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Ann. I. H. Poincaré – AN 32 (2015) 279–305
www.elsevier.com/locate/anihpc
Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model
✩Yihong Du
a,∗, Xing Liang
baSchool of Science and Technology, University of New England, Armidale, NSW 2351, Australia
bWu Wen-Tsun Key Laboratory of Mathematics and School of Mathematical Science, University of Science and Technology of China, Hefei, Anhui 230026, PR China
Received 5 January 2013; received in revised form 1 October 2013; accepted 25 November 2013 Available online 16 December 2013
Abstract
We consider a radially symmetric free boundary problem with logistic nonlinear term. The spatial environment is assumed to be asymptotically periodic at infinity in the radial direction. For such a free boundary problem, it is known from[7]that a spreading-vanishing dichotomy holds. However, when spreading occurs, only upper and lower bounds are obtained in[7]for the asymptotic spreading speed. In this paper, we investigate one-dimensional pulsating semi-waves in spatially periodic media. We prove existence, uniqueness of such pulsating semi-waves, and show that the asymptotic spreading speed of the free boundary problem coincides with the speed of the corresponding pulsating semi-wave.
©2013 Elsevier Masson SAS. All rights reserved.
MSC:35K20; 35R35; 35J60; 92B05
Keywords:Diffusive logistic equation; Free boundary; Periodic environment; Pulsating semi-wave; Spreading speed
1. Introduction
We are interested in the evolution of the positive solution u(t, r) (r= |x|, x ∈RN,N 1), governed by the following diffusive logistic equation with a free boundary:
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
ut−du=u
α(r)−β(r)u
, t >0, 0< r < h(t), ur(t,0)=0, u
t, h(t )
=0, t >0, h(t)= −μur
t, h(t )
, t >0,
h(0)=h0, u(0, r)=u0(r), 0rh0,
(1.1)
✩ We thank the referee for a very careful reading of the paper and useful suggestions. This work was supported by the Australian Research Council and NSFC 11171319. Part of this research was carried out during a visit of X. Liang to the Univ. of New England in Feb.–March 2011 and a visit of Y. Du to the Univ. of Sci. and Tech. of China in Sept.–Oct. 2012.
* Corresponding author.
E-mail addresses:[email protected](Y. Du),[email protected](X. Liang).
0294-1449/$ – see front matter ©2013 Elsevier Masson SAS. All rights reserved.
http://dx.doi.org/10.1016/j.anihpc.2013.11.004
whereu=urr+N−r1ur;r=h(t)is the free boundary to be determined;h0,μandd are given positive constants;
u0∈C2([0, h0])is positive in[0, h0)andu0(0)=u0(h0)=0; the functionsα(r)andβ(r)are positive and satisfy the following conditions:
⎧⎪
⎨
⎪⎩
(i) α, β∈Cν0 [0,∞)
for someν0∈(0,1),
(ii) there exist positiveL-periodic functionsaandbinCν0(R)such that limr→+∞α(r)−a(r)+β(r)−b(r)=0.
(1.2)
Problem(1.1)may be viewed as describing the spreading of a new or invasive species with population density u(t,|x|) over an N-dimensional habitat, which is radially symmetric, heterogeneous and asymptotically space- periodic near infinity in the radial direction. The initial functionu0(|x|)stands for the population in its early stage of introduction. Its spreading front is represented by the free boundary|x| =h(t), which is a sphere∂Bh(t )with radius h(t ) growing at a speed proportional to the gradient of the population density at the front:h(t)= −μur(t, h(t)).
(A deduction of this condition based on ecological considerations can be found in [6].) The coefficient functions α(|x|)andβ(|x|)represent the intrinsic growth rate of the species and its intra-specific competition respectively, and d is the random diffusion rate.
Problem(1.1)was studied recently in[7], and whenα,β are positive constants and the space dimension is one, this problem was considered earlier in[10]. In both cases, it was shown that a unique solution pair(u, h)exists, with u(t, r) >0 andh(t) >0 fort >0 and 0r < h(t ), and a spreading-vanishing dichotomy holds, namely, a spatial barrierr=R∗exists, such that either
• Spreading: the free boundary breaks the barrier at some finite time (i.e.,h(t0)R∗for somet00), and then the free boundary goes to infinity ast→ ∞(i.e., limt→∞h(t)= ∞), and the population spreads to the entire space and stabilizes at its positive steady-state, or
• Vanishing: the free boundary never breaks the barrier (h(t) < R∗ for allt >0), and the population vanishes (limt→∞u(t, r)=0).
Moreover, when spreading occurs, it follows from Theorem 3.6 of[7]that lim inf
t→∞
h(t)
t k∗, lim sup
t→∞
h(t) t k∗
for some positive constantsk∗andk∗determined by the pairs(α∞, β∞)and(α∞, β∞), respectively, where α∞:=lim inf
r→∞ α(r), β∞:=lim sup
r→∞ β(r), α∞:=lim sup
r→∞ α(r), β∞:=lim inf
r→∞ β(r).
It follows that if both limr→∞α(r)and limr→∞β(r)exist, then limt→∞h(t )
t =kexists, and one may regardkas the asymptotic spreading speed.
The main purpose of this paper is to show that under condition (1.2), limt→∞h(t )
t also exists, and we will use pulsating semi-waves (to be defined below) induced by(1.1)to determine this limit. These semi-waves are solutions of the one-dimensional problem
ut−duxx=u
a(x)−b(x)u
, t∈R, −∞< x < h(t), u
t, h(t )
=0, h(t)= −μux t, h(t )
, t∈R. (1.3)
Asymptotic spreading in spatially periodic environment based on Cauchy problem models has received extensive study recently. The spreading speed in such models is usually determined by the so called pulsating fronts, whose existence, uniqueness and other properties have been investigated by many authors; see[1–5,12,14,16,22]and the references therein for more details. In particular, a pulsating front of the reaction diffusion equation
ut−duxx=u
a(x)−b(x)u
, (t, x)∈R2, (1.4)
is a solution to this equation of the formu(t, x)=Ψ (x−ct, x), wherec(the speed) is a positive constant and the functionΨ (ξ, x) (the profile) is L-periodic in x; moreover, limt→−∞u(t, x)=0, limt→+∞u(t, x) is the unique
positive steady state of(1.4). It can be shown thatu(t, x)is strictly increasing int. Let us also observe thatu(t+
L
c, x)=u(t, x−L).
By a well-known result of Berestycki, Hamel and Roques[5], there is a minimal speedc∗>0 such that for each cc∗,(1.4)has a pulsating front with average speedc, and no pulsating front exists with average speedc < c∗. Moreover, it is well known (see[2,13,22]) that this minimal average speedc∗is the spreading speed for the Cauchy problem
vt−dvxx=v
a(x)−b(x)v
, x∈R, t >0; v(0, x)=v0(x), x∈R, where the initial functionv0(x)is nonnegative with nonempty compact support.
In contrast to(1.4), we will show that there is only one average speedC=C(μ)for which(1.3)has a pulsating semi-wave, and such a semi-wave is unique up to translations int (seeTheorem 1.2). Moreover, as mentioned above, this average speed is the spreading speed for the free boundary model(1.1)when spreading happens (Theorem 1.3).
Furthermore, we will show that asμ→ ∞,C(μ)increases toc∗(Theorem 4.2).
We now describe our results more precisely. Our definition below for pulsating semi-waves to(1.3)is motivated by the notion of pulsating fronts, and ideas in[18,21].
Definition 1.1.We call(u(t, x), h(t ))apulsating semi-waveof(1.3)if it solves(1.3)and (i) u(t, x)=U (h(t), h(t)−x) >0 fort∈R,x < h(t ),
(ii) there existsT >0 such thath(t)is a positiveT-periodic function andh(t+T )−h(t)=L, (iii) U (τ, ξ )∈C1,2(R× [0,+∞))isL-periodic inτ.
It will become clear below that C :=L/T is the (average) speed of the semi-wave. Let us also observe that u(t+T , x)=u(t, x−L).
Theorem 1.2. Problem (1.3)always has a pulsating semi-wave (u,˜ h). The pulsating semi-wave is unique up to˜ translations int. Furthermore,limt→±∞h(t)/t˜ =L/T,u˜t(t, x) >0, andu(t, x)˜ →φ(x)ast→ +∞uniformly in any interval of the form(−∞, M],M∈R, whereφis the unique positive solution of
−dφxx=φ
a(x)−b(x)φ
, x∈R1.
Note that the existence and uniqueness ofφis a consequence of Theorem 2.3 of[11]; more general results can be found in[4,20]. UsingTheorem 1.2, we can deduce the following result on the asymptotic spreading speed determined by(1.1).
Theorem 1.3.Suppose that(1.2)holds,(u, h)is the unique solution of(1.1)andlimt→∞h(t)= ∞;then
tlim→∞
h(t )
t =L/T , whereL/T is the average speed of the semi-wave inTheorem1.2.
We remark thatTheorem 1.3only gives the asymptotic speed of the free boundary|x| =h(t). However, from its proof, one sees that for anyσ∈(0, a/b), wherea=minra(r)anda=maxrb(r), the set{x: u(x, t ) > σ}expands toRN with asymptotic speedL/T. Therefore this agrees with the spreading speed in the usual sense.
The existence part ofTheorem 1.2will be proved in Section2, while Section3is devoted to the proof of the rest of Theorem 1.2as well as some further basic properties of the pulsating semi-wave. The proof ofTheorem 1.3is given in Section4, where we also show that the spreading speedL/T increases strictly inμ, and asμ→ ∞, the spreading speed converges toc∗, which is the minimal speed of the pulsating fronts to(1.4).
This paper is a sequel to[9], where the time-periodic case of the free boundary problem was considered. It turns out that very different techniques have to be used to handle the space-periodic case, though some ideas in the time- periodic case can be borrowed. Similarly to the approach in[9], we prove the existence of a pulsating semi-wave via a fixed point argument. However, the techniques here are completely new. The proof for uniqueness of the pulsating semi-wave is based on ideas introduced in[9], with considerable changes in the arguments. In[23], independently, (1.3)and its corresponding initial value problem are investigated by a completely different method, which is based on the approach developed in[12].
Our analysis in this paper carries over easily when the logistic nonlinearityu[α(r)−β(r)u]is replaced by a more general Fisher–KPP type functionf (r, u), which is smooth and satisfies
(i) f (r,0)≡0,f (r, q(r))≡0, whereq(r)is bounded from above and below by positive constants, (ii) limr→∞[f (r, u)−g(r, u)] =0 locally uniformly inu∈ [0,∞), withgperiodic inr,
(iii) f (r, u)/uis strictly decreasing inu, for everyr0.
With care, one could further extend the results to the case thatduis replaced by div(d(|x|)∇u), andμis replaced byμ(r), with suitable conditions ond(r)andμ(r).
2. Existence of pulsating semi-waves
We use CLν(R)to denote the set of allL-periodicCν functions and suppose thatp, a, b∈CLν(R)for someν∈ (0,1), with bothaandbpositive. In order to prove the existence of a pulsating semi-wave, we consider the following problem
p(τ )(Uτ+Uξ)−dUξ ξ=U
a(τ−ξ )−b(τ−ξ )U
, (τ, ξ )∈R×(0,∞),
U (τ,0)=0, τ ∈R. (2.1)
The relationship between(2.1)and(1.3)is given in the following result.
Proposition 2.1.If(2.1)has a positive solutionU (τ, ξ )which isL-periodic inτ, and satisfiesμUξ(τ,0)≡p(τ ) >0 forτ∈R, then(u(t, x), h(t))given by
u(t, x)=U
h(t), h(t)−x , t=
h(t )
0
1
p(τ )dτ (2.2)
is a pulsating semi-wave to(1.3).
Proof. Clearly 1=h(t) 1
p(h(t )), i.e., h(t)=p h(t)
. We calculate to obtain
ut−duxx=h(t)(Uτ+Uξ)−dUξ ξ
=p h(t)
Uτ
h(t), h(t)−x +Uξ
h(t), h(t)−x
−dUξ ξ
=U
a(x)−b(x)U
=u
a(x)−b(x)u , and
h(t)=p h(t)
=μUξ
h(t),0
= −μux
t, h(t ) . It is evident thatu(t, h(t ))=U (h(t),0)=0.
It remains to show thath(t)isT-periodic for someT >0 andh(T )−h(0)=L. We prove thath(t)isT-periodic with
T :=
L 0
ds p(s). Indeed, from
t=
h(t )
0
ds
p(s) and t+T =
h(t+T ) 0
ds p(s)
we obtain T =
h(t+T ) h(t )
ds p(s).
Sincep(τ )isL-periodic and positive, we have
T = L 0
ds p(s) =
h(t )+L h(t )
ds p(s). Hence
h(t+T ) h(t )
ds p(s)=
h(t )+L h(t )
ds p(s), which implies that
h(t+T )=h(t )+L, h(t+T )=h(t).
Thus(u(t, x), h(t))is indeed a pulsating semi-wave of(1.3). 2
Let us note that the pulsating semi-wave given by(2.2)satisfiesh(0)=0. It is easily seen that for any fixedt0∈R, (u(t+t0, x), h(t+t0))is also a pulsating semi-wave to(1.3). It will be shown that the pulsating semi-wave is unique subject to this kind of time shifts.
To prove the existence of a function pair(p(τ ), U (τ, ξ ))such that(2.1)holds andp(τ )=μUξ(τ,0), we break the argument into two major steps. In step one, we show that for any given positivep∈CνL(R),(2.1)has a unique maximal nonnegative solutionUpwhich isL-periodic inτ. This defines a mappingT :p→μUξp(·,0). In the second step, we show thatT has a fixed pointp∗, and thus obtain the required solution pair(p∗, Up∗).
In order to apply suitable fixed point theorems to the operatorT, it is convenient to consider nonnegativep, but h(t )is not well defined for such p. To avoid this difficulty, we use a perturbation approach. For small >0, we replace the original problem(2.1)by
p(τ )(Uτ+Uξ)−dUξ ξ=U
a(τ−ξ )−b(τ−ξ )U
, (τ, ξ )∈R×(0,+∞),
U (τ,0)=0, τ∈R, (2.3)
where
p(τ )=max
p(τ ), .
We will show that(2.3)has a unique maximal nonnegative solutionUp,(τ, ξ )that isL-periodic inτ, and the operator T:p→μUξp,(·,0)has a fixed pointp. Moreover, we will show that there existsδ >0 such thatpδfor all small. Hence(p, Up,)solves the original unperturbed problem when∈(0, δ], andUp,is a positive solution.
2.1. Existence ofUp,
In this subsection we show that for anyp∈CLν(R)and >0,(2.3)has a maximal nonnegative solutionUp,. We also determine exactly when this solution is positive. To this end, we need the following eigenvalue problem
LU=λa(τ−ξ )U, (τ, ξ )∈R2,
whereLU=p(τ )(Uτ+Uξ)−dUξ ξ. (2.4)
Proposition 2.2.The following conclusions hold:
(1) For any nonnegative p∈CLν(R), and >0, α∈R, there exists a unique λ(α, p, )∈R with corresponding positive functionφα,p,=φα,p,(τ, ξ )∈C1,2(R×R), which isL-periodic both inτ andξ, such that(λ, U )= (λ(α, p, ), eαξφα,p,)satisfies(2.4).
(2) Let
λ1(p, )=sup
λ∈R: ∃φ∈C1,2(R×R), φ(τ, ξ )is positive andL-periodic inτ, andLφλa(τ−ξ )φ
.
Then there is a uniqueα0∈Rsuch thatλ1(p, )=λ(α0, p, ). Moreover,α0>0.
(3) Supposeλ= ˜λ(l, p, )is the principal eigenvalue of the eigenvalue problem LU=λa(τ−ξ )U, (τ, ξ )∈R×(0, l),
U (τ, ξ )=U (τ+L, ξ ), U (τ,0)=U (τ, l)=0. (2.5)
Thenλ(l, p, )˜ decreases toλ1(p, )asl→ +∞.
(4) λ(l, p, )˜ is continuous in(l, p, )andλ(α, p, )is continuous in(α, p, ), wherel, >0,α∈Randp∈CLν(R).
(5) λ(α, p, )is concave inα.
Proof. All the conclusions here follow from the main results of[19], except that we need to prove thatα0>0 in conclusion (2). By conclusions (4) and (5),λ(α, p, )is continuous and concave inα. Moreover, it is obvious that λ(0, p, )=0 (with correspondingφ0,p,a positive constant). Hence, we only need to prove that there is someα>0 such that λ(α, p, )=0. To show this, due to the uniqueness of the principal eigenvalue proved in[19], it suffices to find a positive functionφ0(τ, ξ )which isL-periodic inτ, and a positive numberα, such thatU0:=eαξφ0(τ, ξ ) satisfiesLU0=0. We now look for such a functionφ0of the special formφ0(τ, ξ )=ψ (τ ). ThenLU0=0 reduces
to
p(τ )ψ(τ )+
p(τ )α−d α2
ψ (τ )
eαξ=0.
Hence ψ=α
d
p(τ )α−1
ψ.
Sinceψ (τ )isL-periodic, we have α
L 0
d
p(τ )dτ=L, which impliesα>0. Clearly
ψ (τ )=exp
α dα
p(τ )−1
dτ >0.
This completes our proof. 2
We also need the following auxiliary logistic problem p(τ )(Uτ+Uξ)−dUξ ξ=U
a(τ−ξ )−b(τ−ξ )U
, (τ, ξ )∈R2. (2.6)
The following result is contained in[20].
Proposition 2.3. Problem(2.6)has a positive solution Up, =Up,(τ, ξ )which is L-periodic both inτ and ξ if and only if λ(0, p, ) <1. Moreover, for any nonnegative L-periodic continuous initial function ψ=ψ (ξ )≡0, the solution of (2.6)withU (0, ξ )=ψ (ξ ), denoted byU (τ, ξ, ψ ), satisfieslimτ→+∞|U (τ, ξ, ψ )−Up,(τ, ξ )| =0 uniformly forξ∈R.
Remark 2.4.Ifp(τ )forτ∈R, then by the transformation inProposition 2.1,u(t, x)=Up,(h(t), h(t)−x)is a positive entire solution of(1.4), which isL-periodic inxandL
0 1/p(τ ) dτ-periodic int. Hence it must coincide with φ(x), the unique positive periodic solution of−duxx=u[a(x)−b(x)u],x∈R.
Now, we are ready to show the main result of this subsection.
Proposition 2.5.The following conclusions hold:
(1) For any fixedp∈CLν(R)and any positive constant, problem(2.3)admits a maximal nonnegativeL-periodic in τ solutionUp,=Up,(τ, ξ )in the sense that any other nonnegativeL-periodic inτ solution is bounded from above byUp,. Furthermore,Up,≡0if and only ifλ1(p, )1.
(2) For anyp∈CνL(R)and >0withλ1(p, ) <1,Up, is the unique positiveL-periodic inτ solution of (2.3).
Moreover,limξ→+∞|Up,(τ, ξ )−Up,(τ, ξ )| =0uniformly forτ ∈R.
Proof. First, we make use of some simple facts whose easy proof can be found in Step 1 of the proof of Proposition 2.1 in[9]. Suppose thatul is the unique positive solution of
−duξ ξ=u(a−bu) in(0, ),
u(0)=0, u()= ∞, (2.7)
where
a=max
r a(r), b=min
r b(r),
andu∞ is the limit of ul as l→ +∞. Thenu∞=u∞(ξ )maxa/minb, and for every constant M1, Mu∞ is a supersolution of(2.3). By a sweeping argument, anyL-periodic inτ solution of(2.3)is bounded from above by supu∞.
Let us now prove conclusion (1). Whenλ1(p, )1, byProposition 2.2, there is someα0>0 such thatλ1(p, )= λ(α0, p, ). It is easily seen that the corresponding functioneα0ξφα0,p,and its product with any positive constant are L-periodic inτ supersolutions of(2.3). Suppose that(2.3)has a positiveL-periodic inτ solutionU. Then by the boundedness ofU, we can suppose thatMeα0ξφα0,p,(τ, ξ )U (τ, ξ )onR× [0,+∞). We may assume thatM >0 is the minimal constant such that the above inequality holds. Sinceα0>0, the equality must hold at some(τ0, ξ0)in the above inequality. But the strong maximum principle shows that it is impossible. Hence the maximal nonnegative L-periodic solutionUp, is identically 0 whenλ1(p, )1.
In the remaining caseλ1(p, ) <1, byProposition 2.2, there is some largelsuch thatλ(l, p, ) <˜ 1. Suppose that
˜
φl,p,>0 is the corresponding eigenfunction of(2.5)normalized by maxφ˜l,p,=1. Then for any sufficiently small constantγ,
ψ (τ, ξ )=
γφ˜l,p,(τ, ξ−l), τ∈R, ξ∈ [l,2l], 0, τ∈R, ξ∈R\ [l,2l]
is a subsolution of(2.3). Therefore, we can find a maximal positiveL-periodic inτ solutionUp,in the order interval [ψ, u∞]. Since any positive solution of(2.3)is bounded from above byu∞,Up, is the maximal positive solution.
Finally we prove conclusion (2). Letψ (τ, ξ )be defined as above, withl=kLfor some large positive integerk, so that it is anL-periodic inτ subsolution of(2.3)for smallγ. For each nonnegative integeriwe define
ψi(τ, ξ )=ψ (τ, ξ−ikL).
It is easily seen thatψi is a subsolution of(2.3). LetU=U (τ, ξ )be a positiveL-periodic inτ solution of(2.3). For eachi0, there existsσi∈(0,1)such thatU (τ, ξ ) > σiψi(τ, ξ )forτ ∈Randξ 0. For anyσ ∈ [σi,1],σ ψi is a subsolution of(2.3). We claim thatU (τ, ξ ) > σ ψi(τ, ξ )for allτ ∈R,ξ 0 andσ∈ [σi,1]. Otherwise, there exists σ∗∈(σi,1]and(τ∗, ξ∗)∈(0, L] ×((i+1)kL, (i+2)kL)such that
U (τ, ξ )σ∗ψi(τ, ξ ) forτ ∈R, ξ0; U (τ∗, ξ∗)=σ∗ψi(τ∗, ξ∗).
We may then apply the strong maximum principle to conclude thatU≡σ∗ψi, a contradiction. This proves the claim, which implies in particular,
U (τ, ξ ) > ψi(τ, ξ ) forτ∈R, ξ0. (2.8)
Let{ξi}be an arbitrary sequence increasing to+∞asi→ +∞, and defineUi(τ, ξ )=U (τ, ξ+ξi). By applying standardLptheory (see[17]) to the equation satisfied byUiand Sobolev embedding (see Lemma 3.3 in[15]), one sees
that by passing to a subsequence,Ui → ˇU asi→ ∞locally uniformly inR2, andUˇ isL-periodic inτ. Moreover, if we writeξi=niL+ ˜ξi withξ˜i ∈ [0, L), and assume that ξ˜i → ˜ξ0, thenU (τ, ξˇ − ˜ξ0)solves(2.6). Furthermore, due to(2.8), we always have sup(τ,ξ )∈R2U (τ, ξˇ − ˜ξ0)γ. Therefore, U (τ, ξˇ − ˜ξ0)is a positiveL-periodic in τ solution of(2.6). HenceU (τ, ξˇ − ˜ξ0)≡Up,(τ, ξ ). Clearly the above discussion implies that limξ→+∞|U (τ, ξ )− Up,(τ, ξ )| =0 uniformly forτ ∈R.
It remains to show the uniqueness ofU. We follow a standard argument. If there is another positiveL-periodic in τ solutionU˜ ≡U, then from the conclusion proved above, we have
ξ→+∞lim U (τ, ξ )
U (τ, ξ )˜ =1 uniformly inτ∈R. (2.9)
Moreover, by the Hopf boundary lemma, Uξ(τ,0)and U˜ξ(τ,0)are bounded away from 0 and ∞ for all τ ∈R. Therefore there is someM∈(0,1)such thatU (τ, ξ )˜ MU (τ, ξ )onR× [0,+∞), and we may assume thatMtakes the maximal value such that this inequality holds (we may interchangeU andU˜ to guaranteeM <1). Due to(2.9), eitherU (τ˜ 0, ξ0)=MU (τ0, ξ0)at someτ0∈R, ξ0∈(0,+∞)or ∂∂ξU˜(τ0,0)=M∂U∂ξ(τ0,0), and the strong maximum principle then impliesU˜ ≡MU. ButM <1 implies thatMU is not a solution of(2.3). This contradiction completes our proof. 2
We may now define the mappingTonCLν(R)byT(p)=μUξp,(·,0).
2.2. T has a fixed point
We are going to use Schauder’s fixed point theorem to conclude thatThas a fixed point.
Lemma 2.6.T is completely continuous onCLν(R).
Proof. First, we show that for any positiveδ∈(0,1), the norm ofUξp,(·,0)inCLδ(R)can be controlled bypC0
L(R)
when all other parameters are fixed.
Set
s=f (τ ):=
τ 0
β(t) dt, V (s, ξ ):=Up,
f−1(s), ξ
whereβ(t)=p1(t). ThenV (s, ξ )is periodic inswith periodL
0 β(t) dt, and is a positive solution of Vs+p
f−1(s)
Vξ−dVξ ξ=V a
f−1(s)−ξ
−b
f−1(s)−ξ V
, s∈R, ξ >0,
V (s,0)=0, s∈R. (2.10)
Since 0V u∞maxa/minb, we can apply theLpestimates (see, for example, Theorem 7.15 of[17]) to(2.10) to conclude thatV (s0+ ·,·)W1,2
q ([0,1]×[0,l])Cfor alls0∈R,q >1,l >0 and some constantCdepending only on pC0
L(R),landq. By Sobolev embedding (see, e.g. Lemma 3.3 of[15]), we obtain, for everyδ∈(0,1), V (s0+ ·,·)
C1+δ2 ,1+δ
([0,1]×[0,l])Cδ,l ∀s0∈R. Therefore, we have
|Uξp,(t1,0)−Uξp,(t2,0)|
|t1−t2|δ =|Vξ(f (t1),0)−Vξ(f (t2),0)|
|f (t1)−f (t2)|δ ·|f (t1)−f (t2)|δ
|t1−t2|δ Cβ(·)δ
CL0(R)C−δ.
This implies that, for anyM >0,{Uξp,(·,0): pCLν(R)M}is bounded inCLδ(R)for someδ∈(ν,1), and hence it is pre-compact inCLν(R). This proves thatTmaps any bounded set ofCLν(R)into a pre-compact set in this space.
Moreover, ifpn→p0inCLν(R), then by compactness, by passing to a subsequence,Upn,(t, x)→ ˜Up0,(t, x)in C
1+ν 2 ,1+ν
loc (R× [0,+∞)), whereU˜p, is some nonnegativeL-periodic inτ solution of(2.3)withp=p0. It follows thatUξpn,(·,0)→ ˜Uξp0,(·,0)inCLν(R). What we need to show is thatU˜p0,is justUp0,. Ifλ1(p0, )1, then 0 is the unique nonnegativeL-periodic inτ solution of(2.3), and our conclusion holds. Ifλ1(p0, ) <1, we can find a sufficiently small constantγ independent ofpnsuch that
ψn(τ, ξ )=
γφ˜l,pn,(τ, ξ ), τ∈R, ξ∈ [0, l],
0, τ∈R, ξ∈(l,+∞)
is a subsolution of (2.3) with p =pn, where ˜φl,pn,C0(R×[0,l]) =1. It follows that Upn, ψn and hence Upn,C0(R×[0,+∞))γ >0, which infers ˜Up0,C0(R×[0,+∞)) >0. Thus U˜p0,≡Up0,. This shows that T is continuous. 2
In order to use Schauder’s fixed point theorem to prove the existence of a fixed point ofT, we look for an invariant set ofT which is bounded, closed and convex. In the proof ofProposition 2.5, we have shown thatUp,(τ, ξ ) u∞(ξ )onR× [0,+∞)for anyp∈CLν(R). We note thatUp,(τ,0)=u∞(0)=0.
Lemma 2.7.There existsM=M>0such thatE:= {p∈CLν(R): pCνL(R)M, 0p(τ )μ(u∞)(0)∀τ ∈R}
is an invariant set ofT.
Proof. First, for anyp∈CLν(R), 0Uξp,(τ,0)(u∞)(0). Moreover, forp(·)∈ ˜E:= {p∈CLν(R): 0p(τ ) μ(u∞)(0)∀τ ∈R},p∞μ(u∞)(0), and from (2.3)we see, by standard Lp theory for parabolic equations (see [17]) that {Up,} has a bound in Wq1,2([0, L] × [0,1]) (∀q >1) that is independent ofp∈ ˜E. By Sobolev embedding we know that{Up,}has a bound inC1+2ν,1+ν([0, L] × [0,1]) (ν∈(0,1))that is independent ofp∈ ˜E.
Therefore{Uξp,(·,0): p∈ ˜E}is bounded inCLν(R), sayUξp,(·,0)CνL(R)M for allp∈ ˜E. DefineE:= {p∈ E:˜ pCLν(R)M}. Then clearlyEis invariant underT. 2
Thus by Schauder’s fixed point theorem we obtain
Proposition 2.8.For any >0,Thas a fixed pointp¯inE. 2.3. p¯(τ )for all small >0
Proposition 2.9. There is some0>0 such that p¯0(τ ) > 0 on R and hence for p= ¯p0, (2.1)has a positive L-periodic inτ solutionUp¯0,0 such thatp¯0(τ )=μUξp¯0,0(τ,0).
Proof. We only need to show that lim inf→0minτ∈Rp¯(τ ) >0. We use two steps to prove this conclusion.
Step 1.We show that lim inf→0 ¯pC0
L(R)>0.
Recall that
a=mina, a=maxa, b=minb, b=maxb.
First, we can find someδ0small andl0>0 large such that for eachδ∈(0, δ0]andll0, the problem
−dU+δU=U[a−bU] in(0, l), U (0)=U(l)=0
has a unique positive solutionUl, andUl satisfiesUl(ξ ) >0 in [0, l). (The existence follows from a simple upper and lower solution argument, and the uniqueness is a consequence of the concavity of the nonlinearity. The fact that Ul>0 in[0, l)follows fromUl(a−bUl) >0 in[0, l)andUl(l)=0.)
DefineU (ξ )=Ul(ξ )in[0, l]andU (ξ )=Ul(2l−ξ )forξ∈(l,2l], andδ(ξ )=δsgn(l−ξ ). Then it is easily seen thatUis a weak solution of
−dU+δ(ξ )U=U (a−bU ) in(0,2l), U (0)=U (2l)=0. (2.11)
Ifp(τ ) < δ inR, then clearly p(τ )U(ξ )δ(ξ )U(ξ )in[0,2l]for allτ. It follows easily from the comparison principle thatUp,(τ, ξ )U (ξ )for allτ∈Randξ∈ [0,2l]. HenceUξp,(τ,0)U(0)for allτ.
If lim inf→0 ¯pC0
L(R)=0, then we can find some0<min{δ, U(0)}such thatp¯0(τ ) <min{δ, μU(0)}for allτ. This andUp¯0,0(τ, ξ )U (ξ )onR× [0,+∞)imply thatμUξp¯0,0(τ,0)μU(0) >p¯0(τ ), contradicting
¯
p0(τ )=μUξp¯0,0(τ,0).
Step 2.We show that lim inf→0minτ∈Rp¯(τ ) >0. Suppose by way of contradiction that this limit is 0. We are going to derive a contradiction. Since 2θ:=lim inf→0 ¯pCLν(R)>0, for every small >0, there is someτ∈ [0, L] such thatp¯(τ)=θ. Set
s=f(τ ):=
τ τ
β(t ) dt withβ(t )= 1 max{ ¯p(t ), }.
It is easy to check thatf−1(0)=τ,p¯(f−1(0))=θ, and if the functiong(τ )isL-periodic theng(f−1(s))is periodic with periodS:=L
0 β(t ) dt. Denote
V(s, ξ ):=Up¯,
f−1(s), ξ
, q(s)= ¯p f−1(s)
, and
η1(s, ξ )=a
f−1(s)−ξ
, η2(s, ξ )=b
f−1(s)−ξ . ThenV(s, ξ )is a positive solution of
Us+max
q(s),
Uξ−dUξ ξ=U
η1(s, ξ )−η2(s, ξ )U
, (s, ξ )∈R×(0,∞),
U (s,0)=0, U (s, ξ )=U (s+S, ξ ), (s, ξ )∈R×(0,∞). (2.12) Moreover,
q(s)=μVξ(s,0) ∀s∈R.
Letn→0 be such that minτ∈Rp¯n(τ )→0. Then from 0qnμ
u∞
(0), aη1na, bη2nb we see that, by passing to a subsequence,
qn→q weakly inL2(K1), η1n→η1 and η2n→η2 weakly inL2(K1×K2), for every compact setK1⊂Rand every compact setK2⊂ [0,∞). Clearly
0qμ u∞
(0), aη1a, bη2b.
Moreover, applyingLp estimates to the equation ofVn we find that{Vn}is bounded inWp1,2(K1×K2) (∀p >1) with compact setsK1andK2as described above. Therefore by Sobolev embedding we may assume, by passing to a subsequence, thatVn→V inC
1+ν 2 ,1+ν
loc (R× [0,+∞))with 0< ν <1. It follows thatq(s)=μVξ(s,0)converges toq(s)inClocν (R)along=n. Lettingn→ ∞in the equation forVnwe find thatV is a nonnegative weak solution
of
Vs+q(s)Vξ−dVξ ξ=V
η1(s, ξ )−η2(s, ξ )V
, (s, ξ )∈R×(0,∞),
V (s,0)=0, s∈R. (2.13)
We also have
q(s)=μVξ(s,0), 0V (s, ξ )u∞(ξ )a/b∀s∈R, ∀ξ0.
SinceVξ(0,0)= ¯p(f−1(0))=θ, we haveVξ(0,0)=θ >0. Therefore by the strong maximum principleV must be a positive solution. We consider the following two cases:
Case 1. {Sn} is bounded. Without loss of the generality, suppose that limn→∞Sn= ˜L <+∞. In this case, q(s), η1(s, ξ ), η2(s, ξ )andV (s, ξ )are allL-periodic in˜ s. SinceV is a positive solution of(2.13), the Hopf lemma implies that Vξ(s,0)σ0>0 for all s∈ [0,L˜]. Hence Vξn(s,0)σ0/2 for all s∈R and all largen, and thus
¯
pn(τ )=μVξn(fn(τ ),0)μσ0/2 for allτ and all largen, contradicting the choice ofn.
Case 2.{Sn}is unbounded. Without loss of the generality, suppose that limn→∞Sn= +∞. Since
S
0
q(s) ds=
S
0
¯ p
f−1(s)
ds
S
0
1
β(f−1(s))ds≡L, we deduce that
∞ 0
q(s) dsL.
ApplyingLpestimates to(2.13)we easily see thatV (s0+ ·,·)
C1+2ν,1+ν([0,1]×[0,1])is uniformly bounded with respect tos0∈R. It follows that 0q(s)=μVξ(s,0)is uniformly continuous ins. Hence from∞
0 q(s) ds <+∞we can conclude thatq(s)→0 ass→ +∞.
Recall thatU is the unique positive solution of(2.11). Lets0>0 be chosen such thatq(s) < δforss0. Then chooseσ >0 such thatσ U (ξ )V (s0, ξ )forξ∈ [0,2l]. Since
q(s)Uξ(ξ )δ(ξ )Uξ(ξ ) ∀ξ∈ [0,2l], we find thatU˜ =σ Usatisfies
U˜s+q(s)U˜ξ−dU˜ξ ξU˜
η1(s, ξ )−η2(s, ξ )U˜
∀(s, ξ )∈R×(0,∞).
ClearlyU (0)˜ =0=V (s,0),U (2l)˜ =0< V (s,2l) andU (ξ )˜ V (s0, ξ )for allξ ∈(0,2l). Hence we can apply the comparison principle to conclude thatV (s, ξ )U (ξ )˜ for allss0andξ ∈ [0,2l]. It follows that
Vξ(s,0)U˜ξ(0) >0 ∀ss0. On the other hand,
Vξ(s,0)=μ−1q(s)→0 ass→ ∞. This contradiction completes our proof. 2
3. Uniqueness of the pulsating semi-wave and other basic properties
If(u(t, x), h(t ))is a pulsating semi-wave to(1.3), then(U (τ, ξ ), p(τ ))given by U (τ, ξ )=u
h−1(τ ), τ−ξ
, p(τ )= −μux
h−1(τ ), τ solves(2.1)andp(τ )=μUξ(τ,0)∈CLν(R)is positive. So
h(t)=p h(t )
min
τ∈Rp(τ ) >0, and lim
t→±∞h(t)= ±∞. Hence, due to theL-periodicity ofp(τ ), we have
tlim→∞
h(t ) t =L
L 0
ds p(s)
−1
.
Recalling thath(t)isT-periodic withT =L
0 ds
p(s), we obtain
tlim→∞h(t )/t=L/T .
