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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}$

*i.e.*solutions of the problem \begin{eqnarray*} \rho (x) \partial_{t} u = \Delta u^{m}, \quad in \quad Q:= R^n \times R_+, \\ u(x,0)=u_{0}(x), \quad in\quad R^n, \end{eqnarray*} where $m>1$, $u_0\in L^1(R^n, \rho(x)dx)$ and $n\ge 3$. We assume that $\rho (x)\sim C|x|^{-2}$ as $|x|\to\infty$ in $R^n$. Such a decay rate turns out to be critical. We show that the limit behavior can be described in terms of a family of source-type solutions of the associated singular equation $|x|^{-2}u_t = \Delta u^{m}$. The latter have a self-similar structure and exhibit a logarithmic singularity at the origin.

(**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 $E=\int \rho(x)u_0 dx$.

^{[12]}, one cannot expect to have (generically) more than five solutions to the semilinear boundary value problem

$\mathbf{(P)}\qquad \left\{\begin{array}{ ll}-\Delta u =f(u, x)\quad&\text{in}B:=\{x\in\mathbb{R}^n:\, |x|<1\}, \\u =0&\text{on}\partial B\end{array}\right.$ |

^{[12]}can be taken "arbitrarily close" to autonomous, we prove that the dependence of $f$ on $x$ is indeed essential in the arguments. More precisely, we show that

**(P)**with $f$ independent of $x$ and satisfying the assumptions in

^{[12]}has at least six, and generically seven solutions. Under more stringent conditions on the non-linearity, we prove that there are up to nine solutions. Importantly, we do not assume any symmetry on $f$ for any of our results.

We show that instantaneous blow-up at space infinity takes place when $p\le1$.

We also briefly discuss the case $2<\sigma< N$: we prove that the Fujita exponent in this case does not depend on $\sigma$, $\tilde{p}_c=1+\frac2{N-2}$, and for initial values not too small at infinity a phenomenon of instantaneous complete blow-up occurs in the range $1< p < \tilde{p}_c$

$ \rho(x)\, \partial_t u= \Delta u^m $ in $Q$:$=\mathbb{R}^N\times\mathbb{R}_+$

$u(x, 0)=u_0 $ in $\mathbb{R}^N$

in dimensions $ N\ge 3$. We assume that $ m> 1 $ and $
\rho(x) $ is positive and bounded with $ \rho(x)\le
C|x|^{-\gamma} $ as $ |x|\to\infty$ with $\gamma>2$. The
initial data $u_0$ are nonnegative and have finite energy, i.e.,
$ \int \rho(x)u_0 dx< \infty$.

We show that in this case nontrivial solutions to the problem have
a long-time universal behavior in separate variables of the form

$u(x,t)$~$ t^{-1/(m-1)}W(x),$

where $V=W^m$ is the unique bounded, positive solution of the
sublinear elliptic equation $-\Delta V=c\,\rho(x)V^{1/m}$, in
$\mathbb{R}^N$ vanishing as $|x|\to\infty$; $c=1/(m-1)$. Such a
behavior of $u$ is typical of Dirichlet problems on bounded
domains with zero boundary data. It strongly departs from the
behavior in the case of slowly decaying densities, $\rho(x)$~$
|x|^{-\gamma}$ as $|x|\to\infty$ with 0 ≤ $ \gamma<2$, previously
studied by the authors.

If $\rho(x)$ has an intermediate decay, $\rho$~$ |x|^{-\gamma}$
as $|x|\to\infty$ with $2<\gamma<\gamma_2$:$=N-(N-2)/m$, solutions
still enjoy the finite propagation property (as in the case of
lower $\gamma$). In this range a more precise description may be
given at the diffusive scale in terms of source-type solutions
$U(x,t)$ of the related singular equation $|x|^{-\gamma}u_t=
\Delta u^m$. Thus in this range we have * two* different
space-time scales in which the behavior of solutions is
non-trivial. The corresponding results complement each other and
agree in the intermediate region where both apply, thus providing
an example of matched asymptotics.

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