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

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.

1. Introduction

This paper deals with the reaction-diffusion equation: 𝑣_𝑡+ℎ(𝑣)𝑣_𝑥=𝐷(𝑣)𝑣_𝑥_𝑥+𝑓(𝑣),𝑡≥0,𝑥∈ℝ,(1.1) 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 function ℎ, 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 𝑓(𝑢)>0for𝑢∈(0,1),𝑓(0)=𝑓(1)=0.(1.2) 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 lim_𝜉→𝑎+𝐷(𝑢(𝜉))𝑢^(𝜉)=lim_{𝜉→𝑏−}𝐷(𝑢(𝜉))𝑢^(𝜉)=0,(1.3) 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 𝑣_𝑡=−𝑣𝑘₀1−𝑣𝑤𝑣^𝑥_𝑥+𝜇𝑣_𝑥𝑥+𝑓(𝑣),(1.4) 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 𝐷(𝑢)>0for𝑢∈(0,𝛼),𝐷(𝑢)<0,for𝑢∈(𝛼,1).(1.5) 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 𝐷̇(𝛼)<0,(1.6) 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: 𝑣_𝑡+ℎ(𝑣;𝑘)𝑣_𝑥=𝐷(𝑣;𝑘)𝑣_𝑥_𝑥+𝑓(𝑣;𝑘),𝑡≥0,𝑥∈ℝ,(1.7) 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 𝑓(𝑢;𝑘)>0for𝑢∈(0,1),𝑓(0;𝑘)=𝑓(1;𝑘)=0,(1.8) for all 𝑘. Further, we assume the existence of a continuous function 𝛼∶[0,1]→[0,1] such that 𝐷(𝑢;𝛼𝑘)(1>)0=0f,or𝛼𝐷𝑢̇(∈(0𝛼)((=0𝑘,)1𝛼;,𝑘(𝑘))<)𝛼,0(𝑘,𝐷)∈f(o𝑢r(;0𝑘𝑘,)1∈<),(00,,1f)ofo.rr0𝑢<∈𝑘(𝛼<(1𝑘,),1),((11(..111.019))) 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 profile 𝑢(𝜉) of (1.1), having wave speed 𝑐, corresponds to a regular solution of the boundary value problem: 𝐷(𝑢)𝑢𝑢^𝑎^+(𝑐−ℎ(𝑢))𝑢^+𝑓(𝑢)=0,₊=1,𝑢(𝑏⁻)=0,(1.12) 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 Section 4. Put 𝜑(𝑢)∶=𝐷(𝑢)𝑓(𝑢); in the whole section we assume that 𝜑isdifferentiableat𝑢=0and1.(2.1) 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: ℎ(0)+2√𝜑̇(0)≤𝑐^∗≤max_[]𝑢∈0,1ℎ(𝑢)+2sup_𝑢∈(0,1]𝜑(𝑢)𝑢,if𝛼=1,(2.2)ℎ(1)+2√𝜑̇(1)≤𝑐^∗≤max_[]𝑢∈0,1ℎ(𝑢)+2sup_[𝑢∈0,1)𝜑(𝑢)𝑢−1,if𝛼=0,(2.3) and, if 0<𝛼<1, max2√𝜑̇(0)+ℎ(0),2√𝜑̇(1)+ℎ(1)≤𝑐^∗≤max2sup_{]𝑢∈(0,𝛼}𝜑(𝑢)𝑢+max^{[]𝑢∈0,𝛼}ℎ(𝑢),2sup_{[𝑢∈𝛼,1)}𝜑(𝑢)𝑢−1+max^{[]𝑢∈𝛼,1}ℎ(𝑢).(2.4)

As stated in the Introduction section, the main aim of the paper is the study of the continuous dependence, both of the threshold value 𝑐^∗ 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 function 𝛼∶[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 value 𝑐^∗ 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 function 𝑐^∗(𝑘), the following result will be proven.

Theorem 2.2. Let 𝛼∶[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 some 𝑘₀∈(0,1) we have 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,(2.5) then 𝑐^∗ is continuous at 𝑘₀.
With regards to the behavior of 𝑐^∗ at the endpoints 𝑘=0 and 𝑘=1, if limsup_{(𝑢,𝑘)→(1,0)}2𝜑𝑢(−𝑢;1𝑘)−𝜑̇(1;𝑘)≤0,(2.6)limsup_𝑘→0𝜑̇(1;𝑘)+ℎ(1;0)≤2√𝜑̇(0;0)+ℎ(0;0)(2.7) then 𝑐^∗(𝑘) is continuous at 𝑘=0, and, similarly, if limsup_{(𝑢,𝑘)→(0,1)}2𝜑(𝑢𝑢;𝑘)−𝜑̇(0;𝑘)≤0,(2.8)limsup_𝑘→1𝜑̇(0;𝑘)+ℎ(0;1)≤2√𝜑̇(1;1)+ℎ(1;1)(2.9) then 𝑐^∗(𝑘) is continuous at 𝑘=1.

Remark 2.3. Assume now that 𝜑̇(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 ℎℎ((10;;01))≤≤22√√𝜑̇(0;0)+ℎ(0;0),(2.10)𝜑̇(1;1)+ℎ(1;1).(2.11) 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 Theorem 2.2, if 2limsup_𝑘→0𝜑̇(1;𝑘)+ℎ(1;0)>2sup_]𝑢∈(0,1𝜑(𝑢;0)𝑢+max^[]𝑢∈0,1ℎ(𝑢;0),(2.12) then 𝑐^∗(𝑘) is not continuous at 𝑘=0. Similarly, if 2limsup_𝑘→1𝜑̇(0;𝑘)+ℎ(0;1)>2sup_[𝑢∈0,1)𝜑(𝑢;1)𝑢−1+max^[]𝑢∈0,1ℎ(𝑢;1),(2.13) then 𝑐^∗(𝑘) is not continuous at 𝑘=1.

Example 2.5. Let 𝛼(𝑘)∶=1−𝑘 and 𝐷(𝑢;𝑘)∶=𝛼(𝑘)−𝑢=1−𝑘−𝑢, and let 𝑓(𝑢;𝑘)∶=⎧⎪⎨⎪⎩𝑢√min1−𝑢𝑢√,if𝑘=0,1−𝑢,2𝑘(1−𝑢),if𝑘≠0.(2.14) 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 𝑢√1−𝑢<(2/𝑘)(1−𝑢), and then 𝑓(𝑢;𝑘) is differentiable at 𝑢=𝛼(𝑘). Since lim_𝑢→1𝜑(𝑢;𝑘)𝑢−1=lim^𝑢→11−𝑘−𝑢𝑢−12(1𝑘−𝑢)=2,forevery𝑘≠0,(2.15) we have lim_𝑘→0𝜑̇(1;𝑘)=2, whereas sup_𝑢∈(0,1]𝜑(𝑢;0)𝑢=sup_]𝑢∈(0,1(1−𝑢)^3/2=1,(2.16) 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. Let 𝛼∶[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.
Let 𝑈_𝑘(𝜉)∶=𝑈(𝜉;𝛾(𝑘),𝑘) be the profile of the corresponding travelling wave solution of (1.1) with speed 𝛾(𝑘) such that 𝑈_𝑘(0)=𝑢₀, for some fixed 𝑢₀∈(0,1). Then 𝑈_𝑘(𝜉) converges to 𝑈_𝑘0(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 functions ℎ,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 𝑧̇(𝑢)=ℎ(𝑢)−𝑐−𝜑(𝑢)𝑧𝑧0(𝑢),forevery𝑢∈(0,1)⧵{𝛼},₊𝑧(𝑢)𝜑(𝑢)<0,=∀𝑧𝑢(1∈⁻)(=0,01,),𝑢≠𝛼,(3.1) 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. Let ℎ,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: 2𝜑̇ℓ₁+ℎℓ₁≤𝑐^∗≤2sup_{ℓ𝑢∈1,ℓ2}𝜑(𝑢)𝑢−ℓ₁+max_{ℓ𝑢∈1,ℓ2}ℎ(𝑢),(3.2) such that the problem 𝑧̇(𝑢)=ℎ(𝑢)−𝑐−𝜑(𝑢)𝑧(𝑢),forevery𝑢∈ℓ¹𝑧ℓ₊₁,ℓ₂,𝑧(𝑢)𝜑(𝑢)<0=,𝑧∀ℓ𝑢⁻²∈=ℓ0₁,,ℓ₂,(3.3) admits solutions if and only if 𝑐≥𝑐^∗. Moreover, the solution is unique.

As far as the regularity of the solution 𝑧 at the endpoints, the following result holds.

Corollary 3.2. Under the same assumptions of Proposition 3.1, if 𝜑̇(ℓ₂) exists finite, then 𝑧(̇ℓ₂) exists finite too, and 𝑧̇ℓ₂=12ℎℓ²−𝑐+ℎℓ₂−𝑐²−4𝜑̇ℓ₂.(3.4)

Proof. Assume by contradiction that 0≤𝜆⁻∶=liminf_{𝑢→ℓ−2}𝑧(𝑢)𝑢−ℓ₂<limsup_{𝑢→ℓ−2}𝑧(𝑢)𝑢−ℓ₂=Λ⁻≤+∞.(3.5) For a fixed 𝛾∈(𝜆⁻,Λ⁻), let {𝑢_𝑛}_𝑛⊂(0,ℓ₂) be a sequence converging to ℓ₂ and such that 𝑧𝑢_𝑛𝑢_𝑛−ℓ₂=𝛾,dd𝑢𝑢𝑧−(𝑢ℓ)₂_{𝑢=𝑢𝑛}≤0.(3.6) Since dd𝑢𝑢𝑧−(𝑢ℓ)₂_{𝑢=𝑢𝑛}=1𝑢_𝑛−ℓ₂𝑧̇𝑢_𝑛−𝛾,(3.7) we obtain 𝑧(̇𝑢_𝑛)≥𝛾. Replacing this inequality in (3.3) evaluated for 𝑢=𝑢_𝑛, and passing to the limit, we have 𝛾²−ℎℓ₂−𝑐𝛾+𝜑̇ℓ₂≤0.(3.8) Similarly, taking a sequence {𝑣_𝑛}_𝑛⊂(0,ℓ₂) converging to ℓ₂ and such that 𝑧𝑣_𝑛𝑣_𝑛−ℓ₂=𝛾,dd𝑢𝑢𝑧−(𝑢ℓ)₂_{𝑢=𝑣𝑛}≥0,(3.9) 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, 𝜆⁻=Λ⁻=lim_{𝑢→ℓ−2}𝑧(𝑢)𝑢−ℓ₂.(3.10) Since (3.3) can be rewritten as follows: 𝑧̇(𝑢)=ℎ(𝑢)−𝑐−𝜑(𝑢)𝑢−ℓ₂𝑢−ℓ₂𝑧(𝑢),(3.11) then 𝜆⁻=+∞ would imply that 𝑧(̇𝑢)→ℎ(ℓ₂)−𝑐 as 𝑢→ℓ⁻², a contradiction. Hence, 𝜆⁻ is real, nonnegative.
If 𝜆⁻>0, then from (3.11) we get that 𝑧(̇𝑢) admits limit as 𝑢→ℓ⁻², and this limit must coincide with 𝜆⁻. Passing to the limit in (3.11) we get that 𝜆⁻ is a root of 𝜆²+𝑐−ℎℓ₂𝜆+𝜑̇ℓ₂=0.(3.12) If 𝜑̇(ℓ₂)=0, we immediately have 𝑧(̇𝑢)→𝜆⁻=ℎ(ℓ₂)−𝑐>0 as 𝑢→ℓ⁻², since 𝜆⁻>0. If 𝜑̇(ℓ₂)<0, since the above trinomial has two discordant zeros, we deduce that 𝜆⁻=12ℎℓ²−𝑐+ℎℓ₂−𝑐²−4𝜑̇ℓ₂.(3.13) Summarizing, if 𝜆⁻>0, then (3.13) holds.
Assume now that 𝜆⁻=0. If 𝜑̇(ℓ₂)<0; from (3.11) we get 𝑧(̇𝑢)→+∞ as 𝑢→ℓ⁻², a contradiction. Hence, 𝑧(̇ℓ₂)=0 implies 𝜑̇(ℓ₂)=0. Moreover, observe that ℎℓ₂−𝑐=lim_{𝑢→ℓ−2}ℎ(𝑢)−𝑐≤liminf_{𝑢→ℓ−2}ℎ(𝑢)−𝑐−𝜑𝑧((𝑢𝑢))=liminf^{𝑢→ℓ−2}𝑧̇(𝑢)≤𝜆⁻=0.(3.14) Thus, 𝜆⁻ satisfies (3.13) also in the case 𝜆⁻=0.

By a simple change of variable (𝑣=ℓ₁+ℓ₂−𝑢 and 𝑧(̃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. Let ℎ,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: 2𝜑̇ℓ₂+ℎℓ₂≤𝑐^∗≤2sup_{ℓ𝑢∈1,ℓ2}𝜑(𝑢)𝑢−ℓ₂+max_{ℓ𝑢∈1,ℓ2}ℎ(𝑢),(3.15) 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 Proposition 3.3, if 𝜑̇(ℓ₁) exists finite, then also 𝑧(̇ℓ₁) exists finite, and 𝑧̇ℓ₁=12ℎℓ¹−𝑐+ℎℓ₁−𝑐²−4𝜑̇ℓ₁.(3.16)

Combining the previous results, we are able to prove an existence theorem for problem (3.1).

Theorem 3.5. Let ℎ,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 𝜑(𝑢)(𝛼−𝑢)>0,∀𝑢∈(0,1)⧵{𝛼},(3.17) for a given constant 𝛼∈(0,1).
Then, there exists a value 𝑐^∗, 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 𝑐^∗1, 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≥𝑐^∗1. Similarly, considering the problem (3.3) for ℓ₁=𝛼,ℓ₂=1, by Proposition 3.3, there is a threshold value 𝑐^∗2, satisfying (3.15), such that this problem is uniquely solvable if and only if 𝑐≥𝑐^∗2. Let 𝑐^∗∶=max{𝑐^∗1,𝑐^∗2}. 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 fix 𝑐≥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 of 𝜑 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. Let 𝑢(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 𝜉>𝜉^∗ such that 𝑢(𝜉)=1 and 𝑢′(𝜉)=0. Integrating (1.12) in (𝑎,𝜉), we obtain 0=𝐷𝑢𝜉𝑢^𝜉−𝐷(𝑢(𝑎))𝑢^(𝑎)=−^𝜉𝑎𝑓(𝑢(𝜉))𝑑𝜉,(3.18) 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 𝜉<𝜉^∗ such that 𝑢(𝜉)=0 and 𝑢′(𝜉)=0. Integrating (1.12) in (𝜉,𝑏), we get again a contradiction. Therefore, 𝑢′(𝜉)<0 whenever 𝑢(𝜉)∈(0,1), 𝑢(𝜉)≠𝛼.
Put 𝜉₁∶=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 𝑢(̃𝜉)∶=𝑢(𝜉) if 𝜉≤𝜉₁; 𝑢(̃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 𝐼∶=(𝑎,𝑏−𝜉₂+𝜉₁) and satisfies 𝑢^(𝜉)<0, for every 𝜉∈𝐼, 𝜉≠𝜉₁. Hence, 𝑢̃ is strictly decreasing and invertible in 𝐼. 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 that 0<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: 𝑢^=𝑧(𝑢)𝐷𝑢(𝑢(0),)=0𝛼<𝑢<𝛼,2,(3.19) defined on its maximal existence interval (𝑡₁,𝑡₂). Since 𝑢^(𝜉)<0 whenever 𝜉∈(𝑡₁,𝑡₂), we have that 𝑢(𝑡⁻²)=0 and 𝑢(𝑡⁺¹)=𝛼. Moreover, according to (1.6) and Corollary 3.2, the limit lim_𝜉→𝑡⁺¹𝑢^(𝜉) exists and it is not zero, implying that 𝑡₁∈ℝ. Consider now the unique solution 𝑢₂(𝜉) of the initial value problem: 𝑢^=𝑧(𝑢)𝑢𝐷(0(𝑢))=,1𝛼+<𝛼𝑢<1,2,(3.20) on its maximal existence interval (𝜏₂,𝜏₁). Again, 𝑢^2(𝜉)<0 for 𝜉∈(𝜏₁,𝜏₂) implies 𝑢(𝜏⁺¹)=1 and 𝑢(𝜏⁻²)=𝛼. Moreover, according to (1.6) and Corollary 3.4, we obtain that lim_𝜉→𝜏⁻²𝑢^2(𝜉)=lim_𝜉→𝑡⁺¹𝑢₁(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 𝑢_𝑖(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).
When 𝛼=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, assume ℎ≡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]), lim_{𝜉→𝑡+1}𝑢^(𝜉)=lim_{𝜉→𝜏−2}𝑢^(𝜉)=2𝑓(𝛼)𝑐+√𝑐²−4𝑓(𝛼)𝐷̇(𝛼).(3.21) Hence, both limits are finite and nonzero independently on the value of 𝐷̇(𝛼). So condition (1.6) can be removed.

Remark 3.8. If 𝐷(0)>0, then 𝑏=+∞. Indeed, 𝑧(̇𝑢)>ℎ(𝑢)−𝑐, for 𝑢∈(0,𝛼), implies 𝑧(𝑢)>∫^𝑢0[ℎ(𝑠)−𝑐]d𝑠. Therefore, if 𝑢(𝑡₀)=𝛼/2 for some 𝑡₀∈ℝ, we have that 𝑏−𝑡₀=^0𝛼/2𝑡̇(𝑢)d𝑢=^0𝛼/2𝐷(𝑢)−𝑧(𝑢)d𝑢≥^0𝛼/2𝐷(𝑢)∫^𝑢0[𝑐−ℎ(𝑠)]d𝑠d𝑢.(3.22) Being an admissible value, the wave speed 𝑐 satisfies condition (2.4); hence, there exists 𝜎>0 such that ∫^𝑢0[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, Theorem 2.1 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 parameter 𝑘 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. Let ℎ,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]limsup_{(𝑢,𝑘)→0,𝑘0}𝜑(𝑢;𝑘)−𝜑𝑢;𝑘₀𝑢≤0,(4.1) then 𝑐^∗ is continuous at 𝑘=𝑘₀.

By means of a change of variable, in the aggregative case, that is, in case 𝐷(𝑢;𝑘)<0 for every 𝑢∈(0,1), it can be proved that the analogous result holds.

Theorem 4.2. Let ℎ,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]liminf_{(𝑢,𝑘)→1,𝑘0}𝜑(𝑢;𝑘)−𝜑𝑢;𝑘₀1−𝑢≥0,(4.2) then 𝑐^∗ is continuous at 𝑘=𝑘₀.

Combining the results of Section 3 with Theorems 4.1 and 4.2, we are able to prove Theorem 2.2.

Proof of Theorem 2.2. In view of the equivalence proved in Theorem 3.7, we analyze the continuous dependence of the threshold value 𝑐^∗(𝑘) for the problem: 𝑧̇(𝑢)=ℎ(𝑢;𝑘)−𝑐−𝜑(𝑢;𝑘)𝑧𝑧0(𝑢),forevery𝑢∈(0,1)⧵{𝛼(𝑘)},₊𝑧(𝑢)𝜑(𝑢)<0,=∀𝑢𝑧∈(1(⁻0),=1)0,,𝑢≠𝛼(𝑘),(4.3) where 𝑐 is an unknown real constant.
For every 𝑘∈[0,1) we have 0<𝛼(𝑘)≤1; hence, we can consider the boundary value problem: 𝑧̇(𝑢)=ℎ(𝑢;𝑘)−𝑐−𝜑(𝑢;𝑘)𝑧0𝑧(𝑢),𝑢∈(0,𝛼(𝑘)),₊𝑧(𝑢)<0,=𝑧∀𝑢(𝛼∈(𝑘)(⁻0),𝛼=(𝑘0),).(𝑃^1,𝑘)
Put ℎ(𝑢;𝑘)∶=𝛼(𝑘)ℎ(𝛼(𝑘)𝑢;𝑘) and 𝜑(𝑢;𝑘)∶=𝛼(𝑘)𝜑(𝛼(𝑘)𝑢;𝑘), problem (3.22) is equivalent to the normalized one: 𝑤̇(𝑢)=ℎ(𝑢;𝑘)−𝑐𝛼(𝑘)−𝜑(𝑢;𝑘)𝑤0𝑤(𝑢),𝑢∈(0,1),₊𝑤(𝑢)<0=,𝑤∀𝑢(1∈⁻)(=0,01,).(𝑃^1,𝑘) 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 (𝑃_1,𝑘). Hence, if 𝑐^∗1(𝑘) and 𝑐̃^∗1(𝑘) denote, respectively, the minimal values of the parameter 𝑐 for which (3.22) and (𝑃_1,𝑘) are solvable, we have 𝑐^∗1(𝑘)=1𝛼(𝑘)𝑐̃^∗1(𝑘).(4.4) Since 𝛼(𝑘) is continuous and positive in [0,1), from Theorem 4.1 we have that 𝑐^∗1(𝑘) is lower semicontinuous at 𝑘₀ for every 𝑘₀∈[0,1).
Similarly, for every 𝑘∈(0,1], consider the problem: 𝑧̇(𝑢)=ℎ(𝑢;𝑘)−𝑐−𝜑(𝑢;𝑘)𝑧𝛼(𝑘)₊𝑧(𝑢),𝑢∈(𝛼(𝑘),1),𝑧(𝑢)>0,∀=𝑢𝑧∈(1(⁻𝛼)(=𝑘)0,,1).(𝑃^2,𝑘) 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 𝑐^∗2(𝑘) for problem (𝑃_1,𝑘) is a lower semicontinuous function at every 𝑘₀∈(0,1].
Let 𝑘∈(0,1) and let 𝑐^∗ be the threshold value for problem (4.3). Since 𝑐^∗(𝑘)=max{𝑐^∗1(𝑘),𝑐^∗2(𝑘)} (see the proof of Theorem 3.5), with 𝑐^∗1,𝑐^∗2 lower semicontinuous functions, then 𝑐^∗(𝑘) is lower semicontinuous at every 𝑘₀∈(0,1).
If 𝑘₀=0, then it results in 𝑐^∗(0)=𝑐^∗1(0), and liminf_𝑘→0𝑐^∗(𝑘)=liminf_𝑘→0max𝑐^∗1(𝑘),𝑐^∗2(𝑘)≥liminf_𝑘→0𝑐^∗1(𝑘)≥𝑐^∗1(0)=𝑐^∗(0).(4.5) 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 of 𝑐^∗ 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 (𝑃_1,𝑘). Notice that, if 𝜑 satisfies condition (4.1), then also the function 𝜑 does. Indeed, we have limsup_{(𝑢,𝑘)→(0,𝑘0)}𝜑(𝑢;𝑘)−𝜑𝑢;𝑘₀𝑢=limsup_{(𝑢,𝑘)→0,𝑘0}𝛼(𝑘)𝜑(𝛼(𝑘)𝑢;𝑘)−𝛼𝑘₀𝜑𝛼𝑘₀𝑢;𝑘₀=limsup_{(𝑣,𝑘)→0,𝑘0}𝑢𝛼²(𝑘)𝜑(𝑣;𝑘)−𝜑𝑣;𝑘⁰+lim_{(𝑢,𝑘)→0,𝑘0}𝑣𝛼²(𝑘)𝜑𝛼(𝑘)𝑢;𝑘⁰𝑢𝛼(𝑘)−𝛼²𝑘₀𝜑𝛼𝑘⁰𝑢;𝑘₀𝑢𝛼𝑘₀≤0,(4.6) 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 𝑐̃^∗1(𝑘) is continuous at 𝑘₀, and in force of (4.4) we conclude that 𝑐^∗1(𝑘) is continuous too. Analogously, considering the problem (𝑃_1,𝑘), we can prove that the threshold value 𝑐^∗2(𝑘) is continuous at every 𝑘₀∈(0,1). Since 𝑐^∗(𝑘)=max{𝑐^∗1(𝑘),𝑐^∗2(𝑘)} (see the proof of Theorem 3.5), then 𝑐^∗(𝑘) is continuous at every 𝑘₀∈(0,1).
It remains to study the continuity of 𝑐^∗ 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.
Let 𝜖>0 be fixed, and observe that, by assumption (2.6), real values 𝛿=𝛿(𝜖)>0 and 𝑘=𝑘(𝜖)∈(0,1) exist, such that 𝜑(𝑢;𝑘)𝑢−1≤𝜑̇(1;𝑘)+𝜖,forevery𝑢∈(1−𝛿,1),𝑘∈0,𝑘.(4.7) Let 𝑘̃=𝑘̃(𝜖)∈(0,𝑘) be such that 𝛼(𝑘)∈(1−𝛿,1) for every 𝑘∈(0,𝑘̃). Then sup_{[𝑢∈𝛼(𝑘),1)}𝜑(𝑢;𝑘)𝑢−1≤𝜑̇(1;𝑘)+𝜖,forevery𝑘∈0,𝑘̃.(4.8) Taking estimate (3.15) into account, with ℓ₁=𝛼(𝑘) and ℓ₂=1, we get 𝑐^∗2(𝑘)≤2√𝜑̇(1;𝑘)+𝜖+max_{[]𝑢∈𝛼(𝑘),1}ℎ(𝑢;𝑘),forevery𝑘∈0,𝑘̃,(4.9) implying limsup_𝑘→0𝑐^∗2(𝑘)≤2limsup_𝑘→0𝜑̇(1;𝑘)+𝜖+ℎ(1;0).(4.10) By the arbitrariness of 𝜖>0, assumption (2.7), and estimate (3.2) with ℓ₁=0 and ℓ₂=𝛼(𝑘), we have limsup_𝑘→0𝑐^∗2(𝑘)≤2√𝜑̇(0;0)+ℎ(0;0)≤𝑐^∗1(0)=𝑐^∗(0).(4.11) Therefore, since 𝑐^∗1(𝑘)→𝑐^∗1(0)=𝑐^∗(0) as 𝑘→0, we conclude that limsup_𝑘→0𝑐^∗(𝑘)=limsup_𝑘→0max𝑐^∗1(𝑘),𝑐^∗2(𝑘)=𝑐^∗(0),(4.12) 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 Theorem 2.6 about the convergence of the profiles, we need some preliminary results.

Let 𝑧(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. Let 𝛼∶[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 every 𝑘∈(0,1), there exists a real value 𝜆(𝑘) such that the function 𝜓(𝑢;𝑘)∶=⎧⎪⎨⎪⎩𝜆𝑧((𝑘𝑢𝐷;)𝛾,(𝑢(;𝑘𝑘)),𝑘),ffoorr𝑢𝑢∈=𝛼(0(,𝑘1)),𝑢≠𝛼(𝑘),(4.13) 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)). Assume 𝑢₀<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 𝑧_𝑘(𝑢)∶=𝑧(𝑢;𝛾(𝑘),𝑘) and let 𝑤_𝑘(𝑢)∶=𝑤(𝑢;𝛾(𝑘)𝛼(𝑘),𝑘) denote the solution of problem (𝑃_1,𝑘) (see the proof of Theorem 2.2). By applying [11, Theorem  5.1] to the problem (𝑃_1,𝑘) in [𝑎,2𝑏/(𝑏+𝛼(𝑘₀))]⊂(0,1), we get the convergence of 𝑤_𝑘(𝑢) to 𝑤_𝑘0(𝑢) as 𝑘→𝑘₀, uniformly in [𝑎,2𝑏/(𝑏+𝛼(𝑘₀))]; that is, for a fixed 𝜖>0, ||𝑤_𝑘(𝑢)−𝑤_𝑘0(𝑢)||<12𝜖forevery𝑢∈𝑎,𝑏+𝛼2𝑏𝑘₀.(4.14) Taking account of the continuity of 𝛼 and of the uniform continuity of 𝑤_𝑘0, 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 |||||𝑤_𝑘0𝛼(𝑢𝑘)−𝑤^𝑘0𝛼𝑢𝑘₀|||||<12𝜖,forevery𝑢∈[𝑎,𝑏].(4.15) Since 𝑢/𝛼(𝑘)∈[𝑎,2𝑏/(𝑏+𝛼(𝑘₀))] whenever 𝑢∈[𝑎,𝑏] and 𝑘∈(𝑘₀−𝜂,𝑘₀+𝜂), we deduce that ||𝑧_𝑘(𝑢)−𝑧_𝑘0(𝑢)||≤||||𝑤_𝑘𝛼(𝑢𝑘)−𝑤^𝑘0𝛼(𝑢𝑘)||||+|||||𝑤^𝑘0𝛼(𝑢𝑘)−𝑤^𝑘0𝛼𝑢𝑘₀|||||≤𝜖,(4.16) for every 𝑘∈(𝑘₀−𝜂,𝑘₀+𝜂) and 𝑢∈[𝑎,𝑏], that is, the uniform convergence of 𝑧_𝑘 to 𝑧_𝑘0 in [𝑎,𝑏]. Taking account of the uniform continuity of 𝑧_𝑘0 in [0,1], we get 𝑧_𝑘(𝑢)→𝑧_𝑘0(𝑢₀) as (𝑢,𝑘)→(𝑢₀,𝑘₀). Finally, since 𝐷(𝑢₀,𝑘₀)≠0, by the continuity of 𝐷(𝑢;𝑘) we also get lim_{𝑢(𝑢,𝑘)→0,𝑘0}𝑧_𝑘(𝑢)𝐷(𝑢;𝑘)=𝑧^𝑘0𝑢₀𝐷𝑢₀;𝑘₀,(4.17) and the continuity of the function 𝜓 at (𝑢₀,𝑘₀) is proved.
Notice that, for the opposite case 𝑢₀>𝛼(𝑘₀), the usual change of variable 𝑣∶=1−𝑢 allows to apply [11, Theorem  5.1].Case 2 (𝑢₀=𝛼(𝑘₀)). For every 𝑘∈(0,1), let 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).(4.18) 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 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),(4.19) from Corollary 3.2 it follows that 𝑧̇_𝑘(𝛼(𝑘))=𝜆(𝑘)𝐷̇(𝛼(𝑘);𝑘).(4.20) Let us fix a value 𝜖>0 and define Φ𝜉_𝜖(𝑢;𝑘)=𝜆𝑘₀−𝜖𝐷(𝑢;𝑘),_𝜖(𝑢;𝑘)=𝜉̇_𝜖(𝑢;𝑘)−ℎ(𝑢;𝑘)+𝛾(𝑘)+𝑓(𝑢;𝑘)𝐷(𝑢;𝑘)𝜉_𝜖(𝑢;𝑘).(4.21) We have Φ_𝜖(𝑢;𝑘)=𝜆𝑘₀𝐷̇(𝑢;𝑘)−ℎ(𝑢;𝑘)+𝛾(𝑘)+𝑓(𝑢;𝑘)=𝜆𝑘₀𝜆𝑘₀−𝜖−𝜖𝐷̇(𝑢;𝑘)𝐷̇(𝑢;𝑘)−ℎ(𝑢;𝑘)+𝛾(𝑘)+𝑓(𝑢;𝑘)𝜆𝑘₀+𝜆𝑘₀𝜖𝑓𝜆(𝑢;𝑘𝑘₀)−𝜖−𝜖𝐷̇(𝑢;𝑘).(4.22) From the definition of 𝜆(𝑘), we get 𝜆(𝑘)𝐷̇(𝛼(𝑘);𝑘)−ℎ(𝛼(𝑘);𝑘)+𝛾(𝑘)+𝑓(𝛼(𝑘);𝑘)𝜆(𝑘)=0.(4.23) Therefore, lim_{𝛼𝑘(𝑢,𝑘)→0,𝑘0}Φ_𝜖(𝑢;𝑘)=𝜖𝑓𝛼𝑘⁰;𝑘₀𝜆𝑘₀𝜆𝑘₀−𝜖−𝜖𝐷̇𝛼𝑘⁰;𝑘₀=∶2𝜎>0.(4.24) Hence, there exists a value 𝛿₁>0 such that Φ_𝜖(𝑢;𝑘)≥𝜎>0,whenever||𝑢−𝛼𝑘₀||<𝛿₁,||𝑘−𝑘₀||<𝛿₁,(4.25) that is, 𝜉̇_𝜖(𝑢;𝑘)>ℎ(𝑢;𝑘)−𝛾(𝑘)−𝑓(𝑢;𝑘)𝐷(𝑢;𝑘)𝜉_𝜖(𝑢;𝑘),whenever||𝑢−𝛼𝑘⁰||<𝛿₁,||𝑘−𝑘₀||<𝛿₁.(4.26) 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 𝑧̇_𝑘(𝛼(𝑘))<𝜉̇_𝜖(𝛼(𝑘);𝑘). Since 𝑧_𝑘(𝛼(𝑘))=𝜉_𝜖(𝛼(𝑘);𝑘)=0, then for every 𝑘∈(𝑘₀−𝛿,𝑘₀+𝛿), there exists a value 𝜌=𝜌(𝑘)∈(0,𝛼(𝑘)) such that 00><𝑧𝑧_𝑘𝑘(𝑢)>𝜉_𝜖(𝑢;𝑘),forevery𝑢∈(𝛼(𝑘)−𝜌(𝑘),𝛼(𝑘)),(𝑢)<𝜉_𝜖(𝑢;𝑘),forevery𝑢∈(𝛼(𝑘),𝛼(𝑘)+𝜌(𝑘)).(4.27) 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 𝑧_𝑘(𝑣(𝑘))=𝜉_𝜖(𝑣(𝑘);𝑘),𝑧_𝑘(𝑢)>𝜉_𝜖(𝑢;𝑘)in(𝑣(𝑘),𝛼(𝑘)).(4.28) Then, 𝑧̇_𝑘(𝑣(𝑘))=ℎ(𝑣(𝑘);𝑘)−𝛾(𝑘)−𝑓(𝑣(𝑘);𝑘)𝐷(𝑣(𝑘);𝑘)=ℎ(𝑣(𝑘);𝑘)−𝛾(𝑘)−𝑓(𝑣(𝑘)𝑧;_𝑘𝑘()𝑣𝐷(𝑘()𝑣)(𝑘);𝑘)𝜉_𝜖(𝑣(𝑘);𝑘)<𝜉̇^𝜖(𝑣(𝑘);𝑘),(4.29) in contradiction with the definition of 𝑣(𝑘). Therefore, we deduce that 𝜉_𝜖(𝑢;𝑘)<𝑧_𝑘(𝑢)<0,forevery𝑢∈𝛼𝑘₀−𝛿₁,𝛼(𝑘),𝑘∈𝑘₀−𝛿,𝑘₀+𝛿.(4.30) Similarly, we can prove that 𝜉_𝜖(𝑢;𝑘)>𝑧_𝑘(𝑢)>0,forevery𝑢∈𝛼(𝑘),𝛼𝑘₀+𝛿₁,𝑘∈𝑘₀−𝛿,𝑘₀+𝛿.(4.31) The previous conditions (4.30) and (4.31) imply that 𝑧_𝑘(𝑢)𝐷(𝑢;𝑘)>𝜉^𝜖(𝑢;𝑘)𝐷(𝑢;𝑘)=𝜆𝑘⁰−𝜖,forevery𝑢∈𝛼𝑘₀−𝛿₁,𝛼𝑘₀+𝛿₁⧵{𝛼(𝑘)}.(4.32) Let us now take 𝜖<−𝜆(𝑘₀) and consider the functions Ψ𝜂_𝜖(𝑢;𝑘)∶=𝜆𝑘₀+𝜖𝐷(𝑢;𝑘),_𝜖(𝑢;𝑘)∶=𝜂̇_𝜖(𝑢;𝑘)−ℎ(𝑢;𝑘)+𝛾(𝑘)+𝑓(𝑢;𝑘)𝐷(𝑢;𝑘)𝜂_𝜖(𝑢;𝑘).(4.33) We have Ψ_𝜖(𝑢;𝑘)=𝜆𝑘₀𝐷̇(𝑢;𝑘)−ℎ(𝑢;𝑘)+𝛾(𝑘)+𝑓(𝑢;𝑘)=𝜆𝑘₀𝜆𝑘₀+𝜖+𝜖𝐷̇(𝑢;𝑘)𝐷̇(𝑢;𝑘)−ℎ(𝑢;𝑘)+𝛾(𝑘)+𝑓(𝑢;𝑘)𝜆𝑘₀−𝜆𝑘₀𝜖𝑓𝜆(𝑢;𝑘𝑘₀)+𝜖+𝜖𝐷̇(𝑢;𝑘),(4.34) and, from (4.23), we get lim_{𝛼𝑘(𝑢,𝑘)→0,𝑘0}Ψ_𝜖(𝑢;𝑘)=−𝜖𝑓𝛼𝑘⁰;𝑘₀𝜆𝑘₀𝜆𝑘₀+𝜖+𝜖𝐷̇𝛼𝑘⁰;𝑘₀=∶−2𝜎<0.(4.35) Hence, there exists a value 𝛿̃₁>0 such that Ψ_𝜖(𝑢;𝑘)≤−𝜎<0,whenever||𝑢−𝛼𝑘₀||<𝛿̃₁,||𝑘−𝑘₀||<𝛿̃₁,(4.36) that is, 𝜂̇_𝜖(𝑢;𝑘)<ℎ(𝑢;𝑘)−𝛾(𝑘)−𝑓(𝑢;𝑘)𝐷(𝑢;𝑘)𝜂_𝜖(𝑢;𝑘),whenever||𝑢−𝛼𝑘⁰||<𝛿̃₁,||𝑘−𝑘₀||<𝛿̃₁.(4.37) 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 𝑧̇_𝑘(𝛼(𝑘))>𝜂̇_𝜖(𝛼(𝑘);𝑘). Then, since 𝑧_𝑘(𝛼(𝑘))=𝜂_𝜖(𝛼(𝑘);𝑘)=0, for every 𝑘∈(𝑘₀−𝛿̃,𝑘₀+𝛿̃), there exists a value 𝜌̃=𝜌̃(𝑘)∈(0,𝛼(𝑘)), such that 𝑧𝑧_𝑘(𝑢)<𝜂_𝜖(𝑢;𝑘),forevery𝑢∈(𝛼(𝑘)−𝜌̃(𝑘),𝛼(𝑘)),_𝑘(𝑢)>𝜂_𝜖(𝑢;𝑘),forevery𝑢∈(𝛼(𝑘),𝛼(𝑘)+𝜌̃(𝑘)).(4.38) Assume, by contradiction, the existence of 𝑘∈(𝑘₀−𝛿̃,𝑘₀+𝛿̃) and of 𝑣(̃𝑘)∈[𝛼(𝑘₀)−𝛿̃₁,𝛼(𝑘)) such that 𝑧_𝑘(𝑣̃(𝑘))=𝜂_𝜖(𝑣̃(𝑘);𝑘),𝑧_𝑘(𝑢)<𝜂_𝜖(𝑢;𝑘)in(𝑣̃(𝑘),𝛼(𝑘)).(4.39) Then, 𝑧̇_𝑘(𝑣̃(𝑘))=ℎ(𝑣̃(𝑘);𝑘)−𝛾(𝑘)−𝑓(𝑣̃(𝑘);𝑘)𝐷(𝑣̃(𝑘);𝑘)=ℎ(𝑣̃(𝑘);𝑘)−𝛾(𝑘)−𝑓(𝑣̃(𝑘)𝑧;_𝑘𝑘()𝑣𝐷̃(𝑘()𝑣)̃(𝑘);𝑘)𝜂_𝜖(𝑣̃(𝑘);𝑘)>𝜂̇^𝜖(𝑣̃(𝑘);𝑘),(4.40) in contradiction with the definition of 𝑣(̃𝑘). Therefore, we deduce that 𝑧_𝑘(𝑢)<𝜂_𝜖(𝑢;𝑘)<0,forevery𝑢∈𝛼𝑘₀−𝛿̃₁,𝛼(𝑘),𝑘∈𝑘₀−𝛿̃,𝑘₀+𝛿̃.(4.41) Similarly one can prove that 𝑧_𝑘(𝑢)>𝜂_𝜖(𝑢;𝑘)>0,forevery𝑢∈𝛼(𝑘),𝛼𝑘₀+𝛿̃₁,𝑘∈𝑘₀−𝛿̃,𝑘₀+𝛿̃.(4.42) The previous conditions (4.41) and (4.42) imply that 𝑧_𝑘(𝑢)𝐷(𝑢;𝑘)<𝜂^𝜖(𝑢;𝑘)𝐷(𝑢;𝑘)=𝜆𝑘⁰+𝜖,forevery𝑢∈𝛼𝑘₀−𝛿̃₁,𝛼𝑘₀+𝛿̃₁⧵{𝛼(𝑘)}.(4.43) 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 lim_{𝛼𝑘(𝑢,𝑘)→0,𝑘0}𝜓(𝑢;𝑘)=𝜆𝑘₀=𝜓𝛼𝑘₀;𝑘₀,(4.44) and this concludes the proof.

Remark 4.4. The same reasoning developed in Case 1 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 (𝑔_𝑛)_𝑛≥0, 𝑔_𝑛∶ℝ→[0,1], be a sequence of continuous decreasing functions satisfying lim_{𝑡→−∞}𝑔_𝑛(𝑡)=1,lim_𝑡→+∞𝑔_𝑛(𝑡)=0,𝑛≥0.(4.45) Assume that 𝑔_𝑛(𝑡)→𝑔₀(𝑡) for every 𝑡 in a dense subset of the interval (𝛼₀,𝛽₀)∶={𝑡∈ℝ∶0<𝑔₀(𝑡)<1}. Then 𝑔_𝑛→𝑔₀ uniformly on ℝ.

Proof of Theorem 2.6. Let 𝑘₀∈[0,1]. According to Theorem 2.2, 𝛾(𝑘₀)≥𝑐^∗(𝑘₀), and then the profile 𝑢_𝑘0(𝜉) is well defined.
For every 𝑘∈[0,1], let 𝑧_𝑘(𝑢)=𝑧(𝑢;𝛾(𝑘),𝑘) be the unique solution of problem (4.3), and let (𝑎_𝑘,𝑏_𝑘) be the maximal existence interval of the profile 𝑢_𝑘(𝜉), with −∞≤𝑎_𝑘<0<𝑏_𝑘≤+∞, that is, 𝑢_𝑘(𝑎^+𝑘)=1,𝑢_𝑘(𝑏⁻)=0. We firstly prove the pointwise convergence in every compact interval contained in (𝑎_𝑘0,𝑏_𝑘0). To this aim, let [𝜉₀,𝜉₁]⊂(𝑎_𝑘0,𝑏_𝑘0), with 𝜉₀<0<𝜉₁, and let 𝑢_𝑘0([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],𝑢_𝑘(𝜉) is the unique solution of the initial value problem (see Theorem 3.7): 𝑢^(𝜉)𝑢=(0𝜓)=(𝑢𝑢(𝜉₀);𝑘),,(4.46) where 𝜓(𝑢;𝑘) is the function defined in Lemma 4.3 and 𝜓 is continuous for (𝑢,𝑘)∈(0,1)×(0,1). Further, 𝑢_𝑘 is a decreasing 𝐶¹-function on (𝑎_𝑘,𝑏_𝑘). For every 𝑘 put 𝜏_𝑘∶=max𝜉₀,𝑢^𝑘−1𝑣₀<0,𝑡_𝑘∶=min𝜉₁,𝑢^𝑘−1𝑣₁>0.(4.47) Let us consider the truncated functions 𝑢_𝑘̃(𝜉)∶=max{𝑣₁,min{𝑢_𝑘(𝜉),𝑣₀}}, defined in [𝜉₀,𝜉₁]. Of course, if 𝜏_𝑘>𝜉₀ [𝑡_𝑘<𝜉₁] then 𝑢_𝑘̃ is constant in (𝜉₀,𝜏_𝑘) [(𝑡_𝑘,𝜉₁)].
Since 𝜓∶[𝑣₀,𝑣₁]×[𝑘₀−𝛿,𝑘₀+𝛿]→ℝ is bounded, then the functions 𝑢_𝑘̃(𝜉) are equicontinuous in [𝜉₀,𝜉₁], for every 𝑘∈[𝑘₀−𝛿,𝑘₀+𝛿].

Hence, for a fixed sequence (𝑘_𝑛)_𝑛 converging to 𝑘₀, there exists a subsequence, denoted again (𝑘_𝑛)_𝑛, such that (𝑢̃_𝑘𝑛)_𝑛 is uniformly convergent. Possibly passing to further subsequence, we can also assume that (𝜏_𝑘𝑛)_𝑛 and (𝑡_𝑘𝑛)_𝑛, respectively, converge to some 𝜏̃,𝑡̃∈[𝜉₀,𝜉₁].

Notice that 𝜏̃<0<𝑡̃. Indeed, whenever 𝜏̃_𝑘>𝜉₀, we have 𝑢̃_𝑘𝑛(𝜏_𝑘𝑛)=𝑣₀, and, since 𝑢̃_𝑘𝑛(0)=𝑢₀ for every 𝑛, we get 𝜏̃<0, owing to the continuity of the limit function of the sequence (𝑢̃_𝑘𝑛)_𝑛. Similarly one can prove that 𝑡̃>0.

Observe now that for a fixed 𝜉∈(𝜏̃,𝑡)̃, we have 𝑢̃_𝑘𝑛(𝜉)=𝑢_𝑘𝑛(𝜉)=𝑢₀+^𝜉0𝜓𝑢_𝑘𝑛(𝑡);𝑘_𝑛d𝑡,(4.48) for 𝑛 large enough; so from the dominated convergence theorem it follows that (𝑢̃_𝑘𝑛)_𝑛 uniformly converges to 𝑢_𝑘0 in (𝜏̃,𝑡)̃.

If 𝜏̃>𝜉₀, then 𝑢̃_𝑘𝑛(𝜏_𝑘𝑛)=𝑣₀ for every 𝑛 sufficiently large; hence, by the above proved uniform convergence, we get 𝑢_𝑘0(𝜏̃)=𝑣₀, a contradiction since 𝑢_𝑘0 is strictly decreasing and 𝑢_𝑘0(𝜉₀)=𝑣₀. Hence, we have 𝜏̃=𝜉₀ and similarly we can prove that 𝑡̃=𝜉₁. Therefore, we conclude that (𝑢̃_𝑘𝑛(𝜉))_𝑛=(𝑢_𝑘𝑛(𝜉))_𝑛 uniformly converges to 𝑢_𝑘0 in [𝜉₀,𝜉₁].

Now we assume 𝑘₀=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 𝑢_𝑘(𝜉) is the unique solution of the initial value problem (see Theorem 3.7): 𝑢^(𝜉)=𝑧^𝑘(𝑢)𝑢(0)𝐷=(𝑢𝑢₀;𝑘),.(4.49) According to Remark 4.4, 𝑧_𝑘(𝑢)/𝐷(𝑢;𝑘) converges to 𝑧₀(𝑢)/𝐷(𝑢;0) as 𝑘→0 for all 𝑢∈[𝑣₁,𝑣₀]. Hence, |𝑧_𝑘(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 𝑢_𝑘(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 𝑢_𝑘(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 𝑢_𝑘(𝜉) to 𝑢₁(𝜉) in [𝜉₀,𝜉₁].

The uniform convergence on the all real line follows from Lemma 4.5.

Acknowledgment

This research is partially supported by the M.U.R.S.T. (Italy) as the PRIN Project “Equazioni differenziali ordinarie e applicazioni”.