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$\rho(x)\partial_t u= \Delta u^m\qquad$ in $Q$:$=\mathbb R^n\times\mathbb R_+$

$u(x, 0)=u_0$

in dimensions $n\ge 3$. We deal with a class of solutions having finite energy

$E(t)=\int_{\mathbb R^n} \rho(x)u(x,t) dx$

for all $t\ge 0$. We assume that $m> 1$ (slow diffusion) and
the density $\rho(x)$ is positive, bounded and smooth. We
prove existence of weak solutions starting from data $u_0\ge 0$
with finite energy. We show that uniqueness takes place if $\rho$
has a moderate decay as $|x|\to\infty$ that essentially amounts to
the condition $\rho\notin L^1(\mathbb R^n)$. We also identify
conditions on the density that guarantee finite speed of
propagation and energy conservation, $E(t)=$const. Our
results are based on a new *a priori* estimate of the
solutions.

The equation is posed in the whole space $\mathbb R^n$ , $n \geq 2$, for $0 < t < \infty$; $p(x)$ is a positive and bounded function with a certain behaviour at infinity. We take initial data $u(x,0) = u_0(x) \geq 0$, and prove that this problem is well-posed in the class of solutions with finite "energy", that is, in the weighted space $L^1_p$, thus completing previous work of several authors on the issue. Indeed, it generates a contraction semigroup.

We also study the asymptotic behaviour of solutions in two space dimensions when $p$ decays like a non-integrable power as $|x| \rightarrow \infty$ : $p(x)$ $|x|^\alpha$ ~ $1$ with $\alpha \epsilon (0,2)$ (infinite mass medium). We show that the intermediate asymptotics is given by the unique selfsimilar solution $U_2(x, t; E)$ of the singular problem

$ |x|^{- \alpha} u(x,0) = E\delta(x), E = ||u_0||_{L^1_p}$

A first part of the course introduces the subject and discusses the classical questions addressed by the blow-up theory. We propose a list of main questions that extends and hopefully updates on the existing literature. We also introduce extinction problems as a parallel subject.

In the main bulk of the paper we describe in some detail the developments in which we have been involved in recent years, like rates of growth and pattern formation before blow-up, the characterization of complete blow-up, the occurrence of instantaneous blow-up (i.e., immediately after the initial moment) and the construction of transient blow-up patterns (peaking solutions), as well as similar questions for extinction.

In a final part we have tried to give an idea of interesting lines of current research. The survey concludes with an extensive list of references. Due to the varied and intense activity in the field both aspects are partial, and reflect necessarily the authors' tastes.

After establishing the general existence and uniqueness theory, the main properties are described: positivity, regularity, continuous dependence, a priori estimates, Schwarz symmetrization, among others. Self-similar solutions are constructed (fractional Barenblatt solutions) and they explain the asymptotic behaviour of a large class of solutions. In the fast diffusion range we study extinction in finite time and we find suitable special solutions. We discuss KPP type propagation. We also examine some related equations that extend the model and briefly comment on current work.

(**P**)
$\qquad \rho(x) \partial_t u= \Delta u^m\qquad$
in $Q$:$=\mathbb R^n\times\mathbb R_+$

$u(x, 0)=u_0$

in dimensions $n\ge 3$. We assume that $m> 1$ (slow
diffusion) and $\rho(x)$ is positive, bounded and behaves
like $\rho(x)$~$|x|^{-\gamma}$ as $|x|\to\infty$, with
$0\le \gamma<2$. The data $u_0$ are assumed to be nonnegative
and such that $\int \rho(x)u_0 dx< \infty$.

Our asymptotic analysis leads to the associated
singular equation $|x|^{-\gamma}u_t= \Delta u^m,$ which
admits a one-parameter family of selfsimilar solutions $
U_E(x,t)=t^{-\alpha}F_E(xt^{-\beta})$, $E>0$, which are
source-type in the sense that $|x|^{-\gamma}u(x,0)=E\delta(x)$. We
show that these solutions provide the first term in the asymptotic
expansion of generic solutions to problem (**P**)
for large
times, both in the weighted $L^1$ sense

$u(t)=U_E(t)+o(1)\qquad$ in $L^1_\rho$

and in the uniform sense $u(t)=U_E(t)+o(t^{-\alpha})$ in $L^\infty $ as $t\to \infty$ for the explicit rate $\alpha=\alpha(m,n,\gamma)>0$ which is precisely the time-decay rate of $U_E$. For a given solution, the proper choice of the parameter is $E=\int \rho(x)u_0 dx$.

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