\documentclass[twoside]{article} \usepackage{amsfonts} % used for R in Real numbers \pagestyle{myheadings} \markboth{ Convergence to Equilibrium } { J\'er\^ome Busca } \begin{document} \setcounter{page}{45} \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)} \vspace{\bigskipamount} \\ % 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} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{lemma}[theorem]{Lemma} \renewcommand{\theequation}{\thesection.\arabic{equation}} \catcode@=11 \@addtoreset{equation}{section} \catcode@=12 \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 $$\label{stationary} \begin{array}{c} \Delta w - w + w^p = 0\quad\hbox{in }{\mathbb R}^N\\[2pt] w>0,\quad w(x)\to 0\quad\hbox{as } |x|\to\infty \end{array}$$ with $p$ subcritical, i.e. $10} \overline{\mathop{\cup}\limits_{t\ge T}\left\{u(\cdot,t)\right\}} in the parabolic case. In the elliptic case, some care is needed. One introduces the$\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: $$\label{energy_decreases} \frac{d}{dt}\left\{\int \left\{\frac12 |\nabla u|^2 - F(u)\right\}dx\right\} = -\int u_t^2dx.$$ 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$\forall r_0\ge R_0\,\exists\delta >0\;\exists C>0\;\forall T>0\; \forall r\ge r_00\le H^T(r)\le C\left(1+H^T(r_0)\right) e^{ -2\delta r}$. 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: $$\label{expr_der} \frac{d}{dt}\widetilde{E_i}(u)=\int u_t u_{i} dx$$ 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). \begin{thebibliography}{99} {\frenchspacing \bibitem{beres_pll} Berestycki H., Lions P.-L. Non linear scalar field equations I. Existence of a ground state. {\em Arch. Rational Mech. Anal.} {\bf 82}, 313-345 (1983). \bibitem{busca} Busca J. Symmetry and nonexistence results for Emden-Fowler equations in cones. 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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 \end{document}