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Abstract and Applied Analysis
Volume 2011, Article ID 986738, 22 pages
Research Article

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

1Department of Engineering Sciences and Methods, University of Modena and Reggio Emilia, 42122 Reggio Emilia, Italy
2Department of Mathematical Sciences, Polytechnic University of Marche, 60131 Ancona, Italy
3Department of Electronic and Telecommunications, University of Florence, 50139 Florence, Italy

Received 26 July 2011; Revised 18 October 2011; Accepted 18 October 2011

Academic Editor: Michiel Bertsch

Copyright © 2011 Luisa Malaguti et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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: 𝑣𝑡+(𝑣)𝑣𝑥=𝐷(𝑣)𝑣𝑥𝑥+𝑓(𝑣),𝑡0,𝑥,(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 [13] 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 , 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), 𝑣0 and 𝑣1 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 𝑢𝐶(𝑐𝑙(𝑎,𝑏))𝐶2(𝑎,𝑏), satisfying lim𝜉𝑎+𝐷(𝑢(𝜉))𝑢(𝜉)=lim𝜉𝑏𝐷(𝑢(𝜉))𝑢(𝜉)=0,(1.3) such that 𝑣(𝑡,𝑥)=𝑢(𝑥𝑐𝑡) is a solution of (1.1) for all 𝑡(𝑎,𝑏). The symbol 𝑐𝑙(𝑎,𝑏) denotes the closure in of (𝑎,𝑏), and the quantity 𝜉=𝑥𝑐𝑡 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,,𝐷 defined in such an interval and, throughout the paper, we take ,𝑓𝐶[0,1] and 𝐷𝐶1[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 𝑣𝑡=𝑣𝑘0𝑣1𝑤𝑣𝑥𝑥+𝜇𝑣𝑥𝑥+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)=𝜇𝑣𝑘0(1𝑣/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 ̇𝐷(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 𝑐𝑐, 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: 𝑣𝑡+(𝑣;𝑘)𝑣𝑥=𝐷(𝑣;𝑘)𝑣𝑥𝑥+𝑓(𝑣;𝑘),𝑡0,𝑥,(1.7) which continuously depends on the real parameter 𝑘[0,1]. We take , 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 ̇𝛼(1)=0,𝛼(0)=1,𝛼(𝑘)(0,1),for0<𝑘<1,(1.9)𝐷(𝑢;𝑘)>0for𝑢(0,𝛼(𝑘)),𝐷(𝑢;𝑘)<0,for𝑢(𝛼(𝑘),1),(1.10)𝐷(𝛼(𝑘);𝑘)<0,for𝑘(0,1).(1.11) 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)𝑢+(𝑐(𝑢))𝑢𝑢𝑎+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 𝑢𝐶(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 𝜑isdierentiableat𝑢=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 𝑐𝑐, and the profile is unique, up to shifts. Moreover, the threshold value 𝑐 satisfies the following estimates, depending on the value 𝛼: (0)+2̇𝜑(0)𝑐max[]𝑢0,1(𝑢)+2sup𝑢(0,1]𝜑(𝑢)𝑢,if𝛼=1,(2.2)(1)+2̇𝜑(1)𝑐max[]𝑢0,1(𝑢)+2sup[𝑢0,1)𝜑(𝑢)𝑢1,if𝛼=0,(2.3) and, if 0<𝛼<1, 2maẋ𝜑(0)+(0),2̇𝜑(1)+(1)𝑐2maxsup]𝑢(0,𝛼𝜑(𝑢)𝑢+max[]𝑢0,𝛼(𝑢),2sup[𝑢𝛼,1)𝜑(𝑢)𝑢1+max[]𝑢𝛼,1.(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 , 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 (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 𝑐𝑐(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 ,𝐷,𝑓[0,1]×[0,1] 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,𝑘0𝜑(𝑢;𝑘)𝜑𝑢;𝑘0𝑢0,liminf(𝑢,𝑘)1,𝑘0𝜑(𝑢;𝑘)𝜑𝑢;𝑘01𝑢0,(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)𝜑(𝑢;𝑘)2𝑢1̇𝜑(1;𝑘)0,(2.6)limsup𝑘0̇𝜑(1;𝑘)+(1;0)2̇𝜑(0;0)+(0;0)(2.7) then 𝑐(𝑘) is continuous at 𝑘=0, and, similarly, if limsup(𝑢,𝑘)(0,1)𝜑(𝑢;𝑘)𝑢2̇𝜑(0;𝑘)0,(2.8)limsup𝑘1̇𝜑(0;𝑘)+(0;1)2̇𝜑(1;1)+(1;1)(2.9) then 𝑐(𝑘) is continuous at 𝑘=1.

Remark 2.3. Assume now that ̇𝜑(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 ̇𝜑(1;0)0 and also ̇𝜑(0;1)0 while ̇𝜑(1;𝑘)0 and ̇𝜑(0;𝑘)0 for 𝑘(0,1). The continuity of ̇𝜑, hence, implies that ̇𝜑(1;0)=̇𝜑(0;1)=0. Consequently, conditions (2.7) and (2.9), respectively, reduce to (1;0)2̇𝜑(0;0)+(0;0),(2.10)(0;1)2̇𝜑(1;1)+(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 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 2limsup𝑘0̇𝜑(1;𝑘)+(1;0)>2sup]𝑢(0,1𝜑(𝑢;0)𝑢+max[]𝑢0,1(𝑢;0),(2.12) then 𝑐(𝑘) is not continuous at 𝑘=0. Similarly, if 2limsup𝑘1̇𝜑(0;𝑘)+(0;1)>2sup[𝑢0,1)𝜑(𝑢;1)𝑢1+max[]𝑢0,1(𝑢;1),(2.13) then 𝑐(𝑘) is not continuous at 𝑘=1.

Example 2.5. Let 𝛼(𝑘)=1𝑘 and 𝐷(𝑢;𝑘)=𝛼(𝑘)𝑢=1𝑘𝑢, and let 𝑢𝑓(𝑢;𝑘)=𝑢1𝑢,if𝑘=0,min21𝑢,𝑘(1𝑢),if𝑘0.(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 𝑘0, and 𝑓 is continuous in [0,1]×[0,1]. Furthermore, also condition (2.1) is satisfied. Indeed, 𝜑(𝑢;0)=𝑢(1𝑢)3/2 is differentiable in [0,1], ̇𝜑(0;𝑘)=(1𝑘) and ̇𝜑(1;𝑘)=2 for every 𝑘(0,1], and finally 𝜑(𝑢;𝑘) is differentiable at 𝑢=𝛼(𝑘) since for 𝑢=𝛼(𝑘)=1𝑘 we have 𝑢1𝑢<(2/𝑘)(1𝑢), and then 𝑓(𝑢;𝑘) is differentiable at 𝑢=𝛼(𝑘). Since lim𝑢1𝜑(𝑢;𝑘)𝑢1=lim𝑢11𝑘𝑢𝑢12(1𝑢)𝑘=2,forevery𝑘0,(2.15) we have lim𝑘0̇𝜑(1;𝑘)=2, whereas sup𝑢(0,1]𝜑(𝑢;0)𝑢=sup]𝑢(0,1(1𝑢)3/2=1,(2.16) then (2.12) holds for every continuous function [0,1]×[0,1] such that (1;0)=max[0,1](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 ,𝐷,𝑓[0,1]×[0,1] 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 𝑈𝑘(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, (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 ,𝐷,𝑓[0,1][0,1], with 𝐷𝐶1[0,1], satisfying conditions (1.2), (1.5), (1.6), and (2.1), let us consider the following singular first-order boundary value problem ̇𝑧(𝑢)=(𝑢)𝑐𝜑(𝑢)𝑧0𝑧(𝑢),forevery𝑢(0,1){𝛼},+=𝑧(1)=0,𝑧(𝑢)𝜑(𝑢)<0,𝑢(0,1),𝑢𝛼,(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), 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 ,𝜑[1,2] be continuous functions such that 𝜑(1)=𝜑(2)=0, 𝜑(𝑢)>0 in (1,2), and ̇𝜑(1) exists finite. Then, there exists a value 𝑐, satisfying the following estimate: 2̇𝜑1+1𝑐2sup𝑢1,2𝜑(𝑢)𝑢1+max𝑢1,2(𝑢),(3.2) such that the problem ̇𝑧(𝑢)=(𝑢)𝑐𝜑(𝑢)𝑧(𝑢),forevery𝑢1,2,𝑧+1=𝑧2𝑧=0,(𝑢)𝜑(𝑢)<0,𝑢1,2,(3.3) admits solutions if and only if 𝑐𝑐. 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 ̇𝜑(2) exists finite, then ̇𝑧(2) exists finite too, and ̇𝑧2=122𝑐+2𝑐24̇𝜑2.(3.4)

Proof. Assume by contradiction that 0𝜆=liminf𝑢2𝑧(𝑢)𝑢2<limsup𝑢2𝑧(𝑢)𝑢2=Λ+.(3.5) For a fixed 𝛾(𝜆,Λ), let {𝑢𝑛}𝑛(0,2) be a sequence converging to 2 and such that 𝑧𝑢𝑛𝑢𝑛2d=𝛾,d𝑢𝑧(𝑢)𝑢2𝑢=𝑢𝑛0.(3.6) Since dd𝑢𝑧(𝑢)𝑢2𝑢=𝑢𝑛=1𝑢𝑛2𝑢̇𝑧𝑛𝛾,(3.7) we obtain ̇𝑧(𝑢𝑛)𝛾. Replacing this inequality in (3.3) evaluated for 𝑢=𝑢𝑛, and passing to the limit, we have 𝛾22𝑐𝛾+̇𝜑20.(3.8) Similarly, taking a sequence {𝑣𝑛}𝑛(0,2) converging to 2 and such that 𝑧𝑣𝑛𝑣𝑛2d=𝛾,d𝑢𝑧(𝑢)𝑢2𝑢=𝑣𝑛0,(3.9) arguing as before, we obtain 𝛾2[(2)𝑐]𝛾+̇𝜑(2)0. Combining with (3.8) we get 𝛾2[(2)𝑐]𝛾+̇𝜑(2)=0, which is impossible, due to the arbitrariness of 𝛾(𝜆,Λ). Then, 𝜆=Λ=lim𝑢2𝑧(𝑢)𝑢2.(3.10) Since (3.3) can be rewritten as follows: ̇𝑧(𝑢)=(𝑢)𝑐𝜑(𝑢)𝑢2𝑢2𝑧(𝑢),(3.11) then 𝜆=+ would imply that ̇𝑧(𝑢)(2)𝑐 as 𝑢2, a contradiction. Hence, 𝜆 is real, nonnegative.
If 𝜆>0, then from (3.11) we get that ̇𝑧(𝑢) admits limit as 𝑢2, and this limit must coincide with 𝜆. Passing to the limit in (3.11) we get that 𝜆 is a root of 𝜆2+𝑐2𝜆+̇𝜑2=0.(3.12) If ̇𝜑(2)=0, we immediately have ̇𝑧(𝑢)𝜆=(2)𝑐>0 as 𝑢2, since 𝜆>0. If ̇𝜑(2)<0, since the above trinomial has two discordant zeros, we deduce that 𝜆=122𝑐+2𝑐24̇𝜑2.(3.13) Summarizing, if 𝜆>0, then (3.13) holds.
Assume now that 𝜆=0. If ̇𝜑(2)<0; from (3.11) we get ̇𝑧(𝑢)+ as 𝑢2, a contradiction. Hence, ̇𝑧(2)=0 implies ̇𝜑(2)=0. Moreover, observe that 2𝑐=lim𝑢2(𝑢)𝑐liminf𝑢2(𝑢)𝑐𝜑(𝑢)𝑧(𝑢)=liminf𝑢2̇𝑧(𝑢)𝜆=0.(3.14) Thus, 𝜆 satisfies (3.13) also in the case 𝜆=0.

By a simple change of variable (𝑣=1+2𝑢 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 (1,2).

Proposition 3.3. Let ,𝜑[1,2] be continuous functions such that 𝜑(1)=𝜑(2)=0, 𝜑(𝑢)<0 in (1,2), and ̇𝜑(2) exists finite. Then, there exists a value 𝑐, satisfying the following estimate: 2̇𝜑2+2𝑐2sup𝑢1,2𝜑(𝑢)𝑢2+max𝑢1,2(𝑢),(3.15) such that problem (3.3) admits solutions if and only if 𝑐𝑐. Moreover, the solution is unique.

Corollary 3.4. Under the same assumptions of Proposition 3.3, if ̇𝜑(1) exists finite, then also ̇𝑧(1) exists finite, and ̇𝑧1=121𝑐+1𝑐24̇𝜑1.(3.16)

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

Theorem 3.5. Let ,𝜑[0,1] be continuous functions, with 𝜑(0)=𝜑(1)=0, 𝜑 differentiable at 𝑢=𝛼, satisfying condition (2.1) and such that 𝜑(𝑢)(𝛼𝑢)>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 𝑐𝑐. Moreover, the solution is unique.

Proof. Consider the boundary value problem (3.3) for 1=0,2=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 1=0 and 2=𝛼, is uniquely solvable if and only if 𝑐𝑐1. Similarly, considering the problem (3.3) for 1=𝛼,2=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 𝑐𝑐2. Let 𝑐=max{𝑐1,𝑐2}. Clearly, 𝑐 satisfies (2.4). Moreover, if problem (3.3), with 1=0and2=1, is solvable for some 𝑐, then also problem (3.3) with 1=0,2=𝛼, and problem (3.3), with 1=𝛼 and 2=1, are solvable for the same value 𝑐. Hence, 𝑐𝑐.
Conversely, let us fix 𝑐𝑐, and let 𝑧𝐶(0,1) be the function obtained gluing the unique solution of problem (3.3), with 1=0 and 2=𝛼 and the unique solution of problem (3.3) with 1=𝛼and2=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 𝜆𝜆+ 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 (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 𝑢0=𝐷𝜉𝑢𝜉𝐷(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{𝜉𝑢(𝜉)=𝛼} and 𝜉2=sup{𝜉𝑢(𝜉)=𝛼}. If 𝜉1<𝜉2 then 𝑢 is constant in (𝜉1,𝜉2), and we can define ̃𝑢(𝜉)=𝑢(𝜉) if 𝜉𝜉1; ̃𝑢(𝜉)=𝑢(𝜉+𝜉2𝜉1) if 𝜉>𝜉1. Of course, since (1.12) is autonomous, ̃𝑢 is a t.w.s. of (1.1) in its existence interval 𝐼=(𝑎,𝑏𝜉2+𝜉1) and satisfies 𝑢(𝜉)<0, for every 𝐼𝜉, 𝜉𝜉1. Hence, ̃𝑢 is strictly decreasing and invertible in 𝐼. Let 𝑢𝑧(𝑢)=𝐷(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)𝛼𝐷(𝑢),0<𝑢<𝛼,𝑢(0)=2,(3.19) defined on its maximal existence interval (𝑡1,𝑡2). Since 𝑢(𝜉)<0 whenever 𝜉(𝑡1,𝑡2), we have that 𝑢(𝑡2)=0 and 𝑢(𝑡+1)=𝛼. Moreover, according to (1.6) and Corollary 3.2, the limit lim𝜉𝑡+1𝑢(𝜉) exists and it is not zero, implying that 𝑡1. Consider now the unique solution 𝑢2(𝜉) of the initial value problem: 𝑢=𝑧(𝑢)𝐷(𝑢),𝛼<𝑢<1,𝑢(0)=1+𝛼2,(3.20) on its maximal existence interval (𝜏2,𝜏1). Again, 𝑢2(𝜉)<0 for 𝜉(𝜏1,𝜏2) implies 𝑢(𝜏+1)=1 and 𝑢(𝜏2)=𝛼. Moreover, according to (1.6) and Corollary 3.4, we obtain that lim𝜉𝜏2𝑢2(𝜉)=lim𝜉𝑡+1𝑢1(𝜉); hence, also the former limit is not zero and 𝜏2. 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), with 𝑢𝐶1((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 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]), lim𝜉𝑡+1𝑢(𝜉)=lim𝜉𝜏2𝑢(𝜉)=2𝑓(𝛼)𝑐+𝑐2̇4𝑓(𝛼)𝐷(𝛼).(3.21) Hence, both limits are finite and nonzero independently on the value of ̇𝐷(𝛼). So condition (1.6) can be removed.

Remark 3.8. If 𝐷(0)>0, then 𝑏=+. Indeed, ̇𝑧(𝑢)>(𝑢)𝑐, for 𝑢(0,𝛼), implies 𝑧(𝑢)>𝑢0[(𝑠)𝑐]d𝑠. Therefore, if 𝑢(𝑡0)=𝛼/2 for some 𝑡0, we have that 𝑏𝑡0=0𝛼/2̇𝑡(𝑢)d𝑢=0𝛼/2𝐷(𝑢)𝑧(𝑢)d𝑢0𝛼/2𝐷(𝑢)𝑢0[]𝑐(𝑠)d𝑠d𝑢.(3.22) Being an admissible value, the wave speed 𝑐 satisfies condition (2.4); hence, there exists 𝜎>0 such that 𝑢0[(𝑠)𝑐]d𝑠𝜎𝑢 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 ,𝐷,𝑓[0,1]×[0,1] 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,𝑘0𝜑(𝑢;𝑘)𝜑𝑢;𝑘0𝑢0,(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 ,𝐷,𝑓[0,1]×[0,1] 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,𝑘0𝜑(𝑢;𝑘)𝜑𝑢;𝑘01𝑢0,(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: ̇𝑧(𝑢)=(𝑢;𝑘)𝑐𝜑(𝑢;𝑘)𝑧0𝑧(𝑢),forevery𝑢(0,1){𝛼(𝑘)},+=𝑧(1)=0,𝑧(𝑢)𝜑(𝑢)<0,𝑢(0,1),𝑢𝛼(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: ̇𝑧(𝑢)=(𝑢;𝑘)𝑐𝜑(𝑢;𝑘)𝑧0𝑧(𝑢),𝑢(0,𝛼(𝑘)),+=𝑧(𝛼(𝑘))=0,𝑧(𝑢)<0,𝑢(0,𝛼(𝑘)).(𝑃1,𝑘)
Put (𝑢;𝑘)=𝛼(𝑘)(𝛼(𝑘)𝑢;𝑘) and 𝜑(𝑢;𝑘)=𝛼(𝑘)𝜑(𝛼(𝑘)𝑢;𝑘), problem (3.22) is equivalent to the normalized one: ̇𝑤(𝑢)=(𝑢;𝑘)𝑐𝛼(𝑘)𝜑(𝑢;𝑘)𝑤0𝑤(𝑢),𝑢(0,1),+=𝑤(1(𝑃)=0,𝑤(𝑢)<0,𝑢(0,1).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 (𝑃1,𝑘). Hence, if 𝑐1(𝑘) and ̃𝑐1(𝑘) denote, respectively, the minimal values of the parameter 𝑐 for which (3.22) and (𝑃1,𝑘) are solvable, we have 𝑐11(𝑘)=𝛼(𝑘)̃𝑐1(𝑘).(4.4) 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 every 𝑘(0,1], consider the problem: ̇𝑧(𝑢)=(𝑢;𝑘)𝑐𝜑(𝑢;𝑘)𝑧𝑧(𝑢),𝑢(𝛼(𝑘),1),𝛼(𝑘)+=𝑧(1)=0,𝑧(𝑢)>0,𝑢(𝛼(𝑘),1).(𝑃2,𝑘) Using an argument analogous to the previous one and taking into account that, in this case, 0𝛼(𝑘)<1, we deduce that the threshold value 𝑐2(𝑘) for problem (𝑃1,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{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).
If 𝑘0=0, then it results in 𝑐(0)=𝑐1(0), and liminf𝑘0𝑐(𝑘)=liminf𝑘0𝑐max1(𝑘),𝑐2(𝑘)liminf𝑘0𝑐1(𝑘)𝑐1(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 (𝑃1,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)𝜑𝑢;𝑘0𝑢=limsup(𝑢,𝑘)0,𝑘0𝛼𝑘(𝑘)𝜑(𝛼(𝑘)𝑢;𝑘)𝛼0𝜑𝛼𝑘0𝑢;𝑘0𝑢=limsup(𝑣,𝑘)0,𝑘0𝛼2(𝑘)𝜑(𝑣;𝑘)𝜑𝑣;𝑘0𝑣+lim(𝑢,𝑘)0,𝑘0𝛼2𝜑(𝑘)𝛼(𝑘)𝑢;𝑘0𝑢𝛼(𝑘)𝛼2𝑘0𝜑𝛼𝑘0𝑢;𝑘0𝑘𝑢𝛼00,(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 ̃𝑐1(𝑘) is continuous at 𝑘0, and in force of (4.4) we conclude that 𝑐1(𝑘) is continuous too. Analogously, considering the problem (𝑃1,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 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)𝑢1̇𝜑(1;𝑘)+𝜖,forevery𝑢(1𝛿,1),𝑘0,𝑘.(4.7) Let ̃̃𝑘=𝑘(𝜖)(0,𝑘) be such that 𝛼(𝑘)(1𝛿,1) for every ̃𝑘(0,𝑘). Then sup[𝑢𝛼(𝑘),1)𝜑(𝑢;𝑘)̃𝑘𝑢1̇𝜑(1;𝑘)+𝜖,forevery𝑘0,.(4.8) Taking estimate (3.15) into account, with 1=𝛼(𝑘) and 2=1, we get 𝑐2(𝑘)2̇𝜑(1;𝑘)+𝜖+max[]𝑢𝛼(𝑘),1̃𝑘(𝑢;𝑘),forevery𝑘0,,(4.9) implying limsup𝑘0𝑐2(𝑘)2limsup𝑘0̇𝜑(1;𝑘)+𝜖+(1;0).(4.10) By the arbitrariness of 𝜖>0, assumption (2.7), and estimate (3.2) with 1=0 and 2=𝛼(𝑘), we have limsup𝑘0𝑐2(𝑘)2̇𝜑(0;0)+(0;0)𝑐1(0)=𝑐(0).(4.11) Therefore, since 𝑐1(𝑘)𝑐1(0)=𝑐(0) as 𝑘0, we conclude that limsup𝑘0𝑐(𝑘)=limsup𝑘0𝑐max1(𝑘),𝑐2(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 ,𝐷,𝑓[0,1]×[0,1] 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)=𝑧(𝑢;𝛾(𝑘),𝑘)𝐷(𝑢;𝑘),for𝑢(0,1),𝑢𝛼(𝑘),𝜆(𝑘),for𝑢=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 (𝑢0𝛼(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 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 𝑘𝑘0, uniformly in [𝑎,2𝑏/(𝑏+𝛼(𝑘0))]; that is, for a fixed 𝜖>0, ||𝑤𝑘(𝑢)𝑤𝑘0||<1(𝑢)2𝜖forevery𝑢𝑎,2𝑏𝑘𝑏+𝛼0.(4.14) 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(𝑘0𝜂,𝑘0+𝜂) we have 𝛼(𝑘)>(1/2)(𝑏+𝛼(𝑘0)) and |||||𝑤𝑘0𝑢𝛼(𝑘)𝑤𝑘0𝑢𝛼𝑘0|||||<12[]𝜖,forevery𝑢𝑎,𝑏.(4.15) Since 𝑢/𝛼(𝑘)[𝑎,2𝑏/(𝑏+𝛼(𝑘0))] whenever 𝑢[𝑎,𝑏] and 𝑘(𝑘0𝜂,𝑘0+𝜂), we deduce that ||𝑧𝑘(𝑢)𝑧𝑘0||||||𝑤(𝑢)𝑘𝑢𝛼(𝑘)𝑤𝑘0𝑢||||+|||||𝑤𝛼(𝑘)𝑘0𝑢𝛼(𝑘)𝑤𝑘0𝑢𝛼𝑘0|||||𝜖,(4.16) for every 𝑘(𝑘0𝜂,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 lim𝑢(𝑢,𝑘)0,𝑘0𝑧𝑘(𝑢)=𝑧𝐷(𝑢;𝑘)𝑘0𝑢0𝐷𝑢0;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=1𝑢 allows to apply [11, Theorem  5.1].
Case 2 (𝑢0=𝛼(𝑘0)). For every 𝑘(0,1), let 𝜆(𝑘)=2𝑓(𝛼(𝑘);𝑘)(𝛼(𝑘);𝑘)+𝛾(𝑘)+[](𝛼(𝑘);𝑘)𝛾(𝑘)2̇4𝑓(𝛼(𝑘);𝑘)𝐷(𝛼(𝑘);𝑘).(4.18) Notice that the function 𝜆 is well defined since by assumption (1.6) we have ̇𝐷(𝛼(𝑘);𝑘)<0. Moreover, 𝜆 is continuous and negative for every 𝑘(0,1). Since 𝜆(𝑘)=(𝛼(𝑘);𝑘)𝛾(𝑘)+[](𝛼(𝑘);𝑘)𝛾(𝑘)2̇4𝑓(𝛼(𝑘);𝑘)𝐷(𝛼(𝑘);𝑘)2̇𝐷(𝛼(𝑘);𝑘),(4.19) from Corollary 3.2 it follows that ̇𝑧𝑘̇(𝛼(𝑘))=𝜆(𝑘)𝐷(𝛼(𝑘);𝑘).(4.20) Let us fix a value 𝜖>0 and define 𝜉𝜖𝜆𝑘(𝑢;𝑘)=0𝐷Φ𝜖(𝑢;𝑘),𝜖̇𝜉(𝑢;𝑘)=𝜖(𝑢;𝑘)(𝑢;𝑘)+𝛾(𝑘)+𝑓(𝑢;𝑘)𝐷(𝑢;𝑘)𝜉𝜖.(𝑢;𝑘)(4.21) We have Φ𝜖(𝑘𝑢;𝑘)=𝜆0̇𝐷(𝑢;𝑘)(𝑢;𝑘)+𝛾(𝑘)+𝑓(𝑢;𝑘)𝜆𝑘0̇𝑘𝜖𝜖𝐷(𝑢;𝑘)=𝜆0̇𝑓𝐷(𝑢;𝑘)(𝑢;𝑘)+𝛾(𝑘)+(𝑢;𝑘)𝜆𝑘0+𝜖𝑓(𝑢;𝑘)𝜆𝑘0𝜆𝑘0̇𝜖𝜖𝐷(𝑢;𝑘).(4.22) From the definition of 𝜆(𝑘), we get ̇𝜆(𝑘)𝐷(𝛼(𝑘);𝑘)(𝛼(𝑘);𝑘)+𝛾(𝑘)+𝑓(𝛼(𝑘);𝑘)𝜆(𝑘)=0.(4.23) Therefore, lim𝛼𝑘(𝑢,𝑘)0,𝑘0Φ𝜖𝑓𝛼𝑘(𝑢;𝑘)=𝜖0;𝑘0𝜆𝑘0𝜆𝑘0̇𝐷𝛼𝑘𝜖𝜖0;𝑘0=2𝜎>0.(4.24) Hence, there exists a value 𝛿1>0 such that Φ𝜖||𝑘(𝑢;𝑘)𝜎>0,whenever𝑢𝛼0||<𝛿1,||𝑘𝑘0||<𝛿1,(4.25) that is, ̇𝜉𝜖(𝑢;𝑘)>(𝑢;𝑘)𝛾(𝑘)𝑓(𝑢;𝑘)𝐷(𝑢;𝑘)𝜉𝜖||𝑘(𝑢;𝑘),whenever𝑢𝛼0||<𝛿1,||𝑘𝑘0||<𝛿1.(4.26) Take now 𝛿(0,𝛿1] such that |𝛼(𝑘)𝛼(𝑘0)|<(1/2)𝛿1 and 𝜆(𝑘)>𝜆(𝑘0)𝜖 whenever |𝑘𝑘0|<𝛿. Observe that, by (4.20), we have ̇𝑧𝑘̇𝜉(𝛼(𝑘))<𝜖(𝛼(𝑘);𝑘). Since 𝑧𝑘(𝛼(𝑘))=𝜉𝜖(𝛼(𝑘);𝑘)=0, then for every 𝑘(𝑘0𝛿,𝑘0+𝛿), there exists a value 𝜌=𝜌(𝑘)(0,𝛼(𝑘)) such that 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)).(4.27) Assume, by contradiction, the existence of a value 𝑘(𝑘0𝛿,𝑘0+𝛿) and of a point 𝑣(𝑘)[𝛼(𝑘0)𝛿1,𝛼(𝑘)) such that 𝑧𝑘(𝑣(𝑘))=𝜉𝜖(𝑣(𝑘);𝑘),𝑧𝑘(𝑢)>𝜉𝜖(𝑢;𝑘)in(𝑣(𝑘),𝛼(𝑘)).(4.28) Then, ̇𝑧𝑘(𝑣(𝑘))=(𝑣(𝑘);𝑘)𝛾(𝑘)𝑓(𝑣(𝑘);𝑘)𝐷(𝑣(𝑘);𝑘)𝑧𝑘(𝑣(𝑘))=(𝑣(𝑘);𝑘)𝛾(𝑘)𝑓(𝑣(𝑘);𝑘)𝐷(𝑣(𝑘);𝑘)𝜉𝜖<̇𝜉(𝑣(𝑘);𝑘)𝜖(𝑣(𝑘);𝑘),(4.29) in contradiction with the definition of 𝑣(𝑘). Therefore, we deduce that 𝜉𝜖(𝑢;𝑘)<𝑧𝑘𝛼𝑘(𝑢)<0,forevery𝑢0𝛿1𝑘,𝛼(𝑘),𝑘0𝛿,𝑘0+𝛿.(4.30) Similarly, we can prove that 𝜉𝜖(𝑢;𝑘)>𝑧𝑘𝛼𝑘(𝑢)>0,forevery𝑢(𝑘),𝛼0+𝛿1𝑘,𝑘0𝛿,𝑘0+𝛿.(4.31) The previous conditions (4.30) and (4.31) imply that 𝑧𝑘(𝑢)>𝜉𝐷(𝑢;𝑘)𝜖(𝑢;𝑘)𝑘𝐷(𝑢;𝑘)=𝜆0𝛼𝑘𝜖,forevery𝑢0𝛿1𝑘,𝛼0+𝛿1{𝛼(𝑘)}.(4.32) Let us now take 𝜖<𝜆(𝑘0) and consider the functions 𝜂𝜖𝜆𝑘(𝑢;𝑘)=0𝐷Ψ+𝜖(𝑢;𝑘),𝜖(𝑢;𝑘)=̇𝜂𝜖(𝑢;𝑘)(𝑢;𝑘)+𝛾(𝑘)+𝑓(𝑢;𝑘)𝐷(𝑢;𝑘)𝜂𝜖.(𝑢;𝑘)(4.33) We have Ψ𝜖(𝑘𝑢;𝑘)=𝜆0̇𝐷(𝑢;𝑘)(𝑢;𝑘)+𝛾(𝑘)+𝑓(𝑢;𝑘)𝜆𝑘0̇𝑘+𝜖+𝜖𝐷(𝑢;𝑘)=𝜆0̇𝑓𝐷(𝑢;𝑘)(𝑢;𝑘)+𝛾(𝑘)+(𝑢;𝑘)𝜆𝑘0𝜖𝑓(𝑢;𝑘)𝜆𝑘0𝜆𝑘0̇+𝜖+𝜖𝐷(𝑢;𝑘),(4.34) and, from (4.23), we get lim𝛼𝑘(𝑢,𝑘)0,𝑘0Ψ𝜖𝑓𝛼𝑘(𝑢;𝑘)=𝜖0;𝑘0𝜆𝑘0𝜆𝑘0̇𝐷𝛼𝑘+𝜖+𝜖0;𝑘0=2𝜎<0.(4.35) Hence, there exists a value ̃𝛿1>0 such that Ψ𝜖||𝑘(𝑢;𝑘)𝜎<0,whenever𝑢𝛼0||<̃𝛿1,||𝑘𝑘0||<̃𝛿1,(4.36) that is, ̇𝜂𝜖(𝑢;𝑘)<(𝑢;𝑘)𝛾(𝑘)𝑓(𝑢;𝑘)𝐷(𝑢;𝑘)𝜂𝜖||𝑘(𝑢;𝑘),whenever𝑢𝛼0||<̃𝛿1,||𝑘𝑘0||<̃𝛿1.(4.37) Let ̃̃𝛿𝛿(0,1] satisfying |𝛼(𝑘)𝛼(𝑘0̃𝛿)|<(1/2)1 and 𝜆(𝑘)<𝜆(𝑘0)+𝜖 for |𝑘𝑘0̃𝛿|<. Observe that, by (4.20), we have ̇𝑧𝑘(𝛼(𝑘))>̇𝜂𝜖(𝛼(𝑘);𝑘). Then, since 𝑧𝑘(𝛼(𝑘))=𝜂𝜖(𝛼(𝑘);𝑘)=0, for every 𝑘(𝑘0̃𝛿,𝑘0+̃𝛿), there exists a value ̃𝜌=̃𝜌(𝑘)(0,𝛼(𝑘)), such that 𝑧𝑘(𝑢)<𝜂𝜖𝑧(𝑢;𝑘),forevery𝑢(𝛼(𝑘)̃𝜌(𝑘),𝛼(𝑘)),𝑘(𝑢)>𝜂𝜖(𝑢;𝑘),forevery𝑢(𝛼(𝑘),𝛼(𝑘)+̃𝜌(𝑘)).(4.38) Assume, by contradiction, the existence of 𝑘(𝑘0̃𝛿,𝑘0+̃𝛿) and of ̃𝑣(𝑘)[𝛼(𝑘0̃𝛿)1,𝛼(𝑘)) such that 𝑧𝑘(̃𝑣(𝑘))=𝜂𝜖(̃𝑣(𝑘);𝑘),𝑧𝑘(𝑢)<𝜂𝜖̃(𝑢;𝑘)in(𝑣(𝑘),𝛼(𝑘)).(4.39) Then, ̇𝑧𝑘(̃̃̃̃𝑣(𝑘))=(𝑣(𝑘);𝑘)𝛾(𝑘)𝑓(𝑣(𝑘);𝑘)𝐷(𝑣(𝑘);𝑘)𝑧𝑘(̃̃̃̃𝑣(𝑘))=(𝑣(𝑘);𝑘)𝛾(𝑘)𝑓(𝑣(𝑘);𝑘)𝐷(𝑣(𝑘);𝑘)𝜂𝜖(̃𝑣(𝑘);𝑘)>̇𝜂𝜖(̃𝑣(𝑘);𝑘),(4.40) in contradiction with the definition of ̃𝑣(𝑘). Therefore, we deduce that 𝑧𝑘(𝑢)<𝜂𝜖𝛼𝑘(𝑢;𝑘)<0,forevery𝑢0̃𝛿1𝑘,𝛼(𝑘),𝑘0̃𝛿,𝑘0+̃𝛿.(4.41) Similarly one can prove that 𝑧𝑘(𝑢)>𝜂𝜖𝛼𝑘(𝑢;𝑘)>0,forevery𝑢(𝑘),𝛼0+̃𝛿1𝑘,𝑘0̃𝛿,𝑘0+̃𝛿.(4.42) The previous conditions (4.41) and (4.42) imply that 𝑧𝑘(𝑢)<𝜂𝐷(𝑢;𝑘)𝜖(𝑢;𝑘)𝑘𝐷(𝑢;𝑘)=𝜆0𝛼𝑘+𝜖,forevery𝑢0̃𝛿1𝑘,𝛼0+̃𝛿1{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 lim𝛼𝑘(𝑢,𝑘)0,𝑘0𝜓𝑘(𝑢;𝑘)=𝜆0𝛼𝑘=𝜓0;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