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\markboth{ Convergence to Equilibrium }
{ J\'er\^ome Busca }
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\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
USA-Chile Workshop on Nonlinear Analysis, \newline
Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 45--53. \newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)}
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First moments of energy \\ and
 convergence to equilibrium 
%
\thanks{{\em Mathematics Subject Classifications:}  35B50, 35A05.
 \hfil\break\indent 
{\em Key words:} Parabolic Equations, Equilibrium, Convergence.
 \hfil\break\indent
\copyright 2001 Southwest Texas State University. 
\hfil\break\indent 
Published January 8, 2001. \hfil\break\indent
Partially supported by ECOS/CONICYT project C99E06 } }

\date{}
\author{J\'er\^ome Busca}

\maketitle
\begin{abstract} 
A basic question is to establish 
convergence to equilibrium for globally defined 
solutions to evolution problems. The purpose here 
is to emphasize the role of symmetry. In particular, 
it is proved that in some cases the {\em first moments of energy} are 
constant on the $\omega$-limit set of the solution. This 
key property is used to prove convergence in two model 
evolution problems. This communication 
is based on two joint works with P. Felmer \cite{busca_felmer} 
and M.A. Jendoubi, P. Polacik \cite{busca_jendoubi}.
\end{abstract}

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\section{Introduction and Main Results} 

 A basic question in the study of evolution problems 
is the following: do globally defined in time solutions  
converge to an equilibrium? In case the problem is  {\em dissipative}, 
one can typically prove that the $\omega-$limit set (i.e. the set of all accumulation 
points of the solution $u$) $\omega (u)$ is included in the set of the solutions 
of some limiting stationary equation ({\em steady states}). This 
is usually done thanks to some 
appropriate Lyapunov energy functional.

If the set of steady states contains a continuum, then the convergence issue is 
whether 
the solution actually selects one of them as $t\to +\infty$, 
that 
is, whether $\omega(u)$ is a singleton.
  
 In this generality, or even if one specializes to 
nonlinear parabolic evolution problems for instance, the question is still open, and 
appears to be surprinsingly difficult. Couterexamples 
in the non-autonomous case suggest that limitations do exist (see \cite{polacik}). 
Partial results are available for instance in the analytic setting 
\cite{jendoubi1} \cite{jendoubi2} \cite{simon}, in 
one dimension \cite{matano} \cite{zeleniak}, or under assumptions 
on the linearized operator, either explicitly stated as such, or resulting from the 
nature of the specific problem \cite{hale_raugel} \cite{haraux_polacik} \cite{pll}. 
 
 There is a large literature devoted to these questions, and I will not 
attempt to give any review of the results. For a very clear account 
on this, I refer to \cite{cazenave_haraux} 
and \cite{jendoubi_these}. 
 
 It is my purpose here to show that when the solutions enjoy some symmetry, 
the convergence question can be solved. The simplest nontrivial case is 
probably when the problem is posed in the whole space, is translation and 
rotationally invariant, and the 
set of positive steady states is 
made of all translates of a single radial solution. In this case, 
proving convergence of positive solutions of the evolution problem is tantamount 
to proving that $\omega(u)$ cannot contains more than one of such translates. To this purpose, 
I intend to introduce a method that makes use of {\em first moments of the energy}, 
a tool which appears to be new. These moments will be shown to assume constant 
values on the $\omega-$limit set, just as energy does. However, unlike the latter, 
they are able to discriminate, meaning taking different values on, distinct 
translates. 
 
 For the interested reader I mention references \cite{matano1} \cite{matano2} 
where a thorough investigation of the links between symmetry and convergence 
is given, is the context of stable equilibria (different from the one I address 
here). 
 
 Rather than elaborating on this in full generality, let me select two simple 
instances where 
one can easily highlight the main underlying idea. These examples are taken from joint works with 
P. Felmer \cite{busca_felmer} 
and M.A. Jendoubi, P. Polacik \cite{busca_jendoubi}. 
These two evolutions problems will turn out to 
share the same stationary equation, namely the so-called scalar field equation
\begin{equation}\label{stationary}
\begin{array}{c}
\Delta w - w + w^p = 0\quad\hbox{in }{\mathbb R}^N\\[2pt]
w>0,\quad w(x)\to 0\quad\hbox{as } |x|\to\infty
\end{array}
\end{equation}
with $p$ subcritical, i.e. $1<p<(N+2)/(N-2)$, $N\ge 3$, an assumption 
that I make throughout this paper. 
By the well-known results of Berestycki-Lions \cite{beres_pll}, 
Gidas, Ni and Nirenberg \cite{gnn} and 
Kwong \cite{k} we know that the set of solutions to (\ref{stationary}) is made 
of all translates of a unique positive radial solution. Note that the 
{\em ground state} condition 
\begin{equation}\label{ground_state}
w(x)\to 0\quad\hbox{as }  |x|\to\infty
\end{equation}
is an essential piece of information here. 

 I now turn to the description of the two model problems.


\paragraph{1) A dissipative case: a nonlinear parabolic equation.}
\par Let us consider a globally defined in time 
weak positive solution $u = u(x,t) \in C\left([0,+\infty),H^1({\mathbb R}^N)\right)$ to the following problem:
\begin{equation}\label{parabolic}
\begin{array}{c}
u_t = \Delta u - u + u^p  \quad\hbox{in }{\mathbb R}^N\times(0,+\infty)\\[2pt]
u(\cdot,0) = u_0(\cdot)\in C^\infty_0({\mathbb R}^N)
\end{array}
\end{equation} 
We have the following convergence result: 

\begin{theorem}[\cite{busca_jendoubi, CdPE,fp}]  \label{theo1}
Under the above assumptions, $u(\cdot,t)$ converges (in the $H^1({\mathbb R}^N)$ sense) either to zero or 
to a solution of (\ref{stationary}).
\end{theorem}

\paragraph{2) A conservative case: a nonlinear elliptic equation.}
\par Let us now consider an entire positive solution $u = u(x)$, 
$x = (x_1,\cdots,x_{N+1})$ to the scalar field equation 
$$
\Delta_{{\mathbb R}^{N+1}} u - u +u^p = 0
$$ 
in ${\mathbb R}^{N+1}$, and suppose we are interested in solutions which are not necessarily 
ground states in the sense of (\ref{ground_state}). The simplest case is 
to assume that $u$ goes to zero in a cylindrical set of directions. This leaves out 
one variable ($x_{N+1}$, say) in the direction of which one wants to study the 
possible asymptotic behaviour of $u$. For that reason, it is convenient 
to think of $x_{N+1}$ as {\em time}, and recast the problem as: 
\begin{equation}\label{elliptic}
\begin{array}{c}
u_{tt} + \Delta u - u + u^p = 0\quad\hbox{in }{\mathbb R}^N\times(-\infty,+\infty)\\[2pt]
u(\cdot,t) \to 0\quad\hbox{as }  |x|\to\infty,\ \hbox{uniformy in $t\in{\mathbb R}$},
\end{array}  
\end{equation}
with $(x,t)= (x_1,\cdots, x_N,x_{N+1})$ and 
$\displaystyle\Delta = \sum_{1\le i\le N} \frac{\partial^2}{\partial x_i^2}$, 
the Laplace operator in ${\mathbb R}^N$.  
 
 Since this problem is conservative, soliton-like solution may exists, so  
it is natural to assume that 
\begin{equation}\label{dissipation} 
u_t(x,t)\to 0 \quad\hbox{as }  t\to +\infty,\ \hbox{for all } x\in{\mathbb R}^N.
\end{equation}
Under these assumptions, one has the following convergence results:

\begin{theorem}\label{theo2} (\cite{busca_felmer}) 
Let $u$ be a bounded weak solution to (\ref{elliptic}) satisfying (\ref{dissipation}). 
Then $u(\cdot,t)$ converges (in the $H^1({\mathbb R}^N)$ sense) either to zero or 
to a solution of (\ref{stationary}) as $t\to +\infty$.
\end{theorem}

Since time is reversible in (\ref{elliptic}), it is straightforward 
to get the same result as $t\to-\infty$ if one assumes the equivalent of 
(\ref{dissipation}) as $t\to -\infty$. Moreover, it is an 
interesting fact that the right- and left-hand limits actually turn out to coincide. 
I refer to \cite{busca_felmer} for a proof of this.
 
 
 Theorem \ref{theo1} has been obtained in earlier independent 
works by Cortazar, Elgueta, del Pino \cite{CdPE} and 
Feireisl and Petzeltov\'a \cite{fp} by different methods. 
Theorem \ref{theo1} and Theorem \ref{theo2} are quite particular cases of 
the results in \cite{busca_jendoubi} and \cite{busca_felmer} respectively. 
I simplified the setting here 
in order to draw a parallel between these two results. 
 
 In the sequel, I denote 
by case 1, or parabolic case (resp. 2 or elliptic case) the situation 
prevailing in Theorem \ref{theo1} (resp. \ref{theo2}). 

\section{Sketch of the proofs}

 The key role is played by the energy functional 
\begin{equation}\label{energy}
E(u(\cdot,t)) = \int\left\{\frac{1}{2}|\nabla u|^2 - F(u)\right\}\, dx,
\end{equation}
where $\displaystyle F(s) = \int_0^s f(\zeta)\, d\zeta = - \frac{1}{2}s^2 + \frac{1}{p+1} s^{p+1}$, 
$f(\zeta) = -\zeta + \zeta^p$, and its first ``moments'' (for want of a better name): 
\begin{equation}\label{moments}
E_i(u(\cdot,t))  = \int x_i
\left\{\frac{1}{2}|\nabla u|^2 - F(u)\right\}\, dx,
\end{equation}
$i = 1,\cdots, N$. As for the $\omega-$limit set, it is defined as usual by:
\begin{equation}\label{omega_limit_parabolic}
\omega(u) = \mathop{\cap}\limits_{T >0}
\overline{\mathop{\cup}\limits_{t\ge T}\left\{u(\cdot,t)\right\}}
\end{equation}
in the parabolic case. In the elliptic case, some care is needed. 
One introduces the $\left\{v(t)\right\}_{t\in{\mathbb R}}$, 
defined as $v(t)(x,\tau)=u(x,t+\tau)$, for all $(x,\tau)\in {\mathbb R}^N\times[0,1]$. 
The correct notion 
of (right-hand) $\omega-$limit set turns out to be in this case:
$$
\omega(u) = \mathop{\cap}\limits_{T > 0}
\overline{\mathop{\cup}\limits_{ t\ge T }\left\{v(t)\right\}}.
$$
Here the closures are taken for instance in the $C^2({\mathbb R}^N)$ 
(resp. ${C^2 ({\mathbb R}^{N}\times [0,1])}$) topology.

The relevant properties of these functions are summarized in the following 
result.

\begin{proposition}\label{prop}
In both cases 1 and 2
 we have:  \begin{itemize} 
\item[a)] $\omega(u)$ is 
either $\{0\}$ or made of positive steady states, 
i.e. solutions to (\ref{stationary}) 
\item[b)] $E$ is constant on $\omega(u)$
\item[c)] Each function $E_1,\cdots, E_N$ 
assumes a constant value on $\omega(u)$.
\end{itemize}
\end{proposition}
Note that in case 2, it is part of result a) that the functions in 
$\omega(u)$ do not depend on $\tau\in [0,1]$. 
 
 Theorem \ref{theo1} and \ref{theo2} are simple consequences 
of this proposition. Indeed, suppose for contradiction that 
$\omega(u)$ were to contain two 
distinct translates of the radial solution of (\ref{stationary}), say $w_1$ and $w_2$. 
Up to a Euclidean change 
in co-ordinates, $w_1(x) = w_1(|x|)$ and 
$w_2(x) = w_2(|x-\alpha e_1|)$ with $\alpha\neq 0$. Now $E_1(w_1) = E_1(w_2)$ yields:
\begin{eqnarray*}
0&=&E_1(w_1) = E_1(w_2) \\
&=& \int x_1\left\{\frac{1}{2} |\nabla w_2|^2 - F(w_2)\right\}\,dx \\
 &=& \int (x_1-\alpha)\left\{\frac{1}{2} |\nabla w_2(|x-\alpha e_1|)|^2 - 
F(w_2(|x-\alpha e_1|))\right\}\,dx +\alpha E(w_2)\\
&=& \alpha E(w_2).
\end{eqnarray*}
Here and in the sequel, unless otherwise specified, all integrals in space 
range over ${\mathbb R}^N$. 
 
 Multiplying the stationary equation in (\ref{stationary}) by $w_2$ 
and integrating by parts, it is straightforward to see that 
$E(w_2)\neq 0$, a contradiction.  
 Convergence then follows easily from the fact that 
$\omega(u)$ is a singleton, by 
compactness arguments that I do not reproduce here. 

\begin{lemma}\label{uniform_estimates}
$\exists \varepsilon_0 > 0$ $\exists C>0$ such that $\forall \alpha = (\alpha_1,\cdots, \alpha_N)
\in{\mathbb N}^N$, $\forall k\in\{0,1\}$ in case 1 ($\{0,1,2\}$ in case 2), 
$|\alpha| + k\le 2$, 
$$
\left| \partial_x^\alpha \partial_t^k u(x,t) \right|\le C e^{-\varepsilon_0 |x|}\quad 
\hbox{for all } (x,t)\in{\mathbb R}^N\times {\mathbb R}_+ \ (\hbox{resp. }{\mathbb R}).
$$
\end{lemma}
\noindent {\bf Proof:} It results from Corollary 3.1 in \cite{CdPE} and Lemma 2.1 
in \cite{busca_felmer}, to which I refer. 
The proof is based on comparison principles, together 
with Harnack inequality and blow-up arguments in the parabolic case. \hfill\hfill
\smallskip
\par We now turn to the proof of part a) and b) in Proposition \ref{prop}.

\smallskip 
 
\noindent {\bf Case 1:} It is well-known that $t\mapsto  E(u(\cdot,t))$ is
decreasing since:
\begin{equation}\label{energy_decreases}
\frac{d}{dt}\left\{\int \left\{\frac12 |\nabla u|^2 - F(u)\right\}dx\right\} 
= -\int u_t^2dx.
\end{equation}
Hence $\displaystyle \int_0^{+\infty} \!\! dt \int  u_t^2 dx <\infty$. Making use of 
Lemma \ref{uniform_estimates}, it is standard to infer part a) and b) in Proposition 
\ref{prop}. 

\smallskip 
 
\noindent {\bf Case 2:} Let us test equation (\ref{elliptic}) with $u_t$:
$$
\int_t^{t'}ds\int\left\{ u_t u_{tt} + \Delta u u_t + f(u) u_t\right\} dx = 0.
$$
Integrating by part (see Lemma \ref{uniform_estimates}) results in: 
$$ 
E(u(\cdot,t')) - E(u(\cdot,t)) = \left.\frac12 \int u_t^2(x,s) dx\right|_{s=t}^{t'}.
$$ 
Hence by assumption (\ref{dissipation}) and Lemma \ref{uniform_estimates}, clearly 
$\exists \lim\limits_{t\to+\infty} E(u(\cdot,t))$, Now, testing 
(\ref{elliptic}) with $\phi\in C^\infty_0 ({\mathbb R}^N)$ and integrating by 
parts results in:
\begin{eqnarray*}
\int_t^{t+1} ds\int \left\{u_{tt} + \Delta u + f(u)\right\} \phi(x) dx &=& 0\\
\left.\int u_t(s,x) \phi(x)dx\right|_{s=t}^{t+1} + \int_t^{t+1} ds \int\left\{ u
\Delta \phi 
+ f(u) \phi  \right\}dx &=& 0.
\end{eqnarray*}
Since the first term in this last expression goes to zero as $t\to+\infty$ 
by assumption (\ref{dissipation}) and Lemma \ref{uniform_estimates}, it is clear 
from the definition of $\omega(u)$, Lemma \ref{uniform_estimates}, and (\ref{dissipation}) 
again that any $v = v(x,\tau)\in \omega(u)$ is actually independent of $\tau$ 
and satisfies $\forall\phi\in  C^\infty_0 ({\mathbb R}^N)$ $\int\left\{ 
v\Delta \phi + f(v)\phi\right\} dx = 0$. Since $v\in C^2$, it implies that 
$v$ is a steady state. This completes the proof of part a) and b) 
in Proposition \ref{prop}.     \hfill\hfil
\smallskip

\noindent I shall know sketch the proof of part c) in Proposition \ref{prop} 
in Case 1 and 2.
\smallskip

\noindent {\bf Case 1:} That the $E_i$'s are constant on $\omega(u)$ rely on 
the following lemma:

\begin{lemma}
\begin{itemize}
\item[i)] We have $\displaystyle \frac{d}{dt}E_i\left(u(\cdot,t)\right) = 
-\int_{{\mathbb R}^N} x_i u_t^2\,dx$
\item[ii)] There exists $\delta > 0$ such that
$$
\int_0^{+\infty}dt \int e^{\delta |x|} u_t^2\,dx < +\infty.
$$
Hence the limits $\lim_{t\to+\infty} E_i\left(u(\cdot,t)\right)$, 
$i=1,\cdots,N$, are well-defined.
\end{itemize}
\end{lemma}

\noindent {\bf Proof of i) } Denoting by $u_i$ the derivative of 
$u$ with respect to $x_i$ one has:
\begin{eqnarray*}
\frac{d}{dt} E_i\left(u(\cdot,t)\right) &=& 
-\int u_t\left\{\nabla\cdot\left(x_i\nabla u\right) + x_i f(u)\right\}\, dx\\
&=& -\int x_i u_t^2\, dx - \int u_t u_i\, dx
\end{eqnarray*}
\begin{eqnarray*}
\int u_t u_i\, dx &=& \int\left\{\Delta u + f(u)\right) u_i\, dx\\
&=& -\int \nabla u\cdot\nabla u_i\, dx + \int f(u) u_i\, dx = 0\,. 
\end{eqnarray*}
\ \hfill

\noindent {\bf Proof of ii) }
 Taking polar co-ordinates $x = (r,\theta)$, define for $r>0$: 
$$
 H^T(r) = \frac{1}{2}\int_0^T\, dt\int_{{\cal S}^{N-1}}
u_t^2(r,\theta,t)\,d\theta.
$$
Denoting by $\displaystyle  \Delta_r = \frac{\partial^2}{\partial r^2} + \frac{(N-1)}{r}
\frac{\partial}{\partial r}$ the radial Laplace operator, we have:
We have:
\begin{eqnarray*}
\Delta_r H^T &=& \int_0^{T}dt \int_{{\cal S}^{N-1}} u_t\Delta_r u_t\,d\theta + 
\int_0^{T}\, dt \int_{{\cal S}^{N-1}} |\nabla_r u_t|^2\,d\theta\\
&\ge& -\frac{1}{2}\int_{{\cal S}^{N-1}} u_t^2(0,r,\theta)\, d\theta - 
\int_0^T\, dt \int_{{\cal S}^{N-1}} f' (u) u_t^2\,d\theta,
\end{eqnarray*}
hence: 
$$\Delta_r H^T (r)\ge - \psi(r)  + \alpha H^T (r)\quad\forall r\ge R_0,$$ 
for some positive constants $\alpha,R_0$ and some $\psi\in C_0^\infty ({\mathbb R}^N)$, 
$\psi\ge 0$. 
 
 Now a simple comparison argument with the solutions of $\Delta_r g_0 - \alpha g_0 
= -\psi$, $g_0(r_0) = 0$, $g_0(r)\to 0$ as $r\to \infty$ and 
$\Delta_r g_1 - \alpha g_1 = 0$, 
$g_1(r_0) = 1$, $g_1(r)\to 0$ as $r\to \infty$ implies 
$\forall r_0\ge R_0\,\exists\delta >0\;\exists C>0\;\forall  T>0\;
\forall r\ge r_0$ 
$0\le H^T(r)\le C\left(1+H^T(r_0)\right) e^{ -2\delta r}$. Since 
$\int_0^{+\infty}\! dt\int u_t^2\, dx<+\infty$ by Fubini's Theorem 
$H^T(r_0) \to H^\infty (r_0)<\infty$ as $T\to\infty$ for a.e. $r_0$. 
Hence $H^\infty (r)\le Ce^{-2 \delta r} $ $\forall r\ge r_0$.\hfill\hfill

\smallskip

\noindent {\bf Case 2:} Let us define 
$$
\widetilde{E_i} = E_i - \frac12  \int x_i u_t^2 dx. 
$$
Differentating $\widetilde{E_i}$ results in: 
$$
\frac{d}{dt}\widetilde{E_i}(u)= 
-\int u_t\left\{\nabla\cdot\left(x_i\nabla u\right) + x_i f(u)\right\}\, dx
 - \int x_i u_t u_{tt}\, dx
$$
integrating by parts:
\begin{equation}\label{expr_der}
\frac{d}{dt}\widetilde{E_i}(u)=\int u_t u_{i} dx
\end{equation}
differentiating once more:
\begin{eqnarray*}
\frac{d^2}{ dt^2 }\widetilde{E_i}(u)&=&\int\left\{ \Delta u + f(u)\right\} 
u_{i} dx - \int u_t u_{it} dx\\
&=& - \int \nabla u\cdot\nabla u_{i} - f(u) u_{i} + u_t u_{i}\\ 
&=& - \int \frac{\partial}{\partial x_i} \left\{ \frac12  
|\nabla u |^2 - F(u) + \frac{u_t^2}{2}\right\} dx  = 0\,,
\end{eqnarray*}
thanks to Fubini's Lemma. Note that I have repeatedly used 
Lemma \ref{uniform_estimates} here. Hence $\widetilde{E_i}(u) = \alpha t + \beta$ 
for constants $\alpha$, $\beta$. Now (\ref{expr_der}) together with 
assumption (\ref{dissipation}) imply $\displaystyle\lim_{t\to\infty}\frac{d}{dt}
\widetilde{E_i}(u(\cdot,t)) = 0$. 
Thus $ \alpha = 0 $, hence $t\mapsto\widetilde{E_i}(u(\cdot,t))$ is 
constant. Hence $E_i$ is constant on $\omega(u)$ by (\ref{dissipation}). 
This completes 
the proof of part c) in Proposition \ref{prop}, hence that 
of Theorem \ref{theo2}.\hfill\hfill

\paragraph{Acknowledgments} It is my pleasure to thank the organizers 
of the USA-Chile Congress on Nonlinear Analysis for their invitation. 
This paper is based on a contribution to 
the Nonlinear Analysis 2000 meeting (Courant Institute of Mathematical 
Sciences, New York, May 2000). 

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\bibitem{zeleniak} Zelenyak T.J. Stabilization of solutions of boundary 
value problems for a second-order parabolic equation with one space variable 
{\em Differentsial'nye Uravneniya} {\bf 4} (1968), 17-22.

}\end{thebibliography}

\noindent{\sc J\'er\^ome Busca} \\
Laboratoire de Math\'ematiques et de Physique Th\'eorique \\
Universit\'e Francois Rabelais \\
Parc de Grandmont \\
37200 Tours,  France \\
email: busca@univ-tours.fr


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