# Continuous Dependence in Front Propagation for Convective Reaction-Diffusion Models with Aggregative Movements

Authors: Luisa Malaguti; Cristina Marcelli; Serena Matucci
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

The paper deals with a degenerate reaction-diffusion equation, including aggregative movements and convective terms. The model also incorporates a real parameter causing the change from a purely diffusive to a diffusive-aggregative and to a purely aggregative regime. Existence and qualitative properties of traveling wave solutions are investigated, and estimates of their threshold speeds are furnished. Further, the continuous dependence of the threshold wave speed and of the wave profiles on a real parameter is studied, both when the process maintains its diffusion-aggregation nature and when it switches from it to another regime.


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## Body

## 1. Introduction

This paper deals with the reaction-diffusion equation:(1.1)vt+h(v)vx=(D(v)vx)x+f(v),t≥0,x∈R,
which provides an interesting model in several frameworks such as the population dispersal, ecology, nerve pulses, chemical processes, epidemiology, cancer growth, chemotaxis processes etc. The great variety of contexts where it meets justifies its interest by the scientific community. We refer to the books [1–3] for an updated illustration of the main properties and applications of (1.1).In many cases the presence of a convective flux, expressed by the functionh, is a fundamental ingredient of these models. Indeed, it accounts for external forces acting into the process such as, for instance, control strategies. We assume that f is a monostable, that is, of Fisher-type, reaction term; that is, it satisfies  (1.2)f(u)>0foru∈(0,1),f(0)=f(1)=0.
According to (1.2), v≡0 and v≡1 are stationary solutions of (1.1). Since v frequently represents a density, of main interest is the investigation of solutions taking values in the interval [0,1] and connecting these two stationary states. The family of traveling wave solutions (t.w.s.) is an important class of functions which share this property. It was showed in fact that, at least in some sense, t.w.s. are able to capture the main features of all the solutions taking values in [0,1] (see, e.g., [4, 5]). We recall that a function v=v(t,x) is said to be a t.w.s. of (1.1) with wave speed c if there exist a maximal interval  (a,b), with -∞≤a<b≤+∞, and a function  u∈C(cl(a,b))∩C2(a,b), satisfying  (1.3)lim⁡ξ→a+D(u(ξ))u′(ξ)=lim⁡ξ→b-D(u(ξ))u′(ξ)=0,
such that v(t,x)=u(x-ct) is a solution of (1.1) for all t∈(a,b). The symbol cl(a,b) denotes the closure in ℝ of (a,b), and the quantity ξ=x-ct is usually called the wave coordinate.Since we restrict our analysis to those solutions of (1.1) with values in [0,1], we assume f,h,D defined in such an interval and, throughout the paper, we take h,f∈C[0,1] and D∈C1[0,1].A population in a hostile environment usually increases its chances to survive if it is able to produce aggregative movements. Therefore, when (1.1) is a model for population dynamics, it has to take into account also this important aspect. As a prototype of equation (1.1) in this framework, we can consider (1.4)vt=-[vk0(1-vw)vx]x+μvxx+f(v),
where μ, k0, and w denote positive constants while the function f accounts for the net rate of growth of the individuals and it satisfies (1.2). In this case D(v)=μ-vk0(1-v/w), and (1.4) is a generalized version of a model proposed by Turchin [6] in order to describe the aggregative movements of Aphis varians. Under suitable conditions on the parameters, it is straightforward to show the existence of a value v0∈(0,1) such that D(v)>0 when v∈[0,v0) while D(v)<0 in (v0,1]. This interesting situation is motivated by the remark that it is highly unlikely to find conspecific in the vicinity at low population densities. It seems then more reasonable to expect that the tendency of a population to aggregate, modeled by negative values of D, appears after a certain threshold density level. Therefore, in order to include this type of aggregative behavior into the model, we assume the existence of α∈[0,1] satisfying  (1.5)D(u)>0foru∈(0,α),D(u)<0,foru∈(α,1).
The extremal cases α=1 and α=0 respectively correspond to a purely diffusive and a purely aggregative term. In the former case, that is, α=1, the presence of t.w.s. in these models and their main qualitative properties have been investigated since a long time, and we refer to [2, 7, 8] for details. The latter case; that is, α=0, was recently discussed in [9]. As far as we know, the first detailed investigation concerning the existence of t.w.s. of (1.1) when D satisfies (1.5) and α∈(0,1) appeared in [10], but there was no convective effect included. Under the additional assumption that (1.6)Ḋ(α)<0,
we complete in Theorem 2.1 the analysis started in [10] and show that also (1.1), incorporating a convective behavior, is able to support a continuum of t.w.s., parameterized by their wave speeds c which satisfy c≥c*, and we give an estimate of the threshold speed c*.The very nature of aggregative processes causes ill posedness of our model (1.1) when α∈(0,1) and v∈[α,1] (see, e.g., [10] and the references there contained). However, discrete models underlying (1.1) are well posed, and some numerical computations (see [6]) seem in good agreement with the information obtained in the discrete setting. Moreover, even in this possible ill-posedness context, the t.w.s. above defined are regular solutions for (1.1), and this increases the interest in studying them.As it is clear also from the prototype equation (1.4), very naturally these models include real parameters which frequently cause, on their varying, the transition of the process a diffusive to a diffusive-aggregative regime or from a diffusive-aggregative to a purely aggregative one. The main aim of this paper lies in the analysis of these behaviors, so we consider the more general dynamic: (1.7)vt+h(v;k)vx=(D(v;k)vx)x+f(v;k),t≥0,x∈R,
which continuously depends on the real parameter k∈[0,1]. We take h, D, and f in C([0,1]2) with D(·;k)∈C1[0,1] and (1.8)f(u;k)>0foru∈(0,1),f(0;k)=f(1;k)=0,
for all k. Further, we assume the existence of a continuous function α:[0,1]→[0,1] such that (1.9)α(1)=0,α(0)=1,α(k)∈(0,1),for0<k<1,(1.10)D(u;k)>0foru∈(0,α(k)),D(u;k)<0,foru∈(α(k),1),(1.11)Ḋ(α(k);k)<0,fork∈(0,1).
We discuss both the case k∈(0,1), corresponding to a diffusive-aggregative dynamic, as well as the case k=0 or 1, where (1.7) is, respectively, a purely diffusive reaction-convection equation and a purely aggregative one. In Theorem 2.2 we prove that the threshold value c*(k) is always a lower semicontinuous function (l.s.c.) on the whole interval [0,1]; hence, in particular, the lower semicontinuity holds when the process (1.7) switches from a purely diffusive to a diffusive-aggregative and from the latter to a purely aggregative behavior. We provide quite general conditions (see Theorem 2.2 and Proposition 2.4) either guaranteeing that c*(k) is continuous on [0,1] or that it fails to be continuous for k=0 or k=1. Two simple illustrative examples complete this discussion. In Theorem 2.6 we show the continuous dependence, on the parameter k, of any family of profiles u(ξ;k) corresponding to a continuum of wave speeds γ(k). These are the main results of the paper; their statements appear in Section 2 while their proofs can be found in Section 4. Notice that the function α can be extended in a continuous way outside the interval [0,1], so that the model also includes the purely diffusive or aggregative dynamic. However, the study of the continuity in the former case has been carried on in [11], while the latter case can be treated by a suitable change of variable, and this discussion is included in Theorem 4.2.As for the methodology, it is easy to see that every wave profileu(ξ) of (1.1), having wave speed c, corresponds to a regular solution of the boundary value problem:  (1.12)(D(u)u′)′+(c-h(u))u′+f(u)=0,u(a+)=1,u(b-)=0,
where -∞≤a<b≤∞. More precisely, a solution of (1.12) in (a,b) is a function   u∈C(cl(a,b))∩C2(a,b) satisfying the boundary conditions (1.3).Since, in addition,u is strictly monotone in every interval where 0<u(ξ)<1 (see, e.g., Theorem 3.7), the investigation of t.w.s. can be reduced to the study of a first-order singular boundary value problem. In order to investigate this problem, we mainly use comparison-type techniques, that is, suitable upper and lower solutions. In Section 3 we report and complete the original discussion developed in [8] and show the equivalence between the first-order singular problem and the existence of t.w.s. for (1.1).


## 2. Statements of the Main Results

Here we present the main results of the paper, which will be proved in Section4. Put φ(u):=D(u)f(u); in the whole section we assume that  (2.1)φisdifferentiableatu=0and1.
The first result concerns the existence of t.w.s. for (1.1).Theorem 2.1.
Assume that conditions (1.2), (1.5), (1.6) and (2.1) hold. Then, there exists a value  c* such that (1.1) supports t.w.s. if and only if  c≥c*, and the profile is unique, up to shifts. Moreover, the threshold value  c* satisfies the following estimates, depending on the value  α:
(2.2)h(0)+2φ̇(0)≤c*≤max⁡u∈[0,1]h(u)+2sup⁡u∈(0,1]φ(u)u,ifα=1,(2.3)h(1)+2φ̇(1)≤c*≤max⁡u∈[0,1]h(u)+2sup⁡u∈[0,1)φ(u)u-1,ifα=0,
and, if 0<α<1,
(2.4)max⁡{2φ̇(0)+h(0),2φ̇(1)+h(1)}≤c*≤max⁡{2sup⁡u∈(0,α]φ(u)u+max⁡u∈[0,α]h(u),2sup⁡u∈[α,1)φ(u)u-1+max⁡u∈[α,1]h(u)}.As stated in the Introduction section, the main aim of the paper is the study of the continuous dependence, both of the threshold valuec* and of the wave profiles u, with respect to the coefficients h, D, and f appearing in (1.1). More precisely, we are mainly interested in studying the continuous dependence when a change in the type of dynamics occurs: from a purely diffusive to a diffusive-aggregative one or from a diffusive-aggregative to a purely aggregative one.To this aim, we consider a continuous functionα:[0,1]→[0,1] satisfying conditions (1.9), and we introduce a real parameter k varying in [0,1], inside the coefficients of (1.1), in such a way that h(u;k), D(u;k), and f(u;k) are continuous functions of two variables. We further assume (1.8), (1.10), and (1.11) and suppose that φ(u;·) satisfy (2.1) for each k∈[0,1].In this framework, the threshold valuec* is a function of the parameter k, say c*=c*(k), and, for every c≥c*(k), the profile u (modulo shifts) of the t.w.s. having speed c is a function of (ξ,k), where ξ is the wave variable, say u=u(ξ;k).As for the functionc*(k), the following result will be proven.Theorem 2.2.
Letα:[0,1]→[0,1] and  h,D,f:[0,1]×[0,1]→ℝ be continuous functions satisfying (1.8), (1.9), (1.10), and (1.11). For all  k∈[0,1], assume that condition (2.1) holds for  φ(·;k), and let  D(·;k)∈C1([0,1]). Then,  c*(k) is l.s.c..
Further, if for somek0∈(0,1) we have
(2.5)limsup⁡(u,k)→(0,k0)φ(u;k)-φ(u;k0)u≤0,liminf⁡(u,k)→(1,k0)φ(u;k)-φ(u;k0)1-u≥0,
then c* is continuous at k0.
With regards to the behavior ofc* at the endpoints k=0 and k=1, if
(2.6)limsup⁡(u,k)→(1,0){φ(u;k)u-1-φ̇(1;k)}≤0,(2.7)2limsup⁡k→0φ̇(1;k)+h(1;0)≤2φ̇(0;0)+h(0;0)
then c*(k) is continuous at k=0, and, similarly, if
(2.8)limsup⁡(u,k)→(0,1){φ(u;k)u-φ̇(0;k)}≤0,(2.9)2limsup⁡k→1φ̇(0;k)+h(0;1)≤2φ̇(1;1)+h(1;1)
then c*(k) is continuous at k=1.Remark 2.3.
Assume now thatφ̇(u;k) exists and it is continuous on [0,1]×[0,1]. In this case it is easy to verify that all conditions in (2.5), (2.6), and (2.8) are satisfied as equalities. Furthermore, according to (1.8) and (1.10), we obtain that φ̇(1;0)≤0 and also φ̇(0;1)≤0 while φ̇(1;k)≥0 and φ̇(0;k)≥0 for k∈(0,1). The continuity of φ̇, hence, implies that φ̇(1;0)=φ̇(0;1)=0. Consequently, conditions (2.7) and (2.9), respectively, reduce to
(2.10)h(1;0)≤2φ̇(0;0)+h(0;0),(2.11)h(0;1)≤2φ̇(1;1)+h(1;1).
Therefore, under conditions (2.10) and (2.11), the function c*(k) is continuous on the whole interval [0,1]. Of course, when h≡0, then (2.10) and (2.11) trivially hold, and c*(k) is continuous on [0,1].As an immediate consequence of the estimates (2.2), (2.3), and (2.4), if the relation (2.7) or (2.9) in the previous theorem does not hold, then the function c*(k) could be discontinuous at the endpoints, as the following result states.Proposition 2.4.
Under the same conditions of Theorem2.2, if
(2.12)2limsup⁡k→0φ̇(1;k)+h(1;0)>2sup⁡u∈(0,1]φ(u;0)u+max⁡u∈[0,1]h(u;0),
then c*(k) is not continuous at k=0. Similarly, if
(2.13)2limsup⁡k→1φ̇(0;k)+h(0;1)>2sup⁡u∈[0,1)φ(u;1)u-1+max⁡u∈[0,1]h(u;1),
then c*(k) is not continuous at k=1.Example 2.5.
Letα(k):=1-k  and  D(u;k):=α(k)-u=1-k-u, and let
(2.14)f(u;k):={u1-u,ifk=0,min⁡{u1-u,2k(1-u)},ifk≠0.
Observe that conditions (1.8), (1.9), and (1.10) are trivially satisfied. Moreover, f(u;k) uniformly converges to f(u;0) as k→0, and f is continuous in [0,1]×[0,1]. Furthermore, also condition (2.1) is satisfied. Indeed, φ(u;0)=u(1-u)3/2 is differentiable in [0,1], φ̇(0;k)=(1-k) and φ̇(1;k)=2 for every k∈(0,1], and finally φ(u;k) is differentiable at u=α(k) since for u=α(k)=1-k we have u1-u<(2/k)(1-u), and then f(u;k) is differentiable at u=α(k). Since
(2.15)lim⁡u→1φ(u;k)u-1=lim⁡u→11-k-uu-12(1-u)k=2,foreveryk≠0,
we have limk→0φ̇(1;k)=2, whereas
(2.16)sup⁡u∈(0,1]φ(u;0)u=sup⁡u∈(0,1](1-u)3/2=1,
then (2.12) holds for every continuous function h:[0,1]×[0,1]→ℝ such that h(1;0)=max[0,1]h(u;0). Therefore, the function c* is not continuous at k=0; that is, c* looses its continuity in the transition from a diffusive-aggregative dynamic to a purely diffusive one.The last result concerns the continuous dependence of the profiles of the t.w.s..Theorem 2.6.
Letα:[0,1]→[0,1] and  h,D,f:[0,1]×[0,1]→ℝ be continuous functions satisfying (1.8), (1.9), (1.10), and (1.11). For all  k∈[0,1], assume that condition (2.1) holds for  φ(·;k), and let  D(·;k)∈C1([0,1]). Take  γ:[0,1]→[0,1] continuous and such that  γ(k)≥c*(k) for every  k∈[0,1], and let  k0∈[0,1] be fixed.
LetUk(ξ):=U(ξ;γ(k),k) be the profile of the corresponding travelling wave solution of (1.1) with speed γ(k) such that Uk(0)=u0, for some fixed u0∈(0,1). Then  Uk(ξ) converges to Uk0(ξ) uniformly on all the real line.Remark 2.7.
In all the results stated in this section, if convective effects are not present; that is,h(u;k)≡0, then assumptions (1.6) and (1.11) can be removed, as we show in the end of the proof of Theorem 3.7.


## 3. Reduction to a Singular First-Order Equation

Given continuous functionsh,D,f:[0,1]→[0,1], with D∈C1[0,1], satisfying conditions (1.2), (1.5), (1.6), and (2.1), let us consider the following singular first-order boundary value problem (3.1)ż(u)=h(u)-c-φ(u)z(u),foreveryu∈(0,1)∖{α},z(0+)=z(1-)=0,z(u)φ(u)<0,∀u∈(0,1),u≠α,
where c is an unknown real constant. By a solution of the b.v.p. (3.1), we mean a differentiable function z:(0,1)→ℝ, satisfying all the conditions in (3.1). Of course, any possible solution of (3.1) is of class C1 in any compact interval [a,b]⊂(0,1) not containing the value α.As we will show at the end of this section, the solvability of problem (3.1) is equivalent to the existence of solutions of (1.12). For this reason, we now study the existence of solutions to problem (3.1). To this aim, in [12] it was proved the existence of solutions to the singular boundary value problem (3.1) in the case when φ is everywhere positive. In particular, the following Proposition is a consequence of [12, Lemma  2.2] combined to [12, Theorem  1.4].Proposition 3.1.
Leth,φ:[ℓ1,ℓ2]→ℝ be continuous functions such that  φ(ℓ1)=φ(ℓ2)=0,  φ(u)>0 in  (ℓ1,ℓ2), and  φ̇(ℓ1) exists finite. Then, there exists a value  c*, satisfying the following estimate:
(3.2)2φ̇(l1)+h(l1)≤c*≤2sup⁡u∈(l1,l2]φ(u)u-l1+max⁡u∈[l1,l2]h(u),
such that the problem
(3.3)ż(u)=h(u)-c-φ(u)z(u),foreveryu∈(l1,l2),z(l1+)=z(l2-)=0,z(u)φ(u)<0,∀u∈(l1,l2),
admits solutions if and only if c≥c*. Moreover, the solution is unique.As far as the regularity of the solutionz at the endpoints, the following result holds.Corollary 3.2.
Under the same assumptions of Proposition3.1, if  φ̇(ℓ2) exists finite, then  ż(ℓ2) exists finite too, and
(3.4)ż(l2)=12(h(l2)-c+(h(l2)-c)2-4φ̇(l2)).Proof.
Assume by contradiction that(3.5)0≤λ-:=liminf⁡u→l2-z(u)u-l2<limsup⁡u→l2-z(u)u-l2=Λ-≤+∞.
For a fixed  γ∈(λ-,Λ-), let {un}n⊂(0,ℓ2) be a sequence converging to ℓ2 and such that
(3.6)z(un)un-l2=γ,ddu(z(u)u-l2)u=un≤0.
Since
(3.7)ddu(z(u)u-l2)u=un=1un-l2[ż(un)-γ],
we obtain ż(un)≥γ. Replacing this inequality in (3.3) evaluated for u=un, and passing to the limit, we have
(3.8)γ2-[h(l2)-c]γ+φ̇(l2)≤0.
Similarly, taking a sequence {vn}n⊂(0,ℓ2) converging to ℓ2 and such that
(3.9)z(vn)vn-l2=γ,ddu(z(u)u-l2)u=vn≥0,
arguing as before, we obtain γ2-[h(ℓ2)-c]γ+φ̇(ℓ2)≥0. Combining with (3.8) we get γ2-[h(ℓ2)-c]γ+φ̇(ℓ2)=0, which is impossible, due to the arbitrariness of γ∈(λ-,Λ-). Then,
(3.10)λ-=Λ-=lim⁡u→l2-z(u)u-l2.
Since (3.3) can be rewritten as follows:
(3.11)ż(u)=h(u)-c-φ(u)u-l2u-l2z(u),
then λ-=+∞ would imply that ż(u)→h(ℓ2)-c as u→ℓ2-, a contradiction. Hence, λ- is real, nonnegative.
Ifλ->0, then from (3.11) we get that ż(u) admits limit as u→ℓ2-, and this limit must coincide with λ-. Passing to the limit in (3.11) we get that λ- is a root of
(3.12)λ2+(c-h(l2))λ+φ̇(l2)=0.
If φ̇(ℓ2)=0, we immediately have ż(u)→λ-=h(ℓ2)-c>0 as u→ℓ2-, since λ->0. If φ̇(ℓ2)<0, since the above trinomial has two discordant zeros, we deduce that
(3.13)λ-=12(h(l2)-c+(h(l2)-c)2-4φ̇(l2)).
Summarizing, if λ->0, then (3.13) holds.
Assume now thatλ-=0. If φ̇(ℓ2)<0; from (3.11) we get ż(u)→+∞ as u→ℓ2-, a contradiction. Hence, ż(ℓ2)=0 implies φ̇(ℓ2)=0. Moreover, observe that
(3.14)h(l2)-c=lim⁡u→l2-h(u)-c≤liminf⁡u→l2-(h(u)-c-φ(u)z(u))=liminf⁡u→l2-ż(u)≤λ-=0.
Thus, λ- satisfies (3.13) also in the case λ-=0.By a simple change of variable (v=ℓ1+ℓ2-u and z̃(v)=-z(u)), it is easy to check the validity of the following results in the case when φ is everywhere negative in (ℓ1,ℓ2).Proposition 3.3.
Leth,φ:[ℓ1,ℓ2]→ℝ be continuous functions such that  φ(ℓ1)=φ(ℓ2)=0,  φ(u)<0 in  (ℓ1,ℓ2), and  φ̇(ℓ2) exists finite. Then, there exists a value  c*, satisfying the following estimate:
(3.15)2φ̇(l2)+h(l2)≤c*≤2sup⁡u∈[l1,l2)φ(u)u-l2+max⁡u∈[l1,l2]h(u),
such that problem (3.3) admits solutions if and only if c≥c*. Moreover, the solution is unique.Corollary 3.4.
Under the same assumptions of Proposition3.3, if  φ̇(ℓ1) exists finite, then also  ż(ℓ1) exists finite, and
(3.16)ż(l1)=12(h(l1)-c+(h(l1)-c)2-4φ̇(l1)).Combining the previous results, we are able to prove an existence theorem for problem (3.1).Theorem 3.5.
Leth,φ:[0,1]→ℝ be continuous functions, with  φ(0)=φ(1)=0,  φ differentiable at  u=α, satisfying condition (2.1) and such that
(3.17)φ(u)(α-u)>0,∀u∈(0,1)∖{α},
for a given constant α∈(0,1).
Then, there exists a valuec*, satisfying estimate (2.4), such that problem (3.1) admits solutions if and only if c≥c*. Moreover, the solution is unique.Proof.
Consider the boundary value problem (3.3) for ℓ1=0,ℓ2=α. Since all the assumptions in Proposition 3.1 are satisfied, there exists a threshold value c1*, satisfying (3.2), such that the b.v.p (3.3), with ℓ1=0 and ℓ2=α, is uniquely solvable if and only if c≥c1*. Similarly, considering the problem (3.3) for ℓ1=α,ℓ2=1, by Proposition 3.3, there is a threshold value c2*, satisfying (3.15), such that this problem is uniquely solvable if and only if c≥c2*. Let c*:=max{c1*,c2*}. Clearly, c* satisfies (2.4). Moreover, if problem (3.3), with ℓ1=0andℓ2=1, is solvable for some c, then also problem (3.3) with ℓ1=0,ℓ2=α, and problem (3.3), with ℓ1=α and ℓ2=1, are solvable for the same value c. Hence, c≥c*.
Conversely, let us fixc≥c*, and let z∈C(0,1) be the function obtained gluing the unique solution of problem (3.3), with ℓ1=0 and ℓ2=α and the unique solution of problem (3.3) with ℓ1=αandℓ2=1. In order to prove the assertion, it suffices to observe that, from Corollaries 3.2 and 3.4, the glued function z is differentiable at the point α.Remark 3.6.
Notice that in the previous theorem the required differentiability ofφ at α cannot be dropped. Indeed, in view of the proof, if the right derivative of φ at α does not coincide with the left one, then λ-≠λ+ and z is not differentiable at α.Concerning the equivalence between the solvability of problem (3.1) and the existence of t.w.s., the following result holds.Theorem 3.7.
Let the assumptions (1.2), (1.5), (1.6), and (2.1) be satisfied. The existence of solutions to (1.12) for some  c is equivalent to the solvability of problem (3.1) with the same  c. If  h(u)≡0, then condition (1.6) can be removed.Proof.
Letu(ξ) be a t.w.s. of (1.1) with wave speed c, that is, a solution of problem (1.12) satisfying condition (1.3). Observe that if u′(ξ0)=0 with 0<u(ξ0)<1, for some ξ0∈(a,b), then since (D(u(ξ0)u′(ξ0)))′=-f(u(ξ0))<0, we have that ξ0 is a point of proper local maximum for u when u(ξ0)∈(0,α), while ξ0 is a point of proper local minimum when u(ξ0)∈(α,1). Hence, assume now by contradiction the existence of a value ξ*∈(a,b) such that α<u(ξ*)<1 and u′(ξ*)>0. By the boundary condition u(b-)=0, we get the existence of a value ξ̅>ξ* such that u(ξ̅)=1 and u′(ξ̅)=0. Integrating (1.12) in (a,ξ̅), we obtain
(3.18)0=D(u(ξ̅))u′(ξ̅)-D(u(a))u′(a)=-∫aξ̅f(u(ξ))dξ,
in contradiction with condition (1.2). Similarly, if u′(ξ*)>0 with 0<u(ξ*)<α for some ξ*∈(a,b), then there exists a value ξ̅<ξ* such that u(ξ̅)=0 and u′(ξ̅)=0. Integrating (1.12) in (ξ̅,b), we get again a contradiction. Therefore, u′(ξ)<0 whenever u(ξ)∈(0,1), u(ξ)≠α.
Putξ1:=inf{ξ:u(ξ)=α} and ξ2:=sup{ξ:u(ξ)=α}. If ξ1<ξ2 then u is constant in (ξ1,ξ2), and we can define ũ(ξ):=u(ξ) if ξ≤ξ1; ũ(ξ):=u(ξ+ξ2-ξ1) if ξ>ξ1. Of course, since (1.12) is autonomous, ũ is a t.w.s. of (1.1) in its existence interval Ĩ:=(a,b-ξ2+ξ1) and satisfies u′̃(ξ)<0, for every ξ∈Ĩ, ξ≠ξ1. Hence, ũ is strictly decreasing and invertible in Ĩ. Let z(u):=D(u)u′̃(ξ(u)), u∈(0,1). It is easy to see that the function z(u) is a solution of (3.1) for the same value of c.
Now assume that0<α<1, and let z(u) be a solution of (3.1) for some admissible real value c. Let u1(ξ) be the unique solution of the Cauchy problem:
(3.19)u′=z(u)D(u),0<u<α,u(0)=α2,
defined on its maximal existence interval (t1,t2). Since u′(ξ)<0 whenever ξ∈(t1,t2), we have that u(t2-)=0 and u(t1+)=α. Moreover, according to (1.6) and Corollary 3.2, the limit limξ→t1+u′(ξ) exists and it is not zero, implying that t1∈ℝ. Consider now the unique solution u2(ξ) of the initial value problem:
(3.20)u′=z(u)D(u),α<u<1,u(0)=1+α2,
on its maximal existence interval (τ2,τ1). Again, u2′(ξ)<0 for ξ∈(τ1,τ2) implies u(τ1+)=1 and u(τ2-)=α. Moreover, according to (1.6) and Corollary 3.4, we obtain that limξ→τ2-u2′(ξ)=limξ→t1+u1(ξ); hence, also the former limit is not zero and τ2∈ℝ. Then, as we made above, with a suitable shift, it is possible to glue the two functions ui(ξ),i=1,2, in such a way to have a unique u=u(ξ), defined on some interval (a,b)⊆ℝ, with u∈C1((a,b)), u(ξ)∈(0,1) on the whole (a,b), and u satisfying (1.12) and (1.3).
Whenα=0 or α=1 one considers just one of the Cauchy problems above defined, whose solution is a t.w.s. for (1.1).
Finally, assumeh≡0. It is well known (see, e.g., [2]) that c* is always strictly positive in this case. Moreover, (see, e.g., [2, Lemma  1]),
(3.21)lim⁡ξ→t1+u′(ξ)=lim⁡ξ→τ2-u′(ξ)=2f(α)c+c2-4f(α)Ḋ(α).
Hence, both limits are finite and nonzero independently on the value of Ḋ(α). So condition (1.6) can be removed.Remark 3.8.
IfD(0)>0, then b=+∞. Indeed, ż(u)>h(u)-c, for u∈(0,α), implies z(u)>∫0u[h(s)-c]ds. Therefore, if u(t0)=α/2 for some t0∈ℝ, we have that
(3.22)b-t0=∫α/20ṫ(u)du=∫0α/2D(u)-z(u)du≥∫0α/2D(u)∫0u[c-h(s)]dsdu.
Being an admissible value, the wave speed c satisfies condition (2.4); hence, there exists σ>0 such that ∫0u[h(s)-c]ds≥σu for u∈(0,α), implying b=+∞. With a similar argument, it can be showed that D(1)<0 implies a=-∞.In light of the above equivalence result, Theorem2.1 is an immediate consequence of Propositions 3.1, 3.3 and Theorem 3.5.


## 4. The Continuous Dependence Results

As we mentioned in the Introduction section, both the dependence on the parameterk of the minimal wave speed c*(k) and of the wave profile U(ξ;c,k) corresponding to the speed c were already studied in [11] for the diffusive case, that is, in case D(u;k)≥0, for every u∈(0,1). More in detail, the following result holds (see [11, Theorems  4.1 and 4.2]).Theorem 4.1.
Leth,D,f:[0,1]×[0,1]→ℝ be continuous functions, with  D(·;k)∈C1[0,1] for every  k∈[0,1] and  D(u;k)>0 for every  (u,k)∈(0,1)×[0,1]. Assume that (1.8) and (2.1) are satisfied for all  k∈[0,1]. Then, the threshold value  c* is a l.s.c. function in  [0,1]. Moreover, if for some  k0∈[0,1](4.1)limsup⁡(u,k)→(0,k0)φ(u;k)-φ(u;k0)u≤0,
then c* is continuous at k=k0.By means of a change of variable, in the aggregative case, that is, in caseD(u;k)<0 for every u∈(0,1), it can be proved that the analogous result holds.Theorem 4.2.
Leth,D,f:[0,1]×[0,1]→ℝ be continuous functions, with  D(·;k)∈C1[0,1] for every  k∈[0,1] and  D(u;k)<0 for every  (u,k)∈(0,1)×[0,1]. Assume that (1.8) and (2.1) are satisfied for all  k∈[0,1]. Then, the threshold value  c* is a l.s.c. function in  [0,1]. Moreover, if for some  k0∈[0,1](4.2)liminf⁡(u,k)→(1,k0)φ(u;k)-φ(u;k0)1-u≥0,
then c* is continuous at k=k0.Combining the results of Section3 with Theorems 4.1 and 4.2, we are able to prove Theorem 2.2.Proof of Theorem2.2.
In view of the equivalence proved in Theorem3.7, we analyze the continuous dependence of the threshold value c*(k) for the problem:
(4.3)ż(u)=h(u;k)-c-φ(u;k)z(u),foreveryu∈(0,1)∖{α(k)},z(0+)=z(1-)=0,z(u)φ(u)<0,∀u∈(0,1),u≠α(k),
where c is an unknown real constant.
For everyk∈[0,1) we have 0<α(k)≤1; hence, we can consider the boundary value problem:
(P1,k)ż(u)=h(u;k)-c-φ(u;k)z(u),u∈(0,α(k)),z(0+)=z(α(k)-)=0,z(u)<0,∀u∈(0,α(k)).
Puth̃(u;k):=α(k)h(α(k)u;k) and φ̃(u;k):=α(k)φ(α(k)u;k), problem (3.22) is equivalent to the normalized one:
(P̃1,k)ẇ(u)=h̃(u;k)-cα(k)-φ̃(u;k)w(u),u∈(0,1),w(0+)=w(1-)=0,w(u)<0,∀u∈(0,1).
Indeed, a function z(u) is a solution of (3.22) if and only if the function w(u):=z(α(k)u) is a solution of (P1,k). Hence, if c1*(k) and c̃1*(k) denote, respectively, the minimal values of the parameter c for which (3.22) and (P1,k) are solvable, we have
(4.4)c1*(k)=1α(k)c̃1*(k).
Since α(k) is continuous and positive in [0,1), from Theorem 4.1 we have that c1*(k) is lower semicontinuous at k0 for every k0∈[0,1).
Similarly, for everyk∈(0,1], consider the problem:
(P2,k)ż(u)=h(u;k)-c-φ(u;k)z(u),u∈(α(k),1),z(α(k)+)=z(1-)=0,z(u)>0,∀u∈(α(k),1).
Using an argument analogous to the previous one and taking into account that, in this case, 0≤α(k)<1, we deduce that the threshold value c2*(k) for problem (P̃1,k) is a lower semicontinuous function at every k0∈(0,1].
Letk∈(0,1) and let c* be the threshold value for problem (4.3). Since c*(k)=max{c1*(k),c2*(k)} (see the proof of Theorem 3.5), with c1*,c2* lower semicontinuous functions, then c*(k) is lower semicontinuous at every k0∈(0,1).
Ifk0=0, then it results in c*(0)=c1*(0), and
(4.5)liminf⁡k→0c*(k)=liminf⁡k→0(max⁡{c1*(k),c2*(k)})≥liminf⁡k→0c1*(k)≥c1*(0)=c*(0).
The case k0=1 can be treated in a similar way, and the semicontinuity of c* in [0,1] is proved.
Concerning the study of the continuity ofc* under condition (2.5), let us first consider a value k0∈(0,1). Analogously to the arguments above developed, we can consider the problem (3.22) and the normalized one (P1,k). Notice that, if φ satisfies condition (4.1), then also the function φ̃ does. Indeed, we have
(4.6)limsup⁡(u,k)→(0,k0)φ̃(u;k)-φ̃(u;k0)u=limsup⁡(u,k)→(0,k0)α(k)φ(α(k)u;k)-α(k0)φ(α(k0)u;k0)u=limsup⁡(v,k)→(0,k0)α2(k)φ(v;k)-φ(v;k0)v+lim⁡(u,k)→(0,k0)(α2(k)φ(α(k)u;k0)uα(k)-α2(k0)φ(α(k0)u;k0uα(k0))≤0,
by the positivity and the continuity of α(k) and the differentiability of φ(u;k0) at 0. Hence, Theorem 4.1 assures that the function c̃1*(k) is continuous at k0, and in force of (4.4) we conclude that c1*(k) is continuous too. Analogously, considering the problem (P̃1,k), we can prove that the threshold value c2*(k) is continuous at every k0∈(0,1). Since c*(k)=max{c1*(k),c2*(k)} (see the proof of Theorem 3.5), then c*(k) is continuous at every k0∈(0,1).
It remains to study the continuity ofc* at the values k0=0 and k0=1, where the dynamic change its nature. We limit ourselves in considering the value k0=0, since the proof for k0=1 is analogous.
Letϵ>0 be fixed, and observe that, by assumption (2.6), real values δ=δ(ϵ)>0 and k̅=k̅(ϵ)∈(0,1) exist, such that
(4.7)φ(u;k)u-1≤φ̇(1;k)+ϵ,foreveryu∈(1-δ,1),k∈(0,k̅).
Let k̃=k̃(ϵ)∈(0,k̅) be such that α(k)∈(1-δ,1) for every k∈(0,k̃). Then
(4.8)sup⁡u∈[α(k),1)φ(u;k)u-1≤φ̇(1;k)+ϵ,foreveryk∈(0,k̃).
Taking estimate (3.15) into account, with ℓ1=α(k) and ℓ2=1, we get
(4.9)c2*(k)≤2φ̇(1;k)+ϵ+max⁡u∈[α(k),1]h(u;k),foreveryk∈(0,k̃),
implying
(4.10)limsup⁡k→0c2*(k)≤2limsup⁡k→0φ̇(1;k)+ϵ+h(1;0).
By the arbitrariness of ϵ>0, assumption (2.7), and estimate (3.2) with ℓ1=0 and ℓ2=α(k), we have
(4.11)limsup⁡k→0c2*(k)≤2φ̇(0;0)+h(0;0)≤c1*(0)=c*(0).
Therefore, since c1*(k)→c1*(0)=c*(0) as k→0, we conclude that
(4.12)limsup⁡k→0c*(k)=limsup⁡k→0max⁡{c1*(k),c2*(k)}=c*(0),
and the function c* is upper semicontinuous in k0=0. Taking into account the lower semicontinuity of c* in the whole interval [0,1], it follows the continuity in k0=0. The proof for the value k0=1 is similar.In order to prove Theorem2.6 about the convergence of the profiles, we need some preliminary results.Letz(u;c,k) denote the unique solution of problem (4.3), emphasizing the dependence on the parameters c and k. The following result holds.Lemma 4.3.
Letα:[0,1]→[0,1] and  h,D,f:[0,1]×[0,1]→ℝ be continuous functions satisfying conditions (1.8), (1.9), (1.10), and (2.1) for every  k∈[0,1]. Let  γ:[0,1]→[0,1] be a continuous function such that  γ(k)≥c*(k) for every  k∈[0,1]. Finally, let  z(u;γ(k),k) be the unique solution of problem (4.3).
Then, for everyk∈(0,1), there exists a real value λ(k) such that the function
(4.13)ψ(u;k):={z(u;γ(k),k)D(u;k),foru∈(0,1),u≠α(k),λ(k),foru=α(k)
is continuous in (0,1)×(0,1).Proof.
Let(u0,k0)∈(0,1)×(0,1) be fixed. We divide the proof in two cases. Case 1 (u0≠α(k0)).
Assumeu0<α(k0) (the proof in the other case is analogous), and let [a,b]⊂(0,α(k0)) be a compact interval containing the value u0. Put zk(u):=z(u;γ(k),k) and let wk(u):=w(u;γ(k)α(k),k) denote the solution of problem (P1,k) (see the proof of Theorem 2.2). By applying [11, Theorem  5.1] to the problem (P1,k) in [a,2b/(b+α(k0))]⊂(0,1), we get the convergence of wk(u) to wk0(u) as k→k0, uniformly in [a,2b/(b+α(k0))]; that is, for a fixed ϵ>0,
(4.14)|wk(u)-wk0(u)|<12ϵforeveryu∈[a,2bb+α(k0)].
Taking account of the continuity of α and of the uniform continuity of wk0, there exists a value η>0 such that for every k∈(k0-η,k0+η) we have α(k)>(1/2)(b+α(k0)) and
(4.15)|wk0(uα(k))-wk0(uα(k0))|<12ϵ,foreveryu∈[a,b].
Since u/α(k)∈[a,2b/(b+α(k0))] whenever u∈[a,b] and k∈(k0-η,k0+η), we deduce that
(4.16)|zk(u)-zk0(u)|≤|wk(uα(k))-wk0(uα(k))|+|wk0(uα(k))-wk0(uα(k0))|≤ϵ,
for every k∈(k0-η,k0+η) and u∈[a,b], that is, the uniform convergence of zk to zk0 in [a,b]. Taking account of the uniform continuity of zk0 in [0,1], we get zk(u)→zk0(u0) as (u,k)→(u0,k0). Finally, since D(u0,k0)≠0, by the continuity of D(u;k) we also get
(4.17)lim⁡(u,k)→(u0,k0)zk(u)D(u;k)=zk0(u0)D(u0;k0),
and the continuity of the function ψ at (u0,k0) is proved.
Notice that, for the opposite caseu0>α(k0), the usual change of variable v:=1-u allows to apply [11, Theorem  5.1].Case 2(u0=α(k0)).
For everyk∈(0,1), let
(4.18)λ(k)=-2f(α(k);k)-h(α(k);k)+γ(k)+[h(α(k);k)-γ(k)]2-4f(α(k);k)Ḋ(α(k);k).
Notice that the function λ is well defined since by assumption (1.6) we have Ḋ(α(k);k)<0. Moreover, λ is continuous and negative for every k∈(0,1). Since
(4.19)λ(k)=h(α(k);k)-γ(k)+[h(α(k);k)-γ(k)]2-4f(α(k);k)Ḋ(α(k);k)2Ḋ(α(k);k),
from Corollary 3.2 it follows that
(4.20)żk(α(k))=λ(k)Ḋ(α(k);k).
Let us fix a value ϵ>0 and define
(4.21)ξϵ(u;k)=(λ(k0)-ϵ)D(u;k),Φϵ(u;k)=ξ̇ϵ(u;k)-h(u;k)+γ(k)+f(u;k)D(u;k)ξϵ(u;k).
We have
(4.22)Φϵ(u;k)=λ(k0)Ḋ(u;k)-h(u;k)+γ(k)+f(u;k)λ(k0)-ϵ-ϵḊ(u;k)=λ(k0)Ḋ(u;k)-h(u;k)+γ(k)+f(u;k)λ(k0)+ϵf(u;k)λ(k0)(λ(k0)-ϵ)-ϵḊ(u;k).
From the definition of λ(k), we get
(4.23)λ(k)Ḋ(α(k);k)-h(α(k);k)+γ(k)+f(α(k);k)λ(k)=0.
Therefore,
(4.24)lim⁡(u,k)→(α(k0),k0)Φϵ(u;k)=ϵf(α(k0);k0)λ(k0)(λ(k0)-ϵ)-ϵḊ(α(k0);k0)=:2σ>0.
Hence, there exists a value δ1>0 such that
(4.25)Φϵ(u;k)≥σ>0,whenever|u-α(k0)|<δ1,|k-k0|<δ1,
that is,
(4.26)ξ̇ϵ(u;k)>h(u;k)-γ(k)-f(u;k)D(u;k)ξϵ(u;k),whenever|u-α(k0)|<δ1,|k-k0|<δ1.
Take now δ∈(0,δ1] such that |α(k)-α(k0)|<(1/2)δ1 and λ(k)>λ(k0)-ϵ whenever |k-k0|<δ. Observe that, by (4.20), we have żk(α(k))<ξ̇ϵ(α(k);k). Since zk(α(k))=ξϵ(α(k);k)=0, then for every k∈(k0-δ,k0+δ), there exists a value ρ=ρ(k)∈(0,α(k)) such that
(4.27)0>zk(u)>ξϵ(u;k),foreveryu∈(α(k)-ρ(k),α(k)),0<zk(u)<ξϵ(u;k),foreveryu∈(α(k),α(k)+ρ(k)).
Assume, by contradiction, the existence of a value k∈(k0-δ,k0+δ) and of a point v(k)∈[α(k0)-δ1,α(k)) such that
(4.28)zk(v(k))=ξϵ(v(k);k),zk(u)>ξϵ(u;k)in(v(k),α(k)).
Then,
(4.29)żk(v(k))=h(v(k);k)-γ(k)-f(v(k);k)D(v(k);k)zk(v(k))=h(v(k);k)-γ(k)-f(v(k);k)D(v(k);k)ξϵ(v(k);k)<ξ̇ϵ(v(k);k),
in contradiction with the definition of v(k). Therefore, we deduce that
(4.30)ξϵ(u;k)<zk(u)<0,foreveryu∈[α(k0)-δ1,α(k)),k∈[k0-δ,k0+δ].
Similarly, we can prove that
(4.31)ξϵ(u;k)>zk(u)>0,foreveryu∈(α(k),α(k0)+δ1],k∈[k0-δ,k0+δ].
The previous conditions (4.30) and (4.31) imply that
(4.32)zk(u)D(u;k)>ξϵ(u;k)D(u;k)=λ(k0)-ϵ,foreveryu∈[α(k0)-δ1,α(k0)+δ1]∖{α(k)}.
Let us now take ϵ<-λ(k0) and consider the functions
(4.33)ηϵ(u;k):=(λ(k0)+ϵ)D(u;k),Ψϵ(u;k):=η̇ϵ(u;k)-h(u;k)+γ(k)+f(u;k)D(u;k)ηϵ(u;k).
We have
(4.34)Ψϵ(u;k)=λ(k0)Ḋ(u;k)-h(u;k)+γ(k)+f(u;k)λ(k0)+ϵ+ϵḊ(u;k)=λ(k0)Ḋ(u;k)-h(u;k)+γ(k)+f(u;k)λ(k0)-ϵf(u;k)λ(k0)(λ(k0)+ϵ)+ϵḊ(u;k),
and, from (4.23), we get
(4.35)lim⁡(u,k)→(α(k0),k0)Ψϵ(u;k)=-ϵf(α(k0);k0)λ(k0)(λ(k0)+ϵ)+ϵḊ(α(k0);k0)=:-2σ̃<0.
Hence, there exists a value δ̃1>0 such that
(4.36)Ψϵ(u;k)≤-σ<0,whenever|u-α(k0)|<δ̃1,|k-k0|<δ̃1,
that is,
(4.37)η̇ϵ(u;k)<h(u;k)-γ(k)-f(u;k)D(u;k)ηϵ(u;k),whenever|u-α(k0)|<δ̃1,|k-k0|<δ̃1.
Let δ̃∈(0,δ̃1] satisfying |α(k)-α(k0)|<(1/2)δ̃1 and λ(k)<λ(k0)+ϵ for |k-k0|<δ̃. Observe that, by (4.20), we have żk(α(k))>η̇ϵ(α(k);k). Then, since zk(α(k))=ηϵ(α(k);k)=0, for every k∈(k0-δ̃,k0+δ̃), there exists a value ρ̃=ρ̃(k)∈(0,α(k)), such that
(4.38)zk(u)<ηϵ(u;k),foreveryu∈(α(k)-ρ̃(k),α(k)),zk(u)>ηϵ(u;k),foreveryu∈(α(k),α(k)+ρ̃(k)).
Assume, by contradiction, the existence of k∈(k0-δ̃,k0+δ̃) and of ṽ(k)∈[α(k0)-δ̃1,α(k)) such that
(4.39)zk(ṽ(k))=ηϵ(ṽ(k);k),zk(u)<ηϵ(u;k)in(ṽ(k),α(k)).
Then,
(4.40)żk(ṽ(k))=h(ṽ(k);k)-γ(k)-f(ṽ(k);k)D(ṽ(k);k)zk(ṽ(k))=h(ṽ(k);k)-γ(k)-f(ṽ(k);k)D(ṽ(k);k)ηϵ(ṽ(k);k)>η̇ϵ(ṽ(k);k),
in contradiction with the definition of ṽ(k). Therefore, we deduce that
(4.41)zk(u)<ηϵ(u;k)<0,foreveryu∈[α(k0)-δ̃1,α(k)),k∈[k0-δ̃,k0+δ̃].
Similarly one can prove that
(4.42)zk(u)>ηϵ(u;k)>0,foreveryu∈(α(k),α(k0)+δ̃1],k∈[k0-δ̃,k0+δ̃].
The previous conditions (4.41) and (4.42) imply that
(4.43)zk(u)D(u;k)<ηϵ(u;k)D(u;k)=λ(k0)+ϵ,foreveryu∈[α(k0)-δ̃1,α(k0)+δ̃1]∖{α(k)}.
Taking the definition of function ψ into account, since the function λ(k) is continuous, by (4.32), (4.43), and the arbitrariness of ϵ>0, we deduce that
(4.44)lim⁡(u,k)→(α(k0),k0)ψ(u;k)=λ(k0)=ψ(α(k0);k0),
and this concludes the proof.Remark 4.4.
The same reasoning developed in Case1 of the proof of Lemma 4.3 also allows to discuss the continuity of ψ in all points (u,0) or (u,1) with u∈(0,1). More precisely, ψ is continuous in every set [a,b]×[α,β] where [a,b]⊂(0,1), [α,β]⊂[0,1] and α(k)∉[a,b] for all k∈[α,β].Lemma 4.5 (see [13, Lemma  2.5]).
Let(gn)n≥0,  gn:ℝ→[0,1], be a sequence of continuous decreasing functions satisfying
(4.45)lim⁡t→-∞gn(t)=1,lim⁡t→+∞gn(t)=0,n≥0.
Assume that gn(t)→g0(t) for every t in a dense subset of the interval (α0,β0):={t∈ℝ:0<g0(t)<1}. Then gn→g0 uniformly on ℝ.Proof of Theorem2.6.
Letk0∈[0,1]. According to Theorem 2.2, γ(k0)≥c*(k0), and then the profile uk0(ξ) is well defined.
For everyk∈[0,1], let zk(u)=z(u;γ(k),k) be the unique solution of problem (4.3), and let (ak,bk) be the maximal existence interval of the profile uk(ξ), with -∞≤ak<0<bk≤+∞, that is, uk(ak+)=1,uk(b-)=0. We firstly prove the pointwise convergence in every compact interval contained in (ak0,bk0). To this aim, let [ξ0,ξ1]⊂(ak0,bk0), with ξ0<0<ξ1, and let uk0([ξ0,ξ1])=:[v1,v0]⊂(0,1). First we consider the case when k0∈(0,1) and fix δ>0 such that [k0-δ,k0+δ]⊂(0,1). For k∈[k0-δ,k0+δ],uk(ξ) is the unique solution of the initial value problem (see Theorem 3.7):
(4.46)u′(ξ)=ψ(u(ξ);k),u(0)=u0,
where ψ(u;k) is the function defined in Lemma 4.3 and ψ is continuous for (u,k)∈(0,1)×(0,1). Further, uk is a decreasing C1-function on (ak,bk). For every k put
(4.47)τk:=max⁡{ξ0,uk-1(v0)}<0,tk:=min⁡{ξ1,uk-1(v1)}>0.
Let us consider the truncated functions ũk(ξ):=max{v1,min{uk(ξ),v0}}, defined in [ξ0,ξ1]. Of course, if τk>ξ0  [tk<ξ1] then ũk is constant in (ξ0,τk)  [(tk,ξ1)].
Sinceψ:[v0,v1]×[k0-δ,k0+δ]→ℝ is bounded, then the functions ũk(ξ) are equicontinuous in [ξ0,ξ1], for every k∈[k0-δ,k0+δ].Hence, for a fixed sequence(kn)n converging to k0, there exists a subsequence, denoted again (kn)n, such that (ũkn)n is uniformly convergent. Possibly passing to further subsequence, we can also assume that (τkn)n and (tkn)n, respectively, converge to some τ̃,t̃∈[ξ0,ξ1].Notice thatτ̃<0<t̃. Indeed, whenever τ̃k>ξ0, we have  ũkn(τkn)=v0, and, since ũkn(0)=u0 for every n, we get τ̃<0, owing to the continuity of the limit function of the sequence (ũkn)n. Similarly one can prove that t̃>0.Observe now that for a fixedξ∈(τ̃,t̃), we have
(4.48)ũkn(ξ)=ukn(ξ)=u0+∫0ξψ(ukn(t);kn)dt,
for n large enough; so from the dominated convergence theorem it follows that (ũkn)n uniformly converges to uk0 in (τ̃,t̃).Ifτ̃>ξ0, then ũkn(τkn)=v0 for every n sufficiently large; hence, by the above proved uniform convergence, we get uk0(τ̃)=v0, a contradiction since uk0 is strictly decreasing and uk0(ξ0)=v0. Hence, we have τ̃=ξ0 and similarly we can prove that t̃=ξ1. Therefore, we conclude that (ũkn(ξ))n=(ukn(ξ))n uniformly converges to uk0 in [ξ0,ξ1].Now we assumek0=0 and let δ>0 be such that α(k)∈(v0,1) for k∈(0,δ]. For k∈[0,δ] and ξ∈[ξ0,ξ1], the profile uk(ξ) is the unique solution of the initial value problem (see Theorem 3.7):
(4.49)u′(ξ)=zk(u)D(u;k),u(0)=u0.
According to Remark 4.4, zk(u)/D(u;k) converges to z0(u)/D(u;0) as k→0 for all u∈[v1,v0]. Hence, |zk(u)/D(u;k)| is bounded for (u,k)∈[v1,v0]×[0,δ], and the same reasoning as before can be repeated in order to prove the pointwise convergence of uk(ξ) to u0(ξ) in [ξ0,ξ1]. Similarly, when k0=1, we take δ>0 such that α(k)∈(0,v1) for k∈[1-δ,1). Since uk(ξ) is again the unique solution of (4.49) for k∈[1-δ,1] and ξ∈[ξ0,ξ1], we obtain also the pointwise convergence of uk(ξ) to u1(ξ) in [ξ0,ξ1].The uniform convergence on the all real line follows from Lemma4.5.



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*Source: 986738-2011-12-20.xml*