ηε(ϕ) = q(ϕ)
zε(ϕ)zˆ(ϕ)ηε(ϕ), ϕ∈(0, 1). Thus,
˙ ηε(ϕ)
ηε(ϕ) = q(ϕ)
zε(ϕ)zˆ(ϕ), ϕ∈ (0, 1) and hence, for any 0<τ< ϕ,
log(ηε(ϕ))−log(ηε(τ)) =
Z ϕ
τ
q(s)
zε(s)zˆ(s)ds≤
Z 1
τ
q(s)
zβ(s)zˆ(s)ds. (8.8) Notice, from (6.4)2it follows that we can apply (8.6) with ˙q(0) =0 (because of (6.3)) and obtain zβ(s)zˆ(s) = (h(0)−c)2s2+o(s2), ass →0+. Hence, from (6.4)1,
sup
τ>0
Z 1
τ
q(s)
zβ(s)zˆ(s)ds=:C< +∞.
From (8.8), by taking the limit asτ→0+we deduceηε(ϕ)≤εeC,ϕ∈[0, 1), and then lim
ε→0+zε(ϕ) =zˆ(ϕ), ϕ∈[0, 1). (8.9) We now apply Lemma3.3 to deduce thatzε converges (uniformly on[0, 1]) to a solution ¯zof (3.1)1 in (0, 1)such that ¯z < 0 in (0, 1)and ¯z(0) = 0. Since zε < zβ and zβ lies below every solution of (3.1), by the very definition of zβ, we conclude that ¯z coincides with zβ, that is limε→0+zε(ϕ) =zβ(ϕ), ϕ∈ [0, 1]. From this formula and (8.9) we clearly have zβ =z.ˆ
9 Strongly non-unique strict semi-wavefronts
We now apply the previous results to study semi-wavefronts of Equation (1.1) whenDandg satisfy (D1), (g0) and (1.2); in particular, we prove Theorem2.2 and Corollary9.4. Indeed, all the results obtained in Sections4–8apply when we set
q:=Dg, (9.1)
since such qfulfills (q). Throughout this section, byc∗ we always intend the threshold given by Proposition4.2forqas in (9.1), for which it holds (2.2), as observed in Remark5.5.
Lemma 9.1. Assume(D1), (g0)and(1.2). Consider c ≥ c∗ and let z be the solution of (3.12)when
Proof. First, observe that Proposition7.1applies to the current case.
If either ˙D(1) < 0 or ˙D(1) = 0 and c < h(1), then ˙z(1) > 0, by (7.2), because ˙q(1) =
which, together with (7.2), implies both (9.2)1and the first half of (9.2)2.
If ˙D(1) = 0 and c≥ h(1), we need a refined argument based on strict upper- and lower-solutions of (3.11)1. We split the proof in two subcases.
(i)Assume first ˙D(1) =0 andc>h(1). Fix ε>0 and defineω =ω(ϕ)by (3.11)1 applied toω, we obtain
h(ϕ)−c− D(ϕ)g(ϕ)
Hence, there existsnsuch thatω(ϕn)>z(ϕn)forn ≥n. Without loss of generality we assume that n=1. We claim that
ω(ϕ)> z(ϕ), for ϕ∈ (ϕ1, 1). (9.6) We reason by contradiction, see Figure 9.1. Suppose that there exists ˜ϕ ∈ (ϕ1, 1) such that ω(ϕ˜) ≤ z(ϕ˜). There existsn ∈ N for which ˜ϕ ∈ (ϕn,ϕn+1). Since ω(ϕn) > z(ϕn) and ω(ϕn+1) > z(ϕn+1), the existence of such a ˜ϕimplies that the function (ω−z)in (ϕn,ϕn+1) admits a non-positive minimum at ˜ϕ2∈(ϕn,ϕn+1), that is ˙ω(ϕ˜2) =z˙(ϕ˜2)andω(ϕ˜2)≤z(ϕ˜2). Thus, from (3.11)1and (9.4) we have that
h(ϕ˜2)−c− (Dg)(ϕ˜2)
z(ϕ˜2) =z˙(ϕ˜2) =ω˙(ϕ˜2)< h(ϕ˜2)−c− (Dg)(ϕ˜2) ω(ϕ˜2) ,
which in turn implies 1/z(ϕ˜2)> 1/ω(ϕ˜2)because of (Dg)(ϕ˜2) > 0. Hence, z(ϕ˜2) < ω(ϕ˜2) which contradicts the existence of ˜ϕ2. Then (9.6) is proved. At last, we have
D(ϕ)
z(ϕ) > D(ϕ)
ω(ϕ) = −c−h(1)
g(1) −ε, ϕ∈ (ϕ1, 1). (9.7)
ϕ z
z ω
ϕn ϕn+1 1
ϕ1 ϕ˜ ϕ˜2
Figure 9.1: A detail of the plots of functions ωandzin case(i).
Analogously, forε>0 small enough to satisfy c>h(1) +εg(1), we defineη=η(ϕ)by η(ϕ):=− g(1)
c−h(1)−εg(1)D(ϕ), ϕ∈(0, 1).
By arguing as above when we considered ω in (9.3), we deduce that η is a (strict) upper-solution of (3.11)1in [σ2, 1)for someσ2 ∈(0, 1). Proceeding as we did to obtain (9.7), we now get η(ϕ)< z(ϕ)forϕ∈(ϕ1, 1), for someϕ1> σ2. Thus,
D(ϕ)
z(ϕ) < D(ϕ)
η(ϕ) = −c−h(1)
g(1) +ε, ϕ∈ (ϕ1, 1). (9.8) Finally, putting together (9.7) and (9.8), sinceε>0 is arbitrary, we deduce
lim
ϕ→1−
D(ϕ)
z(ϕ) = h(1)−c
g(1) . (9.9)
Thus, we proved (9.2)2 withc>h(1).
(ii)Now, we consider the case ˙D(1) =0 andc= h(1). Fix ε>0. Set ω(ϕ):=−D(ϕ)
ε , ϕ∈(0, 1), (9.10)
which coincides with (9.3) in the current case. By proceeding exactly as in the case (ii), we obtain (9.3) for ω defined as in (9.10), namely 0 > ω(ϕ) > z(ϕ), for ϕ ∈ (ϕ1, 1), for some ϕ1 ∈(0, 1). This implies, as in (9.7),
0> D(ϕ)
z(ϕ) > D(ϕ)
ω(ϕ) =−ε, ϕ∈ (ϕ1, 1). (9.11) Then (9.11) impliesD(ϕ)/z(ϕ)→0−as ϕ→1−, which is (9.2)2in the casec=h(1).
Remark 9.2. Let c≥ c∗ and z be any solution of (3.11). We infer that z ∈ C1(0, 1]. In fact, if z(1) = b < 0, in the proof of case (i) of Proposition 7.1 we already checked that this is true, since limϕ→1−z˙(ϕ) =z˙(1). Ifz(1) =0, from (9.2) it follows that the right-hand side of (3.11)1 still has a finite limit, asϕ→1−. As observed, this means thatz∈ C1(0, 1].
We now prove Theorem2.2.
Proof of Theorem2.2. We first prove that there exists a semi-wavefront to 0 of (1.1) if c ≥ c∗. Forq= Dg, consider one of the solutionsz= z(ϕ)of (3.11), provided by Propositions4.2and 5.1. Consider the Cauchy problem
(
ϕ0 = z(ϕ)
D(ϕ),
ϕ(0) = 12. (9.12)
The right-hand side of (9.12)1 is of class C1 in a neighborhood of 12, and then there exists a unique solution ϕ in its maximal-existence interval (a,ξ0), for −∞ ≤ a < ξ0 ≤ ∞. Since z(ϕ)/D(ϕ) < 0 for ϕ ∈ (0, 1), we deduce that ϕ is decreasing and then limξ→a+ ϕ(ξ) = 1, limξ→ξ−
0 ϕ(ξ) = 0. By (9.12)1, the profile ϕsatisfies (1.3) in (a,ξ0). We show that, if ξ0 ∈ R, we can extend ϕand obtain a solution of (1.3), in the sense of Definition 2.1, defined in the half-line(a,+∞).
Assumeξ0 ∈ R and set ϕ(ξ) = 0, for anyξ ≥ ξ0. The new function (which without any ambiguity we still call ϕ) is clearly of classC0(a,+∞)∩C2((a,+∞)\ {ξ0})and is a classical solution of (1.3) in (a,+∞)\ {ξ0}. Moreover, observe that, as a consequence of both the fact thatzsatisfies (3.11)3, and (9.12)1, we have
lim
ξ→ξ0−
D(ϕ(ξ))ϕ0(ξ) =0. (9.13)
This implies thatD(ϕ)ϕ0 ∈ L1loc(a,+∞).
To show that ϕis a solution of (1.3) according to Definition2.1, it remains to prove (2.1).
For this purpose, considerψ∈C0∞(a,+∞), and leta< ξ1<ξ2 <∞be such thatψ(ξ) =0, for anyξ ≥ ξ2or ξ ≤ξ1. Our goal is then to prove the following:
Z ξ2
ξ1
D(ϕ)ϕ0− f(ϕ) +cϕ
ψ0−g(ϕ)ψdξ =0. (9.14) Identity (9.14) is obvious if ξ2 < ξ0, since ϕ solves (1.3) in (a,ξ0). Assume ξ2 ≥ ξ0. In the interval(ξ0,ξ2)we have ϕ=0, and sinceg(0) = f(0) =0 we deduce
Z ξ2
ξ0
D(ϕ)ϕ0− f(ϕ) +cϕ
ψ0−g(ϕ)ψdξ =0. (9.15)
In the interval(ξ1,ξ0)we have, by (9.13), Z ξ0
ξ1
D(ϕ)ϕ0− f(ϕ) +cϕ
ψ0−g(ϕ)ψdξ
= lim
ε→0+
Z ξ0−ε
ξ1
D(ϕ)ϕ0− f(ϕ) +cϕ
ψ0−g(ϕ)ψdξ
= lim
ε→0+ D(ϕ)ϕ0− f(ϕ) +cϕ ψ
(ξ0−ε) =0.
(9.16)
Thus, identities (9.15) and (9.16) imply (9.14).
At last, we claim thata∈R, i.e., thatϕisstrict. For this, it is sufficient to prove lim
ξ→a+ϕ0(ξ)<0. (9.17)
We stress that the case limξ→a+ ϕ0(ξ)→ −∞, for short ϕ0(a+) =−∞, is included in (9.17). To prove (9.17), we notice that, from (9.12),
lim
ξ→a+ϕ0(ξ) = lim
ϕ→1−
z(ϕ) D(ϕ).
Thus, (9.17) easily follows from either a direct check, in the casez(1) < 0, or the application of Lemma9.1, in the casez(1) =0. This concludes the first part of the proof.
Conversely, we prove that if there exists a semi-wavefront ϕto 0 defined in(a,+∞), then c≥c∗. Let ¯bbe defined by
b¯ :=sup{ξ >a: ϕ(ξ)>0} ∈(a,+∞]. (9.18) We have 0< ϕ<1 in a, ¯b
and so ϕis a classical solution of (1.3) in a, ¯b
. We claim that lim
ξ→b¯−D(ϕ(ξ))ϕ0(ξ) =0. (9.19) Suppose ¯b ∈R. Takeξ1 > a andξ2 > b. By choosing, in Definition¯ 2.1, ψ∈ C∞0 (a,+∞)with support in(ξ1,ξ2)such thatψ(b¯)6=0, (2.1) reads as (passing to the limit in the integral as in (9.16))
0=
Z ξ2
ξ1
D(ϕ)ϕ0+cϕ− f(ϕ)ψ0−g(ϕ)ψdξ
=
Z b¯ ξ1
D(ϕ)ϕ0+cϕ− f(ϕ)ψ0−g(ϕ)ψdξ = D(ϕ)ϕ0
(b¯−)ψ(b¯).
Then we got (9.19) in this case. If ¯b= +∞, by integrating (1.3) in[η,ξ]⊂(a,¯ +∞), we have D(ϕ(ξ))ϕ0(ξ)
=D(ϕ(η))ϕ0(η)−c(ϕ(ξ)−ϕ(η)) + (f(ϕ(ξ))− f(ϕ(η)))−
Z ξ
η
g(ϕ(σ)) dσ. (9.20) Since the function
ξ 7→
Z ξ
η
g(ϕ(σ))dσ
is increasing (because g > 0 in(0, 1)), then limξ→∞D(ϕ(ξ))ϕ0(ξ) = ` for some` ∈ [−∞, 0].
If` <0, then, ϕ0(ξ)tends either to some negative value or to −∞asξ →+∞. In both cases,
this contradicts the boundedness of ϕ, and so (9.19) is proved.
We show now (2.3). Suppose by contradiction that (2.3) does not occur, there exists ξ0 ∈ (a, ¯b), with 0 < ϕ(ξ0) < 1, such that ϕ0(ξ0) = 0. Then (1.3) implies ϕ00(ξ0) =
−g(ϕ(ξ0))/D(ϕ(ξ0)) < 0 and hence ξ0 is a local maximum point of ϕ. It is plain to see that, in turn, this implies that there existsa < ξ1 < ξ0 which is a local minimum point of ϕ.
From what we said aboutξ0, we necessarily have ϕ(ξ1) = ϕ0(ξ1) =0.
Takeξ ∈ (ξ1, ¯b). Integrating (1.3) in [ξ1,ξ] gives (9.20) with ξ1 replacingη. By passing to the limit forξ →b¯−, from (9.19) we obtain the contradiction 0<0. This proves (2.3).
From (2.3), we can define the functionz=z(ϕ), forϕ∈(0, 1), by
z(ϕ):= D(ϕ)ϕ0(ξ(ϕ)), (9.21) whereξ =ξ(ϕ)is the inverse function of ϕ. Again by (2.3), it follows also thatz <0 in(0, 1). From (9.19), we clearly havez(0+) =0; furthermore, a direct computation shows thatzsolves equation (1.6)1. Thus,zsolves problem (1.6), which is (3.11) withq=Dg. At last, Proposition 6.5impliesc≥c∗.
Remark 9.3. The proof of Theorem 2.2 provides a formula for ϕ0(a+). If z(1) < 0, then ϕ0(a+) =−∞. Ifz(1) =0, Lemma9.1leads to
lim
ξ→a+ϕ0(ξ) =
2g(1) h(1)−c−
√
(h(1)−c)2−4 ˙D(1)g(1) if ˙D(1)<0,
g(1)
h(1)−c if ˙D(1) =0 andc>h(1),
−∞ if ˙D(1) =0 andc≤h(1).
(9.22)
We now investigate the qualitative properties of the profiles when they reach the equi-librium 0. The classification is complete, apart from some cases corresponding to c∗ = h(0), when further assumptions are needed, see Remark 10.1. Below the existence of the limξ→a+ D(ϕ(ξ))ϕ0(ξ)is a consequence of the definition (9.21) and Lemma3.1.
Corollary 9.4. Under the assumptions of Theorem2.2, let c ≥c∗ andϕbe a strict semi-wavefront to 0of (1.1), connecting1to0, defined in its maximal-existence interval(a,+∞). Then, for c>c∗, there existsβˆ(c)∈[β(c), 0]such that the following results hold.
(i) D(0)>0implies that ϕis classical and strictly decreasing.
(ii) D(0) =0, c>c∗ and
lim
ξ→a+D(ϕ(ξ))ϕ0(ξ)>βˆ(c), (9.23) imply thatϕis classical; moreover,ϕreaches0at someξ0> a if
c>h(0) +lim sup
ϕ→0+
g(ϕ)
ϕ . (9.24)
(iii) D(0) =0, c∗ >h(0)and
either c= c∗ or lim
ξ→a+D(ϕ(ξ))ϕ0(ξ)≤ βˆ(c) (9.25)
imply that ϕis sharp at0(reached at someξ0>a) with
Notice that β is related to the existence of the semi-wavefronts while ˆβ deals with their smoothness (see Figure 9.2). The two thresholds coincide under the assumptions of Proposi-tion8.4.
Figure 9.2: Examples of profiles occurring in Corollary 9.4. From the left to the right, they depict, respectively, what stated in Parts(i),(ii)and(iii).
Proof of Corollary9.4. Defineξ0:=sup{ξ >a :ϕ(ξ)>0} ∈(a,+∞]. We assume without loss we deduce that (9.27) does not hold. Then,ξ0= +∞and so ϕis strictly decreasing. This, and the fact that ϕis of classC2 whenϕ ∈ (0, 1), imply ϕ ∈ C2(a,+∞), hence ϕis classical. Part
Therefore ηis a strict upper-solution of (1.6)1 in (0,δ], for some δ > 0. Also, since ˙z(0) = 0, Thus, by means of (9.24), we get
lim inf
Therefore, ηis a strict upper-solution of (1.6)1 in (0,δ], for some δ > 0. Furthermore, taking the same sequenceϕn →0+as above such that ˙z(ϕn)→0, asn→∞, then we have which concludes the proof of(ii), by means of (9.27).
We show(iii). By (8.5), (8.6),c∗ >h(0)and (9.25) we obtain ˙z(0) =h(0)−c<0. Then, D(σ)
−z(σ) = D˙(0) +o(1)
c−h(0) +o(1) as σ →0+,
and consequently (9.27) is verified. Thus,ξ0 ∈R. Furthermore, from (9.21), lim
which implies thatϕis sharp at 0 and that (9.26) holds.