0$, provided $1

\gamma_p$, while, for $\gamma= \gamma_p$, it decays at infinity as \be f_{\gamma_p}(x) \sim |x|^{\frac{2}{p-1}-n} e^{- \frac{x^2}{4}}. \label{8} \en We prove here that these solutions are stable in two senses: first, there exists a ball in a Banach space of initial data such that the corresponding solutions tend, in the appropriate norm, as $t\rightarrow\infty$, to (\ref{3}). Secondly, any initial data satisfying a suitable positivity condition will give rise to a solution again tending to (\ref{3}). More precisely, let $q>{2\over p-1}$ and consider the Banach space $B$ of $L^\infty$ functions $h$ equipped with the norm (with some abuse of notation!) \be \| h \|_\infty = {\rm ess}\sup_\xi |h(\xi) (1+|\xi|^q)|. \label{19} \en We consider the initial data (taken at time 1 for later convenience) \be u(x,1)=f_\gamma(x)+h(x) \label{89} \en with $h\in B$. We prove the \vs{3mm} \no {\bf Theorem } {\it Let $1 < p < 1 + \frac{2}{n} $. There exist $\varepsilon > 0, C < \infty$ and $\mu > 0$ such that, if the initial data $u(x,1)$ of} (\ref{4}) {\it is given by} (\ref{89}) {\it with $h\in B$ and satisfies either $$ \|h\|_\infty \leq \varepsilon $$ or $$ h(x)\geq 0 $$ (a.e.) then,} (\ref{4}) {\it has a unique classical solution and, for all $t$, $$ \| t^{\frac{1}{p-1}} u(\cdot t^{\frac{1}{2}},t) - f_\gamma(\cdot) \|_\infty \leq C t^{- \mu}\| h\|_\infty $$} \vs{3mm} \section{Proof} %\setcounter{equation}{0} \medskip Before going to the proof of the Theorem, we will briefly discuss the scale invariant solutions (\ref{3}). These are given by $f_\gamma(x) = \phi_\gamma(|x|)$ and $\phi_\gamma$ solves the ordinary differential equation \be \phi^{''} + \left( \frac{n-1}{\eta} + \frac{\eta}{2} \right) \phi^{'} + \frac{\phi}{p-1} - \phi^{p} = 0 \label{6} \en for $\eta = |x| \in [0, \infty[$. The theory of positive solutions of (\ref{6}) has been developped in \cite{Br,Ga,KP1}. The main result is that, for any $p>1$, there exists smooth, everywhere positive solutions, $\phi_\gamma$, of (\ref{6}) with $\phi_\gamma^{'}(0)=0$ and $\phi_\gamma(0) =\gamma$ for $\gamma$ larger than a certain critical value $\gamma_p$ (but not too large). Actually, for $p < 1 + \frac{2}{n}, \gamma_p > 0$ while $\gamma_p = 0$ for $p \geq 1 + \frac{2}{n}$. The decay at infinity of these solutions is given in (\ref{7}, \ref{8}). The existence of a critical $\gamma_p$ can be understood intuitively by viewing (\ref{6}) as Newton's equation for a particle of mass one, whose ``position" as a function of ``time" is $\phi(\eta)$. The potential is then $U(\phi) = \frac{\phi^{2}}{2(p-1)} - \frac{\phi^{p+1}}{p+1}$ and the ``friction term" $\left( \frac{n-1}{\eta} + \frac{\eta}{2} \right) \phi^{'}$ depends on the ``time" $\eta$. Hence, if $\phi_\gamma^{'} (0) = 0$ and $\phi_\gamma(0)=\gamma$ is large enough, the time it takes to approach zero is long and, by then, the friction term has become sufficiently strong to prevent ``overshooting". However, as $p$ increases, the potential becomes flatter and one therefore expects $\gamma_p$ to decrease with $p$. Given the initial data (\ref{89}), it is convenient to rewrite (\ref{4}) in terms of the variables $\xi = xt^{- \frac{1}{2}}$ and $\tau = \log t$; so, define $v(\xi, \tau)$ by: \be u(x,t) = t^{- \frac{1}{p-1}} (f_\gamma (xt^{- \frac{1}{2}}) + v (xt^{- \frac{1}{2}}, \log t)) \label{90} \en where now \be v(\xi,0) = h(\xi). \label{10} \en Then, (\ref{4}) is equivalent to the equation \be \partial_\tau v = {\cal L} v - \left(|f_\gamma+v|^{p-1}(f_\gamma +v) - f_\gamma^p - pf_\gamma^{p-1} v \right) \equiv {\cal L} v + N(v) \label{123} \en where we used the fact that (\ref{3}) solves (\ref{4}) and gathered the linear terms in $$ {\cal L} = {\cal L}_0 + V_\gamma, $$ with \be {\cal L}_0=\Delta + \frac{\xi}{2} \cdot \nabla + \frac{1}{p-1}, \label{11} \en and \be V_\gamma(\xi)=-pf_\gamma^{p-1}(\xi). \label{124} \en To prove that the solution $t^{- \frac{1}{p-1}} (f_\gamma (xt^{- \frac{1}{2}}) )$ is stable means to find a class of initial data $v(\xi,0)$ such that the corresponding solution of (\ref{123}) goes to zero as $\tau \rightarrow \infty$, in a suitable norm. The Theorem of Section 1 reads now in terms of $v$ as \vs{3mm} \no{\bf Proposition 1.} {\it With the assumptions of the Theorem,} (\ref{123}), {\it with initial data (11) ,has a unique classical solution and} $$ \|v(\cdot,\tau)\|_\infty\leq Ce^{-\mu\tau}\|h\|_\infty $$ \vs{3mm} The main input in the proof is the following estimate on the semigroup $e^{\tau{\cal L}}$: \vs{3mm} \no{\bf Proposition 2.} {\it The operator $e^{\tau{\cal L}}$ is a bounded operator in the Banach space $ B$, and its norm satisfies $$ \|e^{\tau{\cal L}} \|\leq Ce^{-\mu\tau} $$ for some $\mu>0$, $C<\infty$.} \vs{3mm} There are two important ingredients in the proof of Proposition 2. The first is the fact that $e^{\tau{\cal L}}$ is a contraction in a suitable Hilbert space of rapidly decreasing functions. To see this, note first that ${\cal L}_0$ is conjugate to the Schr\"odinger operator \be e^{\frac{\xi^2}{8}} {\cal L}_0 e^{- \frac{\xi^2}{8}} = \Delta - \frac{\xi^2}{16} - \frac{n}{4} + \frac{1}{p-1} \label{12} \en i.e. the harmonic oscillator. Thus ${\cal L}_0$ is self-adjoint on its domain ${\cal D}({\cal L}_0) \subset L^2 ({\bf R}^n, d\mu)$, where $$ d\mu(\xi)=e^{\frac{\xi^2}{4}}d\xi. $$ ${\cal L}_0$ has a pure point spectrum $\{{1\over p-1}-{n\over 2} -{m\over 2}\;|\; m=0,1,\dots\}$ and the largest eigenvalue $ {1\over p-1}-{n\over 2}$ is {\it positive} if $1

1+{2\over n}$ case).
Remarkably, it is possible to prove that ${\cal L}< -E <0$ without a
detailed study of the function $f_\gamma$, but only using equation
(\ref{6}). We have the
\vs{3mm}
\no{\bf Lemma 1.} {\it The operator $e^{\tau{\cal L}}$ is a bounded
operator in the Hilbert space $L^2 ({\bf R}^n, d\mu)$ and its norm
satisfies
$$
\|e^{\tau{\cal L}} \|\leq e^{-E\tau}
$$
for some $E>0$.}
\vs{3mm}
\no{\bf Proof.} Since $V_\gamma$ is bounded, $\cal L$ is self-adjoint and,
as for ${\cal L}_0$, its resolvent is compact and, therefore, its spectrum
is
pure point. Let $-E_\gamma$ be the largest eigenvalue.
First note that $-E_\gamma\leq -E_{\gamma_p}$. Indeed, this holds since
$V_\gamma\leq V_{\gamma_p}$, because
$f_{\gamma}\geq f_{\gamma_p}$, which in turn follows from the fact that
$\gamma_p$ is the smallest
allowed value of $\phi_\gamma(0)=\gamma$ in (\ref{6}), and that two
solutions of (\ref{6}), both with initial conditions $\phi_\gamma'(0)=0$,
will not cross.
Hence it suffices to prove the claim for $\gamma=\gamma_p$. Let us write $E
\equiv E_{\gamma_p}$.
Next we note that, by the Feynman-Kac formula \cite{Si}, $e^{\tau{\cal L}}$
has a strictly positive kernel; indeed, since $-C