Travelling waves for a non-monotone bistable equation with delay: existence and oscillations

0.38 MB·

by  Matthieu Alfaro

Travelling waves for a non-monotone bistable equation with
delay: existence and oscillations
7
1 Matthieu Alfaro 1, Arnaud Ducrot 2 and Thomas Giletti 3.
0
2
Contents
n
a
J 1 Introduction 2

3
2 2 Assumptions and results 2

] 3 Construction of a travelling wave 4
P
A 3.1 A priori estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.2 Construction in the box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
.
h 3.3 Construction of a travelling wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
t
a
m 4 Convergence to 1 14
4.1 Under an additional bound on f . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
[
4.2 When the delay is small . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1
v 5 Oscillations around 1 16
4
5.1 Preliminary on related monotone problems . . . . . . . . . . . . . . . . . . . . . . 16
9
5.2 Proof of Theorem 2.6 on oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3
6 5.3 Convergence to a wavetrain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
0
.
1
Abstract
0
7 Weconsiderabistable(0<θ<1beingthethreeconstantsteadystates)delayedreaction
1 diffusion equation, which serves as a model in population dynamics. The problem does not
:
v admit any comparison principle. This prevents the use of classical technics and, as a conse-
i quence,itisfarfromobvioustounderstandthebehaviourofapossibletravellingwavein+∞.
X
Combining refined a priori estimates and a Leray Schauder topological degree argument, we
r construct a travelling wave connecting 0 in −∞ to “something” which is strictly above the
a unstable equilibrium θ in +∞. Furthemore, we present situations (additional bound on the
nonlinearityorsmalldelay)wherethewaveconvergesto1in+∞,whereasthewaveisshown
tooscillate around 1 in +∞ when, typically, thedelay is large.
Key Words: delayed reaction diffusion equation, non-monotone equation, bistable nonlinear-
ity,travelling wave, oscillations.
1IMAG,Universit´edeMontpellier,CC051,PlaceEug`eneBataillon,34095MontpellierCedex5,France. E-mail:
[email protected]
2IMBUMRCNRS5251, Universit´edeBordeaux, 3ter, PlacedelaVictoire,33000Bordeaux, France. E-mail:
[email protected]
3IECL, Universit´e de Lorraine, B.P. 70239, 54506 Vandoeuvre-l`es-Nancy Cedex, France. E-mail:
[email protected]
1
1 Introduction
We consider the following delayed reaction-diffusion equation
∂ u(t,x)=∂ u(t,x)+f(u(t τ,x)) u(t,x), t>0, x R, (1)
t xx
− − ∈
whereτ >0isa giventime delay. This equationtypicallydescribesthe spatio-temporalevolution
of a population density u = u(t,x) at time t and location x R. In this context f describes the
∈
birth rate function while the term u represents the (normalized) death rate. The time delay τ
−
allows to take into account a maturity period.
The existence and the properties of travelling wave solutions for problem (1) have received
a lot of attention in the case where the function f is increasing, so that problem (1) generates
a monotone semiflow. The literature can mostly be divided into the so-called monostable and
bistable cases, the distinction being made with respect to the dynamical properties of the ODE
problem
u˙ =F(u):=f(u) u.
−
The monostable case contains, as a special case, the so-called Ricker’s function f(u) = βue u
−
for some parameter β > 1, for which (1) is then referred to as the diffusive Nicholson’s blowflies
equation [27, 28]. On the other hand , a typical bistable situation is given by f(u)=βu2e u, for
−
some parameter β >e, see [22].
For the study of monostable monotone waves we refer to the works of So, Wu and Zou [31],
ThiemeandZhao[32],GourleyandSo[14],Li,RuanandWang[18],LiangandZhao[19],Ma[21],
and the references therein. Monotone waves for bistable problems of the form (1) have also been
investigated and we refer the reader to Schaaf [29], Smith and Zhao [30], Ma and Wu [22], Wang,
Li and Ruan [34, 35], Fang and Zhao [11]. Notice also an analysis of the spreading properties of
problem (1), in a bistable situation, by Lin and Ruan [20].
Note that the aforementioned works are concerned with monotone birth rate functions f or,
more generally, with monotone semiflows. Much less is known when this assumption is removed.
Let us underline the analysis of a non-monotone monostable situation by Trofimchuk et al. [33],
who prove the existence of travelling waves exhibiting oscillations on one side.
Inthisworkweshalldiscusstheexistenceoftravellingwavesolutionsforproblem(1)whenthe
birth rate function f, of the bistable type, is not assumed to be monotone (see below for precise
statements). Our set of assumptions prevents the existence of a comparisonprinciple, and makes
the analysis rather involved.
Roughlyspeakingweprovethat,ifthenonlinearfunctionF(u):=f(u) uisbistablebetween
−
0 and 1 and 1F(u)du > 0, then problem (1) admits a travelling wave solution with a positive
0
wave speed. To prove this result we make use of a Leray-Schauder topological degree argument
R
to construct a solution of a similar problem posed on a bounded domain. Then, using refined
estimates, we are able to pass to the limit as the length of the bounded domain tends to infinity.
We thus get a travelling wave in the sense of Definition 2.2, that is connecting 0 in to
−∞
“something” which is above the intermediate equilibrium in + , meaning that the wave is truly
∞
bistable. Then, in some regimes, we are able to precise their behaviours in + : convergence to
∞
1 under an additional bound on f or when the delay is small, but oscillations around 1 for large
delay (and an additional assumption on the shape of f).
2 Assumptions and results
Throughout the paper, we make the following assumption on the bistable nonlinearity f, whose
three fixed points are 0<θ <1.
Assumption 2.1 (Bistable nonlinearity). The function f : R [0, ) is of the class C1,γ on
→ ∞
[0, ) for some γ >0. There are 0<θ <1 such that
∞
f(u)=0 on ( ,0), f(u)<u on (0,θ), f(u)>u on (θ,1), f(u)<u on (1, ).
−∞ ∞
2
We also denote
M :=maxf 1,
[0,1] ≥
and assume that f >θ in (θ,M] (notice that this is automatically satisfied if M =1). Moreover
1
(f(u) u)du>0. (2)
−
Z0
Let us emphasize again that when f is monotone, the situation is well understood: there is a
unique (up to translation) travelling wave solution (c,U) R C2(R), which moreoversatisfies
∈ × b
U( )=0, U(+ )=1, and U′ >0.
−∞ ∞
The existence follows from the results of Fang and Zhao [11], see also [29], while uniqueness is
ensured by the results of Smith and Zhao [30].
In this work no monotony assumption on the nonlinearity is made. We do however assume
f 0 = f(0), to insure the positivity of travelling waves, and f > θ = f(θ) on (θ,M], to insure
≥
a truly bistable behaviour of travelling waves. Nonetheless our assumption allows the situation
M >1=f(1). In sucha case,oscillationsaround1 may occur whenthe delay is large,see below.
On the other hand, if either M = 1 or the delay is small, we will show that the travelling wave
does convergeto 1 at + . Notice that our hypothesis allows the degenerate situations f (0)=1,
′
∞
f (θ)=1, f (1)=1.
′ ′
From the mathematical point of view, assumption (2) insures that we can work with positive
speeds. Hence, when considering the travelling wave equation, the information is to be searched
“on the left”. If we cannot insure positivity of speeds, then the analysis becomes much more
difficult (if possible). Notice that, from the modelling point of view, (2) is hardly a restriction
since, having in mind a Allee effect, the unstable zero θ is usually small.
For simplicity we make the smoothness assumptionf C1,γ([0, )) but this could be relaxed
∈ ∞
to the conditions that f is Lipschitz and satisfies some kind of Ho¨lder regularity at 0 and 1, that
is the conditions used in [6] whose some of the techniques will be used later on.
Definition2.2(Bistabletravellingwave). A(bistable) travelling wavefor equation (1)is acouple
(c,U) R C2(R), with U >0 on R, and such that
∈ ×
U +cU =f(U( cτ)) U in R,
′′ ′
− ·− −
(U( )=0, liminfx + U(x)>θ.
−∞ → ∞
Notice that the boundary condition as x + is understood in a “weak” sense in Defini-
→ ∞
tion2.2,whichisquiteclassicalwhennonlocaleffectsthatmaydestabilizethe expectedboundary
condition (1 in the present case) are concerned; see [4], [2] in a KPP context or [3] in a bistable
situation.
Our first main result consists in constructing such a bistable travelling wave.
Theorem 2.3 (Constructionofabistabletravellingwave). Let Assumption 2.1 hold. Then, there
exists a bistable travelling wave such that c>0, U <M, and U C2(R), with the normalization
∈ b
U(0)=θ.
Furthermore, U is increasing on ( ,cτ] and satisfies U >θ on [cτ,+ ).
−∞ ∞
Inthesequelweenquireonthebehaviouroftravellingwavesasx + . Ournextresultpro-
→ ∞
videstwosufficientconditionsunderwhichthebistabletravellingwaveconstructedinTheorem2.3
does converge to 1 in + .
∞
Theorem 2.4 (Convergence to 1). Let (c,U) R C2(R) be a bistable travelling wave. Assume
∈ × b
either
M :=maxf =1 (additional bound on f), (3)
[0,1]
or
τkf′kL∞(0,kUk∞) <1 (small delay). (4)
Then U(+ )=1.
∞
3
As far as travelling waves of the nonlocal Fisher-KPP equation
∂ u=∂ u+u(1 φ u)
t xx
− ∗
are concerned the convergence to 1 is known to hold not only for focused competition kernels φ,
see [4], but also for large speeds, and this even if the state 1 is driven unstable by the kernel φ,
see [10] and [1]. Also, for cases of convergence to 1, we refer to [13] for a Fisher-KPP equation
with delay and to [3] for a nonlocal bistable equation.
Now, we wonder if there are travelling waves that connect 0 in to a wavetrain oscillating
−∞
around1 in+ . Suchconnectionszero-wavetrainareknownto existinsome nonlocalequations.
∞
Forinstance,wereferto[26]forasimplifiedmodelwhichallowstoperformexplicitcomputations.
See alsothe works[33], [8], [13], [17], [9] inKPPsituations. Inourbistable context, ourlastmain
result shows that such a connection zero-wavetrain can also exist: roughly speaking, when both
the time delay τ and the slope of the nonlinearity f at u = 1 are large enough, the travelling
wave provided by Theorem 2.3 does not converge to 1 in + , and oscillates around 1. In order
∞
to capture this behaviour, we need to strengthen Assumption 2.1 as follows.
Assumption 2.5 (Bistablenonlinearityallowingoscillations). In addition to Assumption 2.1, we
suppose that the following properties hold true.
(i) There exist two values α and β such that θ <α<β <1 and
f >0 on (0,β),
′
f <0 on (β,M :=f(β)),
 ′
f(α)=1 and f(M)>α.

f(M)
(ii) (min f(u),f(M) u)du>0.
{ }−
Z0
Our result on oscillating travelling waves then reads as follows.
Theorem 2.6 (Oscillations around 1 for large delay). Let Assumption 2.5 hold. If f (1) is
′
| |
large enough then there exists τ >0 such that, for all τ τ , the travelling wave solution (c,U)
0 0
≥
of (1) provided by Theorem 2.3 is such that the profile U does not converge to 1 as x + , and
→ ∞
oscillates around 1.
Noticethat,underanadditionalassumption,wecanshowthattheabovenonconvergenceto1
is actually a convergence to a wavetrain. This will be explored and precised in subsection 5.3.
The organization of the paper is as follows. Section 3 is devoted to the construction of a
travelling wave thanks to a Leray-Schauder toplogical degree argument, that is we prove Theo-
rem2.3. InSection4westudy twocaseswherethe travellingwaveconvergesto1 in+ ,proving
∞
Theorem 2.4. Last, in Section 5, we consider a case where oscillations occur, that is we prove
Theorem 2.6.
3 Construction of a travelling wave
This section is devoted to the proof of Theorem 2.3. The strategy is to first construct a solution
in a box by a Leray-Schauder topological degree argument and then to let the box tend to the
whole line, with enough estimates to guarantee a true bistable behaviour in the limit.
For a>0 and 0 σ 1, we consider the problem of finding a speed c=ca R and a profile
u=ua :[ a,a] R≤such≤that σ ∈
σ − →
u +cu =f(u¯( σcτ)) u in ( a,a)
′′ ′
P (a) − ·− − −
σ

u( a)=0, u(a)=1,
 −

4
to which we shall sometimes add the normalization
θ
max u(x)= , (5)
a x 0 2
− ≤ ≤
andwhereu¯ denotestheextensionofuequalto0on( , a)and1on(a, )(inthe sequel,for
−∞ − ∞
ease of notation, we always write u in place of u¯). This realizes a homotopy from a local problem
(σ =0)to our nonlocaldelayedproblem(σ =1)in the box ( a,a). We shallconstructa solution
−
to P (a) by using a Leray-Schauder topological degree argument. To make this scheme rigorous
1
we will need several a priori bounds which are proved in the following subsection.
3.1 A priori estimates
Here we prove several lemmas on any profile u and speed c solving P (a) together with the
σ
normalization (5).
Lemma 3.1 (A priori bounds for profiles). For all a>0 and 0 σ 1, any solution (ca,ua)=
≤ ≤ σ σ
(c,u) of P (a) satisfies
σ
0 <u<M on ( a,a).
−
Proof. If u reaches its minimum at some point x ( a,a) then the equation yields u(x )
0 0
∈ − ≥
f(u(x σcτ)) 0. In a similar fashion, if u reaches its maximum at some point x ( a,a),
0 1
− ≥ ∈ −
then u(x ) f(u(x σcτ)). On the one hand, if u(x σcτ) > 1, then f(u(x σcτ)) <
1 1 1 1
≤ − − −
u(x σcτ) u(x ), which is a contradiction. On the other hand, if u(x σcτ) 1 then we
1 1 1
− ≤ − ≤
conclude that u(x ) max f =M.
1 [0,1]
≤
Thisprovesthat0 u M on( a,a). NotethatM isalsothemaximumoff ontheinterval
≤ ≤ −
[0,M]. Thus
0 u +cu +u=f(u( σcτ)) M,
′′ ′
≤− ·− ≤
and the strong maximum principle yields 0<u<M on ( a,a).
−
In the next lemma, we prove some a priori monotonicity property. This will not only provide
information on the shape of the travelling wave, but will also be quite useful in order to bound
the speed.
Lemma 3.2 (A priori monotony for a while). Let a >0 and 0 σ 1 be given, and (ca,ua)=
≤ ≤ σ σ
(c,u)be a solution of P (a) with normalization (5). If c 0 then u is increasing on [ a,min(x +
σ a
≥ −
σcτ,a)], where
x :=min x R:u(x)=θ >0. (6)
a
{ ∈ }
Proof. Notice that the positivity of x is an immediate consequence of the normalization (5), so
a
that we only need to deal with the monotonicity of the solution.
Assume first that σ =0 or c=0, so that
u +cu =f(u) u in ( a,a).
′′ ′
− − −
Then the problem is local and one may proceed as in the seminal work of Berestycki and Niren-
berg[6]to showthatthe solutionis actuallyincreasingonthe wholeinterval[ a,a]. We omitthe
−
full details but, for the sake of clarity, briefly sketch the argument. First, arguing as in Lemma
3.1, we see that 0<u<1 on ( a,a). One can therefore use a sliding method and define
−
h :=min h [0,2a] : u(x+ζ) u(x), x [ a,a ζ], ζ [h,2a] [0,2a).
∗
{ ∈ ≥ ∀ ∈ − − ∀ ∈ }∈
Assume by contradiction that h > 0. Since u( a+h ) >u( a)= 0 and u(a)= 1> u(a h ),
∗ ∗ ∗
− − −
one can use the strong maximum principle to get that u( +h ) > u(), which contradicts the
∗
· ·
definition of h . Hence h = 0 and u is nondecreasing in x. The strict monotonicity is finally
∗ ∗
obtained by the strong maximum principle.
5
Now assume that 0 < σ 1 and c > 0. We first claim that u is increasing on the interval
≤
[ a, a+σcτ]. Indeed, if not, then there exists x ( a, a+σcτ) such that u(x ) = 0 and
0 ′ 0
− − ∈ − −
u (x ) 0. The travelling wave equation yields u(x ) f(u(x σcτ)) = 0, as x σcτ a
′′ 0 0 0 0
≤ ≤ − − ≤ −
implies u(x σcτ)=0. This contradicts u>0 and proves the claim. Hence, we can define
0
−
γ∗ :=sup γ >0: u is increasing on [ a, a+γ] [σcτ,2a].
{ − − }∈
Assume by contradiction that
γ <min(a+x +σcτ,2a).
∗ a
Thenu( a+γ )=0andu ( a+γ ) 0sothatthetravellingwaveequationyieldsu( a+γ )
′ ∗ ′′ ∗ ∗
− − ≤ − ≤
f(u( a+γ σcτ)). But a a+γ σcτ x and the definition (6) of x implies that
∗ ∗ a a
− − − ≤ − − ≤
0 u( a + γ σcτ) θ, hence f(u( a + γ σcτ)) u( a + γ σcτ). As a result
∗ ∗ ∗
≤ − − ≤ − − ≤ − −
u( a+γ ) u( a+γ σcτ). Since σcτ >0, this contradicts the definition of γ . The lemma
∗ ∗ ∗
− ≤ − −
is proved.
We are now in the position to prove a priori estimates on the speed c for which P (a) admits
σ
a solution with normalization (5).
Lemma 3.3 (A priori bounds for speeds). (i) There exist c >0 and a >0 such that, for all
max 0
a>a , all 0 σ 1, any solution (ca,ua)=(c,u) of P (a) with normalization (5) satisfies
0 ≤ ≤ σ σ σ
c=0 and c<c .
max
6
(ii) Furthermore and up to enlarging a , for all a > a , any solution (ca,ua) = (c,u) of the
0 0 0 0
local problem P (a) with normalization (5) satisfies
0
0<c<c .
max
Remark 3.4. Notice that the above lemma does not exclude the possibility that a solution with
c < 0 may exist for 0 < σ 1. Nevertheless, excluding c = 0 will be enough for our purpose.
≤
Indeed, we aim at constructing a travelling wave with positive speed, and in order to apply a
topological degree argument we only need to show that, along the homotopy 0 σ 1, no
≤ ≤
solutioncanescapethroughc=0. See subsection3.2formoredetails,and[2]or[15]for asimilar
trick.
Proof. Let us first exclude the case c = 0 along 0 σ 1. Assume by contradiction that there
≤ ≤
are sequences a , 0 σ 1, and (c = 0,ua) = (c = 0,u) solving P (a) with normalization
→ ∞ ≤ ≤ σ σ
(5). In other words, we are equipped with (c=0,u) solving
u =f(u) u in ( a,a)
′′
P(a) − − −

u( a)=0, u(a)=1,
 −
and, from Lemma 3.2, with the normalization u(0)= θ. Problem P(a) being local, we know (see
2
the proof of Lemma 3.2), that u is actually increasing on the whole interval [ a,a]. Thus, letting
−
a we end up with (c=0,U) solving
→∞
U =f(U) U in R
′′
− −
U( )=0, U(0)= θ, U(+ ) θ,1 , U 0.
 −∞ 2 ∞ ∈{ } ′ ≥
NotethatthelimitsU( )abovesimplyfollowfromthefactthatthesemustbezerosoff(u) u.
Next, by a standart ar±gu∞ment — multiply the above equation by U and integrate over R—−we
′
end up with a contradiction — since θ(f(u) u)du < 0 and 1(f(u) u)du > 0 in view of
0 − 0 −
Assumption 2.1.
R R
6
Let us now turn to the bound from above on c along 0 σ 1. In view of Assumption 2.1,
≤ ≤
we can bound f from above by a linear function. More precisely, there exists K > 0 such that
f(u) Ku for all u 0. We now select c >K. It is then straightforwardto see that
max
≤ ≥
u(x):=ex
satisfies
u′′+cmaxu′ >Ku( σcmaxτ) u.
− ·− −
Let us now show that c c , where c is such that P (a) with normalization (5) admits
max σ
≤
a solution. Proceed by contradiction and assume that c > c . Since u(+ ) = + , one can
max
∞ ∞
select a large shift X =Xa >>1 so that
u( a+X)>M. (7)
−
Going back to the parabolic problem, we define
v(t,x):=u(x+c t+X), (t,x) R2,
max
∈
which satisfies
∂ v(t,x)>∂ v(t,x)+Kv(t στ,x) v(t,x), for (t,x) R2. (8)
t xx
− − ∈
We also define
v(t,x):=u(x+ct), (t,x) R2,
∈
which satisfies
∂ v(t,x) = ∂ v(t,x)+f(v(t στ,x)) v(t,x)
t xx
− −
∂ v(t,x)+Kv(t στ,x) v(t,x), for x+ct <a. (9)
xx
≤ − − | |
Recall that v is well-defined on the whole domain as u is extended by constants outside of the
interval ( a,a).
−
We claim that
V(t,x):=v(t,x) v(t,x)<0, στ t 0, x R. (10)
− − ≤ ≤ ∈
Indeed if x+ct< a then v(t,x)=0 so the inequality is clear. On the other hand if x+ct a
− ≥−
thenthemonotonyofu(noticethat(c c)t 0when στ t 0)and(7)implyv(t,x)>M,
max
− ≥ − ≤ ≤
which proves (10).
Next, since c>c , we can define the first touching time, namely
max
t :=inf t>0 : x R, V(t,x)>0 0.
0
{ ∃ ∈ }≥
In particular V(t , ) 0 on R, and there is x R such that V(t ,x ) = 0. Clearly the
0 0 0 0
· ≤ ∈
touching point (t ,x ) is such that x +ct > a. If (t ,x ) is such that x +ct < a (i.e.
0 0 0 0 0 0 0 0
− | |
in the region where v can be differentiated) then ∂ V(t ,x ) 0 and ∂ V(t ,x ) 0. Then,
t 0 0 xx 0 0
≥ ≤
subtracting (8) from (9) and evaluating at point (t ,x ) leads to a contradiction. If (t ,x ) is
0 0 0 0
such that x +ct >a (i.e. in the region where v 1) then ∂ V(t ,x )= ∂ v(t ,x )<0 which
0 0 t 0 0 t 0 0
≡ −
contradicts the definition of t . It remains to exclude the case x +ct =a. In this case we have
0 0 0
1=v(t ,x )=v(t ,x )=u(x +c t +X) so that
0 0 0 0 0 max 0
θ θ
u(z) , z x +c +X +ln . (11)
0 max
≤ 4 ∀ ≤ 4
(cid:18) (cid:19)
But v(t ,x) v(t ,x) may be rewritten as u(z) u(z+(c c)t +X)=u(z+x +c t +
0 0 max 0 0 max 0
X a). For≤any a> ln θ , we deduce from (1≤1) that −
− − 4
(cid:0) (cid:1) θ
u(z) , z 0.
≤ 4 ∀ ≤
7
This contradicts the normalization (5), and thus we have c c . This concludes the proof of
max
≤
item (i) of the lemma.
Last, we turn to the bound from below in (ii), which is only concerned with the local case
σ = 0. We select δ > 0 small enough and g a bistable type function with δ, θ+δ and 1 δ its
− −
fixed points, such that g <f and
1 δ
−
(g(u) u)du>0.
−
Z−δ
It is well-knownthat there exists a bistable travelling wave(c >0,u ) for the nonlinearity g, i.e.
δ δ
solving
u +c u =g(u ) u ,
− ′δ′ δ ′δ δ − δ
with u ( ) = δ < u () < u (+ ) = 1 δ. Moreover, u is an increasing function of
δ δ δ δ
x R. R−ep∞roducin−gtheabo·veproof—∞thatisu−singashiftedu asasubsolutionoftheparabolic
δ
∈
problem— we find that 0 < c c for all a large enough, which concludes the proof of (ii) and
δ
≤
thus of the lemma.
Nextweprovesometechnicalestimatesonthebehaviourofuontheleftofx . Werecallthat,
a
as defined in (6), x is the leftmost (and, as it will turn out in Lemma 3.7, unique) point where u
a
takes the value θ.
Lemma3.5(Aprioricontrolontheleftofx andoftheslopeatx ). Foralla>0,all0 σ 1,
a a
≤ ≤
any solution (ca,ua)=(c,u) of P (a) with c 0 and normalization (5) satisfies
σ σ σ ≥
ec+√2c2+4xe√c2+4(a+x2a) ec−√2c2+4xe√c2+4x2a
ψ(x) := θ −
e2cxa(e√c2+4(a+xa) 1)
−
u(x) θec(x−xa) =:ϕ(x), (12)
≤ ≤
for all x [ a,x ]. This in turn implies the following: there is s >0 such that for all a>0, all
a 2
∈ −
0 σ 1, any solution (ca,ua)=(c,u) of P (a) with c 0 and normalization (5) satisfies
≤ ≤ σ σ σ ≥
s u(x ) θc 0. (13)
2 ′ a
≥ ≥ ≥
Proof. First, u satisfies u +cu +u 0 on ( a,x ) with the boundary conditions u( a)= 0,
′′ ′ a
− ≥ − −
u(x ) = θ. Since ψ satisfies ψ +cψ +ψ = 0 on ( a,x ) and ψ( a) = 0, ψ(x ) = θ, the
a ′′ ′ a a
− − −
estimate from below in (12) follows from the comparison principle.
From c 0 and the monotonicity Lemma 3.2, we know that u(x cστ) u(x) θ = u(x )
a
≥ − ≤ ≤
for all x [ a,x ], which in turn implies f(u(x cστ)) u(x) u(x cστ) u(x) 0. Hence u
a
∈ − − − ≤ − − ≤
satisfies u +cu 0 on ( a,x ) with the boundary conditions u( a) = 0, u(x ) = θ, so that
′′ ′ a a
− ≤ − −
the estimate from above in (12) follows from the comparison principle.
Last, (12) enforces ψ (x ) u(x ) ϕ(x ). Since
′ a ′ a ′ a
≥ ≥
ψ′(xa)=θ c + √c2+4e(a+xa)√c2+4+1 θcmax+ c2max+4supX≥2a0 eeXX+−11 =:s2,
2 2 e(a+xa)√c2+4 1!≤ p 2
−
and ϕ(x )=θc, estimate (13) is proved.
′ a
3.2 Construction in the box
Equipped with the above a priori estimates, we now use a Leray-Schauder topological degree
argument(for relatedargumentssee e.g. [5], [4]or [2]in a KPP context,[3] ina bistable context)
to construct a solution (c,u) to P (a) with normalization (5).
1
8
Proposition 3.6 (A solution in the box). There exist C > 0 and a > 0 such that, for all
0 0
a a , problem P (a) admits a solution (ca,ua) with normalization (5), which is such that
0 1
≥
ua C .
C2( a,a) 0
k k − ≤
Proof. For a given c R and a given nonnegative function v defined on ( a,a), consider the
∈ −
family 0 σ 1 of linear problems
≤ ≤
u +cu =f(v¯( σcτ)) v in ( a,a)
Pc(a) − ′′ ′ ·− − − (14)
σ
(u( a)=0, u(a)=1,
−
where, as before, v¯ denotes the extension of v by 0 on ( , a] and by 1 on [a, ). Denote by
the mapping of the Banach space X := R v C−1,α∞([ −a,a]) : v( a) = 0 ,∞equipped with
σ
K ×{ ∈ − − }
the norm (c,v) :=max(c, v ), onto itself defined by
X C1,α
k k | | k k
θ
:(c,v) max v(x)+c,uc := the solution of Pc(a) .
Kσ 7→ 2 − a x 0 σ σ
(cid:18) − ≤ ≤ (cid:19)
Constructing a solution (c,u) of P (a) with normalization (5) is equivalent to showing that the
1
kernel of Id is nontrivial. The operator is compact and depends continuously on the
1 σ
−K K
parameter 0 σ 1. Thus the Leray-Schauder topological argument can be applied.
≤ ≤
From Lemma 3.1 and the interior elliptic estimates there is C > 0 such that, for any 0
M
≤
σ 1, any solution (c,u) of P (a) with 0 c c has to satisfy u <C . Let us define
σ max C1,α M
≤ ≤ ≤ k k
the open set
S := (c,v): 0<c<c , v >0 on ( a,a], v ( a)>0, v <C X.
max ′ C1,α M
{ − − k k }⊂
It follows from the above and Lemma 3.3 (i) that, for any a a , any 0 σ 1, any fixed
0
≥ ≤ ≤
point of satisfies u > 0 on ( a,a], u( a) > 0 by the Hopf lemma, c = 0 and c < c , and
σ ′ max
K − − 6
thereforethe operatorId cannotvanishontheboundary∂S. Bythehomotopyinvarianceof
σ
−K
thedegreewethushavedeg(Id ,S,0)=deg(Id ,S,0). Now,asfaras isconcerned,we
1 0 0
−K −K K
knowfromLemma 3.3(ii)thatc 0 is impossible fora fixedpoint. We canthereforeenlargethe
≤
speedintervalinS without changingthe degree,thatis deg(Id ,S,0)=deg(Id ,S(a),0),
0 0
−K −K
where
S(a):= (c,v): c (a)<c<c , v >0, v <C X,
min max C1,α M
{ − k k }⊂
and c (a) 0 is to be selected below.
min
≥
Roughly speaking, the role of (14) was to get rid of the nonlocal term. Now, in order to get
rid of the nonlinearity, we consider the family 0 σ 1 of local problems
≤ ≤
u +cu =σ(f(v) v) in ( a,a)
Pc(a) − ′′ ′ − − (15)
σ
(u( a)=0, u(a)=1,
−
e
and let be the associated solution operator, namely
σ
K
θ
e :(c,v) max v(x)+c,uc := the solution of Pc(a) .
Kσ 7→ 2 − a x 0 σ σ
(cid:18) − ≤ ≤ (cid:19)
It is clear that,efor any 0 σ 1, a fixed pointe(c,u) of satisfies 0e< u < M in ( a,a)
σ
≤ ≤ K −
(reproduce Lemma 3.1). Furthermore, one can also proceed as in the proof of Lemma 3.3 to
check that the inequality c<cmax still holds. Indeed, the samee argument applies noting that the
exponential function ex+cmaxt is a supersolution of the parabolic equation
∂ u=∂ u+σ(f(u) u),
t xx
−
associated with Pc(a), provided as before that c > K where f(u) Ku for all u 0. Also,
σ max ≤ ≥
anysolutionofPc(a)satisfies u +cu σu usothat, bythe comparisonprinciple, u w
σ − ′′ ′ ≥− ≥− ≥
where w solves e
e
w +cw +w =0 on ( a,a), w( a)=0, w(a)=1.
′′ ′
− − −
9
After explicitly computing w we see that, for any a a , w(0) 1 as c so that there is
0
≥ → → −∞
c (a) 0 such that, in order not to miss the normalization,one needs c (a)<c. Then the
min min
≥ −
operator Id cannot vanish on the boundary ∂S(a). Hence, by the homotopy invariance of
σ
−K
the degree and the fact that = , we have deg(Id ,S(a),0)=deg(Id ,S(a),0).
0 1 0 0
K K −K −K
Beforecompeutingdeg(Id ,S(a),0)byusingtwoadditionalhomotopies,letusobservethat
0
−K
the solution of Pc(a) is given by e e
0
e
ecx e ca x+a
e uc(x)= − − if c=0, uc(x)= if c=0. (16)
0 eca e ca 6 0 2a
− −
In particular, uc(0)eis decreasing with respect to c aned, there is a unique c such that uc0(0)= θ.
0 0 0 2
As a result, the operator has a unique fixed point (c ,u ) which moreover belongs to S(a).
K0 0 c0
Let us noweperform two additional homotopies. First, consider, for 0 σ 1, e
≤ ≤
e e
θ
:(c,v) (1 σ) max v(x) σuc(0)+c,uc := the solution of Pc(a) .
Gσ 7→ 2 − − a x 0 − 0 0 0
(cid:18) − ≤ ≤ (cid:19)
Again,Id doesnotvanishonthe boundary∂Se(a). Bytehe homotopyinvarianceeofthe degree
σ
−G
and the fact that = , we have deg(Id ,S(a),0)=deg(Id ,S(a),0). Then, consider,
0 0 0 1
K G −K −G
for 0 σ 1,
≤ ≤
e :(c,v) θ uce(0)+c,σuc +(1 σ)u ,
Hσ 7→ 2 − 0 0 − c0
(cid:18) (cid:19)
where (c ,u )was definedin the previousparagraph. If (c,v)=(c,v) forsome (c,v) ∂S(a),
0 c0 e eHσ e ∈
then it follows from the previous paragraphthat (c,uc) (c ,u ) and therefore (c,v) (c ,u )
0 ≡ 0 c0 ≡ 0 c0
sothat(c,ve)∈∂S(a)solvesthelocalproblemP0c(a)andsatisfiesnormalization(5),whichcannot
be. Therefore Id σ does not vanish on the boundeary ∂S(a)e. Since 1 = 1 we have deg(Ied
−H H G −
,S(a),0)=deg(Id ,S(a),0), where e
1 0
G −H
θ
Id :(c,v) uc(0) ,v u .
−H0 7→ 0 − 2 − c0
(cid:18) (cid:19)
As seen above, uc(0) is strictly decreasing in c seo the degree ofethe first component of the above
0
operator is 1. Clearly the degree of the second one is 1. Hence deg(Id ,S(a),0) = 1 so
0
− −H −
that deg(Id e1,S,0)= 1 and there is a solution (ca,ua) S of P1(a) with normalization (5).
−K − ∈
Together with standard estimates, this concludes the proof of the proposition.
In the sequel, in order to complete the construction of a bistable travelling wave as stated in
Theorem2.3, we actually need to strengthen the above result. More precisely,we show below the
existence of a solution that crosses the value θ only once.
Proposition 3.7 (A solution such that θ is attained only at x = x ). Let C > 0 and a > 0
a 0 0
be as in Proposition 3.6. Then, for all a a , problem P (a) admits a solution (ca,ua) with
0 1
≥
normalization (5), which is such that ua C and
C2( a,a) 0
k k − ≤
ua(x)=θ if and only if x=x ,
a
where x >0 was defined in (6)
a
Proof. Let a a be given. We again work along the homotopy 0 σ 1. From the proof of
0
≥ ≤ ≤
Proposition 3.6, we know that there is a solution (c ,u ) of
σ σ
u +c u =f(u ( σc τ)) u in ( a,a)
− ′σ′ σ ′σ σ ·− σ − σ − (17)
( uσ(−a)=0, uσ(0)= 2θ, uσ(a)=1.
For the localcase σ =0, we also knowfrom the proofofLemma 3.2 that the (unique) solutionu
0
is increasing and therefore satisfies u (x)=θ if and only if x=x .
0 a
10