www.elsevier.com/locate/anihpc
Traveling waves in a one-dimensional heterogeneous medium
James Nolen
a,∗, Lenya Ryzhik
baDepartment of Mathematics, Stanford University, Stanford, CA 94305, USA bDepartment of Mathematics, University of Chicago, Chicago, IL 60637, USA
Received 16 October 2007; accepted 17 February 2009 Available online 17 March 2009
Abstract
We consider solutions of a scalar reaction–diffusion equation of the ignition type with a random, stationary and ergodic reaction rate. We show that solutions of the Cauchy problem spread with a deterministic rate in the long time limit. We also establish existence of generalized random traveling waves and of transition fronts in general heterogeneous media.
©2009 Elsevier Masson SAS. All rights reserved.
1. Introduction
We consider solutions to the equation
ut=u+f (x, u, ω), x∈R, (1.1)
where f (x, u, ω) is a random ignition-type non-linearity that is stationary with respect to translation in x. The functionf has the formf (x, u, ω)=g(x, ω)f0(u). Here,f0(u) is an ignition-type non-linearity with an ignition temperatureθ0∈(0,1):f0(u)is a Lipschitz function, and, in addition,
f0(u)=0 foru∈ [0, θ0] ∪ {1}, f0(u) >0 foru∈(θ0,1), f0(1) <0.
The reaction rateg(x, ω),x∈R, is a stationary, ergodic random field defined over a probability space(Ω,P,F):
there exists a group{πx},x∈R, of measure-preserving transformations acting ergodically on(Ω,P,F)such that g(x+h, ω)=g(x, πhω). We suppose thatg(x, ω)is almost surely Lipschitz continuous with respect tox and that there are deterministic constantsgmin, gmaxsuch that
0< gming(x, ω)gmax<∞ holds almost surely. Thus, we have
fmin(u)f (x, u, ω)fmax(u),
* Corresponding author.
E-mail addresses:nolen@math.stanford.edu (J. Nolen), ryzhik@math.uchicago.edu (L. Ryzhik).
0294-1449/$ – see front matter ©2009 Elsevier Masson SAS. All rights reserved.
doi:10.1016/j.anihpc.2009.02.003
wherefmin(u)=gminf0(u)andfmax(u)=gmaxf0(u)are both ignition-type non-linearities with the same ignition temperature. We assume that the probability spaceΩ=C(R; [gmin, gmax])and thatF contains the Borelσ-algebra generated by the compact open topology (the topology of locally uniform convergence) onC(R; [gmin, gmax]).
We are interested in the following two issues: first, how do solutions of the Cauchy problem for (1.1) with a compactly supported non-negative initial data spread in the long time limit? Second, do there exist special solutions of (1.1) that generalize the notion of a traveling front in the homogeneous case?
It is well known since the pioneering work by Ya. Kanel [14] that in the uniform case:
ut=u+f (u) (1.2)
with an ignition-type non-linearity f (u), all solutions with the initial data u0(x)=u(0, x) in a class I ⊂Cc(R), 0u0(x)1, propagate with the same speedc∗in the sense that
t→+∞lim u(t, ct )=0 for|c|> c∗, (1.3)
and
t→+∞lim u(t, ct )=1 for|c|< c∗. (1.4)
The initial data is restricted to the classI to preclude the possibility of the so-called quenching phenomenon where u→0 uniformly inxast→ ∞. In particular,I contains functions that are larger thanθ0+εon a sufficiently large interval, depending on ε >0. The constant c∗ above is the speed of the unique traveling wave solutionu(t, x)= U (x−c∗t )of (1.2):
−c∗U=U+f (U ), U (−∞)=1, U (+∞)=0.
As far as heterogeneous media are concerned this result has been extended to the periodic case: J. Xin [29,30], and H. Berestycki and F. Hamel [2] have shown that when the functionf (x, u)is periodic inx, Eq. (1.1) admits special solutions of the formu(t, x)=U (x−c∗t, x), called pulsating fronts, which are periodic in the second variable and satisfy
U (s, x)→1 ass→ −∞, and U (s, x)→0 ass→ +∞.
H. Weinberger [28] has proved that solutions with general non-negative compactly supported initial data spread with the speedc∗in the sense of (1.3)–(1.4), though the spreading rates to the left and right may now be different.
The purpose of the present paper is to extend the result of [28] to the stationary random ergodic case, and show that special solutions which generalize the notion of a pulsating front to random media exist.
Deterministic spreading rates
Our first result concerns the asymptotic behavior of solutions to the Cauchy problem for (1.1) with compactly supported initial data. We show that for sufficiently large initial data the solution develops two diverging fronts that propagate with a deterministic asymptotic speed. Specifically, we prove the following.
Theorem 1.1.Letw(t, x, ω) solve(1.1)with compactly supported deterministic initial dataw0(x),0w0(x)1.
Leth∈(θ0,1)and suppose thatw0hon an interval of sizeL >0. There exist deterministic constantsc∗−<0< c+∗ such that for anyε >0, the limits
tlim→∞ inf
c∈[c∗−+ε,c+∗−ε]w(t, ct, ω)=1 and
tlim→∞ sup
c∈(−∞,c∗−−ε]∪[c∗++ε,∞)
w(t, ct, ω)=0
hold almost surely with respect toP, ifLis sufficiently large. The constantsc∗−, c∗+are independent ofhandL.
The condition thatLbe sufficiently large is necessary only to exclude the possibility of uniform convergence to zero [14].
Using large deviation techniques, Freidlin, and Freidlin and Gärtner (see [8, Section 7.4], [7,9–11]) proved a similar asymptotic result in the case thatf0(u)is of KPP-type satisfyingf (u)f(0)u(e.g.f0(u)=u(1−u)). Moreover, the asymptotic speed can be identified by a variational principle that arises from the linearized problem atu=0. This asymptotic spreading result has been extended recently to time-dependent random media in [24,25]. The problem with a KPP non-linearity also admits homogenization, both in the periodic [20] and random [15,19] cases. However, in all aforementioned papers, the KPP conditionf (u)f(0)useems to be essential, and the techniques do not extend to the present case wheref vanishes whenuis close to zero. To the best of our knowledge Theorem 1.1 is the first result on the deterministic spreading rates of solutions of reaction–diffusion equations with a non-KPP non-linearity in a random medium.
Random traveling waves
Two generalizations of the notion of a traveling front in a uniform medium for general (non-periodic) inhomoge- neous media were proposed. Shen in [27], and Berestycki and Hamel in [3,4] have introduced generalized transition fronts (called wave-like solutions in [27]) — these are global in time solutions that, roughly speaking, have an inter- face which “stays together” uniformly in time. On the other hand, H. Matano has defined a generalized traveling wave as a global in time solution whose shape is “a continuous function of the current environment” [22]. These notions are not equivalent: there exist transition fronts of the KPP equation with constant coefficients that are not traveling waves in the usual sense (and hence not generalized traveling waves in the sense of Matano as there is only one environment in the case of a uniform medium and thus only one solution profile) [12,13].
Matano’s definition was formalized by W. Shen in [27] as follows.
Definition 1.2.(See [27, Definition 2.2].) A solutionw(t, x, ω)˜ :R×R×Ω→Rof (1.1) is called arandom traveling waveif the following hold:
(i) For almost everyω∈Ω,w(t, x, ω)˜ is a classical solution of (1.1) for allt∈R.
(ii) The functionw(0, x, ω)˜ is measurable with respect toω.
(ii) 0<w(0, x, ω) <˜ 1,∀x∈R.
(iii) limx→+∞w(0, x, ω)˜ =0.
(iv) limx→−∞w(0, x, ω)˜ =1.
(v) There exists a measurable functionX(t, ω)˜ :R×Ω→Rsuch that
˜
w(t, x, ω)= ˜w
0, x− ˜X(t, ω), πX(t,ω)˜ ω .
The functionW (x, ω):R×Ω→Rdefined byW (x, ω)= ˜w(0, x, ω)is said togeneratethe random traveling wave.
The random functionW (x, ω) is the profile of the wave in the moving reference frame defined by the current front positionX(t, ω). In the pioneering paper [27], Shen has established some general criteria for the existence of a˜ traveling wave in ergodic spatially and temporally varying media and also proved some important properties of the wave. In particular, as shown in [27] (see [27, Theorem B]), Definition 1.2 of a random traveling wave generalizes the notion of a pulsating traveling front in the periodic case. More precisely, iff is actually periodic inx, then a random traveling wave solution of (1.1) is a pulsating traveling front solution in the sense of [2,29,30].
However, the only example provided in [27] where the results of [27] ensure existence of a random traveling wave is a bistable reaction–diffusion equation of the form
ut=uxx+(1−u)(1+u)
u−a(t ) ,
wherea(t )is a stationary ergodic random process. As far as we are aware, no other examples of such traveling waves in non-periodic media have been exhibited. In this paper we construct a Matano–Shen traveling wave in a spatially varying ergodic random medium for (1.1) with an ignition-type non-linearity.
Theorem 1.3.There exists a random traveling wave solutionw(t, x, ω)˜ of(1.1)which is increasing monotonically in time:
˜
wt(t, x, ω) >0 for allt∈Randx∈R.
Moreover, the interfaceX(t, ω)˜ satisfies (i) w(t,˜ X(t, ω), ω)˜ =θ0for allt∈R, and
(ii) X(t˜ +h, ω) >X(t, ω)˜ for allt∈Randh >0, and
(iii) lim
t→∞
X(t, ω)˜
t =c∗+ (1.5)
holds almost surely, wherec∗+is the same constant as in Theorem1.1.
Monotonicity of the wave and the fact that the interface is moving to the right is the direct analog of the corre- sponding properties of the periodic pulsating fronts.
SinceX(t, ω)˜ is increasing int, we may define its inverseT (x, ω)˜ :R×Ω→Rby x= ˜XT (x, ω), ω˜
. (1.6)
This may be interpreted as the time at which the interface reaches the position x∈R. The following corollary says that the statistics of the profile of the wave as the wave passes through the pointξ are invariant with respect toξ: Corollary 1.4.The functionw(˜ T (ξ, ω), x˜ +ξ, ω)is stationary with respect to shifts inξ.
This is a direct analog of the corresponding property of a pulsating front in a periodic medium: the profile of a pulsating front at the timeT (ξ )it passes a pointξ is periodic inξ.
We believe the present article gives the first construction of such a wave in a spatially random medium. To construct the wave, we use a dynamic approach from [27] combined with some analytical estimates needed to show that the construction produces a non-trivial result.
Generalized transition fronts
Our last result concerns existence of the transition fronts for (1.1) in the sense of Berestycki and Hamel, and Shen, in general heterogeneous (non-random) media with the reaction rate uniformly bounded from below and above. Let us recall first the definition of a generalized transition wave.
Definition 1.5.A global in time solutionv(t, x),˜ t∈R,x∈R, of (1.1) is called a transition wave if for anyh, k∈(0,1) withh > k, we have
0θk+(t, ω)−θh−(t, ω)C (1.7)
for allt∈R, where θh−(t, ω)=sup
x∈Rv(t, x˜ , ω) > h, ∀x< x , θk+(t, ω)=inf
x∈Rv(t, x˜ , ω) < k, ∀x> x
(1.8) andC=C(h, k)is a constant independent oftandω.
Roughly speaking, a transition wave is a global in time solution for which there are uniform, global-in-time bounds on the width of the interface. Basic properties of transition waves were investigated in [3,4].
Theorem 1.6.Letf (x, u)be a non-linearity such thatgminf0(u)f (x, u)gmaxf0(u), with the constantsgmin>0, gmax<+∞ and f0(u) an ignition-type non-linearity. Then there exists a transition front solutionu(t, x),t ∈R, x∈R, of(1.1)which is monotonically increasing in time:ut(t, x) >0. In addition, there exists a unique pointX(t ) such thatu(t, X(t ))≡θ0, and a constantp >0so thatux(t, X(t )) <−pfor allt∈R.
As this paper was written we learned about the concurrent work by A. Mellet, J.-M. Roquejoffre, and Y. Sire [23], in which Theorem 1.6 is also proved.
Let us point out that all of the results in this paper extend to the case of a bistable-type non-linearity, under certain restrictions. Specifically, we may letf have the formf (x, u)=g(x)f0(u)withf0(0)=f0(θ0)=f0(1)=0, f0(u) <0 foru∈(0, θ0),f0(u) >0 foru∈(θ0,1), andf0(1) <0. Under the additional condition that
1 0
fmin(u) du= 1 0
min
x∈Rf (x, u) du >0, (1.9)
all of the results in Theorems 1.1, 1.3, and 1.6 apply. This condition is necessary to preclude the phenomenon of wave- blocking, which can occur with a spatially-dependent bistable-type non-linearity (for example, see the work Lewis and Keener [17]). In particular, under the condition (1.9), one can modify our argument to construct the time-monotonic solutions that are the building blocks for the generalized transition fronts.
The paper is organized as follows. In Section 2, we study solutions to (1.1) that are monotone increasing in time and prove Theorem 1.6. The main ingredients in the proof are Propositions 2.3 and 2.5 which show that the interface (the region where ε < u <1−ε, for some ε >0) may not be arbitrarily wide and must move forward with an instantaneous speed that is bounded above and below away from zero. These estimates are also used later in the proof of the asymptotic spreading and in the construction of the random traveling waves. In Section 3 we prove Theorem 1.1, first for monotone increasing solutions and then for general compactly supported data. In Section 4 we construct the random traveling wave and prove Theorem 1.3 and Corollary 1.4.
Throughout the paper we denote byCandKuniversal constants that depend only on the constantsgminandgmax, and the functionf0(u).
2. Existence of a generalized transition front
Monotonic in time solutions
In this section we prove Theorem 1.6. The generalized transition wave is constructed as follows. We consider a sequence of solutionsun(t, x)of (1.1) (withf (x, u, ω)replaced byf (x, u)as in the statement of the theorem) defined fort−n, with the Cauchy data
un(t= −n, x)=ζ x−x0n
, ζ (x):=maxζ (x),ˆ 0
. (2.1)
The choice of the initial shiftx0n is specified below while the functionζ (x)ˆ is positive on an open interval and is a sub-solution for (1.1):
−ˆζ(x)=fminζ (x)ˆ
f (x,ζ ),ˆ (2.2)
withfmin(u)=gminf0(u)f (x, u). It is constructed as follows. For a givenh0∈(θ0,1)andx∈R, letζ (x)ˆ satisfy
−ˆζ(x)=fminζ (x)ˆ
, ζ (0)ˆ =h0, ζˆ(0)=0,
with the convention that fmin(u)=0 foru <0 above. To fix ideas we may seth0=(1+θ0)/2 in (2.2). Let us definez1=min{x >0| ˆζ (x)=θ0}andz2=min{x >0| ˆζ (x)=0}. The functionζˆsatisfies the following elementary properties:
• ˆζ (−x)= ˆζ (x)for allx∈R,
• 0ζ (x)ˆ h0= ˆζ (0)for allx∈ [−z2, z2],
• ˆζ (x)is strictly concave forx∈(−z1, z1),
• ˆζ (−z2)= ˆζ (z2)=0.
As in [21,26] it follows thatun(t, x)un(−n, x)fort−n, andun(t, x)is monotonically increasing in time to
¯ u≡1.
Lemma 2.1.Letun(t, x)solve(1.1)with initial data(2.1)at timet= −n. Then,un(t, x)is strictly increasing int:
∂un
∂t (t, x) >0 for allt >−n, (2.3)
and, moreover,
tlim→∞un(t, x)=1 locally uniformly inx. (2.4)
Proof. Since
−ζxxf (x, ζ ),
the maximum principle implies thatun(t, x)un(−n, x)=ζ (x)for allt >−n. Applying the maximum principle to the functionw(t, x)=un(t+τ, x)−un(t, x), forτ >0 fixed, we see that, asw(−n, x)0, we havew(t, x) >0 for allt >−n; thus,unis monotonically increasing in time and (2.3) holds.
Sinceunis monotone int, the limitu(x)¯ =limt→∞un(t, x)exists and satisfies
¯
uxx= −f (x,u),¯ 0<u(x)¯ 1, max
x u > h¯ 0> θ0. (2.5)
Note thatu¯xx=0 on the set{ ¯u < θ0}, sou¯is linear there. It is easy to see that this implies that this set must be empty because of the lower boundu¯0 and the fact that maxxu > h¯ 0> θ0. Hence, we haveu¯θ0.
Now, (2.5) implies thatu¯is concave. Sinceθ0u¯1, this impliesu¯is constant, so thatf (x,u)¯ = − ¯uxx≡0. This fact and maxxu > h¯ 0implies thatu¯≡1. The local uniformity of the limit follows from standard regularity estimates foru. 2
The initial shift
The initial shiftx0nis normalized by requiring that
un(0,0)=θ0. (2.6)
Lemma 2.2.There existsx0nso thatun(t, x)satisfies(2.6)andlimn→+∞x0n= −∞. Moreover, there existsN0and ε >0so thatxn0<−εnfor alln > N0.
Proof. Letvn(t, x;y)be the solution of (1.1) with the initial datavn(t= −n, x;y)=ζ (x−y)— we are looking forx0 such thatvn(0,0;x0)=θ0. Note that (2.3) impliesvn(0,0;0) > θ0. In addition, the functionψ (x)=exp(−λ(x−ct )) is a super-solution for (1.1) provided that
cλλ2+Mgmax, (2.7)
with the constantM >0 chosen so thatf0(u)Mu. Let us chooseλ >0 andc >0 sufficiently large so that (2.7) holds. The maximum principle implies that there exists a constantC >0 so that
vn(t, x;y)Cexp
−λ
x−y−c(t+n) , and thus
vn(0,0;y)Cexp
λ(y+cn) θ0
2
fory <−cn−Kwith a constantK >0. By continuity ofvn(0,0;y)as a function ofythere existsx0∈(−cn−K,0) such thatvn(0,0;x0)=θ0. In order to see thatx0n→ −∞asn→ +∞, observe thatvn(t, x;y)wn(t, x;y), where wn(t, x;y)is the solution of the Cauchy problem
∂wn
∂t =∂2wn
∂x2 +gminf0(wn), wn(−n, x;y)=ζ (x−y). (2.8)
Note that ify stays uniformly bounded from below asn→ +∞:yK for alln, then, as in (2.4),wn(0,0;y)→1, which contradicts (2.6), thusx0n→ −∞asn→ +∞. The refined estimatex0n<−εnfollows the results of [26] on
the exponential in time convergence of the solution of (2.8) to a sum of two traveling waves of (2.8) moving with a positive speedcmin>0 to the right and left, respectively. In particular, this implies that ify >−ncmin/2 then for nsufficiently large we havewn(0,0;y) > (1+θ0)/2> θ0which is a contradiction. 2
The standard elliptic regularity estimates imply that the sequence of functions un(t, x) is uniformly bounded, together with its derivatives, so that along a suitable subsequence the limitu(t, x)=limk→+∞unk(t, x)is a global in time and space, monotonically increasing solution to (1.1). The main remaining difficulty is to show thatu(t, x)has the correct limits asx→ ±∞and its “interface width” is uniformly bounded in time so that it is indeed a transition front in the sense of Berestycki and Hamel. The rest of the proof of Theorem 1.6 is based on the following estimates for any solution of the Cauchy problem (1.1) with the initial datau(0, x)=ζ (x−x0), with anyx0∈R(we set here the initial timet0=0 for convenience).
The interface width estimate
Forh∈(θ0,1)andk∈(0, θ0), letXlh(t )andXkr(t )be defined by Xhl(t )=max
x > x0u(t, x) > h, ∀x∈ [x0, x) , Xkr(t )=min
x > x0u(t, x) < k, ∀x∈(x,∞)
. (2.9)
Our goal is to show that the width of the front can be bounded by a universal constant depending only on fmax andfmin.
Proposition 2.3.Letu(t, x)be a solution of(1.1)with the initial datau(0, x)=ζ (x−x0)for somex0∈R. For any h∈(θ0,1)andk∈(0, θ0], there are constantsKh0andC0depending only onh,k,fmaxandfminsuch that for anyt > Khwe haveu(t, x0) > h, and
0< Xrk(t )−Xlh(t )C <+∞ (2.10)
for allt > Kh. We can takeKh=0forh∈(θ0, h0).
Let us note that the time delayKhis introduced simply because initially the solution may be belowheverywhere so thatXlh(t )is not defined for small times.
The interface steepness bound
The next crucial estimate provides a lower bound for the steepness of the interface. First, we use the following lemma to define the interface location.
Lemma 2.4.Letu(t, x)be a solution of(1.1)with the initial datau(0, x)=ζ (x−x0)for somex0∈R. For allt >0, there exists a continuous function(the right interface) X(t ),t 0, monotonically increasing in t and satisfying:
x0< X(t ),u(t, X(t ))=θ0andu(t, x) < θ0for allx > X(t ),u(t, x) > θ0for allx∈(x0, X(t )).
Proof. This follows from the strict monotonicity ofuwith respect to time and the maximum principle which pre- cludesX(t )from having jumps sincef (x, u)=0 for 0uθ0. 2
Proposition 2.5.Letu(t, x)be a solution of(1.1)with the initial datau(0, x)=ζ (x−x0)for somex0∈R. Then the following hold.
(i) There are constantsp >0andτ00depending only ongmax,gmin, and the functionf0such that ux
t, X(t )
<−p (2.11)
for allt >0, and u
t, x+X(t )
θ0e−px (2.12)
for allx >0andtτ0.
(ii) There exists a constantδ >0depending only ongmax,gmin, and the functionf0such that ut
t, X(t )
> δ (2.13)
for allt >1. Moreover, for anyt1>0, there are constantsH >0andL >0such that
0< L <X(t ) < H <˙ +∞ for alltt1. (2.14)
The constantsLandH depend only ont1,gmin,gmaxand the functionf0. The end of the proof of Theorem 1.6
Theorem 1.6 is an immediate consequence of Propositions 2.3 and 2.5. Consider the sequence of functionsun(t, x) defined fort−nas solutions of the Cauchy problem for (1.1) with the initial data (2.1) andx0nfixed by normaliza- tion (2.6). As we have mentioned above, the standard elliptic regularity estimates imply that there exists a subsequence nk→ +∞so thatunk(t, x)converge locally uniformly, together with its derivatives, to a limitu(t, x)which is a global in time and space solution to (1.1), monotonically increasing in time. Moreover, the interface locationsXn(t )con- verge to X(t )such that u(t, X(t ))=θ0, 0< L <X(t ) < H˙ , andX(0)=0. The normalization (2.6) implies that u(0,0)=θ0, and, in addition, the bounds (2.11)–(2.13) hold for the limit u(t, x). The upper bound (2.12) implies immediately thatu(t, x+X(t ))→0 asx→ +∞uniformly int. It remains only to check thatu(t, x+X(t ))→1 as x→ −∞uniformly int. To this end, assume that there existsε0>0 and a sequence of pointstm∈R, andxm→ −∞
such that u(tm, xm+X(tm)) <1−ε0. This, however, contradicts (2.10) with k=θ0 andh=1−ε0. Therefore, u(t, x)is, indeed, a transition wave.
A convenient way to restate some of the properties of the functionsun(t, x)we will need later is as follows. Let us define the class of admissible non-linearities
G=
f (x, u)=g(x)f0(u): gming(x)gmax, g(x)∈C(R) .
Lemma 2.6.Given0< gmingmax<+∞and f0(u), there existp >0and a non-increasing inx functionv(x), such thatv(0)=θ0,v(0) <−p,
x→−∞lim v(x)=1, (2.15)
x→+∞lim v(x)=0, (2.16)
and the following holds:given any solutionun(t, x)of(1.1)withf (x, u)∈G, and with the initial data(2.1)which satisfies the normalization(2.6), and for anyR >0, we have
un
t, x+Xn(t )
v(x), ∀x∈ [−R,0], (2.17)
un
t, x+Xn(t )
v(x), ∀x∈ [0,∞], (2.18)
for allt0, ifnis sufficiently large, depending only onR,gmin, andgmax. The functionv(x)depends only on the constantsgmaxandgmin, and the functionf0(u).
Proof. Settingv(x)=θ0e−px forx0, withp >0 as in Proposition 2.5 we see that the upper bound (2.18) follows from (2.12), and (2.16) is obviously satisfied, as well as a strictly negative upper bound forv(0).
In order to definev(x)forx <0 we consider a solution of (1.1) withf∈G, which satisfies (2.6), and with initial data as in (2.1), and chooset1=1 and find the correspondingLas in Proposition 2.5, so thatX˙n(t )Lfort−n+1.
Now,Xn(t )xn0+L(t+n−1), and thus fort0,nNR=1+R/L, andx∈ [−R,0]we have x+Xn(t )xn0+L(t+n−1)−Rxn0.
Forh∈ [θ0,1), letXn,hl (t )be defined by (2.9). We use the convention thatXln,h(t )= −∞ifun(t, x0n) < h. For any h∈ [θ0,1),Xln,h(t )is finite for allt >0 and for allh∈ [θ0, h], ifn > N (h)is sufficiently large, depending only on gminandh. This follows directly from Proposition 2.3. Now, forx <0 andn1, define
vn(x;f )=sup
h∈ [θ0,1): sup
t0
Xn(t )−Xln,h(t ) |x|
. (2.19)
We indicate above explicitly the dependence ofvnon the non-linearityf (x, u). Then we set v(x)= inf
f∈G inf
n1+|x|/Lvn(x;f ).
The set of possible values ofhover which the supremum is taken in (2.19) containsθ0. Therefore,v(x)θ0for all x <0. From (2.19) it is easy to see thatvn(x;f )is non-increasing inx(forx <0) for eachf ∈G, andvn(0;f )=θ0. Hencev(x)is also non-increasing inxandv(0)=θ0.
Next we show thatv(x)→1 asx→ −∞. For anyh∈ [θ0,1], Proposition 2.3 implies that for allf∈Gwe have sup
t0
Xn(t )−Xn,hl (t )
< C(h)
for some finite constant C(h), depending only on gmin and gmax, provided that n > N (h), which ensures that un(0, xn0)h. So, for x such that bothx <−C(h)and 1+ |x|/L > N (h), we have v(x) > h. Since h may be chosen arbitrarily close to 1, it follows that limx→−∞v(x)=1.
Finally, in order to see that (2.17) holds, fixR >0 andf ∈G, and letun(t, x)be the solution of the corresponding Cauchy problem. By definition ofv,
v(x)vn(x;f ) for allx∈ [−R,0], provided thatn1+R/L. Therefore,
Xn(t )−Xln,h(t )−x (2.20)
for allh∈ [θ0, v(x)],n1+R/L, and allt0. Hence,Xln,h(t )x+Xn(t )x0nso that un
t, x+Xn(t )
h for allh∈ [θ0, v(x)], (2.21)
and for allt0 andn1+R/L. This proves (2.17). 2 Bounds for the location of level sets
In order to finish the proof of Theorem 1.6 it remains to prove Propositions 2.3 and 2.5. We need first to establish some simple bounds on the location of the level sets of the functionu(t, x). Letcminandcmaxbe the unique speeds of the traveling wave solutions of the constant coefficient equations
−cqx=qxx+fmin(q), q(−∞)=1, q(+∞)=0, and
−cqx=qxx+fmax(q), q(−∞)=1, q(+∞)=0,
respectively. The next lemma will allow us to relate the positionXhl(t )toXhl(t−s)withs >0 andh< h— this allows us to control the width of the front in the back, whereuis close to 1.
Lemma 2.7.Letδ >0and let0u(t, x)1satisfy(1.1)fort >0. Suppose, in addition, thatu(0, x) > δ+θ0for allx∈ [xL, xR]. Ifσ= |xR−xL|is sufficiently large, depending onδ andfmin, then for anyh∈(θ0,1)there are constantsβ >0andτ1>0, such that
Xhl(t )xR+cmint−β (2.22)
for alltτ1. The constantsβ andτ1depend only onh,δ,σ, andfmin.
Proof. This follows from the comparison principle and the stability results of [26]. Specifically, ifσ is sufficiently large, consider the functionv(t, x)which solves the equation
vt=v+fmin(v)
with the initial data
v(0, x)=(δ+θ0)χ[xL,xR](x)
att=0. Then, as we have mentioned, the results of [26] imply thatv converges ast → +∞to a pair of traveling waves moving to the left and the right with speed cmin>0. The convergence is exponentially fast. Therefore, after some timeτ1, which depends onhand on the convergence rate,v(t, x)hon the set[xR, xR+cmint−β], for some constantβ >0. The maximum principle implies thatu(t, x)v(t, x)and (2.22) follows. 2
Corollary 2.8. Let h∈(θ0,1). Let u(t, x) be as in Propositions2.3 and 2.5. There are constants β0,τ20 depending only onhandfminsuch that
Xlh(t )Xlh(t1)+max
0, cmin(t−t1)−β
(2.23) for alltt1τ2.
Proof. Letδ=h−θ0and letσbe sufficiently large, as required by Lemma 2.7. Sinceuvwherevsolvesvt=v+ fmin(v)with the same initial data, there is a timeτ2>0 depending only onfminandhsuch thatu(t, x)v(t, x)h on the interval[x0, x0+σ], for alltτ2. So,Xlh(t )x0+σ is well-defined and increasing fortτ2. The bound now follows from Lemma 2.7 withxL=x0,xR=Xhl(t1), and replacingt=0 witht1. 2
Lemma 2.9. Suppose thatu(0, x)Ce−cmax(x−x0) for allx ∈R. Then there is a constant η >0 depending only onfmaxandCsuch that
X(t )x0+cmaxt+η, ∀t >0. (2.24)
Proof. This follows from the comparison principle and the stability results of [26]. 2 Corollary 2.8 and Lemma 2.9 immediately imply that
cminlim inf
t→∞
X(t )
t lim sup
t→∞
X(t )
t cmax. (2.25)
The proof of Proposition 2.5(i): (2.11)
We begin the proof of Proposition 2.5 with the proof of (2.11). The strategy of this proof is a version of the sliding method [5]. Supposeε∈(0, cmin)is sufficiently small so that initially at timet=0 we have
ux
0, X(0)
−εθ0 (2.26)
and u
0, x+X(0)
θ0e−εx for allx0. (2.27)
The good times
Let us define the set of good times G when we can control the decay of the solution ahead of the front by an exponential:
G=
t0:u
t, x+X(t )
θ0e−εxfor allx0
. (2.28)
Note that if t ∈Gthen ux(t, X(t ))−εθ0 and thus both (2.11) and (2.12) hold. Our goal is to show thatG= [τ0,+∞)forε >0 sufficiently small, andτ0is a universal constant.
Givens∈Gandy >0 set
ψ (t, x;s, y)=θ0e−ε(x−ε(t−s)−y−X(s).
The difference w=ψ−u satisfies wt wxx in the region R= {(t, x): x > X(t ), t s} and w(s, x) >0 for xX(s). Therefore, fort−ssmall we havew(t, x) >0 forx > X(t ). On the other hand, as lim inft→+∞X(t )/t cmin> ε, there exist a timet andx > X(t )so thatw(t, x) <0. Let us define the first time whenψandutouch:
τy,s=sup
t > s: w(τ, x) >0 for allxX(τ )and allτ∈ [0, t )
. (2.29)
Note thatτy,s> sfor ally >0. The maximum principle implies that the only point where the functionsuandψcan touch is at the boundary, that is, atx=X(τy,s), where
u
τy,s, X(τy,s)
=ψ
τy,s, X(τy,s);s, y
=θ0, so that, in particular,
X(τy,s)−ε(τy,s−s)−y−X(s)=0.
In addition, we havew(τy,s, x+X(τy,s))0 for allx >0. It follows that u
τy,s, x+X(τy,s)
ψ
τy,s, x+X(τy,s);s, y
=θ0e−εx and thusτy,s∈Gis a “good” time for anyy >0.
The bad times
Now, suppose that the setB= [0,∞)\Gof “bad” times is not empty. Note thatt∈B if and only if there exists x >0 so that
u
t, x+X(t )
> θ0e−εx,
and thusB is open. Hence,B is an at most countable union of disjoint open intervals{(tj,t˜j)}∞j=1, withtj,t˜j ∈G.
Observe that tj :=lim
y↓0τy,tjt˜j.
Indeed, asτy,tj> tjfor ally >0, otherwise we could findy >0 so thattj< τy,tj<t˜j, which would be a contradiction sinceτy,tj ∈Gfor ally >0. SinceGis closed,tj∈G.
Let us enlargeBto B=
∞ j=1
tj, tj
⊇B.
We claim that the average front speed on each time interval[tj, tj]is small:
X(t )−X(tj)ε(t−tj), ∀t∈ tj, tj
. (2.30)
Indeed, for anyt∈(tj, tj), anyxX(t ), and anyy >0 we have u(t, x) < ψ (t, x;tj, y)=θ0e−ε(x−ε(t−tj)−y−X(tj)). Passing to the limity↓0 we obtain
u(t, x)ψ (t, x;tj,0)=θ0e−ε(x−ε(t−tj)−X(tj)) for allt∈ tj, tj
,xX(t ) . (2.31)
Evaluating this inequality at(t, X(t )), we obtain (2.30).
The key estimate we will need below is given by the following lemma.
Lemma 2.10.There exists a constantK >0which depends only onf0(u),gminandgmaxso that for allj we have
|tj−tj|K.
We postpone the proof of this lemma for the moment and finish the proof of (2.11) first.
A lower bound for the slope at the front
Using Lemma 2.10 we may prove the lower bound (2.11) for the slope at the front. Note that for each “good” time t∈Gthis bound holds automatically, so we need only to look at “bad” timest∈B. More generally, consider a time t∈Bso thatt∈(tj, tj)for somej. The functionu(t, x)is convex inxforx > X(t )— this is becauseuxx=ut>0 in this region. Therefore, for alll >0 we have
ux t, X(t )
u(t, X(t )+l)−u(t, X(t ))
l .
Let us choosel >0 so thatψ (t, X(t )+l;tj,0)=θ0/2, then, according to (2.31) we have u
t, X(t )+l
ψ
t, X(t )+l;tj,0
=θ0
2, (2.32)
and thus ux
t, X(t ) −θ0
2l. (2.33)
However, the distancelcan be computed explicitly:
X(t )+l−ε(t−tj)−X(tj)=log 2 ε , and thus
l < ε tj−tj
+log 2 ε .
Lemma 2.10 and (2.33) imply now that fort∈Bwe have an estimate ux
t, X(t )
− θ0
2(ε(tj−tj)+ε−1log 2)− θ0
2(εK+ε−1log 2), (2.34)
which is nothing but (2.11).
The proof of Lemma 2.10
Step 1(Reducing to large times).First, we note that if 0tjT thentj is bounded from above. From Corollary 2.8, we have fortj τ2,
X tj
x0+cmin tj −τ2
−β, while for 0tjT,
X tj
X(tj)+ε tj−tj
X(T )+εtj x0+cmaxT +η+εtj,
from Lemma 2.9. It follows thattj −tjtj (cmaxT +η+β+cminτ2)/(cmin−ε)if 0tjT. Hence, for a constantT to be chosen, we will assume thattjT for the rest of the proof of Lemma 2.10.
Step 2(Forming a large plateau).Our next goal is to show that a large plateau develops behind the front sufficiently fast. Without loss of generality, we assumetjT 1, for a constantT to be chosen. Becausetj is a “good” time, we haveux(tj, X(tj))−εθ0. Elliptic regularity implies that there exists a constantMso that|uxx|Mfor allt1.
Thus, att=tj we have a lower bound foru(t, x)immediately behind the front:
u(tj, x)θ0−p
x−X(tj)
−M 2
x−X(tj)2
,
withp= −εθ0. Now defineδ:=4Mp2 and letσδbe the corresponding constant in Lemma 2.7. Evaluating this inequality atxj=X(tj)−p/M we obtain
u
tj, X(tj)− p M
θ0+ p2
2M=θ0+2δ.
Sinceu(t, x)is monotonic int, we conclude that
u(t, xj)θ0+2δ for allttj andxj=X(tj)− p M.
Now takeT sufficiently large so thatX(T )−p/M−10σδ>0. By Corollary 2.8,T may be chosen to depend only ongmin,gmax, andf0. Therefore, forttjT the functionu(t, x)satisfies the following differential inequality and boundary conditions on the interval(xj−10σδ, xj)⊂ [0,∞):
ut−uxx0, u(t, xj−10σδ)θ0, u(xj)θ0+2δ.
It follows that there exists a timeτδ>0 which depends only onδso thatu(tj+τδ, x)θ0+δfor allx∈(xj−σδ, xj).
By Lemma 2.7 this forces the interface to move forward fortsj=tj+τδat the speed of at leastcmin:
X(t )cmin(t−sj)+X(tj)−p/M−β (2.35)
for alltsj.
Step 3(Plateau catching up with the front).We claim that there exists a constantKso that
tj −tjK(1+τδ). (2.36)
Indeed, according to (2.30), the average front speed on the interval (tj, tj) is smaller than ε. Combining this with (2.35), which says that on the interval[sj, tj]the average speed cannot be too small, leads to
X(tj)+ε tj−tj
X
tj cmin
tj−tj−τδ
+X(tj)−p/M−β.
Thus, (2.36) holds in that case as well. Now, the conclusion of Lemma 2.10 follows. Note that sinceε < cmin is arbitrary, the constantKin that lemma can indeed be chosen to depend only onf0(u),gminandgmax.
An upper bound for the front speed
The lower bound (2.11) on the slope of the front implies an upper bound on its speed in Proposition 2.5(ii). Indeed, sinceux(t, X(t ))−p <0 for allt0, regularity estimates imply that for anyt1>0 and anytt1,
X(t )˙ = −ut(t, X(t )) ux(t, X(t ))sup
tt1
ut(·, t )∞
p H, (2.37)
with a constantH >0, depending ont1,gmax,gmin, andf0. The proof of Proposition 2.5(ii)
We may now prove (2.13), the lower bound onut(t, X(t )). The standard elliptic regularity estimates imply that for anyt1>0, there exists a constantM >0 so thatuxx(t,·)∞< M for allt > t1. Therefore, we have forx < X(t ), using (2.11):
u(t, x)u t, X(t )
+ux
t, X(t )
x−X(t )
−1 2M
x−X(t )2
θ0−p
x−X(t )
−1 2M
x−X(t )2
(2.38) for allt > t1. Forh=θ0+p2/2Mandx=X(t )−p/M, this gives us
u(t, x)θ0+ p2
2M=h. (2.39)
Sinceuis monotone in time, this implies that there ist2t1such that for alltt2,
Xhl(t )X(t )−p/M. (2.40)
