Large-time behavior of solutions of parabolic equations on the real line with convergent initial data III: unstable limit at infinity

This is a continuation, and conclusion, of our study of bounded solutions u of the semilinear parabolic equation ut=uxx+f(u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_t=u_{xx}+f(u)$$\end{document} on the real line whose initial data u0=u(·,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_0=u(\cdot ,0)$$\end{document} have finite limits θ±\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta ^\pm $$\end{document} as x→±∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\rightarrow \pm \infty $$\end{document}. We assume that f is a locally Lipschitz function on R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}$$\end{document} satisfying minor nondegeneracy conditions. Our goal is to describe the asymptotic behavior of u(x, t) as t→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\rightarrow \infty $$\end{document}. In the first two parts of this series we mainly considered the cases where either θ-≠θ+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta ^-\ne \theta ^+$$\end{document}; or θ±=θ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta ^\pm =\theta _0$$\end{document} and f(θ0)≠0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(\theta _0)\ne 0$$\end{document}; or else θ±=θ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta ^\pm =\theta _0$$\end{document}, f(θ0)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(\theta _0)=0$$\end{document}, and θ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _0$$\end{document} is a stable equilibrium of the equation ξ˙=f(ξ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\dot{\xi }}}=f(\xi )$$\end{document}. In all these cases we proved that the corresponding solution u is quasiconvergent—if bounded—which is to say that all limit profiles of u(·,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u(\cdot ,t)$$\end{document} as t→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\rightarrow \infty $$\end{document} are steady states. The limit profiles, or accumulation points, are taken in Lloc∞(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty _{loc}(\mathbb {R})$$\end{document}. In the present paper, we take on the case that θ±=θ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta ^\pm =\theta _0$$\end{document}, f(θ0)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(\theta _0)=0$$\end{document}, and θ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _0$$\end{document} is an unstable equilibrium of the equation ξ˙=f(ξ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\dot{\xi }}}=f(\xi )$$\end{document}. Our earlier quasiconvergence theorem in this case involved some restrictive technical conditions on the solution, which we now remove. Our sole condition on u(·,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u(\cdot ,t)$$\end{document} is that it is nonoscillatory (has only finitely many critical points) at some t≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\ge 0$$\end{document}. Since it is known that oscillatory bounded solutions are not always quasiconvergent, our result is nearly optimal.


Introduction and main results
Consider the Cauchy problem where f is a locally Lipschitz function on R and u 0 ∈ C b (R) := C(R) ∩ L ∞ (R).We denote by u(x, t, u 0 ), or simply u(x, t) if there is no danger of confusion, the unique classical solution of (1.1), (1.2)-to ensure the uniqueness, we require classical solutions to satisfy u(•, t) ∈ L ∞ (R) as long as they are defined-and by T (u 0 ) ∈ (0, +∞] its maximal existence time.If u is bounded on R × [0, T (u 0 )), then necessarily T (u 0 ) = +∞, that is, the solution is global.
As in the previous two parts of this paper series, [17,18], we examine solutions with initial data u 0 taken in the space V := {v ∈ C b (R) : the limits v(−∞), v(+∞) ∈ R exist} . (1.3) In other words, we assume that the initial datum u 0 has limits as x → ±∞.It is well known that the space V is invariant for equation (1.1): if u 0 admits finite limits as x → ±∞, then so does u(•, t, u 0 ) for any t ∈ (0, T (u 0 )) (see Lemma 3.9 below; note that the limits may vary with t).So it is natural to consider V as a state space for equation (1.1).Our goal is to understand the large-time behavior of bounded solutions u(•, t) ∈ V and, in particular, to clarify if, in any fixed bounded interval, the shape of u(•, t) at large times is determined by steady states of (1.1).
To express this formally, we introduce the ω-limit set of a bounded solution u: ω(u) := {ϕ ∈ L ∞ (R), u(•, t n ) → ϕ for some sequence t n → ∞} . (1.4) Here the convergence is in the topology of L ∞ loc (R), that is, the locally uniform convergence.By standard parabolic estimates, the trajectory {u(•, t), t ≥ 1} of a bounded solution is relatively compact in L ∞ loc (R).This implies that ω(u) is nonempty, compact, and connected in L ∞ loc (R), and it attracts the solution in (the metric space) L ∞ loc (R): If the ω−limit set reduces to a single element ϕ, then u is convergent: u(•, t) → ϕ in L ∞ loc (R) as t → ∞.Necessarily, ϕ is a steady state of (1.1).If all functions ϕ ∈ ω(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.In both cases, u t (•, t) approaches zero locally uniformly on R as t → ∞.For this reason, it is difficult to numerically distinguish convergence from quasiconvergence (analytically, convergence is characterized by the existence of the improper Riemann integral of u t (x, t) on [1, ∞) for each x).
For analogs of (1.1) on bounded intervals under Dirichlet, Neumann, Robin, or periodic boundary conditions, or sometimes even for (1.1) itself when solutions in suitable energy spaces are considered, quasiconvergence of solutions can be established by means of energy estimates (see, for example, [7]).However, the existence of the limits u 0 (±∞) alone is not sufficient for quasiconvergence.As shown in [19,21], bounded solutions in V which are not quasiconvergent do exist.(We emphasize here that the locally uniform convergence is taken in the definition of the ω-limit set and the corresponding notion of quasiconvergence; if the uniform convergence is taken instead, the existence of bounded solutions which are not quasiconvergent is rather trivial).Moreover, the existence of such solutions is not an exceptional phenomenon at all; it is guaranteed by a robust condition on f , namely the existence of a bistable interval.Note, however, that steady states are not completely irrelevant for the behavior of non-quasiconvergent solutions.A result of [11,12] shows that the ω-limit set of any bounded solution of (1.1) contains at least one steady state.There are convergence and quasiconvergence results for various specific classes of solutions of (1.1), see [5,6,8,14,15,16,24,25,26]; overviews can be found in [22,18]).More recent results include a convergence theorem of [4] for positive solutions of periodic versions of (1.1) with compact initial data and a description of the large-time behavior of entire solutions with localized past [13] (the latter paper deals with equations on R N and its introduction also contains an overview of earlier results for multidimensional parabolic problems).
Our study of solutions in V, which we conclude in this paper, yields a rather complete information on the quasiconvergence property of bounded solutions in this space.In our first result, the main theorem of [17], we proved that if the limits θ ± := u 0 (±∞) are distinct, then the solution u of (1.1), (1.2) is quasiconvergent, if bounded.In [18], we then showed that the same is true if θ − = θ + =: θ 0 , and either f (θ 0 ) = 0 or f (θ 0 ) = 0 and θ 0 is a stable equilibrium of the equation ξ = f (ξ).In this result, we assumed the following nondegeneracy condition on the nonlinearity: (ND) For each γ ∈ f −1 {0}, f is of class C 1 in a neighborhood of γ and f (γ) = 0.
In the remaining case, θ ± = θ 0 , f (θ 0 ) = 0 with θ 0 unstable (f (θ 0 ) > 0), the above quasiconvergence result is not valid without additional conditions on the solution; this is documented by the examples of [19,21], as already mentioned above.It is intriguing, however, that all non-quasiconvergent solutions u found in these examples share a prominent feature: they are oscillatory in the sense that u(•, t) has infinitely many critical points at all times t > 0. This raises a natural question whether without the oscillations the solution is necessarily quasiconvergent, if bounded.More precisely, the question is whether the solution of (1.1), (1.2) is quasiconvergent, provided it is bounded and satisfies the following condition: (NC) There is t > 0 such that u(•, t) has only finitely many critical points.
We remark that if (NC) holds for some t, then it holds for any larger t due to well known properties of the zero number of u x (•, t) (see Section 3.1).In Remark 1.3 below, we mention some sufficient conditions for the validity of (NC) in terms of u 0 .
In [18], we left open the question whether (NC) alone is sufficient for the quasiconvergence of u; we only proved the quasiconvergence assuming (NC) holds together with some additional and somewhat artificial conditions.The main theorem of the present paper gives a positive answer without any extra condition: Theorem 1.1.Assume that (ND) holds, and u 0 ∈ V has both its limits u 0 (±∞) equal to some θ 0 ∈ R with f (θ 0 ) = 0 < f (θ 0 ).Then, if the solution u of (1.1), (1.2) is bounded and satisfies (NC), it is quasiconvergent: ω(u) consists of steady states of (1.1).This theorem, combined with the results of [17,18], gives the following corollary concerning general bonded solutions which are nonoscillatory in the spatial variable: Corollary 1.2.Assume that (ND) holds and let u be a bounded solution of (1.1) such that (NC) holds.Then u is quasiconvergent.
Proof.Choose a large enough t 0 such that (NC) holds with t = t 0 : u(•, t 0 ) has only finitely many critical points.Replacing the initial datum of the solution u by u 0 := u(•, t 0 ), we achieve that u 0 is monotone near ±∞; in particular, u 0 ∈ V.If the limits θ ± := u 0 (±∞) are distinct, or are both equal to θ 0 where either f (θ 0 ) = 0 or θ 0 is a stable equilibrium of ξ = f (ξ), we apply the results of [17] or [18], respectively.If the limits are both equal to an unstable equilibrium of ξ = f (ξ), we apply Theorem 1.1.We thus obtain the quasiconvergence conclusion in all cases.
Remark 1.3.(i) We mention here some simple sufficient conditions, in terms of the initial data u 0 , for the validity of the assumptions on the solution u in Theorem 1.1.A sufficient condition for the boundedness of the solution of (1.1), (1.2) is that u 0 takes values between two constants ξ < η satisfying f (ξ) > 0 > f (η).This follows from the comparison principle.A sufficient condition for (NC) is that u 0 has only finitely many critical points if it is of class C 1 .This is a consequence of the monotonicity of the zero number of u x (•, t).More generally, (NC) holds if there are constants a < b such that the function u 0 is monotone and nonconstant on each of the intervals (−∞, a), (b, ∞).Indeed, if this holds, one shows easily, using the comparison principle (comparing u and its spatial shifts) that for small t > 0 the function is strictly monotone on each of the intervals (−∞, a − 1), (b + 1, ∞); the strong comparison principle then shows that u x (x, t) has no zero in these intervals for small t.Consequently, by properties of the zero number of u x (•, t) (cp.Section 3.1), u(•, t) has only a finite number of critical points for all t > 0.
(ii) As mentioned above, bounded solutions that do not satisfy (NC) are not quasiconvergent in general.In this sense, condition (NC) is optimal.However, some generalization are probably still possible.For example, one may ask if it is sufficient to assume that for some t there is ρ such that u(•, t) has no critical points in at least one of the intervals (−∞, ρ), (ρ, ∞).(Note that, as in (i), if u(•, t) has no critical points in the union of these intervals, then (NC) holds for larger times).Another question is whether condition (NC) can be replaced by the weaker requirement that u(•, t) − θ 0 has only finitely many zeros for some t.Our proof does not apply in these cases and we do not pursue these generalizations.
In the proof of Theorem 1.1, we build on the strategy and some technical results of [18].The strategy 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 ∈ R (and x ∈ R).It is well known that for any ϕ ∈ ω(u) there exists a unique entire solution U (x, t) of (1.1) such that U (•, 0) = ϕ, and this solution satisfies U (•, t) ∈ ω(u) for all t ∈ R.This is how entire solutions are relevant for our problem.The assumption u 0 ∈ V poses some restrictions on the structure of entire solutions that can possibly be contained in ω(u).Using these structural properties in combination with the chain recurrence property of ω(u), we were able to prove in [18], assuming that θ ± = θ 0 and f (θ 0 ) = 0 > f (θ 0 ), that all the entire solutions in ω(u) are necessarily steady states.To prove the same in the present case, θ ± = θ 0 and f (θ 0 ) = 0 < f (θ 0 ), assuming (NC), we need to consider a class of entire solution not covered by the analysis of [18] (see Section 2.2 below for more details on this).We prove a classification result for such entire solution (see Proposition 5.1), after which a general conclusion from [18] becomes applicable and we obtain our quasiconvergent result.
The rest of the paper is organized as follows.In the next section, we define the concepts of a chain of steady states of (1.1) and spatial trajectories of solutions of (1.1).We use these concepts to state a proposition which has Theorem 1.1 as a corollary.The proposition is then proved in Sections 4 and 5.In the preliminary Section 3, we recall several technical results from earlier papers, and discuss the basic properties of α and ω-limit sets and the zero number.
Below, it will be convenient to assume the following additional condition on the nonlinearity: (MF) f is globally Lipschitz and there is κ > 0 such that for all s with |s| > κ one has Since this condition concerns the behavior of f (u) for large values of |u|, it can be assumed with no loss of generality.Indeed, our quasiconvergence theorem deals with an individual bounded solution, thus modifying f outside the range of this solution has no effect on the validity of the theorem.Conditions (ND), (MF), are our standing hypotheses on f .With no loss of generality, shifting f if necessary, we will also assume that θ 0 in Theorem 1.1 is equal to zero.Thus, we henceforth also assume that f (0) = 0, f (0) > 0. (1.5)

Spatial trajectories and chains
As in [18], we employ a geometric technique involving spatial trajectories of solutions of (1.1).Our analysis consists mainly in the examination of how spatial trajectories of entire solutions of (1.1) are related to chains of the planar system corresponding to the equation for the steady states of (1.1): We define the concept of a chain in the next subsection, after recalling some basic properties of the planar trajectories of (2.1).Spatial trajectories of solutions of entire solutions of (1.1) are defined in Subsection 2.2.In that subsection, we state a result concerning entire solutions which implies Theorem 1.1.

Steady states of (1.1) and chains
Consider the planar system associated with equation (2.1).
It is a Hamiltonian system with respect to the energy where F (u) = u 0 f (s) ds.Thus, each orbit of (2.2) is contained in a level set of H.The level sets are symmetric with respect to the u−axis, and our extra hypothesis (MF) implies that they are all bounded.Therefore, all orbits of (2.2) are bounded and there are only four types of them: equilibria (all of which are on the u−axis), nonstationary 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 ϕ of (2.1) is a ground state at level γ if the corresponding solution (ϕ, ϕ x ) of (2.2) is homoclinic to the equilibrium (γ, 0); we say that ϕ is a standing wave of (1.1) connecting γ − and γ + if (ϕ, ϕ x ) is a heteroclinic solution of (2.2) with limit equilibria (γ − , 0) and (γ + , 0).
Each nonstationary periodic orbit O is symmetric about the u−axis and for some p < q one has Let E := {(a, 0) : f (a) = 0} (the set of all equilibria of (2.2)), ) lies on a nonstationary periodic orbit of (2.2)}, P := P 0 ∪ E (the union of all periodic orbits of (2.2), including the equilibria).
The following lemma is the same as [18, Lemma 2.1], which, except for the last two statements in (i), was originally proved in [15,Lemma 3.1].It gives a description of the phase plane portrait of (2.2) without the nonstationary periodic orbits.
Lemma 2.1.The following two statements are valid.
(i) Let Σ be a connected component of R 2 \ P 0 .Then Σ is a compact set contained in a level set of the Hamiltonian H and one has where c is the value of H on Σ and J = [p, q] for some p, q ∈ R with p ≤ q.Moreover, if (u, 0) ∈ Σ and p < u < q, then (u, 0) is an equilibrium.If p < q, the points (p, 0) and (q, 0) lie on homoclinic orbits.If p = q, then Σ = {(p, 0)}, and p is an unstable equilibrium of the equation ξ = f (ξ).
(ii) Each connected component of the set R 2 \ P consists of a single orbit of (2.2), either a homoclinic orbit or a heteroclinic orbit.
We define a chain as any connected component of the set R 2 \ P 0 .Each chain consists of equilibria, homoclinic orbits, and, possibly, heteroclinic orbits of (2.2).We say that a chain is trivial if it consists of a single equilibrium.By a loop we mean a set Λ ⊂ R 2 which is either the union of a homoclinic orbit and its limit equilibrium or the union of two heteroclinic orbits, one reflection of the other around the u axis, and their common limit equilibria.Obviously, every loop Λ is contained in a chain and it can be viewed as a Jordan curve in R 2 .We denote by I(Λ) the interior of Λ (the bounded connected component of R 2 \ Λ).Similarly we define I(O) when O is a nonstationary periodic orbit of (2.2).If Σ is a chain, I(Σ) denotes the union of the interiors of the loops contained in Σ.We also define I(Σ) = I(Σ) ∪ Σ.The set I(Σ) is closed and equal to the closure of I(Σ), except when Σ consists of a single point, in which case I(Σ) = Σ.For a nonstationary periodic orbit O of (2.2), I(O) denotes the closure of I(O).
The following lemma introduces the inner chain and the outer loop associated with a connected component of P 0 (see Figure 1).The lemma is identical with [18,   (iv) There is a zero β of f such that f (β) > 0 and (β, 0) ∈ I(O), for all periodic orbits O ⊂ Π.
(thus, Π is totally ordered by this relation).
We refer to Σ in and Λ out as the inner chain and outer loop associated with Π.If the correspondence to Π is to be explicitly indicated, we denote them by Σ in (Π) and Λ out (Π), respectively.
The inner chain and outer loop associated with a connected component Π of P 0 : Λ out and Σ in form the boundary of Π.The outer loop can be a heteroclinic loop (as in this figure) or a homoclinic loop, and it is part of a chain Σ out .The points p and q are as in Lemma 2.1 for Σ = Σ in .
Below, the connected component of P 0 whose closure contains (0, 0) will play a prominent role.We denote it by Π 0 .Note that Π 0 is well defined, for f (0) > 0 implies that (0, 0) is a center for (2.2), which is to say that it has a neighborhood foliated by periodic orbits.
The shaded region indicates the connected component Π 0 containing the point (0, 0).The corresponding outer loop Λ out is a heteroclinic loop in the figure on the left and a homoclinic loop in the figure of the right.Points p and q indicate the intersections of Λ out with the u-axis.The inner chain is trivial: Σ in = {(0, 0)}.

A key result on spatial trajectories of entire solutions
In this subsection, we introduce spatial trajectories of entire solutions of (1.1).As we explain, Theorem 1.1 follows from a result on entire solutions stated in Proposition 2.3 below.
For any ϕ ∈ C 1 (R), we define and refer to this set as the spatial trajectory (or orbit) of ϕ.Note that if ϕ is a steady state of (1.1), then τ (ϕ) is the usual trajectory of the solution (ϕ, ϕ x ) of the planar system (2.2).If U is an entire solution of (1.1); we refer to the collection is the union of the spatial trajectories of the functions in Y : Assume now that u 0 ∈ C b (R), u 0 (±∞) = 0 (recalling that relations (1.5) are assumed to hold), and the solution u of (1.1), (1.2) is bounded.For a description of ω(u), some results relating the spatial trajectories of entire solutions of (1.1) to the chains of (2.2) are crucial.If one can prove that, for an entire solution U , the spatial trajectories τ (U (•, t)), t ∈ R, are all contained in a chain, then a unique-continuation type result shows that U is a steady state of (1.1) (see Lemma 3.4 below).Thus, the quasiconvergence of u can be proved by showing that τ (ω(u)) is contained in a chain.We now explain a key idea of how this can be accomplished.
Let us first scrutinize the possibility that for some entire solution U with U (•, t) ∈ ω(u) a spatial trajectory τ (U (•, t 0 )) is not contained in any chain for some t 0 ∈ R. It was proved in [18, Proposition 3.2] that then none of the trajectories τ (U (•, t)), t ∈ R, can intersect any chain.This clearly implies that there is a connected component Π of P 0 such that t∈R τ (U (•, t)) ⊂ Π. (2.7) The connected component Π has to be bounded as also shown in [18].Trying to rule (2.7) out, we look for a contradiction.We consider the ω and α-limit sets of the entire solution U , denoted by ω(U ), α(U ), respectively; ω(U ) is defined as in (1.4) and the definition of α(U ) is analogous, with t n → ∞ replaced by t n → −∞.Take the inner chain Σ in (Π) and the outer loop Λ out (Π) associated with Π, as in Lemma 2.2.As in [18, Section 6], a contradiction is obtained if the following relations can be derived from (2.7): The reason why (2.8) leads to a contradiction can intuitively be explained as follows.Relations (2.8) show that there is a specific "direction" of the flow of (1.1) in ω(u): U (•, t) always goes from the inner chain to the outer loop as t increases from −∞ to ∞.However, the existence of such a flow direction is inconsistent with well-known chain-recurrence properties of the ω-limit sets and thus the contradiction (see [18,Section 6] for details).
For reference, we state here the result which implies Theorem 1.1, as explained above.
The case Π = Π 0 differs from the case when Σ in (Π) is a nontrivial chain in several aspects.One key difference is that we need to take into account the possibility that the spatial limits U (±∞, t) of the entire solution U depend on t (this can be ruled out easily if Σ in (Π) is a nontrivial chain, see [18,Lemma 3.9]).Even when U (±∞, t) are independent of t, the case when one or both of them is equal to 0, an unstable equilibrium of ξ = f (ξ), is not encountered in the case Π = Π 0 (the limits U (±∞, t) are always equal to a stable equilibrium of ξ = f (ξ) if Σ in (Π) is a nontrivial chain).On the other hand, assumption (NC) has some consequences on the structure of relevant entire solutions (see Section 4), which we exploit in the proof of (2.8).

Preliminaries
In this section, we first recall basic properties of the zero-number functional and various limit sets of bounded solutions of (1.1) and then state some results from earlier paper that will be referred to in the proof of Proposition 2.3.

Zero number
Consider a linear parabolic equation where −∞ ≤ s < T ≤ ∞ and c is a bounded measurable function.Note that whenever u, ū are bounded solutions of (1.1), their difference v = u − ū satisfies (3.1) with a suitable function c.
Similarly, v = u x and v = u t are solutions of such a linear equation.
For an interval I = (a, b), with −∞ ≤ a < b ≤ ∞, we denote by z I (v(•, t)) the number, possibly infinite, of zeros x ∈ I (counted without their multiplicities) of the function x → v(x, t).
If I = R we usually omit the subscript R: The following intersection-comparison principle holds (see [1,3]).Assume that the following conditions are satisfied: Then the following statements hold true.
(i) For each t ∈ (s, T ) , all zeros of v(•, t) are isolated.In particular, if I is bounded, then (iii) If for some t 0 ∈ (s, T ) the function v(•, t 0 ) has a multiple zero in I and z If (3.2) holds, we say that z I (v(•, t)) drops at t 0 .
We will also use a version of Lemma 3.1 for time-dependent intervals; it is derived easily from Lemma 3.1 (cp.[2, Section 2]).Lemma 3.2.Let v be a nontrivial solution of (3.1) and I(t) = (a(t), b(t)), where −∞ ≤ a(t) < b(t) ≤ ∞ for t ∈ (s, T ).Assume that the following conditions are satisfied: (c1) Either b ≡ ∞ or b is a (finite) continuous function on (s,T).In the latter case, v(b(t), t) = 0 for all t ∈ (s, T ).
(c2) Either a ≡ −∞ or a is a continuous function on (s,T).In the latter case, v(a(t), t) = 0 for all t ∈ (s, T ).
Then statements (i), (ii) of Lemma 3.1 are valid with I, a, b replaced by I(t), a(t), b(t), respectively; and statement (iii) of Lemma 3.1 is valid with all occurrences of z I (v(•, t j )), j = 0, 1, 2, replaced by z The next lemma is a robustness result of [5].
Lemma 3.3.Let w n (x, t) be a sequence of functions converging to w(x, t) in C 1 (I × (s, T )) where I is an open interval.Assume that w(x, t) solves a linear equation (3.1), w ≡ 0, and w(•, t) has a multiple zero x 0 ∈ I for some t 0 ∈ (s, T ).Then there exist sequences x n → x 0 , t n → t 0 such that for all sufficiently large n the function w n (•, t n ) has a multiple zero at x n .

Limit sets and entire solutions
The ω−limit set of a bounded solution u of (1.1) is defined as in (1.4), with the convergence in L ∞ loc (R).As already noted above, it is a nonempty, compact, and connected set in L ∞ loc (R).It is also well known that ω(u) has the following invariance property: for any ϕ ∈ ω(u), there is an entire solution U (x, t) of (1.1) such that , where U is an entire solution of (1.1) satisfying (3.3).This follows by compactness arguments based on parabolic estimates (see [18,Section 3.2] for more details.)Note that the entire solution satisfying U (•, 0) = ϕ is uniquely determined by ϕ; this follows from the uniqueness and backward uniqueness for the Cauchy problem (1.1), (1.2).
The above considerations also imply that ω(u) If U is a bounded entire solution of (1.1), we define its α−limit set by Here, again, the convergence is in L ∞ loc (R), but due to parabolic regularity, it can be taken in C 1 loc (R) with no effect on α(U ).The α-limit set has similar properties as the ω−limit set: it is nonempty, compact and connected in L ∞ loc (R) as well as in C 1 loc (R), and for any ϕ ∈ α(U ) there is an entire solution Ũ such that Ũ (•, 0) = ϕ and Ũ (•, t) ∈ α(U ) for all t ∈ R. The connectivity property of α(U ) implies that the set For a bounded entire solution U of (1.1), we define generalized notions of α and ω-limit sets as follows: The convergence can be taken in L ∞ loc (R) or C 1 loc (R) without altering the sets Ω(U ), A(U ).Both these sets are nonempty, compact and connected in C 1 loc (R), and they have a similar invariance property as ω(U ), α(U ).Also, by their definitions, the sets Ω(U ), A(U ) are translation invariant as well.Further, the definitions and parabolic regularity imply that the sets are connected and compact in R 2 .We remark that the sets τ (ω(u)), τ (α(u)) are both connected (as noted above), but they are not necessarily compact in R 2 .

Further technical results
Throughout this subsection, we assume that u 0 is in C b (R) (not necessarily in V), u is the solution of (1.1), (1.2) and it is bounded.
In view of the invariance property of ω(u) (see (3.3)), the following lemma gives a criterion for an element ϕ ∈ ω(u) to be a steady state.This unique-continuation type result is proved in a more general form in [24, Lemma 6.10].
As already noted above, it is proved in [11] (see also [12]) that the ω-limit set of any bounded solution of (1.1) contains a steady state.We will use this result for entire solutions: Theorem 3.5.If U is a bounded entire solution of (1.1), then each of the sets ω(U ) and α(U ) contains a steady state of (1.1).
The result concerning the α-limit set follows from the result of the ω-limit set via compactness and invariance properties of α(U ): taking an entire solution Ũ with Ũ (•, t) ∈ α(U ) for all t, we have ω( Ũ ) ⊂ α(U ) and ω( Ũ ) contains a steady state.
The following lemma is essentially the same as [18, Lemma 2.11] (see also [2, Proof of Proposition 2.1]).The only difference is that in [18,Lemma 2.11], θ is a constant whereas here we allow θ = θ(t) to continuously depend on t.This makes just a notational difference in the proof given in [18,Lemma 2.11].Lemma 3.6.Let U be a solution of (1.1) on R × J, where J ⊂ R is an open time interval, and let θ(t) be a continuous real function on J. Assume that for each t ∈ J the function U (•, t) − θ(t) has at least one zero and ξ(t) := sup{x : U (x, t) = θ(t)} is finite and depends continuously on t ∈ J.Then, for any t 0 , t 1 ∈ J satisfying the relations t 1 > t 0 and ξ(t 1 ) < ξ(t 0 ), the function U x (•, t 1 ) is of constant sign on the interval (ξ(t 1 ), ξ(t 0 )].
We next state a quasiconvergence result from [17] (cp.[18, Theorem 2.12]).For any λ ∈ R, consider the function V λ u defined by (3.7) Theorem 3.7.Assume that u 0 ∈ V and one of the following conditions holds: Then, u is quasiconvergent.Moreover, ω(u) does not contain any nonconstant periodic function.
The following result is the same as [18,Lemma 2.13].It is a variant of the Squeezing Lemma from [23].
Lemma 3.8.Let U be a bounded entire solution of (1.1) for some N < ∞.Let K be any one of the following subsets of R 2 : Assume that O is a nonstationary periodic orbit of (2.2) such that one of the following inclusions holds: Let Π be the connected component of P 0 containing O.
Finally, we recall the following well known result concerning the solutions in V (the proof can be found in [27, Theorem 5.5.2], for example).Lemma 3.9.Assume that u 0 ∈ V. Then the limits exist for all t > 0 and are solutions of the following initial-value problems: 4 Entire solutions in ω(u) Throughout this section, we assume-in addition to the standing hypotheses (ND), (MF), and (1.5)-that u 0 ∈ V, u 0 (±∞) = 0, and the solution of (1.1), (1.2) is bounded and satisfies (NC).We reserve the symbol u(x, t) for this fixed solution.
In the following lemma, we derive some consequences of the assumption (NC) concerning entire solutions U (•, t) in ω(u).Lemma 4.1.Let U be an entire solution of (1.1) such that U (•, t) ∈ ω(u) for all t ∈ R. Then either U x ≡ 0 or U has the following properties: (i) Each of the functions U (•, t), U x (•, t) has only finitely many zeros, all of them simple, and the number of these zeros is bounded by a constant independent of t.
(ii) For each t ∈ R, the following limits exist: (iii) For each t ∈ R, the spatial trajectory τ (U (•, t)) is a simple curve in R 2 , that is, it has no self-intersections.
(iv) If (2.7) holds with Π = Π 0 , then, for any t ∈ R, the function U (•, t) has no positive local minima and no negative local maxima.
Proof.Assume that U x ≡ 0. We know (cp.Section 3.2) that for some sequence t n → ∞ we have with the convergence in L ∞ loc (R 2 ) in both cases.Moreover, since f is Lipschitz, the function u x is bounded in C 1+α (R × [1, +∞)) for some α ∈ (0, 1).Therefore, possibly after {t n } is replaced by a subsequence, the second convergence in (4.2) takes place in C 1 loc (R 2 ) as well.We now prove that all zeros of U x (•, t) are simple.Suppose for a contradiction that x 0 is multiple zero of U x (•, t 0 ) for some t 0 .It then follows from (4.2) and Lemma 3.3, that there is a sequence τ n → 0 such that u x (•, • + t n + τ n ) has a multiple zero.Thus, by Lemma 3.1, the zero number z(u x (•, t)) drops at t = t n + τ n .Since t n + τ n → ∞, the monotonicity of the zero number gives z(u x (•, t)) = ∞ for all t > 0, in contradiction to (NC).The contradiction proves that the zeros of U x (•, t) are indeed simple for any t ∈ R.
From (4.2) and (NC), it now follows that z(U x (•, t)) is bounded by a constant independent of t.Consequently, by the mean value theorem, z(U (•, t)) is also bounded by a constant independent of t.The proof of statement (i) is complete.
Statement (i) implies that for each t the (bounded) function U (•, t) is monotone near ±∞.This gives (ii).
To prove statement (iii), we go by contradiction.Suppose that for some t 0 ∈ R the curve τ (U (•, t 0 )) is not simple: there are x 0 , η ∈ R with η = 0, such that Consider the function v(x, t) := U (x + η, t) − U (x, t).It is a solution of a linear parabolic equation (3.1), and (4.3) means that x 0 is a multiple zero of v(•, t 0 ).By (4.2) and Lemma 3.3, there is a sequence t n → ∞ such that the function u(• + η, t n ) − u(•, t n ) has a multiple zero (near x 0 ).Therefore, the same arguments as the ones used above for the function u x show that for all t > 0 one has z(u(• + η, t) − u(•, t)) = ∞.By Lemma 3.1, all zeros of u(• + η, t) − u(•, t) are isolated for t > 0. Therefore, given any t > 0, there is a sequence x n with |x n | → ∞ such that u(x n + η, t) − u(x n , t) = 0. Now, for each n, the function u(•, t) has a critical point between x n and x n + η, so it has infinitely many critical points.This contradiction to condition (NC) proves statement (iii).
5 Entire solutions with spatial trajectories in Π 0 In this section, we investigate entire solutions U of (1.1) such that Moreover, we will assume that if U x ≡ 0, then U satisfies the following conditions: (ci) There is an integer m such that and all zeros of U (•, t) and U x (•, t) are simple.
(cii) The following limits exist (ciii) For each t ∈ R, the spatial trajectory τ (U (•, t)) is a simple curve in R 2 (it has no selfintersections).
(civ) For any t ∈ R, the function U (•, t) has no positive local minima and no negative local maxima.
As shown in Lemma 4.1, if the entire solution U satisfying (c0) comes from an ω-limit set: U (•, t) ∈ ω(u), where u is as in Section 4, then either U x ≡ 0 or (ci)-(civ) hold.Therefore the following proposition, which is the main result of this section, implies Proposition 2.3 (and thereby Theorem 1.1).Proposition 5.1.Let U be an entire solution satisfying (c0) such that either U x ≡ 0 or conditions (ci)-(civ) hold.Then Note that, by Lemma 3.4, the second inclusion means that ω(U ) consists of steady states of (1.1) whose trajectories are contained in Λ out (Π 0 ).Thus the proposition says that any entire solution U with the indicated properties is a connection, in L ∞ loc (R), from 0 to a set of steady states with trajectories in the outer loop.In the process of proof of the proposition, we will make this conclusion more precise in some cases.
Although our main purpose of investigating entire solutions U satisfying the assumptions of Proposition 5.1 is to prove Theorem 1.1 concerning the bounded solution u of (1.1), below we make no further reference to the solution u and just examine the entire solutions satisfying conditions (ci)-(civ).Thus, Proposition 5.1 can also be viewed as a statement concerning a class of entire solutions with some additional properties, regardless of whether they belong to an ω-limit set or not.Such a classification result for entire solutions may be of independent interest.
Let us first of all dispose of the trivial case U x ≡ 0. In this case, U , being independent of x, is a solution of ξ = f (ξ).Condition (c0) implies that U is nonconstant and it connects the unstable equilibrium 0 to an equilibrium ζ with (ζ, 0) ∈ Λ out .Thus in this case, (5.4) is proved.
In the remainder of this section, we assume that U is an entire solution of (1.1) satisfying conditions (c0)-(civ) and U x ≡ 0. We continue assuming the standing hypotheses (ND), (MF), and (1.5) on f , and simplify the notation letting There are two possibilities in regard to the structure of Λ out (cp. Figure 2 in Section 2): (A1) Λ out is a homoclinic loop, that is, it is the union of a homoclinic orbit of (2.2) and its limit equilibrium, or, in other words, where f (γ) = 0 and Φ is a ground state of (2.1) at level γ.We choose Φ so that Φ (0) = 0, that is, the only critical point of Φ is x = 0.
We treat these cases in separate subsections, proving (5.4) (and sometimes more) in each of them.The next subsection contains some general lemmas that apply to all three cases.
then, possibly after passing to a subsequence, one has , where ϕ is a steady state of (1.1) with τ (ϕ) ⊂ K.
(ii) There exists a sequence (x n , t n ), n = 1, 2, . . .as in (i) with the additional property that |t n | → ∞.Consequently, there exists a steady state of (1.1) with τ (ϕ) ⊂ K and ϕ ∈ A(U ) ∪ Ω(U ). (5.9) In the previous lemma, Ω(U ), A(U ) are the generalized limit sets of U , as defined in (3.5), (3.6).Statement (i) of the lemma delineates a way spatial trajectories of U can get arbitrarily close to one of the sets {(0, 0)}, Λ out , and statement (ii) shows that the spatial trajectories of U cannot stay away from from both {(0, 0)} and Λ out as |t| → ∞.In the next lemma, we consider the case when the spatial trajectories stay away from one of the points (p, 0), (q, 0) ∈ Λ out (see (5.7) for the definition of p, q).Lemma 5.3.The following statements are valid.
We are making an intentional duplicity in this lemma by including the conclusion α(U ) = {0} in statement (i), although this conclusion is also contained in statement (ii) (which has weaker assumptions).This will allow us to make more straightforward references to one of these statements.
The next lemma gives other sufficient conditions for α(U ) = {0}.It may be useful to reiterate at this point that α(U ) and ω(U ) are considered with respect to the locally uniform convergence.(5.10) (a3) Θ − ≡ p and and there is m ∈ R such that lim sup x∈(−∞,m), t→−∞ U (x, t) ≤ 0.
Proof.We prove the conclusion assuming (a1) holds, all the other cases are analogous.Note that (5.10) in particular implies that To start with, we claim that no nonzero steady state of (1.1) can belong to α(U ).To show this, it is sufficient-because of (c0)-to exclude from α(U ) all steady states ϕ with τ (ϕ) ⊂ Λ out (note that no nonconstant periodic steady state can be contained in α(U ) by (ci)).Relation (5.11) excludes the shifts of the increasing standing wave Φ + (if Λ out a heteroclinic loop as in (A2)), the constant q (if f (q) = 0), and the shift of the ground state Φ in the case (A1) with γ = Φ(±∞) = q.It remains to exclude the constant p, the shifts of the ground state Φ in the case (A1) with γ = p, and the shifts of the decreasing standing wave Φ − in the case (A2).The proofs for all these use very similar comparisons arguments involving periodic steady states, so we give the details for just one of them, say for the ground state Φ when γ = p (and Φ(0) = q).Assume for a contradiction that a shift Φ of Φ is contained in α(U ).Hence, there is sequence . Let x 0 stand for the larger of the two zeros of Φ.

Case (T1): Θ ± (t) = 0 for all t
In this subsection, we assume that the entire solution U , fixed as above, satisfies the relations Θ ± (t) = 0 for all t.
We first show that τ (Ω(U )) ⊂ Λ out (note that this is stronger than the needed conclusion τ (ω(U )) ⊂ Λ out ).In some cases, we even establish the existence of a limit (5.13) (i) If Θ − , Θ + have opposite signs, then necessarily Λ out is a heteroclinic loop as in (A2), and the limit φ in (5.13) exists and is equal to a standing wave -a shift of Φ + or Φ − .
(ii) If the signs of Θ − , Θ + are equal and Λ out is a heteroclinic loop as in (A2), then the limit φ in (5.13) exists and is equal to one of the constants γ − , γ + .
(iii) If the signs of Θ − , Θ + are equal and Λ out is a homoclinic loop as in (A1), then the following statements hold: (a) If the functions Θ − , Θ + are constant (so they are both identical to the constant γ), then the limit φ in (5.13) exists and is equal to the constant γ or a shift of the ground state Φ.
(b) If one (or both) of the functions functions In particular, in all cases, we have τ (ω(U )) ⊂ Λ out .
Remark 5.6.The convergence conclusion as in statement (iii)(a) is likely valid in (iii)(b) as well.However, a result on threshold solutions from [15] that we are using in the proof of (iii)(a) does not seem to be available in general when Θ − , Θ + are nonconstant.
Assume first that Θ − ≡ Θ + ≡ γ.Since (γ, q] is the range of the ground state Φ, F < F (γ) in (γ, q] (here, F (u) = u 0 f (s) ds is as in (2.3)).This is the setup of [15, Theorem 2.5] whose conclusion, translated to the present notation, is the same as the conclusion in (iii)(a).
The following lemma completes the proof of Proposition 5.1 in the case (T1).
For the proof of this result, we need the following lemma.
Lemma 5.8.The following statements are valid: (j) The zero number z(U (•, t)) is independent of t ∈ R.
(jj) If, for some t ∈ R, J is a nodal interval of U (•, t) (that is, U (•, t) is nonzero everywhere in J and vanishes on ∂J), then U (•, t) has at most one critical point in J (the critical point is nondegenerate by (ci)).
Proof of Lemma 5.7.If the functions Θ ± are both constant (so that statements (i),(ii), and (iii)(a) of Lemma 5.5 apply), the conclusion can be proved by nearly the same arguments as those given in the proof of [18,Lemma 4.10].We omit the details in this case, just mention the following simple changes that need to made in the proof: take β − = β + = 0, and replace all references to Lemmas 4.4 and 4.8 of [18] by references to Lemmas 5.8 and 5.3 of the present paper, respectively.We now consider the case when at least one of the functions Θ ± is nonconstant.For definiteness, we assume that Θ + is nonconstant.Here, too, there are two analogous possibilities: Θ + (t) ∈ (p, 0) for all t and Θ + (t) ∈ (0, q) for all t.We just consider the former.Note that, since Θ + (t) converges to a stable equilibrium of ξ = f (ξ) as t → ∞, we necessarily have f (p) = 0.
Whenever (5.17) holds, conditions (civ), (ci), and (T2) imply that the function U (•, t) has exactly two critical points, both nondegenerate; one of them is the global maximum point of U (•, t), further denoted by ξ(t), the other one is the global minimum point of U (•, t), denoted by ξ(t) (cp. Figure 4).The unique zero of U (•, t) is denoted by ξ(t); it is between the two critical points.
By the monotonicity of the zero number, If U (•, t) > 0 for some t, then U (•, t) has a unique critical point, the point of global maximum; we still denote it by ξ(t).If U (•, t) < 0 for some t, then the unique critical point of U (•, t) is the point of global minimum, still denoted by ξ(t).We now prove that the desired conclusion holds when Λ out is a homoclinic loop: Lemma 5.10.
If Λ out is a homoclinic loop as in (A1), then (5.4) holds: Therefore, by the comparison principle, for n > n 0 , In particular, at t = t 0 , we obtain max Since x n → −∞, we have a contradiction to the relation U (−∞, t 0 ) = Θ − = 0.This contradiction completes the proof of (5.20).
We now prove the second needed conclusion: If U (•, t) > 0 for some t, then U (•, t) → γ + as t → ∞ (with the convergence in L ∞ loc (R)) by the well known property of solutions with range in the monostable interval (0, γ + ).Similarly, if U (•, t) < 0 for some t, then U (•, t) → γ − as t → ∞.In these cases we are done.We proceed assuming that U (•, t) changes sign for all t, that is, the equality holds in (5.18) and ξ(t), ξ(t), ξ(t) are defined for all t R. For definiteness, we again assume that (5.21) holds; the case with the reversed inequalities is analogous.For this part of the proof, we adapt some arguments from [18, Proof of Lemma 4.13].
We have shown that the assumption (5.25) leads to a contradiction, which concludes the proof of Lemma 5.11.
First, we prove the desired conclusion regarding the α-limit set of U .
Proof.If U > 0 or U < 0, then the conclusion follows from Lemma 5.3(i).Henceforth we therefore assume that z(U (•, t)) ≥ 1 holds for some t and, consequently, for any sufficiently large negative t.By (ci) and the monotonicity of the zero number, z(U (•, t)) is finite and independent of t for large negative t and all zeros of U (•, t) are simple.We denote by ξ(t) the largest of these zeros.Since U (∞, t) = Θ + (t) = 0, the function U (•, t) is not monotone in (ξ(t), ∞).Therefore, by Lemma 3.6, ξ(t) is bounded from above as t → −∞.Thus, there are t 0 , m ∈ R such that either U (•, t) < 0 on (m, ∞) for all t < t 0 or U (•, t) > 0 on (m, ∞) for all t < t 0 .In either case, an application of Lemma 5.4 gives α(U ) = {0}.

Figure 3 :
Figure 3: The curve Γ ⊂ τ (U (•, t)) whose existence is ruled out in the proof of Lemma 5.9
Lemma 2.2].Let Π be any connected component of P 0 .The following statements hold true.