Large-time behavior of solutions of parabolic equations on the real line with convergent initial data II: equal limits at infinity

Consider the Cauchy problem

	$\displaystyle u_{t}=u_{xx}+f(u),$	$\displaystyle\qquad x\in\mathbb{R},\ t>0,$		(1.1)
	$\displaystyle u(x,0)=u_{0}(x),$	$\displaystyle\qquad x\in\mathbb{R},$		(1.2)

where $f$ is a locally Lipschitz function on $\mathbb{R}$ and $u_{0}\in C_{b}(\mathbb{R}):=C(\mathbb{R})\cap L^{\infty}(\mathbb{R}).$ We denote by $u(\cdot,t,u_{0}),$ or simply $u(\cdot,t)$ if there is no danger of confusion, the unique classical solution of (1.1), (1.2) and by $T(u_{0})\in(0,+\infty]$ its maximal existence time. If $u$ is bounded on $\mathbb{R}\times[0,T(u_{0}))$, then necessarily $T(u_{0})=+\infty,$ that is, the solution is global. In this paper, we are concerned with the behavior of bounded solutions as $t\to\infty$. A basic question we specifically want to address is whether, or to what extent, the large-time behavior of bounded solutions is governed by steady states of (1.1).

If the initial data $u_{0}$ admits limits as $x\to\pm\infty,$ then for all time $t>0,$ the solution $u(\cdot,t)$ of (1.1), (1.2) admits limits as $x\to\pm\infty.$ In other words, the function space

$$\mathcal{V}:=\left\{v\in C_{b}(\mathbb{R}):\textrm{ the limits }v(-\infty),\,v(+\infty)\in\mathbb{R}\textrm{ exist}\right\}$$

(1.3)

is invariant for (1.1). Continuing our study initiated in [29], we examine the large time behavior of bounded solutions in ${\mathcal{V}}$. More specifically, we are interested in the behavior of the solutions in bounded—albeit arbitrarily large—spatial intervals, as $t\to\infty$. For that purpose, we introduce the $\omega-$limit set of a bounded solution $u$, denoted by $\omega(u)$ or $\omega(u_{0})$ with $u_{0}=u(\cdot,0)$, as follows:

$$\omega(u):=\left\{\varphi\in L^{\infty}(\mathbb{R}),\ u(\cdot,t_{n})\to\varphi\textrm{ for some sequence }t_{n}\to\infty\right\}.$$

(1.4)

Here the convergence is in the topology of $L^{\infty}_{loc}(\mathbb{R})$, that is, the locally uniform convergence. By standard parabolic estimates, the trajectory $\{u(\cdot,t),\ t\geq 1\}$ of a bounded solution is relatively compact in $L^{\infty}_{loc}(\mathbb{R}).$ This implies that $\omega(u)$ is nonempty, compact, and connected $L^{\infty}_{loc}(\mathbb{R})$, and it attracts the solution in (the metric space) $L^{\infty}_{loc}(\mathbb{R})$:

$$\textrm{dist}_{L^{\infty}_{loc}(\mathbb{R})}\left(u(\cdot,t),\omega(u)\right)\underset{t\to\infty}{\longrightarrow}0.$$

If the $\omega-$limit set reduces to a single element $\varphi$, then $u$ is convergent: $u(\cdot,t)\to\varphi$ in $L^{\infty}_{loc}(\mathbb{R})$ as $t\to\infty$. Necessarily, $\varphi$ is a steady state of (1.1). If all functions $\varphi\in\omega(u)$ are steady states of (1.1), the solution $u$ is said to be quasiconvergent.

Convergence and quasiconvergence both express a relatively tame character of the solution in question, entailing in particular the property that $u_{t}(\cdot,t)$ approaches zero locally uniformly on $\mathbb{R}$ as $t\to\infty$. In some cases, quasiconvergence can be established by means of energy estimates when bounded solutions in suitable energy spaces are considered (see [12], for example). However, when no particular rate of approach of $u_{0}(x)$ to its limits at $x=\pm\infty$ is assumed, energy techniques typically do not apply. Nonetheless, several quasiconvergence results are available for solutions in ${\mathcal{V}}$ (see [33] for a general overview). These include quasiconvergence theorems of [23] for nonnegative functions $u_{0}$ with $u_{0}(\pm\infty)=0$ when $f(0)=0$—convergence theorems are available under additional conditions on $u_{0}\geq 0$, see [10, 11, 23]; or for generic $f$, see [25]—and a theorem of [35] for functions $u_{0}\in{\mathcal{V}}$ satisfying $u_{0}(-\infty)>u_{0}>u_{0}(\infty)$ or $u_{0}(-\infty)<u_{0}<u_{0}(\infty)$. An improvement over the latter quasiconvergence result was achieved in [29], where we proved that the condition $u_{0}(-\infty)\neq u_{0}(+\infty)$ alone, with no relations involving $u_{0}(x)$ for $x\in\mathbb{R}$, is already sufficient for the quasiconvergence of the solution if it is bounded.

It is also known that the $\omega$-limit set of a bounded solution always contains at least one equilibrium [18, 19]. However, bounded solutions, even those in ${\mathcal{V}}$, are not always quasiconvergent (see [30, 32]). Moreover, as shown in [32], non-quasiconvergent solutions occur in (1.1) in a persistent manner: they exist whenever $f$ is a $C^{1}$ nonlinearity satisfying certain robust conditions (cp. (1.9) below). In view of these results, the following question arises naturally. Given $f$, can one characterize in some way the initial data $u_{0}\in{\mathcal{V}}$ which yield quasiconvergence solutions? Our previous work [29] was our first step in addressing this question: we proved the quasiconvergence in the distinct-limits case: $u_{0}(-\infty)\neq u_{0}(+\infty)$. In the present paper, we consider the case when the limits are equal: $u_{0}(-\infty)=u_{0}(+\infty):=\theta_{0}$. We assume the nonlinearity $f$ to be fixed and satisfy minor nondegeneracy conditions (see the next section).

In our first main theorem, Theorem 1.1, we show that if $f(\theta_{0})\neq 0$, or if $f(\theta_{0})=0$ and $\theta_{0}$ is a stable equilibrium of the equation $\dot{\xi}=f(\xi)$, then the solution $u$ of (1.1), (1.2) is quasiconvergent if bounded. In the examples of non-quasiconvergent solutions with $u_{0}(\pm\infty)=\theta_{0}$, as given in [30, 32], $\theta_{0}$ is an unstable equilibrium of $\dot{\xi}=f(\xi)$. Thus, our theorem shows that this is in fact necessary. Other two results, Theorems 1.3 and 1.4, give sufficient conditions for the quasiconvergence of the solution in the case that $\theta_{0}$ is an unstable equilibrium of the equation $\dot{\xi}=f(\xi)$. A special case of the sufficient condition of Theorem 1.4 is the condition that $u_{0}-\theta_{0}$ has compact support and only finitely many sign changes. Theorem 1.3 has a somewhat surprising result saying that any element $\varphi$ of $\omega(u)$ whose range is not included in the minimal bistable interval containing $\theta_{0}$ is necessarily a steady state.

We give formal statements of these results in Subsection 1.3, after first formulating our hypotheses in Subsection 1.2. In Subsection 1.4, we give an outline of our strategy of proving the quasiconvergence theorems.

A quasiconvergence result closely related to our Theorem 1.1 has recently been proved by Risler. In [36], he considers the Cauchy problem for gradient reaction-diffusion systems on $\mathbb{R}$, where the initial data are assumed to converge at $\pm\infty$ to stable homogeneous steady states contained in the same level of the potential function. Under certain generic conditions on the corresponding stationary system, he proves that bounded solutions of such Cauchy problems are quasiconvergent in a localized topology (in the companion paper [37], the global shape of such solutions at large times is investigated). His approach is variational, which has an advantage that it applies to gradient systems, as opposed to our techniques based on the zero number, which only apply to scalar equations. In the scalar case, our method seems to have some advantages. For example, it allows us to treat to some extent the case when the limit of the initial data at $\pm\infty$ is unstable. Also, in principle, the method can be used under much less stringent nondegeneracy conditions (cp. Subsection 1.2) and, we believe, will eventually allow us to dispose of the nondegeneracy conditions altogether.

A key ingredient of our method of proof of the quasiconvergence theorems is a classification result for a certain type of entire solutions of (1.1), that is, solutions defined for all $t\in\mathbb{R}$. Roughly speaking, the result shows that the entire solutions are either steady states or connections between two disjoint sets of steady states of (1.1) (see Sections 1.4 and 4 for details). Entire solutions play an important role in qualitative analysis of solutions of parabolic equations, as it can usually be proved that the large-time behavior of bounded solutions is governed by entire solutions. In our setting, for example, the $\omega$-limit sets—or their generalized versions, as defined in Section 2.3—of bounded solutions of (1.1) consist of orbits of entire solutions. Entire solutions of (1.1) have been extensively studied and many different types of such solutions have been found. These include, in addition to steady states, spatially periodic heteroclinic orbits between steady states (see [14, 15, 34] for example), traveling waves and many types of “nonlinear superpositions” of traveling waves and other entire solutions (see [4, 5, 8, 20, 21, 27, 28] and references therein), as well entire solutions involving colliding pulses [24]. Unlike for equations on bounded intervals where rather general classification results for entire solutions are available (see [3, 15, 39] and references therein), no such general classification is currently in sight for the vast variety of entire solutions of (1.1). Our result classifying certain entire solutions as connections between two sets of steady state is a modest contribution in this area, exploring the asymptotic behavior of entire solutions as $t\to\pm\infty$ in the topology of $L^{\infty}_{loc}(\mathbb{R})$.

As above, $f$ is a locally Lipschitz function. We also assume the following nondegeneracy condition:

(ND)

For each $\gamma\in f^{-1}\{0\}$, $f$ is of class $C^{1}$ in a neighborhood of $\gamma$ and $f^{\prime}(\gamma)\neq 0.$

Our theorems can be proved under weaker conditions. To give an example of how (ND) can be relaxed, set

$$\displaystyle F(v):=\int_{0}^{v}f(s)ds,$$

(1.5)

so zeros of $f$ are critical points of $F$. The following nondegeneracy conditions can be considered in place of (ND).

(ND1)

Each $\gamma\in f^{-1}\{0\}$ is locally a point of strict maximum or strict minimum for $F.$

(ND2)

If $\gamma_{1}<\gamma_{2}$ are two consecutive local-maximum points of $F$ and $F(\gamma_{1})=F(\gamma_{2}),$ then $\gamma_{1}$, $\gamma_{2}$ are nondegenerate critical points of $F$: $f$ is of class $C^{1}$ in a neighborhood of $\gamma_{1,2}$ and $f^{\prime}(\gamma_{1,2})<0.$

Relaxing (ND) to (ND1), (ND2) does not pose major problems in the proof of our results, but it would obscure the exposition a bit and would require modification of some standard results we refer to. Thus we decided to just work with (ND). All these nondegeneracy conditions are just technical and we believe our theorems can be proved by the same general method without them. Clearly, condition (ND) is generic with respect to “reasonable” topologies. Note, however, that we allow some nongeneric situations, for example, the existence of two consecutive local-maximum points of $F$ at which $F$ takes the same value. The nondegeneracy condition constrains considerably the complexity of possible phase portraits associated with equation for the steady states of (1.1):

$$u_{xx}+f(u)=0,\qquad x\in\mathbb{R}.$$

(1.6)

This is mainly how the nondegeneracy condition is useful in this paper.

We will make another assumption on the nonlinearity. It concerns the behavior of $f(u)$ for large values of $|u|$ and it can be assumed with no loss of generality. Indeed, our main quasiconvergence theorems deal with individual bounded solutions only. Thus we can modify $f$ freely outside the range of the given solution with no effect on the validity of the theorems. It will be convenient to assume that

(MF)

$f$ is globally Lipschitz and there is $\kappa>0$ such that for all $s$ with $|s|>\kappa$ one has $f(s)={s}/{2}.$

Hypotheses (ND) and (MF) are our standing hypotheses on $f$.

Each zero of $f$ is of course an equilibrium of the equation

$$\dot{\xi}(t)=f(\xi).$$

(1.7)

Hypothesis (ND) implies in particular that any such equilibrium is either unstable from above and below, or asymptotically stable (this property would also be implied by (ND1)).

As mentioned above, in this paper we take $u_{0}\in{\mathcal{V}}$, assuming that its limits at $\pm\infty$ are equal. Without loss of generality, we assume the limits to be equal to zero:

$$u_{0}\in\mathcal{V},\qquad u_{0}(-\infty)=u_{0}(+\infty)=0.$$

(1.8)

We distinguish the following two cases:

• (S)

  Either $f(0)\neq 0$, or $f(0)=0$ and 0 is a stable equilibrium for (1.7).

• (U)

  $f(0)=0$ and 0 is an unstable equilibrium for (1.7).

If (S) holds, we have a general quasiconvergence theorem:

If (U) holds, then, as already noted in the introduction, a similar quasiconvergence does not hold in general: the references [30, 32] provide examples of bounded solutions of (1.1), (1.2) with $u_{0}(\pm\infty)=0$ which are not quasiconvergent. More specifically, such solutions exist whenever $f\in C^{1}$ and $0$ belongs to a bistable interval of $f$: there are $\gamma_{1},\gamma_{2}\in\mathbb{R}$ such that

$$\gamma_{1}<0<\gamma_{2},\quad f(\gamma_{1})=f(\gamma_{2})=0,\quad f^{\prime}(\gamma_{1}),f^{\prime}(\gamma_{2})<0,\quad\text{and }f\neq 0\text{ in }(\gamma_{1},0)\cup(0,\gamma_{2}).$$

(1.9)

Whether the bistable nonlinearity $f$ is balanced in $(\gamma_{1},\gamma_{2})$: $F(\gamma_{1})=F(\gamma_{2})$, or unbalanced: $F(\gamma_{1})\neq F(\gamma_{2})$, there always exists a continuous function $u_{0}$ such that $u_{0}(\pm\infty)=0$, $\gamma_{1}\leq u_{0}\leq\gamma_{2}$ and the solution $u$ of (1.1), (1.2) is not quasiconvergent. Obviously, all limit profiles, stationary or not, of the solution $u$ take also values between $\gamma_{1}$, $\gamma_{2}$. One could naturally speculate that when the initial data are not constrained by the assumption $\gamma_{1}\leq u_{0}\leq\gamma_{2}$, the behavior of the corresponding solutions can only get more complicated, with some nonstationary limit profiles possibly occurring outside the interval $[\gamma_{1},\gamma_{2}]$. Surprisingly perhaps, this turns out not to be the case. In our next theorem, we show that any limit profile whose range is not contained in $(\gamma_{1},\gamma_{2})$ is a steady state. Thus it is really the bistable interval $[\gamma_{1},\gamma_{2}]$ which is “responsible” for the nonquasiconvergence of the solutions with $u_{0}(\pm\infty)=0$, regardless of whether the range of $u_{0}$ is contained in $[\gamma_{1},\gamma_{2}]$ or not.

Note that if (U) holds and $\gamma_{1}$, $\gamma_{2}$ are the zeros of $f$ immediately preceding and immediately succeeding 0, respectively, assuming they exist, then the relations in (1.9) are satisfied.

A stronger version of this result will be given in Theorem 6.4 after some needed terminology has been introduced. Obviously, Theorem 1.3 implies that the solution $u$ is quasiconvergent if no function $\varphi\in\omega(u_{0})$ has its range in $(\gamma_{1},\gamma_{2})$.

Another aspect of the examples of non-quasiconvergent solutions given in [30, 32] is that the solutions $u$ found there are always highly oscillatory in space: for all $t>0$ the function $u(\cdot,t)$ has infinitely many critical points and infinitely many zeros. This raises another natural question whether, in the case (U), spatially nonoscillatory solutions are always quasiconvergent. Here, by a spatially nonoscillatory solution we mean a solution satisfying the following condition

(NC)

There is $t>0$ such that $u(\cdot,t)$ has only finitely many critical points.

A sufficient condition for (NC) in terms of $u_{0}$ is that there exist $a<b$ such that the function $u_{0}$ is monotone and nonconstant on each of the intervals $(-\infty,a)$, $(b,\infty)$. For if this holds, then one shows easily, using the comparison principle, that $u_{x}(x,t)\neq 0$ for all $x\in\mathbb{R}$ with $|x|\approx\infty$ and all sufficiently small positive times $t$. Consequently, by standard zero number results (cp. Section 2.1), $u_{x}(\cdot,t)$ has only a finite number of zeros for all $t>0$.

Presently, we are able to prove the quasiconvergence assuming that (NC) holds together with the following technical condition:

(R) There are sequences $a_{n}\to-\infty$, $b_{n}\to\infty$ such that the following holds. For every $\lambda\in\{a_{1},a_{2},\dots\}\cup\{b_{1},b_{2},\dots\}$ there is $t\geq 0$ such that the function $V_{\lambda}u(\cdot,t):=u(2\lambda-\cdot,t)-u(\cdot,t)$ has only finitely many zeros.

Remark 1.5(ii) below gives sufficient conditions for (R) in terms of $u_{0}$.

Our strategy for proving the quasiconvergence theorems consists in careful analysis of a certain type of entire solutions of (1.1). By an entire solution we mean a solution $U(x,t)$ of (1.1) defined for all $t\in\mathbb{R}$ (and $x\in\mathbb{R}$). If is well known (see Section 2.3 for more details) that for any $\varphi\in\omega(u)$ there exists an entire solution $U(x,t)$ of (1.1) such that $U(\cdot,0)=\varphi$ and $U(\cdot,t)\in\omega(u)$ for all $t\in\mathbb{R}$.

In analysis of such entire solutions we employ, as in several earlier papers starting with [31], a geometric technique involving spatial trajectories of solutions of (1.1). This technique, powered by properties of the zero number functional, often allows one to gain insights into the behavior of solutions of equation (1.1) (whose trajectories are in an infinite dimensional space) by examining their spatial trajectories, which are curves in $\mathbb{R}^{2}$. Specifically for any $\varphi\in C^{1}(\mathbb{R})$, we define

$$\tau(\varphi):=\left\{\left(\varphi(x),\varphi_{x}(x)\right):x\in\mathbb{R}\right\}$$

(1.10)

and refer to this set as the spatial trajectory (or orbit) of $\varphi.$ If $Y\subset C^{1}(\mathbb{R})$, $\tau(Y)\subset\mathbb{R}^{2}$ is the union of the spatial trajectories of the functions in $Y$:

$$\tau(Y):=\left\{\left(\varphi(x),\varphi_{x}(x)\right):x\in\mathbb{R},\,\varphi\in Y\right\}.$$

(1.11)

Note that if $\varphi$ is a steady state of (1.1), then $\tau(\varphi)$ is the usual trajectory of the solution $(\varphi,\varphi_{x})$ of the planar system

$$u_{x}=v,\qquad v_{x}=-f(u),$$

(1.12)

associated with equation (1.6).

When considering an entire solution $U$ with $U(\cdot,0)\in\omega(u)$, we want to constrain the locations that the spatial trajectories $\tau(U(\cdot,t))$, $t\in\mathbb{R}$, can occupy in $\mathbb{R}^{2}$ relative to the locations of the spatial trajectories of steady states of (1.1). In this, the concept of a chain is crucial. By definition, a chain is any connected component of the set $\mathbb{R}^{2}\setminus{\mathcal{P}}_{0}$, where ${\mathcal{P}}_{0}$ is the union of trajectories of all nonstationary periodic solutions of (1.12). Any chain consists of equilibria, homoclinic orbits, and, possibly, heteroclinic orbits of (1.12) (see Section 2.2). Our ultimate goal is to prove that the spatial trajectories $\tau(U(\cdot,t))$, $t\in\mathbb{R}$, are all contained in one chain. From this it follows, via a unique-continuation type result (cp. Lemma 2.9 below), that $U$ is a steady state of (1.1), which proves that $\omega(u)$ consists of steady states, as desired.

To achieve our goal, we first show that if the spatial trajectories $\tau(U(\cdot,t))$, $t\in\mathbb{R}$, are not contained in one chain, then none of them can intersect any chain; and there exist two distinct chains $\Sigma_{1}$, $\Sigma_{2}$ such that

$$\tau\left(\alpha(U)\right)\subset\Sigma_{1},\qquad\tau\left(\omega(U)\right)\subset\Sigma_{2}.$$

(1.13)

Here $\omega(U)$, $\alpha(U)$ stand for the $\omega$ and $\alpha$-limit sets of $U$; $\omega(U)$ is defined as in (1.4) and the definition of $\alpha(U)$ is analogous, with $t_{n}\to\infty$ replaced by $t_{n}\to-\infty$. We will also show that the set of all relevant chains, namely, the chains that can possibly intersect $\tau(\omega(u))$, is finite and ordered by a suitable order relation, and that the chains in (1.13) always satisfy $\Sigma_{1}<\Sigma_{2}$ in that relation. As a consequence, we obtain that the sets

$$K:=\{\varphi\in\omega(u):\tau(\varphi)\subset\Sigma\},\quad$$

corresponding to the chains $\Sigma$ with $\Sigma\cap\tau(\omega(u))\neq\emptyset$ constitute a Morse decomposition for the flow of (1.1) in $\omega(u)$ (see [9]). However, the existence of such a Morse decomposition (with at least two Morse sets) contradicts well-known recurrence properties of the flow in $\omega(u)$ (cp. [9, 7, 17]). This contradiction shows that the spatial trajectories $\tau(U(\cdot,t))$, $t\in\mathbb{R}$, must in fact be contained in one chain, as desired.

The detailed proof following the above scenario, which we give below, is rather involved mainly because there are several different possibilities as to how the chains $\Sigma_{1}$, $\Sigma_{2}$ in (1.13) can look like and how the spatial trajectories $\tau(U(\cdot,t))$, $t\in\mathbb{R}$, can fit into the structure of $\Sigma_{1}$, $\Sigma_{2}$. Even though the number of these possibilities is reduced considerably by the nondegeneracy condition (ND), the possibilities that still remain require special considerations and arguments.

We wish to emphasize that we prove (1.13) for a large class of entire solutions of (1.1), regardless of their containment in the $\omega$-limit set of the solution $u$ of (1.1), (1.2), for any $u_{0}$. Accordingly, we have striven to make Section 4, where the entire solutions are studied in detail, completely independent from the other parts of the paper. In particular, no reference is made in that section to the solution $u$ or its limit set $\omega(u)$. Thus the results there can be viewed as a contribution to the general understanding of entire solutions of (1.1). Relations (1.13) can be interpreted as a classification result, characterizing nonstationary entire solutions as connections between two different sets of steady states. We refer the reader to Section 4 for more details.

The rest of the paper is organized as follows. In Section 2, we collect preliminary results on the zero number, steady states of (1.1), entire solutions of (1.1) and their $\alpha$ and $\omega$-limit sets. We also recall there some results from earlier paper that are repeatedly used in the proofs of our main theorems. The proofs themselves comprise Sections 3 and 6. Section 4 is devoted to the classification of entire solutions, as mentioned above.

In this subsection, we consider solutions of a linear parabolic equation

$$v_{t}=v_{xx}+c(x,t)v,\qquad x\in\mathbb{R},\ t\in\left(s,T\right),$$

(2.1)

where $-\infty\leq s<T\leq\infty$ and $c$ is a bounded measurable function. Note that if $u$, $\bar{u}$ are bounded solutions of (1.1), then their difference $v=u-\bar{u}$ satisfies (2.1) with a suitable function $c$. Similarly, $v=u_{x}$ and $v=u_{t}$ are solutions of such a linear equation. These facts are frequently used below, often without notice.

For an interval $I=(a,b),$ with $-\infty\leq a<b\leq\infty,$ we denote by $z_{I}(v(\cdot,t))$ the number, possibly infinite, of zeros $x\in I$ of the function $x\mapsto v(x,t).$ If $I=\mathbb{R}$ we usually omit the subscript $\mathbb{R}$:

$$z(v(\cdot,t)):=z_{\mathbb{R}}(v(\cdot,t)).$$

The following intersection-comparison principle holds (see [1, 6]).

If (2.2) holds, we say that $z_{I}(v(\cdot,t))$ drops in the interval $(t_{1},t_{2}).$

Using the previous remark and the implicit function theorem, we obtain the following corollary.

The following result, which is a version of Lemma 2.1 for time-dependent intervals, is derived easily from Lemma 2.1 (cp. [2, Section 2]).

We will also need the following robustness lemma (see [10, Lemma 2.6]).

In this subsection, we examine the trajectories of the solutions of equation (1.6). The first-order system

$$u_{x}=v,\qquad v_{x}=-f(u),$$

(2.3)

associated with (1.6) is Hamiltonian with respect to the energy

$$H(u,v)=\frac{v^{2}}{2}+F(u)$$

(2.4)

(with $F$ as in (1.5)). Thus, each orbit of (2.3) is contained in a level set of $H.$ The level sets are symmetric with respect to the $v-$axis, and our extra hypothesis (MF) implies that they are all bounded. Therefore, all orbits of (2.3) are bounded and there are only four types of them: equilibria (all of which are on the $u-$axis), non-stationary periodic orbits (by which we mean orbits of nonstationary periodic solutions), homoclinic orbits, and heteroclinic orbits. Following a common terminology, we say that a solution $\varphi$ of (1.6) is a ground state at level $\gamma$ if corresponding solution $(\varphi,\varphi_{x})$ of (2.3) is homoclinic to the equilibrium $(\gamma,0)$; we say that $\varphi$ is a standing wave of (1.1) connecting $\gamma_{-}$ and $\gamma_{+}$ if $(\varphi,\varphi_{x})$ is a heteroclinic solution of (2.3) with limit equilibria $(\gamma_{-},0)$ and $(\gamma_{+},0)$.

Each non-stationary periodic orbit $\mathcal{O}$ is symmetric about the $u-$axis and for some $p<q$ one has

	$\displaystyle\mathcal{O}\cap\{(u,0):u\in\mathbb{R}\}$	$\displaystyle=\left\{(p,0),(q,0)\right\}$
	$\displaystyle\mathcal{O}\cap\left\{(u,v):v>0\right\}$	$\displaystyle=\left\{\left(u,\sqrt{2(F(p)-F(u))}\right):u\in(p,q)\right\}.$		(2.5)

Let

	$\displaystyle\mathcal{E}$	$\displaystyle:=\{(a,0):f(a)=0\}\ \textrm{ (the set of all equilibria of \eqref{eq:sys})},$
	$\displaystyle\mathcal{P}_{0}$	$\displaystyle:=\{(a,b)\in\mathbb{R}^{2}:(a,b)\textrm{ lies on a non-stationary periodic orbit of \eqref{eq:sys}}\},$
	$\displaystyle\mathcal{P}$	$\displaystyle:=\mathcal{P}_{0}\cup\mathcal{E}\ \textrm{ (the set of all periodic orbits of \eqref{eq:sys}, including the equilibria)}.$

The next lemma gives a description of the phase plane portrait of (2.3) with all the non-stationary periodic orbits removed. The following observations will be useful in its proof and at other places below. Let $(p,0)$ be an equilibrium of (2.3). Then $f(p)=0$ and, by (ND), $f^{\prime}(p)\neq 0$. Elementary considerations using the Hamiltonian $H$ show that if $f^{\prime}(p)>0$, then $(p,0)$ is not contained in the closure of any homoclinic or heteroclinic orbit of (2.3). On the other hand, if $f^{\prime}(p)<0$, then (MF) implies that $(p,0)$ is contained in the closure of a homoclinic or heteroclinic orbit contained in the halfplane $\{(u,v):u>p\}$ as well as of another one contained in the halfplane $\{(u,v):u<p\}$.

The above Lemma motivates the following definitions. A chain is any connected component $\Sigma$ of $\mathbb{R}^{2}\setminus\mathcal{P}_{0}.$ We say that a chain is trivial if it consists of a single point.

If $\mathcal{H}$ is a connected component of $\mathbb{R}^{2}\setminus\mathcal{P}$, let $\Lambda(\mathcal{H})$ the set consisting of the closure of $\mathcal{H}$ and the reflection of $\mathcal{H}$ with respect to the $u-$axis. So $\Lambda(\mathcal{H})$ is either the union of a homoclinic orbit and its limit equilibrium, or the union of two heteroclinic orbits and their common limit equilibria. We refer to $\Lambda(\mathcal{H})$ as the loop associated with $\mathcal{H}.$

Hypotheses (ND) and (MF) imply that $f$ has only finitely many zeros. Since any chain or loop contains an equilibrium, there is only a finite number of chains and loops. In particular, any chain is the union of finitely many loops. Also, any chain is a compact subset of $\mathbb{R}^{2}$ and so their (finite) union, that is, the set $\mathbb{R}^{2}\setminus\mathcal{P}_{0}$, is compact. This implies that $\mathcal{P}_{0}$ admits a unique unbounded connected component and all connected components of $\mathcal{P}_{0}$ (as well $\mathcal{P}_{0}$ itself) are open sets.

If $\Sigma$ is a chain, we denote by $\mathcal{I}(\Sigma)$ the union of all bounded connected components of $\mathbb{R}^{2}\setminus\Sigma.$ Thus, $\mathcal{I}(\Sigma)$ is the union of the interiors of the loops, viewed as Jordan curves, contained in $\Sigma$; if $\Sigma$ consists of a single equilibrium (necessarily a center for (2.3)), $\mathcal{I}(\Sigma)=\emptyset$. Since $\Sigma$ is clearly compact in $\mathbb{R}^{2},$ the set $\mathcal{I}(\Sigma)$ is open. We also define ${\overline{\mathcal{I}}(\Sigma)}=\mathcal{I}(\Sigma)\cup\Sigma.$ The set ${\overline{\mathcal{I}}(\Sigma)}$ is closed and equal to the closure of $\mathcal{I}(\Sigma)$, except when $\Sigma$ consists of a single point, in which case $\mathcal{I}(\Sigma)=\Sigma$. In a similar way we define the sets $\mathcal{I}(\Lambda)$, ${\overline{\mathcal{I}}(\Lambda)}$, $\mathcal{I}({\mathcal{O}})$, ${\overline{\mathcal{I}}({\mathcal{O}})}$, when $\Lambda$ is a loop and $\mathcal{O}$ is a non-stationary periodic orbit.

The following lemma introduces two key concepts: the inner chain and the outer loop associated with a connected component of $\mathcal{P}_{0}$ (see also Figure 1).

We refer to $\Sigma_{in}$ and $\Lambda_{out}$ as the inner chain and outer loop associated with $\Pi$; we denote them by $\Sigma_{in}(\Pi)$ and $\Lambda_{out}(\Pi)$ if the correspondence to $\Pi$ is to be explicitly indicated.