\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small 2005-Oujda International Conference on Nonlinear Analysis. \newline {\em Electronic Journal of Differential Equations}, Conference 14, 2006, pp. 119--124.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2006 Texas State University - San Marcos.} \vspace{9mm}} \setcounter{page}{119} \begin{document} \title[\hfilneg EJDE/Conf/14 \hfil Fast and heteroclinic solutions] {Fast and heteroclinic solutions for a second order ODE} \author[M. Arias \hfil EJDE/Conf/14 \hfilneg] {Margarita Arias} \address{Margarita Arias \newline Departamento de Matem\'atica Aplicada \\ Universidad de Granada \\ 18071 Granada, Spain} \email{marias@goliat.ugr.es} \date{} \thanks{Published September 20, 2006.} \subjclass[2000]{34C37, 35K57, 49J35} \keywords{Fisher-Kolmogorov's equation; travelling wave solutions; \hfill\break\indent speed of propagation; variational methods; constrained minimum problem} \begin{abstract} We present some results on the existence of fast and heteroclinic solutions of an ODE connected with travelling wave solutions of a Fisher-Kolmogorov's equation. In particular, we present a variational characterization of the minimum speed of propagation. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \section{Introduction} Some chemical and biological systems can be modelled by an autocatalytic process (see, e.g. \cite{ml,mu}). In many of these process the system can support propagating wavefronts due to a combination of a reaction effect and a molecular diffusion. The pioneering model in this framework is due to Fisher, \cite{f}, who suggested the equation $$u_t=u_{xx}+u(1-u)$$ for studying the spatial spread of a favoured gene in a population. The simplest generalization of that equation is the so called Fisher-Kolmogorov's equation $$u_t=u_{xx}+f(u), \label{1}$$ where $f$ is a given function with two zeroes, say $u=0$ and $u=1$, and positive on $]0,1[$ so that $u=0$ and $u=1$ are the only two stationary states of (\ref{1}). Equations like (\ref{1}) arises in many problems suggested, for instance, by the classical theory of population genetics or by certain flame propagation problems in chemical reactor theory (see, e.g. \cite{aw1}). A \textit{travelling wavefront} or \textit{travelling wave solution} (t.w.s., in short) of (\ref{1}) is a solution $u(t,x)$ having a constant profile, that is, such that $$u(t,x)=\varphi(x-ct),$$ for some fixed $\varphi(\xi)$ (called \textit{the wave shape}) and a constant $c$ (called \textit{the wave speed}). Specially important for the applications are t.w.s. connecting the two stationary states, $u=0$ and $u=1$. A simple calculation shows that if $u(t,x)=\varphi(x-ct)$ is a t.w.s. of (\ref{1}), then the wave shape $\varphi$ is a solution of the ODE $$u''+cu'+f(u)=0. \label{2}$$ When a t.w.s. connects the stationary states, its corresponding wave shape is a positive heteroclinic solution of (\ref{2}) that connects the equilibria 1 and 0, that is, a solution of (\ref{2}) defined on $\mathbb{R}$ and satisfying $$u(t) \in ]0,1[, \quad \forall t \in \mathbb{R}, \quad \lim_{t\to -\infty} u(t)=1, \quad \lim_{t\to +\infty}u(t)=0.$$ There is a vast and rich body of literature dealing with the existence of t.w.s. of (\ref{1}) connecting the stationary states, going from the pioneering work of Kolmogorov, Petrovski and Piskounoff \cite{kpp}, through the remarkable paper of Aronson and Weinberger \cite{aw} up to more recent approaches (see \cite{al,m,m',s}). It is well known (see, e.g. \cite{aw,m}) that there exists a positive number, $c^*$, such that equation (\ref{2}) has a heteroclinic solution connecting 1 and 0 if and only if $c\geq c^*$. In terms of the Fisher-Kolmogorov's equation, that result says that none t.w.s. of (\ref{1}) starting from the stationary state $u=1$ and moving with speed less than $c^*$ reaches the stationary state $u=0$. $c^*$ is called the \textit{minimum propagation speed}. It is clear that the heteroclinic solution, if there exists, is strictly decreasing. When $f$ is differentiable in $[0,1]$, then $c^*\geq 2\sqrt{f'(0)}$ since otherwise the origin cannot acts as an attractor for positive solutions of equation (\ref{2}). It is also proved (see \cite{aw,m,al,s}) that $$c^*\leq 2\sqrt{\sup_{00 approaching (0,0) has slope at the origin$$ \lambda_2= \frac{-c+\sqrt{c^2-4f'(0)}}{2}. $$Moreover, T_c is extremal in the sense that trajectories below it stays bounded away from the origin. Aronson and Weinberger (see \cite{aw}, theorem 4.1) proved that whenever c^*>2\sqrt{f'(0)}, the extremal trajectory T_{c^*} is an heteroclinic solution between 1 and 0. This note is a brief summary of the conference given by the author on the "Colloque International d'Analyse Non lin\'eaire d'Oujda", about some recent results obtained in collaboration with J. Campos, A.M. Robles-P\'erez and L. Sanchez dealing with some variational problems whose solutions are in correspondence with T_c and that, in particular, let us give a variational characterization of c^*. All the presented results with their proofs can be found in \cite{cv}. \section{A variational characterization of fast solutions} We say that a solution u(t) of equation (\ref{2}) is \textit{a fast solution} if its corresponding trajectory is the extremal trajectory T_c. Our purpose is to characterize these solutions in variational terms. In order to do that, we express their speed in approaching 0 by means of an integrability condition: Given c>0, we define the space$$ H_c := \{ u \in H^1_{\rm loc}(0,+\infty) : \int_0^{+\infty} e^{ct}u'(t)^2\,dt < +\infty \mbox{ and } u(+\infty)=0 \} $$with the norm \|u\| = \big(\int_0^{+\infty} e^{ct}u'(t)^2\,dt \big)^{1/2}. This is a Hilbert space and if u\in H_c, u obviously tends quickly'' to 0 as t\to + \infty. We introduce the functional \mathcal{F}:H_c\to\mathbb{R} defined as$$\mathcal{F}(u)=\int_0^{+\infty} e^{ct}(\frac{u'(t)^2}{2}-F(u(t)))\,dt, \quad u\in H_c,$$where F(u):=\int_0^uf(s)\,ds. When \begin{itemize} \item[(H)] f:[0,1]\to\mathbb{R}_+ is a Lipschitz function such that f(0)=0=f(1) and f(u)>0 if 00, and u'(t)<0, for all t\geq0, and that \mathcal{F} has a minimum in \{u\in H_c:u(0)=1\} provided that there exist 0< k <\frac{c^2}{4} with F(u)\leq ku^2/2, for all u \in [0, 1]. Therefore, we have the following result. \begin{proposition} \label{prop1} Assume (H) and there exist 0< k <\frac{c^2}{4} so that F(u)\leq ku^2/2, for all u \in [0, 1]. Then equation \eqref{2} has a fast solution u\in H_c defined on t\geq0 such that u(0)=1 and u'(t)<0, for all t\geq0. \end{proposition} This result is particularly connected to the existence of heteroclinic solutions. Indeed, one can prove that \begin{quote} \textit{If there exists a solution of (\ref{2}) defined on [0, +\infty), with u(0)=1, \; u(t)>0, \; t>0 and u(t)\to 0 as t\to +\infty, then equation (\ref{2}) has an heteroclinic solution.} \end{quote} So, the above proposition proves the existence of heteroclinic solutions whenever \frac{2F(u)}{u^2}\leq \frac{c^2}{4}, for all u\in [0, 1]. Consequently,$$ c^* \leq \inf \{ c>0: \frac{2F(u)}{u^2}\leq \frac{c^2}{4}, \quad \forall u\in [0, 1] \}. $$This upper bound generalizes the estimate in \cite{aw}. \section{Fast heteroclinic solutions} After studying the fast solutions, we ask about heteroclinic connections between the two equilibria u=1 and u=0 of equation (\ref{2}). As in the previous section, we begin by introducing an appropriate space to work. Given c>0, we consider the space$$ X_c:= \{ u \in H^1_{loc}(\mathbb{R}): \int_{_{-\infty}}^{^{+\infty}} e^{ct}u'(t)^2\,dt < +\infty \mbox{ and } u(+\infty)=0 \}, $$with the norm \|u\|_c := \big( \int_{_{-\infty}}^{^{+\infty}} e^{ct}u'(t)^2\,dt\big)^{1/2}. We will say that a solution u, of the equation $$\label{eq41} u''+cu'+\lambda f(u)=0,$$ for some \lambda>0, is a \textit{fast heteroclinic solution} if u\in X_c and u(-\infty)=1. Note that, under assumption (H), any heteroclinic connection u(t) of (\ref{eq41}) between 1 and 0 has the property u(t)\in ]0,1[, u'(t)<0, for all t\in\mathbb{R}. Our aim now is to obtain a variational characterization of the smallest value of \lambda for which equation (\ref{eq41}) has a fast heteroclinic solution. We remark that u(t) is a solution of (\ref{eq41}) for some \lambda >0 if and only if v(t):=u(t/\sqrt{\lambda}) is a solution of (\ref{2}) with c= c/\sqrt{\lambda}. To do that, we introduce two real functionals on X_c:$$ A_c(u) := \int_{-\infty}^{+\infty} e^{ct} \frac{u'(t)^2}{2}\,dt; \quad B_c(u) := \int_{-\infty}^{+\infty} e^{ct}F(u(t))\,dt, $$and we will look for critical points of the restriction of A_c to the set M_c:=\{ u\in X_c: B_c(u)=1 \}. (Note that M_c is non empty as a consequence of the hypothesis on f). We define$$ \lambda(c) :=\inf\{A_c(u): u\in M_c \}. $$It is easy to check that A_c and B_c are C^1-functionals and M_c is a C^1-manifold. By Lagrange multipliers rule, u\in M_c is a critical point of the restriction of A_c to M_c if and only if u\in M_c is a solution of (\ref{eq41}). Playing appropriately with (\ref{eq41}) we are able to prove that \begin{quote} \textit{If \lambda(c) is attained, then equation (\ref{eq41}) with \lambda =\lambda(c) has a fast heteroclinic solution u\in M_c and A_c(u)=\lambda(c).} \end{quote} \begin{remark} \label{rmk0} \rm Given u\in X_c and a \in \mathbb{R}, the function v(t):=u(t-a) belongs to X_c and A_c(v)=e^{ca}A_c(u), B_c(u)=e^{ca}B_c(u). So, if u\in X_c is a critical point of A_c subject to the restriction B_c(u)=1, for all \alpha >0, the function v(t):=u(t-\frac{\ln \alpha}{c}) is a critical point of A_c subject to the restriction B_c(u)=\alpha. Hence, condition B_c(u)=1 is a kind of normalization. \end{remark} The previous result reduces the problem of the existence of fast heteroclinic solutions to prove that \lambda(c) is attained. Using a convenient closed convex set, we show that \lambda(c) is attained when F(u)=o(u^2) as u\to 0^+. Finally, working with an auxiliary functional defined on that closed convex set, we obtain our main result. \begin{theorem} \label{teo44} Assume (H), and also that there exists f'(0) and $$\label{eqH3} \lambda(c) < \frac{c^2}{4f'(0)}.$$ Then, \lambda(c) is attained. In particular, (\ref{3}) with \lambda=\lambda(c) has a fast heteroclinic solution. \end{theorem} Observe that our approach does not require differentiability except at the origin. On the other hand, if there exists f'(0), working with truncations of the function \varepsilon e^{-kt}, \varepsilon \to 0, k\downarrow c/2, one can prove$$ \lambda(c) \leq \frac{c^2}{4f'(0)}, $$and condition (\ref{eqH3}) is almost necessary. Moreover, as a consequence of this result, if there exists f'(0), \lambda(c) is positive. A simple change of variable shows that \lambda(c)=c^2\lambda(1). Hence, condition (\ref{eqH3}) is independent of c and it can be write$$ \lambda(1) < \frac{1}{4f'(0)}. $$\section{A variational characterization of c^*} Theorem \ref{teo44} let us obtain a variational characterization of the \textit{minimum propagation speed} c^*. As we have already said in the introduction,$$ c^*:=\inf\{ c\in \mathbb{R}: (\ref{2}) \mbox{ has an heteroclinic solution.} \} $$Mallaguti and Marcelli \cite{m'} proved that c^* is in fact a minimum and it is positive. We are going to relate this number with the function \lambda(c) introduced in the previous section. In order to do that, let us define$$\bar c :=\frac{1}{\sqrt{\lambda(1)}}.$$Having in mind that \lambda(c) \leq \frac{c^2}{4f'(0)} and \lambda(c)=c^2\lambda(1), one has that \bar c \geq 2\sqrt{f'(0)}. From Theorem \ref{teo44}, if \bar c > 2\sqrt{f'(0)}, equation (\ref{2}) with c=\bar c has a fast heteroclinic solution and, then, \bar c \geq c^*. We can prove the following result. \begin{theorem} \bar{c}=c^*. \end{theorem} The proof of this theorem is based on the following result. \begin{proposition}\label{prop51} Assume that for some c>2\sqrt{f'(0)} there exists an heteroclinic solution. Then, c=c^* if and only if this heteroclinic is fast. \end{proposition} Remark that the previous proposition says: \begin{quote} \textit{At least when c>2\sqrt{f'(0)}, c^* is the only value of the parameter for which the heteroclinic connection between the two equilibria of (\ref{2}) is fast.} \end{quote} The proof of this result follows by interpreting positive decreasing solutions of (\ref{2}) as solutions of a suitable first order equation (as it has been done in \cite{sa,m}). A positive decreasing solution of (\ref{2}) has a trajectory in the second quadrant of the (u,u')-plane. It is about looking at such a trajectory as the graph of a function \phi, so that u'=\phi(u). Putting y(u)=\phi(u)^2, y is a solution of $$\label{eq53} \frac{dy}{du} =2c\sqrt y-2f(u).$$ A heteroclinic solution of (\ref{2}) corresponds to a positive solution of (\ref{eq53}) on ]0,1[ such that y(0)=y(1)=0. (Note that the Cauchy problem for equation (\ref{eq53}) has no uniqueness, but any solution of (\ref{eq53}) can be continued as long as it remains positive.) Summarizing, we obtain$$ c^*= \Big( \inf \Big\{ \int_{-\infty}^{+\infty} e^{ct} \frac{u'(t)^2}{2}\,dt\,: \; u\in X_1, \; \int_{_{-\infty}}^{^{+\infty}} e^{ct}F(u(t))\,dt =1 \Big\}\Big)^{-1}.  Moreover, when $c>c^*$ equation (\ref{2}) has an heteroclinic connection between its equilibria though it is no fast, that is, \textit{the extremal trajectory $T_c$ is not an heteroclinic}, but if $c=c^*>2\sqrt{f'(0)}$, then $T_c$ connects the two equilibria. 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