Abstract and Applied Analysis

Abstract and Applied Analysis / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 986738 | https://doi.org/10.1155/2011/986738

Luisa Malaguti, Cristina Marcelli, Serena Matucci, "Continuous Dependence in Front Propagation for Convective Reaction-Diffusion Models with Aggregative Movements", Abstract and Applied Analysis, vol. 2011, Article ID 986738, 22 pages, 2011. https://doi.org/10.1155/2011/986738

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

Academic Editor: Michiel Bertsch
Received26 Jul 2011
Revised18 Oct 2011
Accepted18 Oct 2011
Published20 Dec 2011


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: 𝑣𝑑+β„Ž(𝑣)𝑣π‘₯=𝐷(𝑣)𝑣π‘₯ξ€Έπ‘₯+𝑓(𝑣),𝑑β‰₯0,π‘₯βˆˆβ„,(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 β„Ž, 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(𝑒)>0forπ‘’βˆˆ(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(𝑑,π‘₯) is said to be a t.w.s. of (1.1) with wave speed 𝑐 if there exist a maximal interval (π‘Ž,𝑏), with βˆ’βˆžβ‰€π‘Ž<𝑏≀+∞, and a function π‘’βˆˆπΆ(𝑐𝑙(π‘Ž,𝑏))∩𝐢2(π‘Ž,𝑏), satisfying limπœ‰β†’π‘Ž+𝐷(𝑒(πœ‰))π‘’ξ…ž(πœ‰)=limπœ‰β†’π‘βˆ’π·(𝑒(πœ‰))π‘’ξ…ž(πœ‰)=0,(1.3) such that 𝑣(𝑑,π‘₯)=𝑒(π‘₯βˆ’π‘π‘‘) is a solution of (1.1) for all π‘‘βˆˆ(π‘Ž,𝑏). The symbol 𝑐𝑙(π‘Ž,𝑏) denotes the closure in ℝ of (π‘Ž,𝑏), and the quantity πœ‰=π‘₯βˆ’π‘π‘‘ 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,β„Ž,𝐷 defined in such an interval and, throughout the paper, we take β„Ž,π‘“βˆˆπΆ[0,1] and 𝐷∈𝐢1[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 𝑣𝑑=βˆ’π‘£π‘˜0𝑣1βˆ’π‘€ξ‚π‘£π‘₯ξ‚„π‘₯+πœ‡π‘£π‘₯π‘₯+𝑓(𝑣),(1.4) where πœ‡, π‘˜0, and 𝑀 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)=πœ‡βˆ’π‘£π‘˜0(1βˆ’π‘£/𝑀), 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 π‘£βˆˆ[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 π›Όβˆˆ[0,1] satisfying 𝐷(𝑒)>0forπ‘’βˆˆ(0,𝛼),𝐷(𝑒)<0,forπ‘’βˆˆ(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 π›Όβˆˆ(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 𝑐β‰₯π‘βˆ—, and we give an estimate of the threshold speed π‘βˆ—.

The very nature of aggregative processes causes ill posedness of our model (1.1) when π›Όβˆˆ(0,1) and π‘£βˆˆ[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: 𝑣𝑑+β„Ž(𝑣;π‘˜)𝑣π‘₯=𝐷(𝑣;π‘˜)𝑣π‘₯ξ€Έπ‘₯+𝑓(𝑣;π‘˜),𝑑β‰₯0,π‘₯βˆˆβ„,(1.7) which continuously depends on the real parameter π‘˜βˆˆ[0,1]. We take β„Ž, 𝐷, and 𝑓 in 𝐢([0,1]2) with 𝐷(β‹…;π‘˜)∈𝐢1[0,1] and 𝑓(𝑒;π‘˜)>0forπ‘’βˆˆ(0,1),𝑓(0;π‘˜)=𝑓(1;π‘˜)=0,(1.8) for all π‘˜. Further, we assume the existence of a continuous function π›ΌβˆΆ[0,1]β†’[0,1] such that ̇𝛼(1)=0,𝛼(0)=1,𝛼(π‘˜)∈(0,1),for0<π‘˜<1,(1.9)𝐷(𝑒;π‘˜)>0forπ‘’βˆˆ(0,𝛼(π‘˜)),𝐷(𝑒;π‘˜)<0,forπ‘’βˆˆ(𝛼(π‘˜),1),(1.10)𝐷(𝛼(π‘˜);π‘˜)<0,forπ‘˜βˆˆ(0,1).(1.11) We discuss both the case π‘˜βˆˆ(0,1), corresponding to a diffusive-aggregative dynamic, as well as the case π‘˜=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 π‘βˆ—(π‘˜) 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 π‘βˆ—(π‘˜) is continuous on [0,1] or that it fails to be continuous for π‘˜=0 or π‘˜=1. Two simple illustrative examples complete this discussion. In Theorem 2.6 we show the continuous dependence, on the parameter π‘˜, of any family of profiles 𝑒(πœ‰;π‘˜) corresponding to a continuum of wave speeds 𝛾(π‘˜). 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 𝑒(πœ‰) of (1.1), having wave speed 𝑐, corresponds to a regular solution of the boundary value problem: 𝐷(𝑒)π‘’ξ…žξ€Έξ…ž+(π‘βˆ’β„Ž(𝑒))π‘’ξ…žπ‘’ξ€·π‘Ž+𝑓(𝑒)=0,+ξ€Έ=1,𝑒(π‘βˆ’)=0,(1.12) where βˆ’βˆžβ‰€π‘Ž<π‘β‰€βˆž. More precisely, a solution of (1.12) in (π‘Ž,𝑏) is a function π‘’βˆˆπΆ(𝑐𝑙(π‘Ž,𝑏))∩𝐢2(π‘Ž,𝑏) satisfying the boundary conditions (1.3).

Since, in addition, 𝑒 is strictly monotone in every interval where 0<𝑒(πœ‰)<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 πœ‘(𝑒)∢=𝐷(𝑒)𝑓(𝑒); in the whole section we assume that πœ‘isdifferentiableat𝑒=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 π‘βˆ— such that (1.1) supports t.w.s. if and only if 𝑐β‰₯π‘βˆ—, and the profile is unique, up to shifts. Moreover, the threshold value π‘βˆ— satisfies the following estimates, depending on the value 𝛼: βˆšβ„Ž(0)+2Μ‡πœ‘(0)β‰€π‘βˆ—β‰€max[]π‘’βˆˆ0,1ξƒŽβ„Ž(𝑒)+2supπ‘’βˆˆ(0,1]πœ‘(𝑒)𝑒,if𝛼=1,(2.2)βˆšβ„Ž(1)+2Μ‡πœ‘(1)β‰€π‘βˆ—β‰€max[]π‘’βˆˆ0,1ξƒŽβ„Ž(𝑒)+2sup[π‘’βˆˆ0,1)πœ‘(𝑒)π‘’βˆ’1,if𝛼=0,(2.3) and, if 0<𝛼<1, 2√maxβˆšΜ‡πœ‘(0)+β„Ž(0),2ξ‚‡Μ‡πœ‘(1)+β„Ž(1)β‰€π‘βˆ—ξƒ―2ξƒŽβ‰€maxsup]π‘’βˆˆ(0,π›Όπœ‘(𝑒)𝑒+max[]π‘’βˆˆ0,π›ΌξƒŽβ„Ž(𝑒),2sup[π‘’βˆˆπ›Ό,1)πœ‘(𝑒)π‘’βˆ’1+max[]π‘’βˆˆπ›Ό,1ξƒ°.β„Ž(𝑒)(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 π‘βˆ— and of the wave profiles 𝑒, with respect to the coefficients β„Ž, 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 π›ΌβˆΆ[0,1]β†’[0,1] satisfying conditions (1.9), and we introduce a real parameter π‘˜ varying in [0,1], inside the coefficients of (1.1), in such a way that β„Ž(𝑒;π‘˜), 𝐷(𝑒;π‘˜), and 𝑓(𝑒;π‘˜) are continuous functions of two variables. We further assume (1.8), (1.10), and (1.11) and suppose that πœ‘(𝑒;β‹…) satisfy (2.1) for each π‘˜βˆˆ[0,1].

In this framework, the threshold value π‘βˆ— is a function of the parameter π‘˜, say π‘βˆ—=π‘βˆ—(π‘˜), and, for every 𝑐β‰₯π‘βˆ—(π‘˜), the profile 𝑒 (modulo shifts) of the t.w.s. having speed 𝑐 is a function of (πœ‰,π‘˜), where πœ‰ is the wave variable, say 𝑒=𝑒(πœ‰;π‘˜).

As for the function π‘βˆ—(π‘˜), the following result will be proven.

Theorem 2.2. Let π›ΌβˆΆ[0,1]β†’[0,1] and β„Ž,𝐷,π‘“βˆΆ[0,1]Γ—[0,1]→ℝ be continuous functions satisfying (1.8), (1.9), (1.10), and (1.11). For all π‘˜βˆˆ[0,1], assume that condition (2.1) holds for πœ‘(β‹…;π‘˜), and let 𝐷(β‹…;π‘˜)∈𝐢1([0,1]). Then, π‘βˆ—(π‘˜) is l.s.c..
Further, if for some π‘˜0∈(0,1) we have limsupξ€·(𝑒,π‘˜)β†’0,π‘˜0ξ€Έπœ‘ξ€·(𝑒;π‘˜)βˆ’πœ‘π‘’;π‘˜0𝑒≀0,liminfξ€·(𝑒,π‘˜)β†’1,π‘˜0ξ€Έπœ‘ξ€·(𝑒;π‘˜)βˆ’πœ‘π‘’;π‘˜0ξ€Έ1βˆ’π‘’β‰₯0,(2.5) then π‘βˆ— is continuous at π‘˜0.
With regards to the behavior of π‘βˆ— at the endpoints π‘˜=0 and π‘˜=1, if limsup(𝑒,π‘˜)β†’(1,0)ξ‚»πœ‘(𝑒;π‘˜)ξ‚Ό2ξ‚™π‘’βˆ’1βˆ’Μ‡πœ‘(1;π‘˜)≀0,(2.6)limsupπ‘˜β†’0βˆšΜ‡πœ‘(1;π‘˜)+β„Ž(1;0)≀2Μ‡πœ‘(0;0)+β„Ž(0;0)(2.7) then π‘βˆ—(π‘˜) is continuous at π‘˜=0, and, similarly, if limsup(𝑒,π‘˜)β†’(0,1)ξ‚»πœ‘(𝑒;π‘˜)𝑒2ξ‚™βˆ’Μ‡πœ‘(0;π‘˜)≀0,(2.8)limsupπ‘˜β†’1βˆšΜ‡πœ‘(0;π‘˜)+β„Ž(0;1)≀2Μ‡πœ‘(1;1)+β„Ž(1;1)(2.9) then π‘βˆ—(π‘˜) is continuous at π‘˜=1.

Remark 2.3. Assume now that Μ‡πœ‘(𝑒;π‘˜) exists and it is continuous on [0,1]Γ—[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 Μ‡πœ‘(1;0)≀0 and also Μ‡πœ‘(0;1)≀0 while Μ‡πœ‘(1;π‘˜)β‰₯0 and Μ‡πœ‘(0;π‘˜)β‰₯0 for π‘˜βˆˆ(0,1). The continuity of Μ‡πœ‘, hence, implies that Μ‡πœ‘(1;0)=Μ‡πœ‘(0;1)=0. Consequently, conditions (2.7) and (2.9), respectively, reduce to βˆšβ„Ž(1;0)≀2βˆšΜ‡πœ‘(0;0)+β„Ž(0;0),(2.10)β„Ž(0;1)≀2Μ‡πœ‘(1;1)+β„Ž(1;1).(2.11) Therefore, under conditions (2.10) and (2.11), the function π‘βˆ—(π‘˜) is continuous on the whole interval [0,1]. Of course, when β„Žβ‰‘0, then (2.10) and (2.11) trivially hold, and π‘βˆ—(π‘˜) 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 π‘βˆ—(π‘˜) could be discontinuous at the endpoints, as the following result states.

Proposition 2.4. Under the same conditions of Theorem 2.2, if 2ξ‚™limsupπ‘˜β†’0ξƒŽΜ‡πœ‘(1;π‘˜)+β„Ž(1;0)>2sup]π‘’βˆˆ(0,1πœ‘(𝑒;0)𝑒+max[]π‘’βˆˆ0,1β„Ž(𝑒;0),(2.12) then π‘βˆ—(π‘˜) is not continuous at π‘˜=0. Similarly, if 2ξ‚™limsupπ‘˜β†’1ξƒŽΜ‡πœ‘(0;π‘˜)+β„Ž(0;1)>2sup[π‘’βˆˆ0,1)πœ‘(𝑒;1)π‘’βˆ’1+max[]π‘’βˆˆ0,1β„Ž(𝑒;1),(2.13) then π‘βˆ—(π‘˜) is not continuous at π‘˜=1.

Example 2.5. Let 𝛼(π‘˜)∢=1βˆ’π‘˜ and 𝐷(𝑒;π‘˜)∢=𝛼(π‘˜)βˆ’π‘’=1βˆ’π‘˜βˆ’π‘’, and let ⎧βŽͺ⎨βŽͺβŽ©π‘’βˆšπ‘“(𝑒;π‘˜)∢=ξ‚†π‘’βˆš1βˆ’π‘’,ifπ‘˜=0,min21βˆ’π‘’,π‘˜ξ‚‡(1βˆ’π‘’),ifπ‘˜β‰ 0.(2.14) Observe that conditions (1.8), (1.9), and (1.10) are trivially satisfied. Moreover, 𝑓(𝑒;π‘˜) uniformly converges to 𝑓(𝑒;0) as π‘˜β†’0, and 𝑓 is continuous in [0,1]Γ—[0,1]. Furthermore, also condition (2.1) is satisfied. Indeed, πœ‘(𝑒;0)=𝑒(1βˆ’π‘’)3/2 is differentiable in [0,1], Μ‡πœ‘(0;π‘˜)=(1βˆ’π‘˜) and Μ‡πœ‘(1;π‘˜)=2 for every π‘˜βˆˆ(0,1], and finally πœ‘(𝑒;π‘˜) is differentiable at 𝑒=𝛼(π‘˜) since for 𝑒=𝛼(π‘˜)=1βˆ’π‘˜ we have π‘’βˆš1βˆ’π‘’<(2/π‘˜)(1βˆ’π‘’), and then 𝑓(𝑒;π‘˜) is differentiable at 𝑒=𝛼(π‘˜). Since lim𝑒→1πœ‘(𝑒;π‘˜)π‘’βˆ’1=lim𝑒→11βˆ’π‘˜βˆ’π‘’π‘’βˆ’12(1βˆ’π‘’)π‘˜=2,foreveryπ‘˜β‰ 0,(2.15) we have limπ‘˜β†’0Μ‡πœ‘(1;π‘˜)=2, whereas supπ‘’βˆˆ(0,1]πœ‘(𝑒;0)𝑒=sup]π‘’βˆˆ(0,1(1βˆ’π‘’)3/2=1,(2.16) then (2.12) holds for every continuous function β„ŽβˆΆ[0,1]Γ—[0,1]→ℝ such that β„Ž(1;0)=max[0,1]β„Ž(𝑒;0). Therefore, the function π‘βˆ— is not continuous at π‘˜=0; that is, π‘βˆ— 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 π›ΌβˆΆ[0,1]β†’[0,1] and β„Ž,𝐷,π‘“βˆΆ[0,1]Γ—[0,1]→ℝ be continuous functions satisfying (1.8), (1.9), (1.10), and (1.11). For all π‘˜βˆˆ[0,1], assume that condition (2.1) holds for πœ‘(β‹…;π‘˜), and let 𝐷(β‹…;π‘˜)∈𝐢1([0,1]). Take π›ΎβˆΆ[0,1]β†’[0,1] continuous and such that 𝛾(π‘˜)β‰₯π‘βˆ—(π‘˜) for every π‘˜βˆˆ[0,1], and let π‘˜0∈[0,1] be fixed.
Let π‘ˆπ‘˜(πœ‰)∢=π‘ˆ(πœ‰;𝛾(π‘˜),π‘˜) be the profile of the corresponding travelling wave solution of (1.1) with speed 𝛾(π‘˜) such that π‘ˆπ‘˜(0)=𝑒0, for some fixed 𝑒0∈(0,1). Then π‘ˆπ‘˜(πœ‰) converges to π‘ˆπ‘˜0(πœ‰) 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, β„Ž(𝑒;π‘˜)≑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 β„Ž,𝐷,π‘“βˆΆ[0,1]β†’[0,1], with 𝐷∈𝐢1[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(𝑒)=β„Ž(𝑒)βˆ’π‘βˆ’πœ‘(𝑒)𝑧0𝑧(𝑒),foreveryπ‘’βˆˆ(0,1)⧡{𝛼},+ξ€Έ=𝑧(1βˆ’)=0,𝑧(𝑒)πœ‘(𝑒)<0,βˆ€π‘’βˆˆ(0,1),𝑒≠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 π‘§βˆΆ(0,1)→ℝ, satisfying all the conditions in (3.1). Of course, any possible solution of (3.1) is of class 𝐢1 in any compact interval [π‘Ž,𝑏]βŠ‚(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 πœ‘ 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 β„Ž,πœ‘βˆΆ[β„“1,β„“2]→ℝ be continuous functions such that πœ‘(β„“1)=πœ‘(β„“2)=0, πœ‘(𝑒)>0 in (β„“1,β„“2), and Μ‡πœ‘(β„“1) exists finite. Then, there exists a value π‘βˆ—, satisfying the following estimate: 2ξ”ξ€·β„“Μ‡πœ‘1ξ€Έξ€·β„“+β„Ž1ξ€Έβ‰€π‘βˆ—ξƒŽβ‰€2supξ€·β„“π‘’βˆˆ1,β„“2ξ€»πœ‘(𝑒)π‘’βˆ’β„“1+maxξ€Ίβ„“π‘’βˆˆ1,β„“2ξ€»β„Ž(𝑒),(3.2) such that the problem ̇𝑧(𝑒)=β„Ž(𝑒)βˆ’π‘βˆ’πœ‘(𝑒)ℓ𝑧(𝑒),foreveryπ‘’βˆˆ1,β„“2ξ€Έ,𝑧ℓ+1ξ€Έξ€·β„“=π‘§βˆ’2𝑧ℓ=0,(𝑒)πœ‘(𝑒)<0,βˆ€π‘’βˆˆ1,β„“2ξ€Έ,(3.3) admits solutions if and only if 𝑐β‰₯π‘βˆ—. 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 Μ‡πœ‘(β„“2) exists finite, then ̇𝑧(β„“2) exists finite too, and ℓ̇𝑧2ξ€Έ=12ξ‚΅β„Žξ€·β„“2ξ€Έξ”βˆ’π‘+ξ€·β„Žξ€·β„“2ξ€Έξ€Έβˆ’π‘2ξ€·β„“βˆ’4Μ‡πœ‘2ξ€Έξ‚Ά.(3.4)

Proof. Assume by contradiction that 0β‰€πœ†βˆ’βˆΆ=liminfπ‘’β†’β„“βˆ’2𝑧(𝑒)π‘’βˆ’β„“2<limsupπ‘’β†’β„“βˆ’2𝑧(𝑒)π‘’βˆ’β„“2=Ξ›βˆ’β‰€+∞.(3.5) For a fixed π›Ύβˆˆ(πœ†βˆ’,Ξ›βˆ’), let {𝑒𝑛}π‘›βŠ‚(0,β„“2) be a sequence converging to β„“2 and such that π‘§ξ€·π‘’π‘›ξ€Έπ‘’π‘›βˆ’β„“2d=𝛾,ξ‚΅d𝑒𝑧(𝑒)π‘’βˆ’β„“2𝑒=𝑒𝑛≀0.(3.6) Since dξ‚΅d𝑒𝑧(𝑒)π‘’βˆ’β„“2𝑒=𝑒𝑛=1π‘’π‘›βˆ’β„“2ξ€Ίξ€·π‘’Μ‡π‘§π‘›ξ€Έξ€»βˆ’π›Ύ,(3.7) we obtain ̇𝑧(𝑒𝑛)β‰₯𝛾. Replacing this inequality in (3.3) evaluated for 𝑒=𝑒𝑛, and passing to the limit, we have 𝛾2βˆ’ξ€Ίβ„Žξ€·β„“2ξ€Έξ€»ξ€·β„“βˆ’π‘π›Ύ+Μ‡πœ‘2≀0.(3.8) Similarly, taking a sequence {𝑣𝑛}π‘›βŠ‚(0,β„“2) converging to β„“2 and such that π‘§ξ€·π‘£π‘›ξ€Έπ‘£π‘›βˆ’β„“2d=𝛾,ξ‚΅d𝑒𝑧(𝑒)π‘’βˆ’β„“2𝑒=𝑣𝑛β‰₯0,(3.9) arguing as before, we obtain 𝛾2βˆ’[β„Ž(β„“2)βˆ’π‘]𝛾+Μ‡πœ‘(β„“2)β‰₯0. Combining with (3.8) we get 𝛾2βˆ’[β„Ž(β„“2)βˆ’π‘]𝛾+Μ‡πœ‘(β„“2)=0, which is impossible, due to the arbitrariness of π›Ύβˆˆ(πœ†βˆ’,Ξ›βˆ’). Then, πœ†βˆ’=Ξ›βˆ’=limπ‘’β†’β„“βˆ’2𝑧(𝑒)π‘’βˆ’β„“2.(3.10) Since (3.3) can be rewritten as follows: ̇𝑧(𝑒)=β„Ž(𝑒)βˆ’π‘βˆ’πœ‘(𝑒)π‘’βˆ’β„“2π‘’βˆ’β„“2𝑧(𝑒),(3.11) then πœ†βˆ’=+∞ would imply that ̇𝑧(𝑒)β†’β„Ž(β„“2)βˆ’π‘ as π‘’β†’β„“βˆ’2, a contradiction. Hence, πœ†βˆ’ is real, nonnegative.
If πœ†βˆ’>0, then from (3.11) we get that ̇𝑧(𝑒) admits limit as π‘’β†’β„“βˆ’2, and this limit must coincide with πœ†βˆ’. Passing to the limit in (3.11) we get that πœ†βˆ’ is a root of πœ†2+ξ€·ξ€·β„“π‘βˆ’β„Ž2ξ€·β„“ξ€Έξ€Έπœ†+Μ‡πœ‘2ξ€Έ=0.(3.12) If Μ‡πœ‘(β„“2)=0, we immediately have ̇𝑧(𝑒)β†’πœ†βˆ’=β„Ž(β„“2)βˆ’π‘>0 as π‘’β†’β„“βˆ’2, since πœ†βˆ’>0. If Μ‡πœ‘(β„“2)<0, since the above trinomial has two discordant zeros, we deduce that πœ†βˆ’=12ξ‚΅β„Žξ€·β„“2ξ€Έξ”βˆ’π‘+ξ€·β„Žξ€·β„“2ξ€Έξ€Έβˆ’π‘2ξ€·β„“βˆ’4Μ‡πœ‘2ξ€Έξ‚Ά.(3.13) Summarizing, if πœ†βˆ’>0, then (3.13) holds.
Assume now that πœ†βˆ’=0. If Μ‡πœ‘(β„“2)<0; from (3.11) we get ̇𝑧(𝑒)β†’+∞ as π‘’β†’β„“βˆ’2, a contradiction. Hence, ̇𝑧(β„“2)=0 implies Μ‡πœ‘(β„“2)=0. Moreover, observe that β„Žξ€·β„“2ξ€Έβˆ’π‘=limπ‘’β†’β„“βˆ’2β„Ž(𝑒)βˆ’π‘β‰€liminfπ‘’β†’β„“βˆ’2ξ‚΅β„Ž(𝑒)βˆ’π‘βˆ’πœ‘(𝑒)𝑧(𝑒)=liminfπ‘’β†’β„“βˆ’2̇𝑧(𝑒)β‰€πœ†βˆ’=0.(3.14) Thus, πœ†βˆ’ satisfies (3.13) also in the case πœ†βˆ’=0.

By a simple change of variable (𝑣=β„“1+β„“2βˆ’π‘’ and ̃𝑧(𝑣)=βˆ’π‘§(𝑒)), it is easy to check the validity of the following results in the case when πœ‘ is everywhere negative in (β„“1,β„“2).

Proposition 3.3. Let β„Ž,πœ‘βˆΆ[β„“1,β„“2]→ℝ be continuous functions such that πœ‘(β„“1)=πœ‘(β„“2)=0, πœ‘(𝑒)<0 in (β„“1,β„“2), and Μ‡πœ‘(β„“2) exists finite. Then, there exists a value π‘βˆ—, satisfying the following estimate: 2ξ”ξ€·β„“Μ‡πœ‘2ξ€Έξ€·β„“+β„Ž2ξ€Έβ‰€π‘βˆ—ξƒŽβ‰€2supξ€Ίβ„“π‘’βˆˆ1,β„“2ξ€Έπœ‘(𝑒)π‘’βˆ’β„“2+maxξ€Ίβ„“π‘’βˆˆ1,β„“2ξ€»β„Ž(𝑒),(3.15) such that problem (3.3) admits solutions if and only if 𝑐β‰₯π‘βˆ—. Moreover, the solution is unique.

Corollary 3.4. Under the same assumptions of Proposition 3.3, if Μ‡πœ‘(β„“1) exists finite, then also ̇𝑧(β„“1) exists finite, and ℓ̇𝑧1ξ€Έ=12ξ‚΅β„Žξ€·β„“1ξ€Έξ”βˆ’π‘+ξ€·β„Žξ€·β„“1ξ€Έξ€Έβˆ’π‘2ξ€·β„“βˆ’4Μ‡πœ‘1ξ€Έξ‚Ά.(3.16)

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

Theorem 3.5. Let β„Ž,πœ‘βˆΆ[0,1]→ℝ be continuous functions, with πœ‘(0)=πœ‘(1)=0, πœ‘ differentiable at 𝑒=𝛼, satisfying condition (2.1) and such that πœ‘(𝑒)(π›Όβˆ’π‘’)>0,βˆ€π‘’βˆˆ(0,1)⧡{𝛼},(3.17) for a given constant π›Όβˆˆ(0,1).
Then, there exists a value π‘βˆ—, satisfying estimate (2.4), such that problem (3.1) admits solutions if and only if 𝑐β‰₯π‘βˆ—. Moreover, the solution is unique.

Proof. Consider the boundary value problem (3.3) for β„“1=0,β„“2=𝛼. Since all the assumptions in Proposition 3.1 are satisfied, there exists a threshold value π‘βˆ—1, satisfying (3.2), such that the b.v.p (3.3), with β„“1=0 and β„“2=𝛼, is uniquely solvable if and only if 𝑐β‰₯π‘βˆ—1. Similarly, considering the problem (3.3) for β„“1=𝛼,β„“2=1, by Proposition 3.3, there is a threshold value π‘βˆ—2, satisfying (3.15), such that this problem is uniquely solvable if and only if 𝑐β‰₯π‘βˆ—2. Let π‘βˆ—βˆΆ=max{π‘βˆ—1,π‘βˆ—2}. Clearly, π‘βˆ— satisfies (2.4). Moreover, if problem (3.3), with β„“1=0andβ„“2=1, is solvable for some 𝑐, then also problem (3.3) with β„“1=0,β„“2=𝛼, and problem (3.3), with β„“1=𝛼 and β„“2=1, are solvable for the same value 𝑐. Hence, 𝑐β‰₯π‘βˆ—.
Conversely, let us fix 𝑐β‰₯π‘βˆ—, and let π‘§βˆˆπΆ(0,1) be the function obtained gluing the unique solution of problem (3.3), with β„“1=0 and β„“2=𝛼 and the unique solution of problem (3.3) with β„“1=𝛼andβ„“2=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 πœ‘ at 𝛼 cannot be dropped. Indeed, in view of the proof, if the right derivative of πœ‘ at 𝛼 does not coincide with the left one, then πœ†βˆ’β‰ πœ†+ 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 β„Ž(𝑒)≑0, then condition (1.6) can be removed.

Proof. Let 𝑒(πœ‰) 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 π‘’ξ…ž(πœ‰0)=0 with 0<𝑒(πœ‰0)<1, for some πœ‰0∈(π‘Ž,𝑏), then since (𝐷(𝑒(πœ‰0)π‘’ξ…ž(πœ‰0)))ξ…ž=βˆ’π‘“(𝑒(πœ‰0))<0, we have that πœ‰0 is a point of proper local maximum for 𝑒 when 𝑒(πœ‰0)∈(0,𝛼), while πœ‰0 is a point of proper local minimum when 𝑒(πœ‰0)∈(𝛼,1). Hence, assume now by contradiction the existence of a value πœ‰βˆ—βˆˆ(π‘Ž,𝑏) such that 𝛼<𝑒(πœ‰βˆ—)<1 and π‘’ξ…ž(πœ‰βˆ—)>0. By the boundary condition 𝑒(π‘βˆ’)=0, we get the existence of a value πœ‰>πœ‰βˆ— such that 𝑒(πœ‰)=1 and 𝑒′(πœ‰)=0. Integrating (1.12) in (π‘Ž,πœ‰), we obtain 𝑒0=π·πœ‰π‘’ξ‚ξ‚ξ…žξ‚€πœ‰ξ‚βˆ’π·(𝑒(π‘Ž))π‘’ξ…žξ€œ(π‘Ž)=βˆ’πœ‰π‘Žπ‘“(𝑒(πœ‰))π‘‘πœ‰,(3.18) in contradiction with condition (1.2). Similarly, if π‘’ξ…ž(πœ‰βˆ—)>0 with 0<𝑒(πœ‰βˆ—)<𝛼 for some πœ‰βˆ—βˆˆ(π‘Ž,𝑏), then there exists a value πœ‰<πœ‰βˆ— such that 𝑒(πœ‰)=0 and 𝑒′(πœ‰)=0. Integrating (1.12) in (πœ‰,𝑏), we get again a contradiction. Therefore, 𝑒′(πœ‰)<0 whenever 𝑒(πœ‰)∈(0,1), 𝑒(πœ‰)≠𝛼.
Put πœ‰1∢=inf{πœ‰βˆΆπ‘’(πœ‰)=𝛼} and πœ‰2∢=sup{πœ‰βˆΆπ‘’(πœ‰)=𝛼}. If πœ‰1<πœ‰2 then 𝑒 is constant in (πœ‰1,πœ‰2), and we can define ̃𝑒(πœ‰)∢=𝑒(πœ‰) if πœ‰β‰€πœ‰1; ̃𝑒(πœ‰)∢=𝑒(πœ‰+πœ‰2βˆ’πœ‰1) if πœ‰>πœ‰1. Of course, since (1.12) is autonomous, ̃𝑒 is a t.w.s. of (1.1) in its existence interval ξ‚πΌβˆΆ=(π‘Ž,π‘βˆ’πœ‰2+πœ‰1) and satisfies ξ‚π‘’ξ…ž(πœ‰)<0, for every ξ‚πΌπœ‰βˆˆ, πœ‰β‰ πœ‰1. Hence, ̃𝑒 is strictly decreasing and invertible in 𝐼. Let 𝑒𝑧(𝑒)∢=𝐷(𝑒)ξ…ž(πœ‰(𝑒)), π‘’βˆˆ(0,1). It is easy to see that the function 𝑧(𝑒) is a solution of (3.1) for the same value of 𝑐.
Now assume that 0<𝛼<1, and let 𝑧(𝑒) be a solution of (3.1) for some admissible real value 𝑐. Let 𝑒1(πœ‰) be the unique solution of the Cauchy problem: π‘’ξ…ž=𝑧(𝑒)𝛼𝐷(𝑒),0<𝑒<𝛼,𝑒(0)=2,(3.19) defined on its maximal existence interval (𝑑1,𝑑2). Since π‘’ξ…ž(πœ‰)<0 whenever πœ‰βˆˆ(𝑑1,𝑑2), we have that 𝑒(π‘‘βˆ’2)=0 and 𝑒(𝑑+1)=𝛼. Moreover, according to (1.6) and Corollary 3.2, the limit limπœ‰β†’π‘‘+1π‘’ξ…ž(πœ‰) exists and it is not zero, implying that 𝑑1βˆˆβ„. Consider now the unique solution 𝑒2(πœ‰) of the initial value problem: π‘’ξ…ž=𝑧(𝑒)𝐷(𝑒),𝛼<𝑒<1,𝑒(0)=1+𝛼2,(3.20) on its maximal existence interval (𝜏2,𝜏1). Again, π‘’ξ…ž2(πœ‰)<0 for πœ‰βˆˆ(𝜏1,𝜏2) implies 𝑒(𝜏+1)=1 and 𝑒(πœβˆ’2)=𝛼. Moreover, according to (1.6) and Corollary 3.4, we obtain that limπœ‰β†’πœβˆ’2π‘’ξ…ž2(πœ‰)=limπœ‰β†’π‘‘+1𝑒1(πœ‰); hence, also the former limit is not zero and 𝜏2βˆˆβ„. Then, as we made above, with a suitable shift, it is possible to glue the two functions 𝑒𝑖(πœ‰),𝑖=1,2, in such a way to have a unique 𝑒=𝑒(πœ‰), defined on some interval (π‘Ž,𝑏)βŠ†β„, with π‘’βˆˆπΆ1((π‘Ž,𝑏)), 𝑒(πœ‰)∈(0,1) on the whole (π‘Ž,𝑏), and 𝑒 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 β„Žβ‰‘0. It is well known (see, e.g., [2]) that π‘βˆ— is always strictly positive in this case. Moreover, (see, e.g., [2, Lemma  1]), limπœ‰β†’π‘‘+1π‘’ξ…ž(πœ‰)=limπœ‰β†’πœβˆ’2π‘’ξ…ž(πœ‰)=2𝑓(𝛼)βˆšπ‘+𝑐2Μ‡βˆ’4𝑓(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(𝑒)>β„Ž(𝑒)βˆ’π‘, for π‘’βˆˆ(0,𝛼), implies βˆ«π‘§(𝑒)>𝑒0[β„Ž(𝑠)βˆ’π‘]d𝑠. Therefore, if 𝑒(𝑑0)=𝛼/2 for some 𝑑0βˆˆβ„, we have that π‘βˆ’π‘‘0=ξ€œ0𝛼/2Μ‡ξ€œπ‘‘(𝑒)d𝑒=0𝛼/2𝐷(𝑒)ξ€œβˆ’π‘§(𝑒)d𝑒β‰₯0𝛼/2𝐷(𝑒)βˆ«π‘’0[]π‘βˆ’β„Ž(𝑠)d𝑠d𝑒.(3.22) Being an admissible value, the wave speed 𝑐 satisfies condition (2.4); hence, there exists 𝜎>0 such that βˆ«π‘’0[β„Ž(𝑠)βˆ’π‘]d𝑠β‰₯πœŽπ‘’ for π‘’βˆˆ(0,𝛼), implying 𝑏=+∞. With a similar argument, it can be showed that 𝐷(1)<0 implies π‘Ž=βˆ’βˆž.

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 π‘˜ of the minimal wave speed π‘βˆ—(π‘˜) and of the wave profile π‘ˆ(πœ‰;𝑐,π‘˜) corresponding to the speed 𝑐 were already studied in [11] for the diffusive case, that is, in case 𝐷(𝑒;π‘˜)β‰₯0, for every π‘’βˆˆ(0,1). More in detail, the following result holds (see [11, Theorems  4.1 and 4.2]).

Theorem 4.1. Let β„Ž,𝐷,π‘“βˆΆ[0,1]Γ—[0,1]→ℝ be continuous functions, with 𝐷(β‹…;π‘˜)∈𝐢1[0,1] for every π‘˜βˆˆ[0,1] and 𝐷(𝑒;π‘˜)>0 for every (𝑒,π‘˜)∈(0,1)Γ—[0,1]. Assume that (1.8) and (2.1) are satisfied for all π‘˜βˆˆ[0,1]. Then, the threshold value π‘βˆ— is a l.s.c. function in [0,1]. Moreover, if for some π‘˜0∈[0,1]limsupξ€·(𝑒,π‘˜)β†’0,π‘˜0ξ€Έπœ‘ξ€·(𝑒;π‘˜)βˆ’πœ‘π‘’;π‘˜0𝑒≀0,(4.1) then π‘βˆ— is continuous at π‘˜=π‘˜0.

By means of a change of variable, in the aggregative case, that is, in case 𝐷(𝑒;π‘˜)<0 for every π‘’βˆˆ(0,1), it can be proved that the analogous result holds.

Theorem 4.2. Let β„Ž,𝐷,π‘“βˆΆ[0,1]Γ—[0,1]→ℝ be continuous functions, with 𝐷(β‹…;π‘˜)∈𝐢1[0,1] for every π‘˜βˆˆ[0,1] and 𝐷(𝑒;π‘˜)<0 for every (𝑒,π‘˜)∈(0,1)Γ—[0,1]. Assume that (1.8) and (2.1) are satisfied for all π‘˜βˆˆ[0,1]. Then, the threshold value π‘βˆ— is a l.s.c. function in [0,1]. Moreover, if for some π‘˜0∈[0,1]liminfξ€·(𝑒,π‘˜)β†’1,π‘˜0ξ€Έπœ‘ξ€·(𝑒;π‘˜)βˆ’πœ‘π‘’;π‘˜0ξ€Έ1βˆ’π‘’β‰₯0,(4.2) then π‘βˆ— is continuous at π‘˜=π‘˜0.

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 π‘βˆ—(π‘˜) for the problem: ̇𝑧(𝑒)=β„Ž(𝑒;π‘˜)βˆ’π‘βˆ’πœ‘(𝑒;π‘˜)𝑧0𝑧(𝑒),foreveryπ‘’βˆˆ(0,1)⧡{𝛼(π‘˜)},+ξ€Έ=𝑧(1βˆ’)=0,𝑧(𝑒)πœ‘(𝑒)<0,βˆ€π‘’βˆˆ(0,1),𝑒≠𝛼(π‘˜),(4.3) where 𝑐 is an unknown real constant.
For every π‘˜βˆˆ[0,1) we have 0<𝛼(π‘˜)≀1; hence, we can consider the boundary value problem: ̇𝑧(𝑒)=β„Ž(𝑒;π‘˜)βˆ’π‘βˆ’πœ‘(𝑒;π‘˜)𝑧0𝑧(𝑒),π‘’βˆˆ(0,𝛼(π‘˜)),+ξ€Έ=𝑧(𝛼(π‘˜)βˆ’)=0,𝑧(𝑒)<0,βˆ€π‘’βˆˆ(0,𝛼(π‘˜)).(𝑃1,π‘˜)
Put ξ‚β„Ž(𝑒;π‘˜)∢=𝛼(π‘˜)β„Ž(𝛼(π‘˜)𝑒;π‘˜) and ξ‚πœ‘(𝑒;π‘˜)∢=𝛼(π‘˜)πœ‘(𝛼(π‘˜)𝑒;π‘˜), problem (3.22) is equivalent to the normalized one: ̇𝑀(𝑒)=β„Ž(𝑒;π‘˜)βˆ’π‘π›Ό(π‘˜)βˆ’ξ‚πœ‘(𝑒;π‘˜)𝑀0𝑀(𝑒),π‘’βˆˆ(0,1),+ξ€Έ=𝑀(1βˆ’(𝑃)=0,𝑀(𝑒)<0,βˆ€π‘’βˆˆ(0,1).1,π‘˜) Indeed, a function 𝑧(𝑒) is a solution of (3.22) if and only if the function 𝑀(𝑒)∢=𝑧(𝛼(π‘˜)𝑒) is a solution of (𝑃1,π‘˜). Hence, if π‘βˆ—1(π‘˜) and Μƒπ‘βˆ—1(π‘˜) denote, respectively, the minimal values of the parameter 𝑐 for which (3.22) and (𝑃1,π‘˜) are solvable, we have π‘βˆ—11(π‘˜)=𝛼(π‘˜)Μƒπ‘βˆ—1(π‘˜).(4.4) Since 𝛼(π‘˜) is continuous and positive in [0,1), from Theorem 4.1 we have that π‘βˆ—1(π‘˜) is lower semicontinuous at π‘˜0 for every π‘˜0∈[0,1).
Similarly, for every π‘˜βˆˆ(0,1], consider the problem: ̇𝑧(𝑒)=β„Ž(𝑒;π‘˜)βˆ’π‘βˆ’πœ‘(𝑒;π‘˜)𝑧𝑧(𝑒),π‘’βˆˆ(𝛼(π‘˜),1),𝛼(π‘˜)+ξ€Έ=𝑧(1βˆ’)=0,𝑧(𝑒)>0,βˆ€π‘’βˆˆ(𝛼(π‘˜),1).(𝑃2,π‘˜) Using an argument analogous to the previous one and taking into account that, in this case, 0≀𝛼(π‘˜)<1, we deduce that the threshold value π‘βˆ—2(π‘˜) for problem (𝑃1,π‘˜) is a lower semicontinuous function at every π‘˜0∈(0,1].
Let π‘˜βˆˆ(0,1) and let π‘βˆ— be the threshold value for problem (4.3). Since π‘βˆ—(π‘˜)=max{π‘βˆ—1(π‘˜),π‘βˆ—2(π‘˜)} (see the proof of Theorem 3.5), with π‘βˆ—1,π‘βˆ—2 lower semicontinuous functions, then π‘βˆ—(π‘˜) is lower semicontinuous at every π‘˜0∈(0,1).
If π‘˜0=0, then it results in π‘βˆ—(0)=π‘βˆ—1(0), and liminfπ‘˜β†’0π‘βˆ—(π‘˜)=liminfπ‘˜β†’0𝑐maxβˆ—1(π‘˜),π‘βˆ—2(π‘˜)ξ€Ύξ€Έβ‰₯liminfπ‘˜β†’0π‘βˆ—1(π‘˜)β‰₯π‘βˆ—1(0)=π‘βˆ—(0).(4.5) The case π‘˜0=1 can be treated in a similar way, and the semicontinuity of π‘βˆ— in [0,1] is proved.
Concerning the study of the continuity of π‘βˆ— under condition (2.5), let us first consider a value π‘˜0∈(0,1). Analogously to the arguments above developed, we can consider the problem (3.22) and the normalized one (𝑃1,π‘˜). Notice that, if πœ‘ satisfies condition (4.1), then also the function ξ‚πœ‘ does. Indeed, we have limsup(𝑒,π‘˜)β†’(0,π‘˜0)ξ€·ξ‚πœ‘(𝑒;π‘˜)βˆ’ξ‚πœ‘π‘’;π‘˜0𝑒=limsupξ€·(𝑒,π‘˜)β†’0,π‘˜0ξ€Έπ›Όξ€·π‘˜(π‘˜)πœ‘(𝛼(π‘˜)𝑒;π‘˜)βˆ’π›Ό0ξ€Έπœ‘ξ€·π›Όξ€·π‘˜0𝑒;π‘˜0𝑒=limsupξ€·(𝑣,π‘˜)β†’0,π‘˜0𝛼2(ξ€·π‘˜)πœ‘(𝑣;π‘˜)βˆ’πœ‘π‘£;π‘˜0𝑣+limξ€·(𝑒,π‘˜)β†’0,π‘˜0𝛼2πœ‘ξ€·(π‘˜)𝛼(π‘˜)𝑒;π‘˜0𝑒𝛼(π‘˜)βˆ’π›Ό2ξ€·π‘˜0ξ€Έπœ‘ξ€·π›Όξ€·π‘˜0𝑒;π‘˜0ξ€·π‘˜π‘’π›Ό0ξ€Έξƒͺ≀0,(4.6) by the positivity and the continuity of 𝛼(π‘˜) and the differentiability of πœ‘(𝑒;π‘˜0) at 0. Hence, Theorem 4.1 assures that the function Μƒπ‘βˆ—1(π‘˜) is continuous at π‘˜0, and in force of (4.4) we conclude that π‘βˆ—1(π‘˜) is continuous too. Analogously, considering the problem (𝑃1,π‘˜), we can prove that the threshold value π‘βˆ—2(π‘˜) is continuous at every π‘˜0∈(0,1). Since π‘βˆ—(π‘˜)=max{π‘βˆ—1(π‘˜),π‘βˆ—2(π‘˜)} (see the proof of Theorem 3.5), then π‘βˆ—(π‘˜) is continuous at every π‘˜0∈(0,1).
It remains to study the continuity of π‘βˆ— at the values π‘˜0=0 and π‘˜0=1, where the dynamic change its nature. We limit ourselves in considering the value π‘˜0=0, since the proof for π‘˜0=1 is analogous.
Let πœ–>0 be fixed, and observe that, by assumption (2.6), real values 𝛿=𝛿(πœ–)>0 and π‘˜=π‘˜(πœ–)∈(0,1) exist, such that πœ‘(𝑒;π‘˜)ξ‚€π‘’βˆ’1β‰€Μ‡πœ‘(1;π‘˜)+πœ–,foreveryπ‘’βˆˆ(1βˆ’π›Ώ,1),π‘˜βˆˆ0,π‘˜ξ‚.(4.7) Let ΜƒΜƒπ‘˜=π‘˜(πœ–)∈(0,π‘˜) be such that 𝛼(π‘˜)∈(1βˆ’π›Ώ,1) for every Μƒπ‘˜βˆˆ(0,π‘˜). Then sup[π‘’βˆˆπ›Ό(π‘˜),1)πœ‘(𝑒;π‘˜)ξ€·Μƒπ‘˜ξ€Έπ‘’βˆ’1β‰€Μ‡πœ‘(1;π‘˜)+πœ–,foreveryπ‘˜βˆˆ0,.(4.8) Taking estimate (3.15) into account, with β„“1=𝛼(π‘˜) and β„“2=1, we get π‘βˆ—2√(π‘˜)≀2Μ‡πœ‘(1;π‘˜)+πœ–+max[]π‘’βˆˆπ›Ό(π‘˜),1ξ€·Μƒπ‘˜ξ€Έβ„Ž(𝑒;π‘˜),foreveryπ‘˜βˆˆ0,,(4.9) implying limsupπ‘˜β†’0π‘βˆ—2ξ‚™(π‘˜)≀2limsupπ‘˜β†’0Μ‡πœ‘(1;π‘˜)+πœ–+β„Ž(1;0).(4.10) By the arbitrariness of πœ–>0, assumption (2.7), and estimate (3.2) with β„“1=0 and β„“2=𝛼(π‘˜), we have limsupπ‘˜β†’0π‘βˆ—2√(π‘˜)≀2Μ‡πœ‘(0;0)+β„Ž(0;0)β‰€π‘βˆ—1(0)=π‘βˆ—(0).(4.11) Therefore, since π‘βˆ—1(π‘˜)β†’π‘βˆ—1(0)=π‘βˆ—(0) as π‘˜β†’0, we conclude that limsupπ‘˜β†’0π‘βˆ—(π‘˜)=limsupπ‘˜β†’0𝑐maxβˆ—1(π‘˜),π‘βˆ—2ξ€Ύ(π‘˜)=π‘βˆ—(0),(4.12) and the function π‘βˆ— is upper semicontinuous in π‘˜0=0. Taking into account the lower semicontinuity of π‘βˆ— in the whole interval [0,1], it follows the continuity in π‘˜0=0. The proof for the value π‘˜0=1 is similar.

In order to prove Theorem 2.6 about the convergence of the profiles, we need some preliminary results.

Let 𝑧(𝑒;𝑐,π‘˜) denote the unique solution of problem (4.3), emphasizing the dependence on the parameters 𝑐 and π‘˜. The following result holds.

Lemma 4.3. Let π›ΌβˆΆ[0,1]β†’[0,1] and β„Ž,𝐷,π‘“βˆΆ[0,1]Γ—[0,1]→ℝ be continuous functions satisfying conditions (1.8), (1.9), (1.10), and (2.1) for every π‘˜βˆˆ[0,1]. Let π›ΎβˆΆ[0,1]β†’[0,1] be a continuous function such that 𝛾(π‘˜)β‰₯π‘βˆ—(π‘˜) for every π‘˜βˆˆ[0,1]. Finally, let 𝑧(𝑒;𝛾(π‘˜),π‘˜) be the unique solution of problem (4.3).
Then, for every π‘˜βˆˆ(0,1), there exists a real value πœ†(π‘˜) such that the function ⎧βŽͺ⎨βŽͺβŽ©πœ“(𝑒;π‘˜)∢=𝑧(𝑒;𝛾(π‘˜),π‘˜)𝐷(𝑒;π‘˜),forπ‘’βˆˆ(0,1),𝑒≠𝛼(π‘˜),πœ†(π‘˜),for𝑒=𝛼(π‘˜)(4.13) is continuous in (0,1)Γ—(0,1).

Proof. Let (𝑒0,π‘˜0)∈(0,1)Γ—(0,1) be fixed. We divide the proof in two cases. Case 1 (𝑒0≠𝛼(π‘˜0)). Assume 𝑒0<𝛼(π‘˜0) (the proof in the other case is analogous), and let [π‘Ž,𝑏]βŠ‚(0,𝛼(π‘˜0)) be a compact interval containing the value 𝑒0. Put π‘§π‘˜(𝑒)∢=𝑧(𝑒;𝛾(π‘˜),π‘˜) and let π‘€π‘˜(𝑒)∢=𝑀(𝑒;𝛾(π‘˜)𝛼(π‘˜),π‘˜) denote the solution of problem (𝑃1,π‘˜) (see the proof of Theorem 2.2). By applying [11, Theorem  5.1] to the problem (𝑃1,π‘˜) in [π‘Ž,2𝑏/(𝑏+𝛼(π‘˜0))]βŠ‚(0,1), we get the convergence of π‘€π‘˜(𝑒) to π‘€π‘˜0(𝑒) as π‘˜β†’π‘˜0, uniformly in [π‘Ž,2𝑏/(𝑏+𝛼(π‘˜0))]; that is, for a fixed πœ–>0, ||π‘€π‘˜(𝑒)βˆ’π‘€π‘˜0||<1(𝑒)2ξƒ¬πœ–foreveryπ‘’βˆˆπ‘Ž,2π‘ξ€·π‘˜π‘+𝛼0ξ€Έξƒ­.(4.14) Taking account of the continuity of 𝛼 and of the uniform continuity of π‘€π‘˜0, there exists a value πœ‚>0 such that for every π‘˜βˆˆ(π‘˜0βˆ’πœ‚,π‘˜0+πœ‚) we have 𝛼(π‘˜)>(1/2)(𝑏+𝛼(π‘˜0)) and |||||π‘€π‘˜0𝑒𝛼(π‘˜)βˆ’π‘€π‘˜0ξƒ©π‘’π›Όξ€·π‘˜0ξ€Έξƒͺ|||||<12[]πœ–,foreveryπ‘’βˆˆπ‘Ž,𝑏.(4.15) Since 𝑒/𝛼(π‘˜)∈[π‘Ž,2𝑏/(𝑏+𝛼(π‘˜0))] whenever π‘’βˆˆ[π‘Ž,𝑏] and π‘˜βˆˆ(π‘˜0βˆ’πœ‚,π‘˜0+πœ‚), we deduce that ||π‘§π‘˜(𝑒)βˆ’π‘§π‘˜0||≀||||𝑀(𝑒)π‘˜ξ‚΅π‘’ξ‚Άπ›Ό(π‘˜)βˆ’π‘€π‘˜0𝑒||||+|||||𝑀𝛼(π‘˜)π‘˜0𝑒𝛼(π‘˜)βˆ’π‘€π‘˜0ξƒ©π‘’π›Όξ€·π‘˜0ξ€Έξƒͺ|||||β‰€πœ–,(4.16) for every π‘˜βˆˆ(π‘˜0βˆ’πœ‚,π‘˜0+πœ‚) and π‘’βˆˆ[π‘Ž,𝑏], that is, the uniform convergence of π‘§π‘˜ to π‘§π‘˜0 in [π‘Ž,𝑏]. Taking account of the uniform continuity of π‘§π‘˜0 in [0,1], we get π‘§π‘˜(𝑒)β†’π‘§π‘˜0(𝑒0) as (𝑒,π‘˜)β†’(𝑒0,π‘˜0). Finally, since 𝐷(𝑒0,π‘˜0)β‰ 0, by the continuity of 𝐷(𝑒;π‘˜) we also get lim𝑒(𝑒,π‘˜)β†’0,π‘˜0ξ€Έπ‘§π‘˜(𝑒)=𝑧𝐷(𝑒;π‘˜)π‘˜0𝑒0𝐷𝑒0;π‘˜0ξ€Έ,(4.17) and the continuity of the function πœ“ at (𝑒0,π‘˜0) is proved.
Notice that, for the opposite case 𝑒0>𝛼(π‘˜0), the usual change of variable π‘£βˆΆ=1βˆ’π‘’ allows to apply [11, Theorem  5.1].
Case 2 (𝑒0=𝛼(π‘˜0)). For every π‘˜βˆˆ(0,1), let πœ†(π‘˜)=βˆ’2𝑓(𝛼(π‘˜);π‘˜)βˆšβˆ’β„Ž(𝛼(π‘˜);π‘˜)+𝛾(π‘˜)+[]β„Ž(𝛼(π‘˜);π‘˜)βˆ’π›Ύ(π‘˜)2Μ‡βˆ’4𝑓(𝛼(π‘˜);π‘˜)𝐷(𝛼(π‘˜);π‘˜).(4.18) Notice that the function πœ† is well defined since by assumption (1.6) we have ̇𝐷(𝛼(π‘˜);π‘˜)<0. Moreover, πœ† is continuous and negative for every π‘˜βˆˆ(0,1). Since βˆšπœ†(π‘˜)=β„Ž(𝛼(π‘˜);π‘˜)βˆ’π›Ύ(π‘˜)+[]β„Ž(𝛼(π‘˜);π‘˜)βˆ’π›Ύ(π‘˜)2Μ‡βˆ’4𝑓(𝛼(π‘˜);π‘˜)𝐷(𝛼(π‘˜);π‘˜)2̇𝐷(𝛼(π‘˜);π‘˜),(4.19) from Corollary 3.2 it follows that Μ‡π‘§π‘˜Μ‡(𝛼(π‘˜))=πœ†(π‘˜)𝐷(𝛼(π‘˜);π‘˜).(4.20) Let us fix a value πœ–>0 and define πœ‰πœ–ξ€·πœ†ξ€·π‘˜(𝑒;π‘˜)=0ξ€Έξ€Έπ·Ξ¦βˆ’πœ–(𝑒;π‘˜),πœ–Μ‡πœ‰(𝑒;π‘˜)=πœ–(𝑒;π‘˜)βˆ’β„Ž(𝑒;π‘˜)+𝛾(π‘˜)+𝑓(𝑒;π‘˜)𝐷(𝑒;π‘˜)πœ‰πœ–.(𝑒;π‘˜)(4.21) We have Ξ¦πœ–(ξ€·π‘˜π‘’;π‘˜)=πœ†0̇𝐷(𝑒;π‘˜)βˆ’β„Ž(𝑒;π‘˜)+𝛾(π‘˜)+𝑓(𝑒;π‘˜)πœ†ξ€·π‘˜0ξ€ΈΜ‡ξ€·π‘˜βˆ’πœ–βˆ’πœ–π·(𝑒;π‘˜)=πœ†0̇𝑓𝐷(𝑒;π‘˜)βˆ’β„Ž(𝑒;π‘˜)+𝛾(π‘˜)+(𝑒;π‘˜)πœ†ξ€·π‘˜0ξ€Έ+πœ–π‘“(𝑒;π‘˜)πœ†ξ€·π‘˜0πœ†ξ€·π‘˜ξ€Έξ€·0ξ€Έξ€ΈΜ‡βˆ’πœ–βˆ’πœ–π·(𝑒;π‘˜).(4.22) From the definition of πœ†(π‘˜), we get Μ‡πœ†(π‘˜)𝐷(𝛼(π‘˜);π‘˜)βˆ’β„Ž(𝛼(π‘˜);π‘˜)+𝛾(π‘˜)+𝑓(𝛼(π‘˜);π‘˜)πœ†(π‘˜)=0.(4.23) Therefore, limξ€·π›Όξ€·π‘˜(𝑒,π‘˜)β†’0ξ€Έ,π‘˜0ξ€ΈΞ¦πœ–π‘“ξ€·π›Όξ€·π‘˜(𝑒;π‘˜)=πœ–0ξ€Έ;π‘˜0ξ€Έπœ†ξ€·π‘˜0πœ†ξ€·π‘˜ξ€Έξ€·0ξ€Έξ€ΈΜ‡π·ξ€·π›Όξ€·π‘˜βˆ’πœ–βˆ’πœ–0ξ€Έ;π‘˜0ξ€Έ=∢2𝜎>0.(4.24) Hence, there exists a value 𝛿1>0 such that Ξ¦πœ–||ξ€·π‘˜(𝑒;π‘˜)β‰₯𝜎>0,wheneverπ‘’βˆ’π›Ό0ξ€Έ||<𝛿1,||π‘˜βˆ’π‘˜0||<𝛿1,(4.25) that is, Μ‡πœ‰πœ–(𝑒;π‘˜)>β„Ž(𝑒;π‘˜)βˆ’π›Ύ(π‘˜)βˆ’π‘“(𝑒;π‘˜)𝐷(𝑒;π‘˜)πœ‰πœ–||ξ€·π‘˜(𝑒;π‘˜),wheneverπ‘’βˆ’π›Ό0ξ€Έ||<𝛿1,||π‘˜βˆ’π‘˜0||<𝛿1.(4.26) Take now π›Ώβˆˆ(0,𝛿1] such that |𝛼(π‘˜)βˆ’π›Ό(π‘˜0)|<(1/2)𝛿1 and πœ†(π‘˜)>πœ†(π‘˜0)βˆ’πœ– whenever |π‘˜βˆ’π‘˜0|<𝛿. Observe that, by (4.20), we have Μ‡π‘§π‘˜Μ‡πœ‰(𝛼(π‘˜))<πœ–(𝛼(π‘˜);π‘˜). Since π‘§π‘˜(𝛼(π‘˜))=πœ‰πœ–(𝛼(π‘˜);π‘˜)=0, then for every π‘˜βˆˆ(π‘˜0βˆ’π›Ώ,π‘˜0+𝛿), there exists a value 𝜌=𝜌(π‘˜)∈(0,𝛼(π‘˜)) such that 0>π‘§π‘˜(𝑒)>πœ‰πœ–(𝑒;π‘˜),foreveryπ‘’βˆˆ(𝛼(π‘˜)βˆ’πœŒ(π‘˜),𝛼(π‘˜)),0<π‘§π‘˜(𝑒)<πœ‰πœ–(𝑒;π‘˜),foreveryπ‘’βˆˆ(𝛼(π‘˜),𝛼(π‘˜)+𝜌(π‘˜)).(4.27) Assume, by contradiction, the existence of a value π‘˜βˆˆ(π‘˜0βˆ’π›Ώ,π‘˜0+𝛿) and of a point 𝑣(π‘˜)∈[𝛼(π‘˜0)βˆ’π›Ώ1,𝛼(π‘˜)) such that π‘§π‘˜(𝑣(π‘˜))=πœ‰πœ–(𝑣(π‘˜);π‘˜),π‘§π‘˜(𝑒)>πœ‰πœ–(𝑒;π‘˜)in(𝑣(π‘˜),𝛼(π‘˜)).(4.28) Then, Μ‡π‘§π‘˜(𝑣(π‘˜))=β„Ž(𝑣(π‘˜);π‘˜)βˆ’π›Ύ(π‘˜)βˆ’π‘“(𝑣(π‘˜);π‘˜)𝐷(𝑣(π‘˜);π‘˜)π‘§π‘˜(𝑣(π‘˜))=β„Ž(𝑣(π‘˜);π‘˜)βˆ’π›Ύ(π‘˜)βˆ’π‘“(𝑣(π‘˜);π‘˜)𝐷(𝑣(π‘˜);π‘˜)πœ‰πœ–<Μ‡πœ‰(𝑣(π‘˜);π‘˜)πœ–(𝑣(π‘˜);π‘˜),(4.29) in contradiction with the definition of 𝑣(π‘˜). Therefore, we deduce that πœ‰πœ–(𝑒;π‘˜)<π‘§π‘˜ξ€Ίπ›Όξ€·π‘˜(𝑒)<0,foreveryπ‘’βˆˆ0ξ€Έβˆ’π›Ώ1ξ€Έξ€Ίπ‘˜,𝛼(π‘˜),π‘˜βˆˆ0βˆ’π›Ώ,π‘˜0ξ€»+𝛿.(4.30) Similarly, we can prove that πœ‰πœ–(𝑒;π‘˜)>π‘§π‘˜ξ€·π›Όξ€·π‘˜(𝑒)>0,foreveryπ‘’βˆˆ(π‘˜),𝛼0ξ€Έ+𝛿1ξ€»ξ€Ίπ‘˜,π‘˜βˆˆ0βˆ’π›Ώ,π‘˜0ξ€»+𝛿.(4.31) The previous conditions (4.30) and (4.31) imply that π‘§π‘˜(𝑒)>πœ‰π·(𝑒;π‘˜)πœ–(𝑒;π‘˜)ξ€·π‘˜π·(𝑒;π‘˜)=πœ†0ξ€Έξ€Ίπ›Όξ€·π‘˜βˆ’πœ–,foreveryπ‘’βˆˆ0ξ€Έβˆ’π›Ώ1ξ€·π‘˜,𝛼0ξ€Έ+𝛿1⧡{𝛼(π‘˜)}.(4.32) Let us now take πœ–<βˆ’πœ†(π‘˜0) and consider the functions πœ‚πœ–ξ€·πœ†ξ€·π‘˜(𝑒;π‘˜)∢=0𝐷Ψ+πœ–(𝑒;π‘˜),πœ–(𝑒;π‘˜)∢=Μ‡πœ‚πœ–(𝑒;π‘˜)βˆ’β„Ž(𝑒;π‘˜)+𝛾(π‘˜)+𝑓(𝑒;π‘˜)𝐷(𝑒;π‘˜)πœ‚πœ–.(𝑒;π‘˜)(4.33) We have Ξ¨πœ–(ξ€·π‘˜π‘’;π‘˜)=πœ†0̇𝐷(𝑒;π‘˜)βˆ’β„Ž(𝑒;π‘˜)+𝛾(π‘˜)+𝑓(𝑒;π‘˜)πœ†ξ€·π‘˜0ξ€ΈΜ‡ξ€·π‘˜+πœ–+πœ–π·(𝑒;π‘˜)=πœ†0̇𝑓𝐷(𝑒;π‘˜)βˆ’β„Ž(𝑒;π‘˜)+𝛾(π‘˜)+(𝑒;π‘˜)πœ†ξ€·π‘˜0ξ€Έβˆ’πœ–π‘“(𝑒;π‘˜)πœ†ξ€·π‘˜0πœ†ξ€·π‘˜ξ€Έξ€·0ξ€Έξ€ΈΜ‡+πœ–+πœ–π·(𝑒;π‘˜),(4.34) and, from (4.23), we get limξ€·π›Όξ€·π‘˜(𝑒,π‘˜)β†’0ξ€Έ,π‘˜0ξ€ΈΞ¨πœ–π‘“ξ€·π›Όξ€·π‘˜(𝑒;π‘˜)=βˆ’πœ–0ξ€Έ;π‘˜0ξ€Έπœ†ξ€·π‘˜