0$ 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 ($2p/(p+1) 2p/(p+1)$ \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 $12$ (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 $\Omega=(0,L)$, $0

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<\epsilon

1$: 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 $p