413--417. \bibitem{Q2} P. {Quittner}, {\it On global existence and stationary solutions for two classes of semilinear parabolic equations}\/, Comment. Math. Univ. Carolinae, 34 (1993), 105--124. \bibitem{Q3} P. {Quittner}, {\it Global solutions in parabolic blowup problems with perturbations\rm, in: Proceedings of the Third European Conference on Elliptic and Parabolic Problems (Pont-\`a-Mousson 1997)}, Pitman Research Notes Series in Mathematics $\sharp$ 384, Addison Wesley Longman, 1998. \bibitem{QS} P. {Quittner} and Ph. {Souplet}, {\it A priori estimates of global solutions of superlinear parabolic problems without variational structure}\/, preprint Univ. Versailles $\#$ 2000-49. \bibitem{R} P. {Rouchon}, {\it Blowup of solutions of nonlinear heat equations in unbounded domains for slowly decaying initial data}\/, Z. Angew. Math. Phys., to appear. \bibitem{SGKM} A. A. {Samarskii}, V. A. {Galaktionov}, S. P. {Kurdyumov} and A. P. {Mikh\-ailov}, Blowup in Quasilinear Parabolic Equations, Nauka, Moscou, 1987~; English translation~: Walter de Gruyter, Berlin, New York, 1995. \bibitem{SYZ} J. {Serrin}, Y. {Yan} and H. {Zou}, {\it A numerical study of the existence and non-existence of ground states and their bifurcations for the equations of Chipot and Weissler}\/, University of Minnesota preprint n$^o$ 93-056 (1993). \bibitem{SZ} J. {Serrin} and H. {Zou}, {\it Existence and non-existence for ground states of quasilinear elliptic equations}\/, Arch. Rational Mech. Anal. 121 (1992), 101--130. \bibitem{SnTW} S. {Snoussi}, S. {Tayachi} and F. B. {Weissler}, {\it Asymptotically self-similar global solutions of a semilinear parabolic equation with a nonlinear gradient terms}\/, Proc. Royal Soc. Edinburgh, 129A (1999), 1291--1307. \bibitem{S1} Ph. {Souplet}, {\it R\'esultats d'explosion en temps fini pour une \' equation de la cha\-leur non lin\'eaire}\/, C. R. Acad. Sc. Paris, 321, S\'erie I (1995), 721--726. \bibitem{S2}Ph. {Souplet}, {\it Finite time blowup for a nonlinear parabolic equation with a gradient term and applications}\/, Math. Methods Appl. Sci., 19 (1996), 1317--1333. \bibitem{S3} Ph. {Souplet}, {\it Geometry of unbounded domains, Poincar\'e inequalities, and stability in semilinear parabolic equations}\/, Commun. Partial Differ. Equations, 24 (1999), 951--973. \bibitem{S4} Ph. {Souplet}, {\it Decay of heat semigroups in $L^\infty$ and applications to nonlinear parabolic problems in unbounded domains}\/, J. Funct. Anal., 173 (2000), 343--360. \bibitem{S5} Ph. {Souplet}, {\it Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions}\/, Differential Integral Equations, to appear. \bibitem{ST} Ph. {Souplet} and S. {Tayachi}, {\it Blowup rates for nonlinear heat equations with gradient terms and for parabolic inequalities}\/, Colloq. Math, 87 (2001), in press. \bibitem{STW} Ph. {Souplet}, S. {Tayachi} and F. B. {Weissler}, {\it Exact self-similar blowup of solutions of a semilinear parabolic equation with a nonlinear gradient term}\/, Indiana Univ. Math. J., 48 (1996), 655--682. \bibitem{SW1} Ph. {Souplet} and F. B. {Weissler}, {\it Self-similar sub-solutions and blowup for nonlinear parabolic equations}\/, J. Math. Anal. Appl., 212 (1997), 60--74. \bibitem{SW2} Ph. {Souplet} and F. B. {Weissler}, {\it Poincar\'e's inequality and global solutions of a nonlinear parabolic equation}\/, Ann. Inst. H. Poincar\'e, Analyse non lin\'eaire, 16 (1999), 337--373. \bibitem{St} B. {Straughan}, Explosive Instabilities in Mechanics, Springer-Verlag, Berlin, 1998. \bibitem{T} S. {Tayachi}, {\it Forward self-similar solutions of a semilinear parabolic equation with a nonlinear gradient term}\/, Differential Integral Equations, 9 (1996), 1107--1117. \bibitem{Ve} J. J. L. {Vel{\'a}zquez}, {\it Blow up for semilinear parabolic equations\rm, in: Research in applied mathematics, Recent advances in partial differential equations}\/, M. A. Herrero and E. Zuazua (eds.), John Wiley \& Sons, 1994, 131--145. \bibitem{Vo} F. {Voirol}, {\it Coexistence of singular and regular solutions for the equation of Chipot and Weissler}\/, Acta Math. Univ. Comenianae, 65 (1996), 53--64. \bibitem{W} F. B. {Weissler}, {\it An $L^\infty$ blowup estimate for a nonlinear heat equation}\/, Commun. Pure Appl. Math., 38 (1985), 291--295. }\end{thebibliography} \noindent{\sc Philippe Souplet } \\ D\'epartement de Math\'ematiques, Universit\'e de Picardie \\ INSSET, 02109 St-Quentin, France \\ and\\ Laboratoire de Math\'ematiques Appliqu\'ees, UMR CNRS 7641 \\ Universit\'e de Versailles, 45 avenue des Etats-Unis, 78035 Versailles, France. \\ e-mail: souplet@math.uvsq.fr \end{document}