\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 \begin{equation} u_t=u_{xx}+f(u), \label{1} \end{equation} 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 \begin{equation} u''+cu'+f(u)=0. \label{2} \end{equation} 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 \}, $$ with the norm $\|u\|_c := \big( \int_{_{-\infty}}^{^{+\infty}} e^{ct}u'(t)^2\,dt\big)^{1/2}$. We will say that a solution $u$, of the equation \begin{equation}\label{eq41} u''+cu'+\lambda f(u)=0, \end{equation} for some $\lambda>0$, is a \textit{fast heteroclinic solution} if $u\in X_c$ and $u(-\infty)=1$. Note that, under assumption (H), any heteroclinic connection $u(t)$ of (\ref{eq41}) between 1 and 0 has the property $u(t)\in ]0,1[$, $u'(t)<0$, for all $t\in\mathbb{R}$. Our aim now is to obtain a variational characterization of the smallest value of $\lambda$ for which equation (\ref{eq41}) has a fast heteroclinic solution. We remark that $u(t)$ is a solution of (\ref{eq41}) for some $\lambda >0$ if and only if $v(t):=u(t/\sqrt{\lambda})$ is a solution of (\ref{2}) with $c= c/\sqrt{\lambda}$. To do that, we introduce two real functionals on $X_c$: $$ 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 \}. $$ It is easy to check that $A_c$ and $B_c$ are $C^1$-functionals and $M_c$ is a $C^1$-manifold. By Lagrange multipliers rule, $u\in M_c$ is a critical point of the restriction of $A_c$ to $M_c$ if and only if $u\in M_c$ is a solution of (\ref{eq41}). Playing appropriately with (\ref{eq41}) we are able to prove that \begin{quote} \textit{If $\lambda(c)$ is attained, then equation (\ref{eq41}) with $\lambda =\lambda(c)$ has a fast heteroclinic solution $u\in M_c$ and $A_c(u)=\lambda(c)$.} \end{quote} \begin{remark} \label{rmk0} \rm Given $u\in X_c$ and $a \in \mathbb{R}$, the function $v(t):=u(t-a)$ belongs to $X_c$ and $A_c(v)=e^{ca}A_c(u$), $B_c(u)=e^{ca}B_c(u)$. So, if $u\in X_c$ is a critical point of $A_c$ subject to the restriction $B_c(u)=1$, for all $\alpha >0$, the function $v(t):=u(t-\frac{\ln \alpha}{c})$ is a critical point of $A_c$ subject to the restriction $B_c(u)=\alpha$. Hence, condition $B_c(u)=1$ is a kind of normalization. \end{remark} The previous result reduces the problem of the existence of fast heteroclinic solutions to prove that $\lambda(c)$ is attained. Using a convenient closed convex set, we show that $\lambda(c)$ is attained when $F(u)=o(u^2)$ as $u\to 0^+$. Finally, working with an auxiliary functional defined on that closed convex set, we obtain our main result. \begin{theorem} \label{teo44} Assume (H), and also that there exists $f'(0)$ and \begin{equation}\label{eqH3} \lambda(c) < \frac{c^2}{4f'(0)}. \end{equation} Then, $\lambda(c)$ is attained. In particular, (\ref{3}) with $\lambda=\lambda(c)$ has a fast heteroclinic solution. \end{theorem} Observe that our approach does not require differentiability except at the origin. On the other hand, if there exists $f'(0)$, working with truncations of the function $\varepsilon e^{-kt}$, $\varepsilon \to 0$, $k\downarrow c/2$, one can prove $$ \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)}}.$$ Having in mind that $\lambda(c) \leq \frac{c^2}{4f'(0)}$ and $\lambda(c)=c^2\lambda(1)$, one has that $\bar c \geq 2\sqrt{f'(0)}$. From Theorem \ref{teo44}, if $\bar c > 2\sqrt{f'(0)}$, equation (\ref{2}) with $c=\bar c$ has a fast heteroclinic solution and, then, $\bar c \geq c^*$. We can prove the following result. \begin{theorem} $\bar{c}=c^*$. \end{theorem} The proof of this theorem is based on the following result. \begin{proposition}\label{prop51} Assume that for some $c>2\sqrt{f'(0)}$ there exists an heteroclinic solution. Then, $c=c^*$ if and only if this heteroclinic is fast. \end{proposition} Remark that the previous proposition says: \begin{quote} \textit{At least when $c>2\sqrt{f'(0)}$, $c^*$ is the only value of the parameter for which the heteroclinic connection between the two equilibria of (\ref{2}) is fast.} \end{quote} The proof of this result follows by interpreting positive decreasing solutions of (\ref{2}) as solutions of a suitable first order equation (as it has been done in \cite{sa,m}). A positive decreasing solution of (\ref{2}) has a trajectory in the second quadrant of the $(u,u')$-plane. It is about looking at such a trajectory as the graph of a function $\phi$, so that $u'=\phi(u)$. Putting $y(u)=\phi(u)^2$, $y$ is a solution of \begin{equation}\label{eq53} \frac{dy}{du} =2c\sqrt y-2f(u). \end{equation} A heteroclinic solution of (\ref{2}) corresponds to a positive solution of (\ref{eq53}) on $]0,1[$ such that $y(0)=y(1)=0$. (Note that the Cauchy problem for equation (\ref{eq53}) has no uniqueness, but any solution of (\ref{eq53}) can be continued as long as it remains positive.) Summarizing, we obtain $$ 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. When $c^*=2\sqrt{f'(0)}$, (\ref{2}) has an heteroclinic connection between its equilibria, but it is an open problem to know if it is or not a fast heteroclinic. \begin{thebibliography}{00} \bibitem{al} \textsc{S. Ahmad and A.C. Lazer, } \emph{An elementary approach to travelling front solutions to a system of N competition-diffusion equations}, Nonlinear Anal. TMA \textbf{16}, 893-901 (1991). \bibitem{cv} \textsc{M. Arias, J. Campos, A.M. Robles-P\'erez and L. Sanchez, } \emph{Fast and heteroclinic solutions for a second order ODE related to Fisher-Kolmogorov's equation.}, Calc. Var. Partial Differential Equations \textbf{21}, no. 3, 319--334, (2004). \bibitem{aw1} \textsc{D.G. Aronson and H.F. Weinberger}, \emph{Nonlinear diffusion in population genetics, commbustion, and nerve propagation}, in "Partial Differential Equations and Related Topics, Lecture Notes in Mathematics",Vol. 446, pp. 5-49, Springer, New York, 1975. \bibitem{aw} \textsc{D.G. Aronson and H.F. Weinberger}, \emph{Multidimensional nonlinear diffusion arising in population genetics}, Adv. Math. \textbf{30}, 33-76 (1987). \bibitem{f} \textsc{R.A. Fisher}, \emph{The wave of advance of advantageous genes}, Ann. Eugenics \textbf{7}, 353-369 (1937). \bibitem{kpp} \textsc{A. Kolmogoroff, I. Petrosky and N. Piscounoff}, \emph{\'{E}tudes de l'equation aved croissance de la quantit\'{e} de mati\`{e}re et son application \`{a} un probl\`{e}me biologique}, Moscow Univ. Bull. Math.,1, 1-25 (1937). \bibitem{m} \textsc{L. Malaguti and C. Marcelli}, \emph{Existence of bounded trajectories via upper and lower solutions}, Discr. Cont. Dyn. Sist., \textbf{6}(3), 575-590 (2000). \bibitem{m'} \textsc{L. Malaguti and C. Marcelli}, \emph{Travelling wavefronts in reaction-diffusion equations with convection effects and non-regular terms}, Math. Nachr., \textbf{242}, 148-164 (2002). \bibitem{ml} \textsc{P.M. Mcabe, J.A. Leach and D.J. Needham}, \emph{The evolution of travelling waves in fractional order autocatalysis with decay. I. Permanent form travelling waves}, SIAM J. Appl. Math., \textbf{59}, 870-899, (1998). \bibitem{mu} \textsc{J.D. Murray}, Mathematical Biology, Springer-Verlag, Berlin, 1993. \bibitem{sa} \textsc{P.L. Sachdev}, Nonlinear ordinary differential equations and their applications, M. Dekker, N. York 1991. \bibitem{s} \textsc{L. Sanchez}, \emph{A note on a nonautonomous O.D.E. related to the Fisher equation}, J. Comp. Appl. Math. \textbf{113}, 201-209 (2000). \end{thebibliography} \end{document}