A variational proof of global stability for bistable travelling waves

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A variational proof of global stability for bistable travelling waves Thierry Gallay Institut Fourier Universite de Grenoble I 38402 Saint-Martin-d'Heres France Emmanuel Risler Institut Camille Jordan INSA de Lyon 69621 Villeurbanne France June 19, 2007 Abstract We give a variational proof of global stability for bistable travelling waves of scalar reaction-diffusion equations on the real line. In particular, we recover some of the classical results by P. Fife and J.B. McLeod (1977) without any use of the maximum principle. The method that is illustrated here in the simplest possi- ble setting has been successfully applied to more general parabolic or hyperbolic gradient-like systems. 1 Introduction The purpose of this work is to revisit the stability theory for travelling waves of reaction- diffusion systems on the real line. We are mainly interested in global stability results which assert that, for a wide class of initial data with a specified behavior at infinity, the solutions approach for large times a travelling wave with nonzero velocity. In the case of scalar reaction-diffusion equations, such properties have been established by Kolmogorov, Petrovski & Piskunov [11], by Kanel [9, 10], and by Fife & McLeod [4, 5] under various assumptions on the nonlinearity. The proofs of all these results use a priori estimates and comparison theorems based on the parabolic maximum principle.

  • scalar equations

  • travelling waves

  • reaction- diffusion systems

  • systems nor

  • corresponding energy

  • gradient

  • global stability


Published : Tuesday, June 19, 2012
Reading/s : 14
Origin : www-fourier.ujf-grenoble.fr
Number of pages: 21
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1
A
variational proof of global stability for bistable travelling waves
Thierry Gallay Institut Fourier
UniversitedeGrenobleI 38402Saint-Martin-dHeres France
Emmanuel Risler Institut Camille Jordan INSA de Lyon 69621 Villeurbanne France
June 19, 2007
Abstract
We give a variational proof of global stability for bistable travelling waves of scalar reaction-di usion equations on the real line. In particular, we recover some of the classical results by P. Fife and J.B. McLeod (1977) without any use of the maximum principle. The method that is illustrated here in the simplest possi-ble setting has been successfully applied to more general parabolic or hyperbolic gradient-like systems.
Introduction
The purpose of this work is to revisit the stability theory for travelling waves of reaction-di usion systems on the real line. We are mainly interested inglobalstability results which assert that, for a wide class of initial data with a speci ed behavior at in nit y, the solutions approach for large times a travelling wave with nonzero velocity. In the case of scalar reaction-di usion equations, such properties have been established by Kolmogorov, Petrovski & Piskunov [11], by Kanel [9, 10], and by Fife & McLeod [4, 5] under various assumptions on the nonlinearity. The proofs of all these results use a priori estimates and comparison theorems based on the parabolic maximum principle. Therefore they cannot be extended to general reaction-di usion systems nor to scalar equations of a di eren t type, such as damped hyperbolic equations or higher-order parabolic equations, for which no maximum principle is available. However, these methods have been successfully applied tomonotonereaction-di usionsystems[15,18],aswellastoscalarequationsonin nite cylinders [14, 16]. Recently,adi erentapproachtotheglobalstabilityofbistabletravellingwaveshas been developped by the second author [13]. The new method is of variational nature and is therefore restricted to systems which admit a gradient structure, but it does not make any use of the maximum principle and is therefore potentially applicable to a wide class
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of problems. The goal of this paper is to explain how this method works in the simplest possible case, namely the scalar parabolic equation ut=uxx F0(u),(1) whereu=u(x, t)R,xR, andtshall thus recover the main result of0. We Fife & McLeod [4] under slightly di eren t assumptions on the nonlinearityF, with a completely di eren t proof. The present article can also serve as an introduction to the more elaborate work [13], where the method is developped in its full generality and applied totheimportantcaseofgradientreaction-di usionsystemsoftheformut=uxx rV(u), withuRnandV:RnR. A further application of our techniques is given in [7], where the global stability of travelling waves is established for the damped hyperbolic equation utt+ut=uxx F0(u), with >0. We thus consider the scalar parabolic equation (1), which models the propagation of fronts in chemical reactions [2], in combustion theory [9, 10], and in population dynamics [1, 6]. We suppose that the “potential”F:RRis a smooth, coercive function with a unique global minimum and at least one additional local minimum. More precisely, we assume thatF∈ C2(R) satis es u)>0.(2) l|iu|minfF0(u In particular,F(u)+as|u| → ∞ also assume that. WeFreaches its global minimum atu= 1: F(1) = A <0, F0(1) = 0, F00(1)>0,(3)
and has in addition a local minimum atu= 0: F(0) =F0(0) = 0, F00(0) = >0.
Finally, we suppose that all the other critical values ofFare positive, namely nuR0(u) = 0, F(u)0o={0 ; 1}. F A typical potential satisfying the above requirements is represented in Fig. 1.
(4)
(5)
Under assumptions (3)-(5), it is well-known that Eq.(1) has a family of travelling waves of the formu(x, t) =h(x ct) connecting the stable equilibriau= 1 andu= 0. More precisely, there exists a unique speedc>0 such that the boundary value problem hh0(0 (y)+)c=h10(,y) h(F+0(h()y))0=0=, yR,(6) ,
has a solutionh:R(0,teehrp ohwcichsa1),inlehitself is unique up to a translation. Moreoverh∈ C3(R),h0(y)<0 for allyR, andh(y) converges exponentially to its limits asy→ ∞. This family of travelling waves plays a major role in the dynamics of Eq.(1), as is shown by the following global convergence result:
2
 A
Fig. 1:The simplest example of a
F(u)
0
1
u
nonlinearityFsatisfying assumptions (2)–(5).
Theorem 1.1LetF∈ C2(R) Then there existsatisfy assumptions (2)–(5). >0such that, for all initial datau0∈ C0(R)with lim sup|u0(x) 1|  ,lim sup|u0(x)|  , x→ ∞x+
>0and
(7)
Eq.(1) has a unique global bounded solution satisfyingu(x,0) =u0(x)for allxR. In addition, there existsx0Rsuch that sxupRu(x, t) h(x ct x0)=O(e  t),ast+.(8)
Theorem1.1was rstprovedbyFife&McLeod[4,5]undertheadditionalassumption that 0u0(x)1 for allxR that case. Inu(x, t)[0,1] for allxRand allt0 by the maximum principle, so that the coercivity assumption (2) is not needed. As is mentioned in [3], the results of [4] can be extended to arbitrary initial data satisfying (7) provided thatuF0(u)>0 for allu /[0,1], a condition that is more restrictive than (2) in the sense thatFis not allowed to have critical points outside the interval [0,1]. The simplest case considered in [4] is whenFexactly one critical point in the open intervalhas (0, However, Fife & McLeod also1), a situation in which condition (5) is clearly met. study the case whereFhas three critical points in the open interval, including a local minimum atu=u(0, this situation there exists a travelling wave solution of1). In (1) with speedc1>0 connectingu=uand also a travelling wave with speed, 1 tou= c2Rconnectingu=utou= 0. Ifc1> c2, which is always the case if (5) holds,  there existsc(0,1) such that (6) has a solutionh:R(0,1), and the conclusion of Theorem 1.1 is still valid. Ifc1< c2, there exists no travelling wave connectingu= 1 to u= 0, and the solution of (1) with initial data satisfying (7) converges ast→ ∞to a superposition of two travelling waves [4]. Theorem 1.1 is a particular case of the general results obtained in [13], see Theorem 4 in Section 9.6 of that reference. Therefore, there is no need to give here a complete proof. Instead we shall prove the convergence result (8) under the additional assumption that the initial datau0(x) decay rapidly to zero asx+. It is intuitively clear that the precise behavior ofu0(x) nearx= +should not play an important role, because the equilibriumu= 0 ahead of the front is stable (this is in sharp contrast with the case of a
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monostable front invading an unstable equilibrium, where the behavior ahead of the front is of crucial importance). However, this restriction allows to shortcut many technicalities and to give a much simpler proof in which the essence of the argument can be easily understood. Our approach is based on the fact that Eq.(1) possesses (at least formally) agradient structureonly in the laboratory frame but also in any frame moving to the right with, not a positive velocity. To see this, we introduce the following notation. Ifu(x, t) is a solution of(1),wede neforanyc >0
v(y, t) =u(y+ct, t),or equivalentlyu(x, t) =v(x 
ct, t).(9)
Settingy=x ctwe see that the new functionv(y, t) satis es vt=vyy+cvy F0(v).(10) We now introduce theenergy functional Ec[v] =ZRecy12vy2+F(v)dy ,(11) and the correspondingenergy dissipation functional Dc[v] =ZRecyvyy+cvy F0(v)2dy .(12) We also denote byHc1(R) the Banach space Hc1(R) =nvL(R)ecy/2vH1(R)o,(13) equipped with the normkvkHc1=kvkL+kecy/2vkH1. Note that anyvHc1(R) decays to zero faster thane cy/2asy+. SinceF(v) v2/2 asv0 by (4), it follows thatEc[v]<for allvHc1(R any). Conversely,vL(R) such thatv(y)0 as y+belongs toHc1(R) as soon asEc[v]<. Ifv(y, t) is a solution of (10) with initial datav0Hc1(R), thenv(, t)Hc1(R) for all t0 and a direct calculation shows that dE[v(, t)] =Dc[v(, t)]0, t >0.(14) dtc In other words, the energyEcis aLyapunov functionof system (10) inHc1(R). This observation is of course not new: in their original proof, Fife & McLeod [4] already used a suitable truncation of the functionalEcfor the particular value c=cto show that the solutionv(y, t However,) of (10) approaches a travelling wave for a sequence of times. the fact that Eq.(1) has a whole family of (nonequivalent) Lyapunov functions has not been fully exploited until recently. The only reference we know where the implications of this rich Lyapunov structure are really discussed is a recent paper by Muratov [12], which contains a lot of interesting observations and a few general results concerning a wider class of systems than Eq.(1), but fails to prove the convergence to travelling waves. The goal of the present article is to show that, in the simple case of Eq.(1), the gradient
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structurealoneis sucien t to establish convergence, at least if we restrict ourselves to solutions which decay to zero rapidly enough asx+so that the energy functionals are properly de ned. The main dicult y of this purely variational approach is that we do not have good a priori estimates on the solutionv(y, t) =u(y+ct, t) in a moving frame with speedc >0. First of all, it is not clear a priori that the energyEc[v(, t)] is bounded from below (this will not be the case typically ifcis too small), and without this information it is dicult to really exploit the dissipation relation (14). Next, if we have a lower bound onEc[v(, t)], we can deduce from (14) that the solutionv(y, t) converges uniformly on compact sets, at least for a sequence of times, towards a stationary solution of (10), but we cannot exclude a priori that this limit is just the trivial equilibriumv0 (this will be the case typically ifcomrcveoodiseheetootsiT.)egralistoideakthetracitsecluamnit,eh position of the front interface in the following way. We  x positive constants 1, 2such that 1< F00(0)< 2, and we chooseε >0 small enough so that 1F00(u) 2,for allu[ 2ε,2ε].(15) Given a continuous solution of (1) satisfying the boundary conditions
limu(x, t) = 1,xli+mu(x, t) = 0, t0,(16) x→ ∞ ∞ wede netheinvasion pointx(t starting from the right where the) as the rst point solutionu(x, t) leaves anε-neighborhood of the equilibriumu= 0: x(t) = maxnxR|u(x, t)| εo.(17) In view of (16), it is clear that<x(t)<for allt0, and that|u(x(t), t)|=ε. A quantity similar to x(t) was also introduced in [12], where it is called the “leading edge”. The strategy of the proof is to show that the solutionu(x, t) converges uniformly on compact sets around the invasion point x(t) towards a suitable translate of the travelling wave (6). Using only the gradient structure, we can prove the following result: Proposition 1.2LetF∈ C2(R)satisfy assumptions (2)–(5). Ifu0Hc1(R)for some sucientlylargec >0andu0 1H1(R ), then the solutionu(x, t)of Eq.(1) with initial datau0 allsatis es, forL >0,
sup|u(x(t) +z, t) hε(z)|t0, z[ L,+)→∞
(18)
wherex(t)is the invasion point (17) andhεis the travelling wave (6) normalized so that hε(0) =ε the map. Moreovert7→x(t)isC1fortlylaientndrgeauscx0(t)s ca t→ ∞.
As is explained above, the assumptionu0Hc1(R) is needed in order to use the energy functionalEcwithout truncating the unbounded exponential factorecy. The proof will showthatisitsucienttotakeherec >2A/ε, whereAis de ned in (3) andεin (15). On the other hand, the assumptionu0 1H1(R ) is just a convenient way to guarantee
5
that the rst condition in (16) is satis ed, but with minor modi cations we can treat the more general case where|u0(x) 1|is assumed to be small for largex <0, as in (7). The local convergence established in Proposition 1.2 is the key step in proof of The-orem 1.1. Once (18) is known, it remains to show that the solutionu(x, t) converges uniformly to 1 in the region far behind the invasion point x(t). Such a “repair” is cer-tainly expected becauseu= 1 is the point where the potentialFreaches its global minimum. A convenient way to prove this is to use a truncated version of the functional EZR12ux2+F(u)dx ,(19) [u] =
whereF(u) =F(u) F(1) this way, we can show that the solution0. Inu(x, t) approaches uniformly onRa travelling wave (at least for a sequence of times), and using in addition the local stability results established in [17] we obtain (8). We thus have:
Corollary 1.3Under the assumptions of Proposition 1.2, there existx0Rand >0 such that (8) holds.
We conclude this introduction with a few comments on the scope of our method. First, it is clear that the assumptions (2)–(5) are not the weakest ones under which Proposition 1.2 holds. A careful examination of the proof reveals that the only hypotheses that we really use are:
H1:For all bounded initial datau0, Eq.(1) has a (unique) global bounded solution.This is certainly true if (2) holds, but it is sucien t to assume, for instance, thatF(u)+as|u| → ∞, or thatuF0(u)>0 whenever|u|is sucien tly large. H2:F(0) =F0(0) = 0, and there existsε >0such thatF00(u)0for allu[ ε, ε]. This is automatically true if (4) holds, butu= 0 need not be a strict local minimum ofF particular Proposition 1.2 holds for the nonlinearities of combustion type. In considered in [9, 10].
H3:There exists a uniquec >0 equationsuch that the di er entialvyy+cvy F0(v) = 0 has a bounded solution satisfying|v(0)|=ε,|v(y)| εfor ally0, andv(y)0 asy+; furthermore, this solution is unique.Under assumptions (3)–(5), we havec=candv=hεwe can assume without loss of generality that general, . Inv is positive and converges to 1 asy→ ∞, so thatF(1)<0 andF0(1) = 0. also It follows thatF(u)0 for allu0 and thatFhas no critical pointu<1 with F(u)<0.
On the other hand, to prove that the solution of (1) given by Proposition 1.2 converges uniformly onRto a travelling wave we need the additional assumption: H4:There existsε0>0ide tfehoionlotuionquatialerentttehnoylobnuedsdsuchtha uxx F0(u) = 0with|u(0) 1| ε0isu1.This requires thatFattains its global minimum atu= 1, and nowhere else.
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Finally, if we want the convergence to be exponential in time as in (8), we need to assume thatF00(1)>0.
Another comment concerns the variational structure of Eq.(1). Due to the exponential weightecy, it is clear that the energy functionalEc fact, for Inis not translation invariant. anyvHc1(R) and any`R, we have the relationEc[v(`)] =ec`Ec[v]. This implies that the in m um ofEc[v] is either 0 or our assumptions on. UnderF, the transition between both regimes occurs precisely at the critical speedcfor which travelling waves exist: vHc1(R)Ec[v] = 0ifficcc<c., inf Indeed, as was observed by Muratov [12], for anyc < c+c2+ 4F00(0) we have the identity cEc[h] = (c c)ZRecyh0(y)2dy , whereh shows in particular that Thisis the solution of (6).Ec[h]<0 whenc < c, hence infEc= fact thatin that case. TheEc0 whenccis not obvious a priori, and will be established in the course of the proof of Proposition 1.2, see Corollary 4.3. Note also thatEc[h] = 0, so that infEc= minEc= 0.    The rest of the paper is organized as follows. In Section 2, we establish the basic inequalities relating the energyEc, the dissipationDc Using, and the invasion point. these relations, we prove in Section 3 that the average speed of the invasion point x(t) has a limitc>0 ast→ ∞. The core of the paper is Section 4, where we show thatc=c and prove Proposition 1.2. The proof of Corollary 1.3 is then performed in the nal Section 5. Acknowledgements.The authors are indebted to S. Heinze, R. Joly, and C.B. Muratov for fruitful discussions.
2 Preliminary estimates
As the potentialFis smooth and coercive, it is well-known that the Cauchy problem for the semilinear equation (1) is globally well-posed in the space of bounded functions, see e.g. [8]. Due to parabolic regularization, the solutions are smooth fort >0 and satisfy (1) in the classical sense. Under assumption (2), one can also show that our system has abounded absorbing setin the following sense:
Lemma 2.1There exists a constantB >0depending only onFsuch that, for all initial datau0L(R), the (unique) solutionu(x, t) large all suciently forof (1) satis es, t0, sup|u(x, t)|+|ux(x, t)|+|uxx(x, t)|B .(20) xR Moreover,u(, t)is bounded inHlsoc(R)for somes >5/2and allt1.
7
The uniform bound on|u(x, t)|follows easily from the maximum principle, but it can also be established using localized energy estimates, see [13, Section 9.1]. The bounds on the derivatives are then obtained in a standard way using parabolic regularization. From now on, we suppose thatu0Hc10(R) for somec0>0 (which will be speci ed later) and thatu0 1H1(R the solution of (1) with initial data). Thenu0satis es  u(, t)Hc10(R) andu(, t) 1H1(R ) for allt0, becauseu= 0 andu= 1 are (stable) equilibria of (1). In particular, the boundary conditions (16) hold for all times, so that one can de ne the invasion point x(t since we are interested in the) by (17). Also, long-time behavior ofu(x, t), we can assume without loss of generality that estimate (20) is valid for allt0. As is explained in the introduction, we shall use the energy functionalsEc(for various values ofc >0) to prove that the solutionu(x, t) converges to a travelling wavehlocally around the invasion point x(t). A technical problem we shall encounter is that the invasion point, as de ned in (17), need not be a continuous function of time and can therefore jump backandforthinanuncontrolledway.Itispossibletoavoidthisdicultyusingamore cleverde nitionthan(17),see[13],butwefollowhereanotherapproachandjustintroduce asecondinvasionpointde nedby X(t) = maxnxR|u(x, t)| 2εo.(21) Clearly,<X(t)<x(t)<+for allt important point is that an0. The information on xat a given time provides an upper bound on Xat later times:
Lemma 2.2There existsT0>0andC0>0such that, for allt00, one has X(t)x(t0) +C0for allt[t0, t0+T0].(22) Proof:Fixt00. The solution of (1) satis es t u(t) =S(t t0)u(t0) Zt0S(t s)F0(u(s)) dsu1(t) +u2(t), tt0, whereS(t) =et∂2x Takeis the heat semigroup.K >0 such that|F0(u)| Kwhenever |u| B, whereBis as in (20). Thenku2(t)kLK(t t0 the other hand, by). On de nition of x, we have|u(x, t0)| εifxx(t0) and|u(x, t0)| Bifxx(t0). Using the explicit form of the heat kernel, we deduce that |u1(x, t)| 4(1t t0)ZRe 4(t t)02|u(y, t0)|dyε+Bfrce2x4 (tx(tt00)),  where erfc(x) = (2/ )Rxe z2dz  rst choose. WeT0>0 such thatKT0< ε/2, and thenC0>0 such thatBerfc(C0/4T0)< ε. Then, for allt[t0, t0+T0] and all xx(t0) +C0we have|u(x, t)|<2ε, which implies that X(t)x(t0) +C0.We now derive the basic estimates on the energy (11) and the energy dissipation (12) which will be used throughout the proof. Givenc(0, c0e ndewe),v(y, t) =u(y+ct, t) as in (9), and we set
Ec(t) =Ec[v(, t)], Dc(t) =Dc[v(, t)], 8
t
0.
(23)
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