Traveling waves for degenerate diffusive equations on networks

  • Received: 01 December 2016 Revised: 01 April 2017
  • Primary: 35K65; Secondary: 35C07, 35K55, 35K57

  • In this paper we consider a scalar parabolic equation on a star graph; the model is quite general but what we have in mind is the description of traffic flows at a crossroad. In particular, we do not necessarily require the continuity of the unknown function at the node of the graph and, moreover, the diffusivity can be degenerate. Our main result concerns a necessary and sufficient algebraic condition for the existence of traveling waves in the graph. We also study in great detail some examples corresponding to quadratic and logarithmic flux functions, for different diffusivities, to which our results apply.

    Citation: Andrea Corli, Lorenzo di Ruvo, Luisa Malaguti, Massimiliano D. Rosini. Traveling waves for degenerate diffusive equations on networks[J]. Networks and Heterogeneous Media, 2017, 12(3): 339-370. doi: 10.3934/nhm.2017015

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  • In this paper we consider a scalar parabolic equation on a star graph; the model is quite general but what we have in mind is the description of traffic flows at a crossroad. In particular, we do not necessarily require the continuity of the unknown function at the node of the graph and, moreover, the diffusivity can be degenerate. Our main result concerns a necessary and sufficient algebraic condition for the existence of traveling waves in the graph. We also study in great detail some examples corresponding to quadratic and logarithmic flux functions, for different diffusivities, to which our results apply.



    Partial differential equations on networks have been considered in the last years by several authors, in particular in the parabolic case; we quote for instance [8,10,11,16,24,30]. According to the modeling in consideration and to the type of equations on the edges of the underlying graph, different conditions at the nodes are imposed. In most of the cases, precise results of existence of solutions are given, even for rather complicated networks.

    In this paper, the main example we have in mind comes from traffic modeling, where the network is constituted by a crossroad connecting m incoming roads with n outgoing roads; the traffic in each road is modeled by the scalar diffusive equation

    ρh,t+fh(ρh)x=(Dh(ρh)ρh,x)x,h=1,,m+n, (1.1)

    where t denotes time and x the position along the road. In this case ρh is a vehicle density; about the diffusivity Dh(ρh)0 we do not exclude that it may vanish at some points. System (1.1) is completed by a condition of flux conservation at the crossroad, which implies the conservation of the total number of cars. Such a model is derived from the famous Lighthill-Whitham-Richards equation [17,25]. We refer to [3,15,17,20,22,27] for several motivations about the introduction of (possibly degenerate) diffusion in traffic flows and in the close field of crowds dynamics. We also refer to the recent books [10,11,26] for more information on the related hyperbolic modeling.

    We focus on a special class of solutions to (1.1), namely, traveling waves. In the case of a single road, traveling waves are considered, for instance, in [21]; in the case of a second-order model without diffusion but including a relaxation term, we refer to [9,28]; for a possibly degenerate diffusion function and in presence of a source term, detailed results are given in [6,7]. In the case of a network, the papers dealing with this subject, to the best of our knowledge, are limited to [30,31] for the semilinear diffusive case and to [19] for the case of a dispersive equation. In these papers, as in most modeling of diffusive or dispersive partial differential equations on networks, both the continuity of the unknown functions and the Kirchhoff condition (or variants of it) are imposed at the nodes. We emphasize that while the classical Kirchhoff condition implies the conservation of the flow and then that of the mass, some variants of this condition are dissipative and, then, imply none of the conservations above. While these assumptions are natural when dealing with heat or fluid flows, they are much less justified in the case of traffic modeling, where the density must be allowed to jump at the node while the conservation of the mass must always hold. Moreover, they impose rather strong conditions on the existence of the profiles, which often amount to proportionality assumptions on the parameters in play.

    In this paper we only require the conservation of the (parabolic) flux at the node, as in [4]; differently from that paper and the other ones quoted above, we do not impose the continuity condition. A strong motivation for dropping this condition comes from the hyperbolic modeling [1,10,11,26]; nevertheless, we show how our results simplify when such a condition is required. In particular, in Sections 6 and 7 we provide explicit conditions for traveling wave solutions which do not satisfy the continuity condition; in some other cases, such a condition is indeed always satisfied. Our main results are essentially of algebraic nature and concern conditions about the end states, flux functions, diffusivities and other parameters which give rise to a traveling wave moving in the network.

    Here follows a plan of the paper. In Section 2 we introduce the model and give some basic definitions; for simplicity we only focus on the case of a star graph. Section 3 deals with a general existence result in the case of a single equation; its proof is provided in Appendix A. Section 4 contains our main theoretical results about traveling waves in a network. In that section we characterize both stationary/non-stationary and degenerate/non-degenerate waves; in particular, Theorem 4.12 contains an important necessary and sufficient condition that we exploit in the following sections. Section 5 focus on the continuity condition; in this case the conditions for the existence of traveling wave solutions are much stricter than in the previous case. Detailed applications of these results are provided in Sections 6 for quadratic fluxes and in Section 7 for logarithmic fluxes; in particular, in subsection 6.2 and in the whole Section 7 the diffusivity is as in [3]. For simplicity, we only deal there with the case of a single ingoing road but we consider both constant and degenerate diffusivities.

    From a theoretical point of view, the extension of our work to more general networks is possible by applying the results in Section 4 to each node iteratively. However, a quick look at Sections 6 and 7 shows that the corresponding results become quickly very technical. In particular, in the general case the whole problem must be recast by using the graph notation, see for instance [18].

    In terms of graph theory, we consider a semi-infinite star-graph with m incoming and n outgoing edges; this means that the incidence vector dRm+n has components di=1 for iI{1,,m} and dj=1 for jJ{m+1,,m+n}. We also denote H{1,,m+n} and refer to Figure 1. For simplicity, having in mind the example in the Introduction, we always refer to the graph as the network, to the node as the crossroad and to the edges as the roads. Then, incoming roads are parametrized by xR(,0] and numbered by the index i, outgoing roads by xR+[0,) and j; the crossroad is located at x=0 for both parameterizations. We denote the generic road by Ωh for hH; then ΩiR for iI and ΩjR+ for jJ. The network is defined as NhHΩh.

    Figure 1.  A star graph.

    Following the above analogy, we understand the unknown functions ρh as vehicular densities in the roads Ωh, hH; ρh ranges in [0,¯ρh], where ¯ρh is the maximal density in the road Ωh. Without loss of generality we assume that ¯ρh=1 for every hH; the general case is easily recovered by a change of variables and modifying (2.2)-(2.3) below for a multiplicative constant. With a slight abuse of notation we denote ρ(ρ1,,ρm+n):R×N[0,1]m+n understanding that ρ(t,x1,,xm+n)=(ρ1(t,x1),,ρm+n(t,xm+n)).

    For each road we assign the functions fh, the hyperbolic flux, and Dh, the diffusivity; we assume for every hH

    (f) fhC1([0,1];R+) is strictly concave with fh(0)=fh(1)=0;

    (D) DhC1([0,1];R+) and Dh(ρ)>0 for any ρ(0,1).

    We emphasize that in (D) we can possibly have either Dh(0)=0 or Dh(1)=0, or even both possibilities at the same time. The evolution of the flow is described by the equations

    ρh,t+fh(ρh)x=(Dh(ρh)ρh,x)x,(t,x)R×Ωh, hH. (2.1)

    Assumption (f) is standard when dealing with traffic flows [2]. More precisely, in that case fh(ρh)=ρhVh(ρh), where Vh is the velocity. Then, assumption (f) is satisfied if, for instance, VhC2([0,1];R+) is either linear or strictly concave, decreasing and satisfying Vh(1)=0, see [17,25]. The prototype of such a velocity satisfying (f) is Vh(ρ)=vh(1ρ) with vh>0, which was introduced in [14]; another example is given in [23]. The simplest model for the diffusivity is then Dh(ρh)=δhρhVh(ρh), where δh is an anticipation length [3,21].

    The coupling among the differential equations in (2.1) occurs by means of suitable conditions at the crossroad. In this paper, having in mind the previous example, we impose a condition on the conservation of the total flow at the crossroad, see [4,5]; in turn, this implies the conservation of the mass. More precisely, we define the parabolic flux by

    Fh(ρh,ρh,x)fh(ρh)Dh(ρh)ρh,x

    and require

    Fj(ρj(t,0+),ρj,x(t,0+))=iIαi,jFi(ρi(t,0),ρi,x(t,0)) for a.e. tR, jJ, (2.2)

    for given constant coefficients αi,j(0,1] satisfying

    jJαi,j=1,iI. (2.3)

    Conditions (2.2) and (2.3) imply

    jJFj(ρj(t,0+),ρj,x(t,0+))=iIFi(ρi(t,0),ρi,x(t,0)) for a.e. tR, (2.4)

    which is the conservation of the total flow at the crossroad. Conditions (2.2) and (2.3) deserve some comments. First, by no means they imply

    ρi(t,0)=ρj(t,0+),tR, (i,j)I×J. (2.5)

    Condition (2.5) is largely used, together with some Kirchhoff conditions, when dealing with parabolic equations in networks and takes the name of continuity condition. Second, above we assumed αi,j>0 for every i and j. The case when αi,j=0 for some i and j would take into account the possibility that some outgoing j roads are not allowed to vehicles coming from some incoming i roads; this could be the case, for instance, if only trucks are allowed in road i but only cars are allowed in road j. For simplicity, we do not consider this possibility. Third, we notice that assumption (2.2) destroys the symmetry of condition (2.4); indeed, with reference to the example of traffic flow, the loss of symmetry is due to the fact that all velocities Vh are positive.

    We point out that condition (2.3) is almost never explicitly exploited in Section 4 and in most of the following: our results hold for every choice of the coefficients αi,j>0. Clearly, condition (2.3) must be imposed to have mathematically (see [11,Lemma 5.1.9]) and physically meaningful solutions; however, this requirement only adds algebraic conditions on the parameters we are dealing with (the end states, the ratios of the traveling-wave speeds, the proportionality coefficients -see (6.11), for instance -and so on) which do not affect the final results.

    Then, we are faced with the system of equations (2.1) that are coupled through (2.2), with the αi,j satisfying (2.3). Solutions to (2.1)-(2.2) are meant in the weak sense, namely ρhC1(R×Ωh;[0,1]) a.e.; see also [2,11] for an analogous definition in the hyperbolic case. We do not impose any initial condition because we only consider traveling waves, which are introduced in the next sections.

    In this section we briefly remind some definitions and results about traveling waves [12] for the single equation

    ρh,t+fh(ρh)x=(Dh(ρh)ρh,x)x,(t,x)R×Ωh, (3.1)

    where we keep for future reference the index h. Equation (3.1) has no source terms (differently from [30,31]) and then any constant is a solution; for simplicity we discard constant solutions in the following analysis.

    Definition 3.1. A weak solution ρh(t,x) to (3.1) is a traveling-wave solution of (3.1) if ρh(t,x)=φh(xcht) for (t,x)R×Ωh, for a non-constant profile φh:R[0,1] and speed chR.

    This definition coincides with that given in [19,29] because we are considering non-constant profiles. The profile must satisfy the equation

    (Fh(φh,φh)chφh)=0, (3.2)

    namely,

    (Dh(φh)φh)gh(φh)φh=0, (3.3)

    in the weak sense, where

    gh(ρ)fh(ρ)chρ (3.4)

    is the reduced flux, see Figure 2.

    Figure 2.  A flux fh satisfying (f), solid curve, and the corresponding reduced flux gh defined in (3.4), dashed curve, in the case ch<0, left, and in the case ch>0, right.

    This means that φhC0(R;[0,1]), Dh(φh)φhL1loc(R;R) and

    R(Dh(φh(ξ))φh(ξ)gh(φh(ξ)))ψ(ξ)dξ=0,

    for every ψCc(R;R). Equation (3.3) is coupled with the limit conditions

    φh(±)=±h, (3.5)

    for ±h[0,1]. Clearly, solutions to (3.3) -(3.5) are determined up to a shift. We define

    Ih{ξR:h<φh(ξ)<+h}. (3.6)

    The existence of profiles is a well-established result [12]; nevertheless, we state for completeness the following theorem, where we point out the qualitative properties of these fronts. The proof is deferred to Appendix A.

    Theorem 3.2. Assume (f) and (D). Equation (3.1) admits a traveling-wave solution ρh with profile φh satisfying (3.5) if and only if

    0h<+h1andch=fh(+h)fh(h)+hh. (3.7)

    We have that φhC2(Ih;(h,+h)) is unique (up to shifts) and φh(ξ)>0 for ξIh; moreover, the following holds true.

    (i) Dh(0)=0=h if and only if there exists νhR such that Ih(νh,) and φh(ξ)=0 for ξνh. In this case

    limξνhφh(ξ)={+hfh(0)fh(+h)+hDh(0)  if  Dh(0)>0,  if  Dh(0)=0, (3.8)
    limξνhDh(φh(ξ))φh(ξ)=0. (3.9)

    (ii) Dh(1)=0=1+h if and only if there exists ν+hR such that Ih(,ν+h) and φh(ξ)=1 for ξν+h. In this case

    limξν+hφh(ξ)={(1h)fh(1)+fh(h)(1h)Dh(1)  if  Dh(1)<0,  if  Dh(1)=0, (3.10)
    limξν+hDh(φh(ξ))φh(ξ)=0. (3.11)

    (iii) In all the other cases Ih=R and

    limξ±φh(ξ)=0. (3.12)

    We observe that for ch given by (3.7), we deduce by (f) that gh(ρ)0 for all ρ[h,+h], see Figure 2. Moreover, we have

    gh(+h)=gh(h)=fh(+h)hfh(h)+h+hh (3.13)

    and no ρ±h makes gh(ρ) equal to that value.

    Theorem 3.2 motivates the following definition.

    Definition 3.3. A traveling-wave solution ρh is stationary if ch=0. It is degenerate if at least one of conditions (ⅰ) or (ⅱ) of Theorem 3.2 holds.

    Remark 3.4. A consequence of assumption (f) is that if ρh is degenerate, then the profile φh is singular either at νh in case (ⅰ) or at ν+h in case (ⅱ), in the sense that φh cannot be extended to the whole of R as a continuous function.

    In case (i) (or (ii)) of Theorem 3.2 does not hold we define νh (respectively, ν+h). In this way the interval (νh,ν+h) is always defined and coincides with the interval Ih defined in (3.6):

    Ih=(νh,ν+h).

    The interval Ih is bounded if and only if both (i) and (ii) hold; in this case ρh is both degenerate and stationary. As a consequence, if ρh is non-stationary then Ih is unbounded and coincides either with a half line (if ρh is degenerate) or with R (if ρh is non-degenerate). At last, ρh is degenerate if and only if either νh or ν+h is finite.

    In the case of non-stationary traveling-wave solutions ρh we use the notation

    ωhmin{c1hνh,c1hν+h}. (3.14)

    Lemma 3.5. Let ρh be a traveling-wave solution of (3.1); then we have the following.

    (a) If ρh is stationary, then it is degenerate if and only if Dh(0)Dh(1)=0 and h=0 (hence +h=1).

    (b) If ρh is non-stationary, then it is degenerate if and only if one of the following equivalent statements hold:

    either Dh(0)=0=h or Dh(1)=0=1+h, but not both;

    ωh is finite.

    In this case the function ξφh(chξ) is singular at ξ=ωh and C1 elsewhere.

    Proof. We recall that ρh is degenerate if and only if either Dh(0)=0=h or Dh(1)=0=1+h. This means that at least one of the end states must be 0 or 1, say 0; but then ch=0 if and only if the other end state is 1. This proves (a) and the first part of (b).

    Now, we prove the second part of (b). Since ch0, ρh is degenerate if and only if exactly one between (i) and (ii) of Theorem 3.2 occurs, namely, exactly one between νh and ν+h is finite. If νh is finite and ν+h=, then ch=fh(+h)/+h>0 and ωh=c1hνh is finite. By Remark 3.4, we know that ξφh(ξ) is singular at ξ=νh and C1 elsewhere, whence the regularity of ξφh(chξ). Analogously, if ν+h is finite and νh=, then ch=fh(h)/(1h)<0 and ωh=c1hν+h is finite. The statement about the smoothness of ξφh(chξ) is proved as above.

    Finally, the converse is straightforward. In fact, if ωh is finite, then either ωh=c1hνh and νh is finite, or ωh=c1hν+h and ν+h is finite; in both cases ρh is degenerate.

    Because of the smoothness properties of the profile proved in Theorem 3.2, we can integrate equation (3.2) in (ξ,ξ)Ih and we obtain

    chφh(ξ)Fh(φh(ξ),φh(ξ))=chφh(ξ)Fh(φh(ξ),φh(ξ)).

    If ξνh in the previous expression, by applying (3.9) or (3.12) we deduce

    Fh(φh(ξ),φh(ξ))=chφh(ξ)+gh(±h),ξIh. (3.15)

    We observe that (3.15) is trivially satisfied in case (i) when ξ<νh and in case (ii) when ξ>ν+h; moreover, by a continuity argument, we deduce from (3.9) and (3.11) that (3.15) is satisfied in case (i) at ξ=νh and in case (ii) at ξ=ν+h, respectively. In conclusion, we have that (3.15) holds in the whole R, namely

    Dh(φh(ξ))φh(ξ)=gh(φh(ξ))gh(±h),ξR. (3.16)

    In this section we consider the traveling-wave solutions of problem (2.1)-(2.2) in the network N. We first introduce the definition of traveling-wave solution in N.

    Definition 4.1. For any hH, let ρh be a traveling-wave solution of (2.1)h in the sense of Definition 3.1 and set ρ(ρ1,,ρm+n). With reference to Definition 3.3, we say that:

    ρ is stationary if each component ρh is stationary;

    ρ is completely non-stationary if none of its components is stationary;

    ρ is degenerate if at least one component ρh is degenerate;

    ρ is completely degenerate if each of its components is degenerate.

    Finally, we say that ρ is a traveling-wave solution of problem (2.1)-(2.2) in the network N if (2.2) holds.

    For brevity, from now on we simply write "traveling wave" for "traveling-wave solution". In analogy to the notation above, we say that φ(φ1,,φm+n) is a profile for ρ if φh is a profile corresponding to ρh for every hH.

    For clarity of exposition, we collect our general results for stationary and non-stationary traveling waves in the following subsections.

    In this subsection, as well as in the following ones, we always assume (f) and (D) without explicitly mentioning it. Moreover, by Definition 4.1 and Theorem 3.2, the end states and the speeds of the profiles must satisfy (3.7) for every hH; both conditions in (3.7) are tacitly assumed as well.

    Proposition 4.2. The function φ is the profile of a traveling wave for (2.1)-(2.2) if and only if φh is a solution to (3.5)-(3.16) for any hH and

    cjφj(cjt)+gj(±j)=iIαi,j(ciφi(cit)+gi(±i)),tR, jJ. (4.1)

    In (4.1) any combination of the signs ± is allowed.

    Proof. By plugging ρh(t,x)=φh(xcht) in (2.2) and recalling that by Theorem 3.2 the profiles are continuous in R, we obtain

    Fj(φj(cjt),φj(cjt))=iIαi,jFi(φi(cit),φi(cit)),tR, jJ,

    which is equivalent to (4.1) by (3.16). At last, we can clearly choose any combination of signs in (4.1) because of (3.13).

    Differently from what specified in Proposition 4.2, in the following the choice of the signs "±" follows the usual rules, i.e., top with top and bottom with bottom.

    Lemma 4.3. Assume that problem (2.1)-(2.2) admits a traveling wave. Then for any jJ we have

    max{fj(j),fj(+j)}=iIαi,jmax{fi(i),fi(+i)}, (4.2)
    min{fj(j),fj(+j)}=iIαi,jmin{fi(i),fi(+i)}. (4.3)

    Proof. Fix jJ. We notice that (4.1) is equivalent to

    Υj(t)=iIαi,jΥi(t),tR, jJ,

    where the map tΥh(t)chφh(cht)+gh(h) is non-decreasing because the profiles are so, by Theorem 3.2. Since we can write Υh(t)=fh(h)+ch[φh(cht)h], we see that Υh ranges between fh(h) and fh(+h) because of (3.7) and the fact that ξφh(ξ) takes values in [h,+h]. As a consequence,

    limtΥh(t)=max{fh(h),fh(+h)},limtΥh(t)=min{fh(h),fh(+h)}.

    Hence, by passing to the limit for t± in (4.1) we obtain (4.2) and (4.3), respectively.

    Lemma 4.4. Assume that problem (2.1)-(2.2) admits a traveling wave. The traveling wave is stationary if and only if one of the following equivalent statements hold:

    (i) there exists jJ such that cj=0;

    (ii) ci=0 for all iI;

    (iii) cj=0 for all jJ.

    Proof. By subtracting (4.3) to (4.2) we obtain

    |fj(+j)fj(j)|=iIαi,j|fi(+i)fi(i)|.

    Since ch=0 if and only if fh(h)=fh(+h), from the above equation we immediately deduce that (i), (ii) and (iii) are equivalent. By the equivalence of (ii) and (iii), a traveling wave is stationary if and only if one of the statements above holds.

    Lemma 4.4 shows that either a traveling wave is stationary, and then ch=0 for every hH, or it is non-stationary, and then

    there exists iI such that ci0 and cj0 for every jJ. (4.4)

    Of course, by Lemma 4.4, ci0 for some iI if and only if cj0 for every jJ.

    Proposition 4.5. Fix ±i[0,1] with i<+i, iI. Then for any jJ there exist ±j[0,1] with j<+j and satisfying (4.2)-(4.3) if and only if

    {max[0,1]fj>iIαi,jmax{fi(i),fi(+i)}   if   c1==cm=0,max[0,1]fjiIαi,jmax{fi(i),fi(+i)}   otherwise. (4.5)

    In this case, the end states ±j are uniquely determined if and only if ci=0 for every iI.

    Proof. Assume that there exist ±j[0,1], with j<+j, which satisfy (4.2)-(4.3). Then clearly we have max[0,1]fjiIαi,jmax{fi(i),fi(+i)}. If ci=0 for every iI, then we have cj=0 for every jJ by Lemma 4.4; the equality max[0,1]fj=f(j)=f(+j) would imply j=+j because of (f), a contradiction, and then max[0,1]fj>f(j)=f(+j). This proves (4.5).

    Conversely, assume (4.5). If ci=0 for every iI, then j<+j are uniquely determined because of the strict concavity of fj. Assume, on the contrary, that ci0 for some iI; then cj0 by Lemma 4.4, i.e., fj(j)fj(+j). Thus (4.2)-(4.3) determine exactly four possible choices of end states ±j with j<+j, see Figure 3.

    Figure 3.  Values max{fj(j),fj(+j)} and min{fj(j),fj(+j)} equal the right-hand side of (4.2) and (4.3), respectively; the lines have slope cj0. Left: cj>0. Right: cj<0.

    By Proposition 4.5 and Lemma 4.4 we deduce that the end states ±j are uniquely determined in terms of the end states ±i if and only if the traveling wave is stationary and the first condition in (4.5) holds.

    We now give an algebraic result about determining the end states of the outgoing profiles in terms of the end states of the ingoing ones. We introduce

    L±i,j{±i if cicj0,i if cicj<0. (4.6)

    Proposition 4.6. Assume that problem (2.1)-(2.2) admits a traveling wave. Then for any jJ we have

    fj(±j)=iIαi,jfi(L±i,j). (4.7)

    Moreover, (4.7) is equivalent to (4.2)-(4.3).

    Proof. Fix jJ. By Lemma 4.3 it is sufficient to prove that (4.7) is equivalent to (4.2)-(4.3). If cj>0, and then fj(+j)>fj(j), by (4.6) we have

    max{fi(i),fi(+i)}={fi(+i)if ci0fi(i)if ci<0=fi(L+i,j),min{fi(i),fi(+i)}={fi(i)if ci0fi(+i)if ci<0=fi(Li,j),

    and therefore (4.7) is equivalent to (4.2)-(4.3). The case cj<0 is analogous. If cj=0, then fj(+j)=fj(j) and by Lemma 4.4 we have fi(+i)=fi(i) for any iI. In this case formulas (4.2)-(4.3) reduce to a single equation, which coincides with (4.7).

    In this short subsection we briefly consider stationary traveling waves.

    Theorem 4.7. Problem (2.1)-(2.2) admits infinitely many stationary traveling waves; such waves are characterized by the conditions on the end states

    fh(+h)=fh(h),fj(j)=iIαi,jfi(i)   for   hH, jJ. (4.8)

    Proof. Clearly, (4.8) is trivially satisfied if h=0 and +h=1 for all hH. We claim that there exist infinitely many choices of ±1,,±m+n satisfying (4.8). To prove the claim, we choose ±i[0,1], with i<+i, such that fi(i)=fi(+i) are sufficiently small to satisfy the first condition in (4.5) for all jJ. Then, by a continuity argument, we can choose ±j[0,1] so that j<+j and fj(j)=fj(+j)=iIαi,jfi(i). This proves the claim.

    With this choice of the end states, by Theorem 3.2 we deduce the existence of a stationary traveling wave in each road satisfying (2.1). At last we notice that, in the stationary case, condition (4.1) is equivalent to the latter condition in (4.8).

    Remark 4.8. The previous result does not use condition (2.3); in case it holds, then (4.8) implies

    jJfj(j)=iIfi(i). (4.9)

    Conversely, consider the stationary case and assume conditions (4.8), (4.9); if moreover fi(i)0 and either jJαi,j1 or jJαi,j1 for every iI, then (2.3) holds. Indeed, (4.8)2 and (4.9) imply

    iIfi(i)=jJfj(j)=(i,j)I×Jαi,jfi(i),

    namely iIfi(i)[1jJαi,j]=0, whence (2.3).

    Clearly, if both Dh(0)0 and Dh(1)0 for every hH, then problem (2.1)-(2.2) admits no degenerate traveling wave. However, even in the general case, the proof of Theorem 4.7 shows that (2.1)-(2.2) admits infinitely many non-degenerate stationary traveling waves: just choose 0h<+h1 satisfying (4.8). Moreover, if there exists hH such that either Dh(0)=0 or Dh(1)=0, then (2.1)-(2.2) admits also infinitely many degenerate stationary traveling waves: just choose h=0=1+h and determine the other end states by (4.8).

    In this subsection we consider non-stationary traveling waves. By Lemma 4.4 this is equivalent to consider the scenario in (4.4): there exists iI such that fi(i)fi(+i) and fj(j)fj(+j) for every jJ. We can therefore introduce the following notation:

    ci,jcicj,Ai,jαi,jci,j,kjiIc0(Ai,jL±i,j)±j,κjcjkj, (4.10)

    where Li,j is defined in (4.6) and

    I0{iI:ci=0}={iI:fi(i)=fi(+i)},Ic0II0.

    We notice that Ic0 by (4.4) and that both I0 and Ic0 depend on the end states ±i, iI, indeed. Moreover, kj is well defined because by (4.7)

    iIc0Ai,j(L+i,jLi,j)=iIc0αi,jc1j(fi(L+i,j)fi(Li,j))=c1j(fj(+j)fj(j))=+jj.

    Finally, by (f) we deduce that

    for no jJ we have both j=0=1+j.

    Proposition 4.9. The function φ is the profile of a non-stationary traveling wave for (2.1)-(2.2) if and only if φh is a solution to (3.5)-(3.16) for any hH and

    φj(ξ)=iIc0(Ai,jφi(ci,jξ))kj,ξR, jJ. (4.11)

    Proof. By Proposition 4.2 it is sufficient to prove that by (4.4) condition (4.1) is equivalent to (4.11). By (3.13) we have gi(+i)=gi(i)=gi(L+i,j)=gi(Li,j) and then by (4.7) we have κj=gj(±j)iIαi,jgi(L±i,j). Hence, by (4.4), with the change of variable ξ=cjt, condition (4.1) is

    cjφj(ξ)=gj(±j)+iIαi,j(ciφi(ci,jξ)+gi(L±i,j))=iIαi,jciφi(ci,jξ)κj,

    that is equivalent to (4.11).

    We observe that kj and (4.11) can be written in a little bit more explicit form by avoiding the use of L±i,j as follows

    kj=iIc0(Ai,ji++i2)j++j2,φj(ξ)=j++j2+iIc0Ai,j(φi(ci,jξ)i++i2). (4.12)

    Proposition 4.9 shows how each outgoing profile φj can be expressed by (4.11) in terms of the ingoing profiles φi, iI. We know a priori that φj is increasing and its end states are contained in the interval [0,1]. Now, we prove a sort of converse implication, which shows that these properties of the profile φj are enjoined by the function defined by the right-hand side of (4.11).

    Lemma 4.10. Let φi, for iI, be the profiles provided by Theorem 3.2 and assume that Ic0; fix jJ and consider any l±j[0,1] satisfying (4.7) and such that, for the corresponding cj, it holds cj0. Then lj<l+j. Moreover, denote by j(ξ) the right-hand side of (4.11); then ξj(ξ) is non-decreasing and j(±)=l±j.

    Proof. Since by Theorem 3.2 we know that i<+i, then by (4.7)

    l+jlj=c1j(fj(l+j)fj(lj))=iIαi,jc1j(fi(L+i,j)fi(Li,j))=iIαi,jci,j(L+i,jLi,j)=iIc0αi,j|ci,j|(+ii)>0.

    By definition of j we have j(ξ)=iIc0αi,jc2i,jφi(ci,jξ) for a.e. ξR, hence ξj(ξ) is non-decreasing since all profiles φi do. Moreover, j(±)=l±j because by the definitions of j and κj we have

    cjj(±)=iIc0(αi,jciL±i,j)κj=cjl±j.

    We notice that Proposition 4.9 exploits condition (2.2) through its expression (4.1) for the profiles; the diffusivities Dh are not involved in (4.11). Indeed, Proposition 4.9 imposes strong necessary conditions on the diffusivities as we discuss now as a preparation to (4.17).

    We notice that if both νh and ν+h are finite, then h=0=1+h and consequently ch=0; therefore either νh or ν+h (possibly both) is infinite for any hIc0J.

    The following result is similar to Lemma 4.4.

    Lemma 4.11. Problem (2.1)-(2.2) admits a degenerate non-stationary traveling wave ρ if and only if at least one of the following conditions holds:

    (A) for some iI0 we have Di(0)Di(1)=0 and i=0 (hence +i=1);

    (B) for every hIc0J we have either Dh(0)=0=h or Dh(1)=0=+h1, but not both. In this case we have

    ωi=ωjω,iIc0, jJ. (4.13)

    Proof. Let us introduce the following conditions:

    (B) for some iIc0 we have either Di(0)=0=i or Di(1)=0=+i1, but not both;

    (B) for some jJ we have either Dj(0)=0=j or Dj(1)=0=+j1, but not both.

    Clearly (B) implies both (B) and (B). Moreover, by Lemma 3.5 and (4.4), problem (2.1)-(2.2) admits a degenerate non-stationary traveling wave ρ if and only if at least one of the conditions (A), (B) and (B) holds. To complete the proof it is therefore sufficient to show that (B), (B) and (B) are equivalent. By Lemma 3.5(b) and (4.4), the conditions (B), (B) and (B) are respectively equivalent to

    (Ⅰ) ωh is finite for every hIc0J,

    (Ⅱ) for some iIc0 we have that ωi is finite,

    (Ⅲ) for some jJ we have that ωj is finite,

    where ωh is defined in (3.14). Differentiating (4.11) gives

    φj(cjξ)=iIc0αi,jc2i,jφi(ciξ) for a.e. ξR, jJ. (4.14)

    More precisely, by Lemma 3.5, formula (4.14) holds for ξR({ωj}iIc0ωi); moreover, by the same lemma we know that ξφh(chξ) is singular at ξ=ωh and C1 elsewhere, for hIc0J. Hence, (4.14) implies (4.13). By (4.13) we have that the above statements (Ⅰ), (Ⅱ) and (Ⅲ) are equivalent and then also (B), (B) and (B) are so.

    As for Lemma 4.4, we notice that Lemma 4.11 implies that a non-stationary traveling wave ρ is either non-degenerate, and then ρh is non-degenerate for every hH, or ρ is degenerate, and then either there exists iI0 such that ρi is degenerate, or ρh is degenerate for all hIc0J. In both cases a non-stationary traveling wave ρ satisfies (4.13).

    When modeling traffic flows it is natural to use different diffusivities, which however share some common properties. For instance, this led to consider in [3,7] the following subcase of (D):

    (D1) Dh satisfies (D) and Dh(0)=0, Dh(1)>0, for every hH.

    The proof of the following result is an immediate consequence of Lemma 4.11 and, hence, omitted.

    Corollary 1. Assume that problem (2.1)-(2.2) has a non-stationary traveling wave ρ and (D1) holds. Then ρ is degenerate if and only at least one of the following conditions holds:

    (A) for some iI0 we have i=0 (hence +i=1);

    (B) for every hIc0J we have h=0 (hence +h1).

    The case when Dh satisfies (D) and Dh(0)=0=Dh(1) for every hH, see [6,7], can be dealt analogously.

    The next result is the most important of this paper; there, we give necessary and sufficient conditions for the existence of non-stationary traveling waves in a network. About its statement, let us recall Theorem 3.2: we have φh(ξ)=0 in case (i) if ξ<νh or in case (ii) if ξ>ν+h. Since φh satisfies equation (3.16), we are led to extend the quotient gh()gh(h)Dh() to the whole of R by defining

    γh(){gh()gh(h)Dh() if Dh()0,0 if Dh()=0. (4.15)

    In fact, when is replaced by φh(ξ), then γh()=φh(ξ) for ξR{νh,ν+h}. We remark that condition Dh()=0 occurs at most when either =0 or =1. To avoid the introduction of the new notation (4.15), in the following we simply keep on writing gh()gh(h)Dh() for γh(). As a consequence, any non-stationary traveling wave of problem (2.1)-(2.2) satisfies

    φh(ξ)=gh(φh(ξ))gh(h)Dh(φh(ξ)),ξR{νh,ν+h}, hH. (4.16)

    Theorem 4.12. Assume conditions (f) and (D). Problem (2.1)-(2.2) admits a non-stationary traveling wave if and only if the following condition holds.

    (T) There exist ±1,,±m[0,1] with i<+i, iI, such that:

    (i) Ic0;

    (ii) for any jJ there exist ±j[0,1] satisfying (4.7) and such that fj(j)fj(+j);

    (iii) for any jJ we have

    gj(j(cjξ))gj(j)Dj(j(cjξ))=iIc0Ai,jci,jgi(φi(ciξ))gi(i)Di(φi(ciξ))   for   a.e.   ξR, (4.17)

    where φ1,,φm are solutions to (3.5)-(3.16) and, for kj as in (4.10),

    j(ξ)iIc0(Ai,jφi(ci,jξ))kj,ξR. (4.18)

    Proof. First, assume that problem (2.1)-(2.2) admits a non-stationary traveling wave ρ with profiles φh, end states ±h and speeds ch, for hH. By Theorem 3.2 we have that ±h and ch satisfy (3.7). By Proposition 4.9 the profiles φh satisfy (3.5)-(3.16) and (4.11). The end states ±j, jJ, satisfy (4.7) by Proposition 4.6. Since ρ is non-stationary we are in the scenario given by (4.4): Ic0 and fj(j)fj(+j) for all jJ. By (4.16) with h=j we have

    φj(cjξ)=gj(φj(cjξ))gj(j)Dj(φj(cjξ)) (4.19)

    for ξR in the non-degenerate case and for ξR{ω} with ω given by (4.13) in the degenerate case. On the other hand, by differentiating (4.11) and applying (4.16) with h=i we deduce

    φj(ξ)=iIc0Ai,jci,jφi(ci,jξ)=iIc0Ai,jci,jgi(φi(ci,jξ))gi(i)Di(φi(ci,jξ)) (4.20)

    for ξR in the non-degenerate case and for ξR{ω} with ω given by (4.13) in the degenerate case. Identity (4.17) follows because jφj by (4.11) and by comparing (4.19), (4.20).

    Conversely, assume that condition (T) holds. We remark that the existence of φi, iI, is assured by Theorem 3.2. Fix jJ. By defining φjj we obtain (4.11). We know by assumption that Ic0, ±j[0,1] satisfy (4.7) and cj0; we can apply therefore Lemma 4.10 and deduce that j<+j, φj is non-decreasing and satisfies (3.5) with h=j. By Proposition 4.9, what remains to prove is that φj satisfies (3.16). But by (4.11) we deduce (4.20) for a.e. ξR, because φ1,,φm satisfy (3.16) and hence, recalling the extension (4.15), also (4.16); then by (4.17) we conclude that φj satisfies (4.16) for a.e. ξR and then (3.16) for a.e. ξR. Finally, (3.16) holds by the regularity ensured by Theorem 3.2 for the profiles.

    Remark 4.13. As previously in Theorem 4.7, also in Theorem 4.12 we do not use condition (2.3). We observe that (2.3) together with (4.7), or equivalently (4.2)-(4.3) by Proposition 4.6, imply

    jJmax{fj(j),fj(+j)}=iImax{fi(i),fi(+i)}. (4.21)

    Conversely, consider the non-stationary case and assume conditions (4.2), (4.21); if moreover max{fi(i),fi(+i)}0 and either jJαi,j1 or jJαi,j1 for every iI, then (2.3) holds. Indeed, (4.2) and (4.21) imply

    iImax{fi(i),fi(+i)}=jJmax{fj(j),fj(+j)}=(i,j)I×Jαi,jmax{fi(i),fi(+i)},

    namely iImax{fi(i),fi(+i)}[1jJαi,j]=0, whence (2.3).

    Remark 4.14. Fix ±i[0,1], iI, so that i<+i and (4.5) holds. We know by Proposition 4.5 that for every jJ there exists (j,+j)[0,1]2, with j<+j, that satisfies (4.7), but it is not unique. If beside (4.7) we impose also (4.17), then we may have three possible scenarios: such (j,+j) either does not exist, or it exists and is unique, or else it exists but is not unique. We refer to Subsections 6.1 and 6.2 for further discussion.

    In this section we discuss the case when solutions to (2.1)-(2.2) are also required to satisfy the continuity condition (2.5); this makes the analysis much easier because (2.5) implies several strong conditions.

    First, we provide the main results about traveling waves satisfying condition (2.5). We point out that some of the consequences below have already been pointed out in [19,30,31] in the case that some Kirchhoff conditions replace the conservation of the total flow (2.2). In order to emphasize the consequences of the continuity condition (2.5), the first two parts of the following lemma do not assume that also condition (2.2) holds.

    Lemma 5.1. For any hH, let ρh be a traveling wave of (2.1)h in the sense of Definition 3.1 and set ρ(ρ1,,ρm+n); then the following holds for every (i,j)I×J and hH.

    (i) ρ satisfies (2.5) if and only if

    φj(cjt)=φi(cit)Φ(t),tR. (5.1)

    (ii) If ρ satisfies (2.5), then either it is stationary (hence (5.1) reduces to φj(0)=φi(0)), or it is completely non-stationary and the speeds ch have the same sign (hence ci,j>0). In the latter case, ρ is either non-degenerate or completely degenerate; moreover

    (c1jIj)=(c1iIi)I, (5.2)
    ±j=±i=L±i,j±, (5.3)
    cjgj()gj(±)Dj()=cigi()gi(±)Di(),       (,+). (5.4)

    (iii) If ρ is non-stationary and satisfies both (2.2) and (2.5), then

    cj=iIαi,jci,       jJcj=iIci,κj=0,iIAi,j=1. (5.5)

    Proof. We split the proof according to the items in the statement.

    (i) Condition (2.5) and (5.1) are clearly equivalent.

    (ii) Since we are discarding constant profiles, by (5.1) we have that either ch=0 for all hH or ch0 for all hH. The stationary case is trivial; therefore we consider below only the non-stationary case and assume that ch0 for all hH. By differentiating (5.1) with respect to t we deduce

    cjφj(cjt)=ciφi(cit) for a.e. tR. (5.6)

    Then (5.6) implies that either ρ is non-degenerate or it is completely degenerate. Moreover (5.6) implies (5.2) because, by Lemma 3.5, we have that ρh is degenerate if and only if the map ξφh(chξ) is singular at ξ=ωhR and C1 elsewhere. By taking tI in (5.6) we deduce that ci and cj have the same sign. As a consequence we have L±i,j=±i and then ±i=±j by letting t± in (5.1). By (5.1), (3.16) and (5.3) we have Dh(Φ(ξ))Φ(ξ)=ch(gh(Φ(ξ))gh(±)) for all hH, whence (5.4) by the extension (4.15).

    (iii) To deduce (5.5)1, we differentiate (4.1) and then exploit (5.6). Formula (5.5)2 follows by summing (5.5)1 with respect to j and by (2.3). By (5.5)1 we have iIAi,j=iIαi,jcic1j=1, which proves (5.5)4. Finally, (5.3) and (5.5)4 imply (5.5)3.

    In the following proposition we deal with stationary traveling waves satisfying condition (2.5).

    Proposition 5.2. Problem (2.1)-(2.2) admits infinitely many stationary traveling waves satisfying (2.5); their end states ±h satisfy (4.8) and are such thatShH(h,+h).

    Proof. By (5.1) condition (2.5) holds in the stationary case if and only if φi(0)=φj(0) for (i,j)I×J. Recalling the proof of Theorem 4.7, it is sufficient to take ±h[0,1] satisfying (4.8) and such that S, 0S and the unique solution φh to (3.5)-(3.16) such that φh(0)=0. There are infinitely many of such profiles because of the arbitrariness of ±h.

    We point out that condition S= can occur if the functions fh assume their maximum values at different points. This is not the case when the following condition (5.10)1 is assumed.

    The following result is analogous to Theorem 4.12 in the case (2.5) holds.

    Theorem 5.3. Assume conditions (f) and (D). Problem (2.1)-(2.2) admits a (completely) non-stationary traveling wave satisfying (2.5) if and only if the following condition holds.

    (Tc) There exist ±[0,1] with <+, such that for any hH, iI and jJ

    fh()fh(+), (5.7)
    fj(±)=iIαi,jfi(±), (5.8)
    cjgj(φj(cjt))gj()Dj(φj(cjt))=cigi(φi(cit))gi()Di(φi(cit))      for   a.e.     tR, (5.9)

    where ch is given by (3.7), φh is a solution to (3.16) such thatφh(±)=± and φ1(0)==φm+n(0).

    Proof. Assume that condition (Tc) holds; the other implication is obvious. We remark that the existence of φ1,,φm+n is assured by Theorem 3.2; indeed, for any 0(,+), up to shifts it is always possible to assume that φh(0)=0, hH. By (5.8) we have (5.5)4 because

    iIAi,j=iIαi,jfi(+)fi()fj(+)fj()=1.

    By (5.7) we have that Ic0=I, I0= and ρ corresponding to the profile φ(φ1,,φm+n) is completely non-stationary. Then (5.5)4 and (4.12) imply (5.5)3, namely kj=0. By Lemma 5.1 (i) and Proposition 4.9 it remains to prove (5.1) and (4.11). We start with (5.1). Clearly (5.1) holds for t=0 because φh(0)=0, hH. Then by the extension (4.15) and (5.9) we have

    ddt(φj(cjt)φi(cit))=cjgj(φj(cjt))gj()Dj(φj(cjt))cigi(φi(cit))gi()Di(φi(cit))=0.

    Therefore we conclude that (5.1) holds. Finally, (4.11) follows immediately from (5.1), (5.5)3 and (5.5)4.

    Consider in particular the case when the functions f and D satisfy (f) and (D), respectively, and assume that

    fh()vhf(),Dh()δhD(),[0,1], (5.10)

    for some constants vh,δh>0. Denote

    vi,jvivj,δi,jδiδj. (5.11)

    We notice that now we have

    vi,j=ci,j. (5.12)

    In the following proposition we apply Theorem 5.3 when (5.10) is assumed; in this case conditions (5.8) and (5.9) no longer depend on the end states and the statement is somewhat simplified.

    Proposition 5.4. Assume (5.10) with f and D satisfying (f) and (D), respectively. Problem (2.1)-(2.2) admits a (completely) non-stationary traveling wave satisfying (2.5) if and only if for every iI and jJ we have

    v2i,j=δi,jandiIαi,jvi,j=1. (5.13)

    Proof. We only need to translate condition (Tc) to the current case. Let ±[0,1] with <+ and f()f(+). By (5.10) it is obvious that (5.7) is satisfied. If =0 or +=1 condition (5.8) is satisfied by (f). In all the other cases (5.8) is equivalent to iIαi,jvi,j=1 by (5.10). Similarly, condition (5.9) reduces to cjvjδ1j=civiδ1i and hence, by (5.12), it is equivalent to v2i,j=δi,j.

    Remark that by (5.12) condition (5.13)2 is equivalent to (5.5)4.

    In this section we assume (5.10) for some constants vh,δh>0, D satisfying (D) and the quadratic flux [14]

    f(ρ)ρ(1ρ),

    with no further mention. The case when only (5.10)1 holds is doable and follows with slight modifications. We use the notation introduced in (5.11).

    For simplicity, in the whole section we focus on the case m=1, see Figure 4, even without explicitly mentioning it. Then I={1}, J={2,,n+1}, H={1,2,,n+1}. The general case m>1 offers no further difficulties than heavier calculations.

    Figure 4.  A network with m=1.

    In this case, condition (3.7) becomes

    0h<+h1 and ch=vh[1+hh]. (6.1)

    In particular, by (6.1)2

     ρh is stationary  +h+h=1. (6.2)

    Moreover, gh()=vh[+h+h] implies

    gh()gh(±h)=vh(+h)(h), (6.3)

    and therefore (3.16) becomes

    δhD(φh(ξ))φh(ξ)=vh(+hφh(ξ))(φh(ξ)h),ξR. (6.4)

    We first consider stationary traveling waves and specify Theorem 4.7 and Proposition 5.2 in the current framework. We define the intervals

    L0j{(0,1/2)if α1,jv1,j1,(0,11α11,jv11,j2)if α1,jv1,j>1,jJ.

    Proposition 6.1. Problem (2.1)-(2.2) admits infinitely many stationary traveling waves; their end states are characterized by the conditions

    1jJL0j,+1+1=1,±j=12(1±14α1,jv1,j+11),jJ.

    Moreover, up to shifts, any stationary traveling wave satisfies (2.5).

    Proof. The first part of the proposition follows from Theorem 4.7. Indeed, conditions (6.2), (3.7)1 and (4.8) are satisfied if and only if for any hH and jJ

    h[0,1/2),+h+h=1,j(1j)=α1,jv1,j1(11);

    then it is sufficient to compute ±j and to observe that the definition of L0j guarantees that they are real numbers.

    The latter part of the proposition is deduced by Proposition 5.2 because 1/2shH(h,+h).

    In the following we treat the existence of non-stationary traveling waves. Since m=1, by Lemma 4.4 this is equivalent to assume ch0 for hH, namely, the traveling wave is completely non-stationary. By (4.7), (6.1) and (6.2), from (4.10) we deduce

    c1,j=v1,j1+111+jj,A1,j=α1,jv1,j1+111+jj,kj=A1,jL±1,j±j,κj=vjj+jα1,jv11+1. (6.5)

    The following result translates Theorem 4.12 to the present case. We define the intervals

    Lcj{[0,1]if α1,jv1,j1,[0,1](11α11,jv11,j2,1+1α11,jv11,j2)if α1,jv1,j>1,jJ.

    Proposition 6.2. Problem (2.1)-(2.2) admits a (completely) non-stationary traveling wave if and only if the following condition holds.

    (Tq) There exist ±1[0,1] with 1<+1 such that:

    (i) +1+11;

    (ii) ±1jJLcj;

    (iii) for any jJ we have

    D()=α1,jδ1,jv1,jD(+kjA1,j),(j,+j), (6.6)

    where kj is defined in (6.5) with ±j being solutions to

    ±j(1±j)=α1,jv1,jL±1,j(1L±1,j). (6.7)

    Proof. The proof consists in showing that, in the present case, condition (T) of Theorem 4.12 is equivalent to (Tq).

    The first item of (T) is clearly equivalent to the first item of (Tq).

    We prove now that the second item of (T) is equivalent to the second item of (Tq).

    '''' Assume that for any jJ there exist ±j[0,1] satisfying (4.7) and such that fj(j)fj(+j). Fix jJ. Clearly (4.7) is equivalent to (6.7).

    If we denote z±1,j4α1,jv1,jL±1,j(1L±1,j), then the ±j-solutions to (6.7) are, see Figure 3,

    {j=12(11z1,j),+j{12(1±1z+1,j)}, if cj>0, (6.8)
    {j{12(1±1z1,j)},+j=12(1+1z+1,j), if cj<0. (6.9)

    The square roots in (6.8)-(6.9) are real numbers if and only if z±1,j1, namely,

    ±1(1±1)(4α1,jv1,j)1.

    It is easy to see that the above estimate is equivalent to require ±1Lcj.

    '''' Assume that ±1jJLcj and fix jJ. The square roots in (6.8)-(6.9) are then real numbers and ±j given in (6.8)-(6.9) satisfy (6.7), namely (4.7). Obviously ±j belong to [0,1]. Finally, since ±j are solutions to (6.7), it is easy to see that fj(+j)fj(j) because f1(+1)f1(1).

    We prove now that (T) implies the last item of (Tq). Since the first two items in (T) are equivalent to the first two items in (Tq), we can assume that +1+11, ±1jJLcj and that for any jJ we have (4.17), namely,

    (+jj(cjξ))(j(cjξ)j)D(j(cjξ))=A1,jc1,jv1,jδ1,j(+1φ1(c1ξ))(φ1(c1ξ)1)D(φ1(c1ξ)) (6.10)

    for a.e. ξR, where φ1 is a solution to (3.5)-(6.4) and

    j(ξ)A1,j(φ1(c1,jξ)L±1,j)+±j,ξR.

    We point out that the above expression of j is deduced from (4.18) by applying (6.5); moreover (6.10) is deduced from (4.17) by applying (6.3). Recall that both fractions in (6.10) are meant as in (4.15). Since

    (+jj(cjξ))(j(cjξ)j)=A21,j(L+1,jφ1(c1ξ))(φ1(c1ξ)L1,j),(+1φ1(c1ξ))(φ1(c1ξ)1)=(L+1,jφ1(c1ξ))(φ1(c1ξ)L1,j),

    we have that (6.10) is equivalent to

    D(j(cjξ))=α1,jδ1,jv1,jD(φ1(c1ξ))for a.e. ξR.

    To conclude now that the above condition is equivalent to (6.6) it is sufficient to recall that by Lemma 4.10 the continuous function ξj(ξ) is increasing and j(±)=±j and that j(ξ)=A1,jφ1(c1,jξ)kj by (6.5).

    Finally, to prove that (Tq) implies the last item of (T) it is enough to trace backwards the proof of the previous item.

    We notice that if D is a polynomial with degree d, then (6.6) is equivalent to d+1 conditions on the parameters, see for instance (6.15) and (6.25).

    Remark 6.3. We point out that by Proposition 5.4 we have that problem (2.1)-(2.2) admits a (completely) non-stationary traveling wave satisfying (2.5) if and only if

    v21,j=δ1,j and α1,jv1,j=1,jJ. (6.11)

    The special cases of constant or linear diffusivities are treated in the following subsections.

    In this subsection we assume

    D1, (6.12)

    and in this case problem (3.5)-(6.4) reduces to

    {δhφh(ξ)=vh(+hφh(ξ))(φh(ξ)h),ξR,φh(±)=±h. (6.13)

    For any hH, the function

    ψh(ξ)+h1+evhδh(+hh)ξ+h1+evhδh(+hh)ξ (6.14)

    solves (6.13) because h<+h; all the other solutions are of the form φh(ξ)=ψh(ξ+σh) for σhR. Notice that ψh(0)=(+h+h)/2.

    We rewrite Proposition 6.2 in the current setting; we emphasize that the shifts appear below because in this case we have the explicit solution (6.14) to problem (6.13).

    Proposition 6.4. Assume (6.12). Problem (2.1)-(2.2) admits a (completely) non-stationary traveling wave if and only if

    α1,jδ1,j=v1,j. (6.15)

    In this case any non-stationary traveling wave ρ has a profile φ of the form

    φ(ξ)=(ψ1(ξ+σ1),,ψn+1(ξ+σn+1)),ξR, (6.16)

    with ±h satisfying (i), (ii) and (6.7) in Proposition 6.2 and σhR, hH, such that

    cjσ1=c1σj,jJ. (6.17)

    Proof. By Theorem 3.2, any solution to (6.13) has the form (6.16) with σhR, hH. Therefore, by Proposition 4.2 it only remains to prove that (4.1) is equivalent to (6.15)-(6.17). Straightforward computations show that in the present case (4.1) can be written as

    fj(+j)ζj(t)+fj(j)1+ζj(t)=α1,jf1(+1)ζ1(t)+f1(1)1+ζ1(t),tR, jJ, (6.18)

    where ζh(t)expzh(t), for zh(t)vhδh(+hh)(cht+σh), hH. By Proposition 4.6 we have

     either fj(±j)=α1,jf1(±1), or fj(±j)=α1,jf1(1).

    In the former case, identity (6.18) is equivalent to

    (fj(+j)fj(j))(ζj(t)ζ1(t))=0,tR, jJ.

    Since by assumption fj(+j)fj(j), it must be ζjζ1, i.e., zj(t)=z1(t), namely

    {vjδj(+jj)cj=v1δ1(+11)c1,vjδj(+jj)σj=v1δ1(+11)σ1,{v1,jδ1,j=fj(+j)fj(j)f1(+1)f1(1)=α1,j,σjcj=σ1c1.

    In the latter case, identity (6.18) is equivalent to

    (fj(+j)fj(j))(ζj(t)ζ1(t)1)=0,tR, jJ.

    Since by assumption fj(+j)fj(j), it must be ζjζ11, i.e. zj(t)=z1(t), namely

    {vjδj(+jj)cj=v1δ1(+11)c1,vjδj(+jj)σj=v1δ1(+11)σ1,{v1,jδ1,j=fj(+j)fj(j)f1(+1)f1(1)=α1,j,σjcj=σ1c1.

    In both cases we proved that (4.1) is equivalent to(6.15)-(6.17); this concludes the proof.

    Remark 6.5. Consider conditions (6.11)1, (6.11)2 and (6.15). Any two of them implies the third one.

    Proposition 6.6. Assume (6.12). Problem (2.1)-(2.2) admits a (completely) non-stationary traveling wave satisfying (2.5) if and only if (6.11) holds true. In this case a non-stationary traveling wave satisfies (2.5) if and only if its end states satisfy (5.3).

    Proof. The first part of the statement is just Remark 6.3. In this case, since (6.11) implies (6.15), by Proposition 6.4 any (completely) non-stationary traveling wave ρ has a profile of the form (6.16)-(6.17).

    The second part of the statement characterizes the end states. If a non-stationary traveling wave ρ satisfies (2.5), then (5.3) holds because of Lemma 5.1. Conversely, if the end states of ρ satisfy (5.3), then long but straightforward computations show that (5.1) holds true, and therefore ρ satisfies (2.5).

    In this subsection we assume

    D(ρ)ρ. (6.19)

    We notice that D degenerates at 0 and this makes the subject more interesting. In this case problem (3.5)-(6.4) reduces to

    {δhφhφh=vh(+hφh)(φhh),ξR,φh(±)=±h. (6.20)

    If h=0, then the function

    ψh(ξ){+h2(2evhδhξ) if ξδhvhln2,0 if ξ<δhvhln2, (6.21)

    solves (6.20) because h<+h; by (3.6) we have Ih=(δhvhln2,). If h>0, then Ih=R and the function ψh implicitly given by

    (2exp(vhδhξ)ψh(ξ)h+hh)h=(2exp(vhδhξ)+hψh(ξ)+hh)+h (6.22)

    solves (6.20) because h<+h. Notice that in both cases ψh(0)=(+h+h)/2 and all the other solutions are of the form φh(ξ)=ψh(ξ+σh) for σhR. Hence, any non-stationary traveling wave ρ has a profile φ of the form

    φ(ξ)=(ψ1(ξ+σ1),,ψn+1(ξ+σn+1)),ξR. (6.23)

    In the sequel we prove that the shifts σh, hH, satisfy (6.17), or equivalently

    v1,jσ1=δ1,jσj,jJ. (6.24)

    Lemma 6.7. Assume (6.19). If +1+11, then condition (6.6) is equivalent to

    v21,jδ1,j=1+jj1+11   and   j+j=α1,jv1,j1+1. (6.25)

    Proof. In the present case, condition (6.6) becomes (c1,jv1,jδ1,j)δ1,jkj=0 for (j,+j): it is satisfied if and only if both c1,jv1,j=δ1,j and kj=0. The former is equivalent to (6.25)1, the latter is equivalent to (6.25)2 by (6.5)4, because κj=0.

    We observe that (6.25)1 and (6.5)1 imply that c1,j=δ1,j/v1,j>0; therefore (6.7) becomes

    ±j(1±j)=α1,jv1,j±1(1±1),jJ. (6.26)

    As a consequence ρ is either non-degenerate or completely degenerate.

    Now, we discuss (completely) non-stationary traveling waves by considering separately the (completely) degenerate and non-degenerate case. We denote

    Δj{α1,jδ1,j,δ1,j,3α1,jδ21,j},jJ.

    Proposition 6.8. Assume (6.19). Problem (2.1)-(2.2) admits a traveling wave that is both (completely) degenerate and (completely) non-stationary if and only if either (6.11) holds true or

    0<v1,j<minΔj   or    v1,j>maxΔj,   jJ,v1,2(δ1,2v21,2)α1,2δ21,2v31,2==v1,n+1(δ1,n+1v21,n+1)α1,n+1δ21,n+1v31,n+1. (6.27)

    In the first case, problem (2.1)-(2.2) has infinitely many of such waves; each of them satisfies (5.3) and (up to shifts) (2.5).

    In the second case, problem (2.1)-(2.2) has a unique (up to shifts) such wave, which does not satisfy (for no shifts) (2.5). Its end states do not satisfy (5.3) and are

    1=0=j,+1=v1,j(δ1,jv21,j)α1,jδ21,jv31,j,+j=α1,jδ1,j(δ1,jv21,j)α1,jδ21,jv31,j,jJ. (6.28)

    In both cases, any degenerate non-stationary traveling wave ρ has a profile φ of the form (6.23) withψh defined by (6.21) and σhR, hH, satisfying (6.24).

    Proof. We claim that the existence of a degenerate non-stationary traveling wave is equivalent to the existence of +h(0,1),hH, such that

    +j=α1,jδ1,jv1,j+1 and (α1,jδ21,jv31,j)+1+v1,j(v21,jδ1,j)=0,jJ. (6.29)

    In fact, by Proposition 6.2 the existence of a non-stationary traveling wave is equivalent to condition (Tq), where (6.6) can be written as (6.25) by Lemma 6.7 and (6.7) as (6.26). Then, (6.25) and (6.26) with h=0, hH, reduce to the relation among the end states

    v21,jδ1,j=1+j1+1,+j(1+j)=α1,jv1,j+1(1+1),jJ. (6.30)

    By (6.30) we obtain v21,j/δ1,j=α1,jv1,j+1/+j and then (6.29)1; by plugging (6.29)1 into (6.30)2 we get (6.29)2 and then the claim.

    Assume there is a degenerate non-stationary traveling wave; then +1 satisfies (6.29)2. As a consequence, we have either α1,jδ21,jv31,j=v21,jδ1,j=0 or α1,jδ21,jv31,j for every jJ. The former case is equivalent to (6.11). In the latter case we can explicitly compute +1 by (6.29)2 for any jJ and impose the constraint 0<+1<1, namely,

    0<v1,j(δ1,jv21,j)α1,jδ21,jv31,j<1.

    A direct computation shows that this is equivalent to (6.27). In conclusion, either condition (6.11) or (6.27) is necessary for the existence of a non-stationary traveling wave with 1=0.

    Conversely, assume condition (6.11). In this case α1,jδ21,j=α1,jv41,j=v31,j. Then (6.29)2 is trivially satisfied for every +1(0,1) and from (6.29)1 we deduce +1=+j. Hence, there is an infinite family of non-stationary traveling waves parameterized by +1(0,1) and satisfying (5.3); as a consequence, they do not coincide up to shifts and they all satisfy (up to shifts) the continuity condition (2.5).

    Assume now condition (6.27). In this case the values for +1 and +j in (6.28) are well defined since v31,jα1,jδ1,j and they are the unique solution to (6.29). In particular, condition (ii) in Proposition 6.2 is automatically satisfied. By the estimates in (6.27) we have +1,+j(0,1) for jJ. Hence, there is a unique (up to shifts) degenerate non-stationary traveling wave and its end states satisfy (6.28). Furthermore, by Lemma 5.1, condition (2.5) implies (5.3), which is precluded by (6.27). Hence, the traveling wave does not satisfy (2.5).

    At last, by Theorem 3.2, any solution to (6.20) has the form (6.23). By (4.14), that in the present case becomes

    φj(cjξ)=α1,jc21,jφ1(c1ξ) for a.e. ξR, jJ,

    and the regularity of ψh defined in (6.21), we have

    1cj(δjvjln2+σj)=1c1(δ1v1ln2+σ1),

    which is equivalent to (6.17) because c1,j=δ1,j/v1,j.

    The following result treats the non-degenerate case.

    Proposition 6.9. Assume (6.19). Problem (2.1)-(2.2) admits a non-degenerate (completely) non-stationary traveling wave if and only if condition (6.11) is satisfied. In this case any non-degenerate non-stationary traveling wave satisfies (up to shifts) (2.5); moreover, it has a profile φ of the form (6.23) with ψh implicitly defined by (6.22) andσhR, hH, satisfying (6.24).

    Proof. Assume that there is a non-degenerate non-stationary traveling wave; then h0 and 1+h+h, hH. Moreover, by Proposition~6.2, condition (Tq) is satisfied, where (6.6) becomes (6.25) by Lemma 6.7 and (6.7) is (6.26). When dividing (6.26) by (6.25)2 we obtain

    (1+j)1=(1+1)j and (1j)+1=(11)+j,jJ.

    By adding the above relations we have 1++1=j++j, hence

    0=+1+j+1j=+11+(1+1)j1+1j=1+111(j1).

    It is now easy to conclude that (5.3) is satisfied and then also (6.11) holds true by (6.25). At last, the traveling wave satisfies (up to shifts) (2.5) by Remark 6.3.

    Conversely, assume (6.11). Then (6.25) and (6.26) write

    +j+j=+1+1,j+j=1+1,±j(1±j)=±1(1±1),jJ.

    The same computations as before give that if we impose h0 and 1+h+h, hH, then the above conditions are equivalent to (5.3); the existence of infinitely many non-degenerate non-stationary traveling waves satisfying (2.5) easily follows.

    At last, by Theorem 3.2, any solution to (6.20) has the form (6.23). Fix jJ. By (2.5) we have (5.1), namely

    ψj(cjt+σj)=ψ1(c1t+σ1),tR.

    This identity together with (6.22) and (5.3) imply

    (2exp(vjδj(cjt+σj))ψ1(c1t+σ1)+)=(2exp(vjδj(cjt+σj))+ψ1(c1t+σ1)+)+,(2exp(v1δ1(c1t+σ1))ψ1(c1t+σ1)+)=(2exp(v1δ1(c1t+σ1))+ψ1(c1t+σ1)+)+.

    By dividing the above equalities and taking the logarithm we get

    (vjδj(cjt+σj)v1δ1(c1t+σ1))=(vjδj(cjt+σj)v1δ1(c1t+σ1))+,tR.

    Since + and c1,j=δ1,j/v1,j, the above equality is equivalently to (6.24).

    In this section we assume (5.10) for some constants vh,δh>0, D1 and the logarithmic flux [13] defined by

    f(ρ)ρln(ρ)

    for ρ(0,1] with f(0)=0 by continuity; in the following we simply write ρln(ρ) for ρ[0,1]. We use the notation introduced in (5.11); then, in the present case the diffusivity Dh coincides with the anticipation length δh of [3], see Section 2. As in Section 6, we focus on the case m=1 and do not mention in the following these assumptions on fh, Dh and m.

    Condition (3.7) becomes

    0h<+h1 and ch=vh+hln(+h)hln(h)+hh. (7.1)

    Moreover we have, for hH,

    gh()=vh(+hln(+h)hln(h)+hhln()),gh()gh(±h)=vh((h)+hln(+h)+(+h)hln(h)+hhln()). (7.2)

    Therefore (3.16) becomes

    φh(ξ)=vhδh[[φh(ξ)h]+hln(+h)+[+hφh(ξ)]hln(h)+hhφh(ξ)ln(φh(ξ))], (7.3)

    for ξR. Let f1:[0,e1][0,e1] and f1r:[0,e1][e1,1] be the inverse functions of the restrictions f and fr of f to [0,e1] and [e1,1], respectively.

    We first consider the case of stationary waves. We define the intervals

    L0j{[0,e1)if α1,jv1,j1,(0,f1(e1α11,jv11,j))if α1,jv1,j>1,jJ.

    Proposition 7.1. Problem (2.1)-(2.2) admits infinitely many stationary traveling waves; their end states are characterized by the conditions

    1jJL0j,+1=f1r(1ln(1)),j=f1(α1,jv1,j1ln(1)),+j=f1r(α1,jv1,j1ln(1)),jJ.

    Moreover, up to shifts, any stationary traveling wave satisfies (2.5).

    Proof. The first part of the proposition follows from Theorem 4.7. Indeed, conditions (3.7)1 and (4.8) are satisfied if and only if for any hH and jJ

    h[0,e1),hln(h)=+hln(+h),jln(j)=α1,jv1,j1ln(1).

    Hence +1=f1r(1ln(1)) and it is sufficient to determine ±j. Observe that the definition of L0j guarantees that they can be uniquely computed. At last, the latter part of the proposition follows by the proof of Proposition 5.2 since e1ShH(h,+h).

    In the following we discuss the existence of non-stationary traveling waves. Since m=1, by Lemma 4.4 this is equivalent to assume that the traveling wave is completely non-stationary. By (7.1)2 we deduce

    c1,j=v1,j +1ln(+1)1ln(1)+jln(+j)jln(j) +jj+11. (7.4)

    The following result translates Theorem 4.12 to the current framework. We define the intervals

    Lcj{[0,1]if α1,jv1,j1,[0,1](f1(e1α11,jv11,j),f1r(e1α11,jv11,j))if α1,jv1,j>1,jJ.

    Proposition 7.2. Problem (2.1)-(2.2) admits a (completely) non-stationary traveling wave if and only if the following condition holds.

    (Tl) There exist ±1[0,1] with 1<+1 such that:

    (i) 1ln(1)+1ln(+1);

    (ii) ±1jJLcj;

    (iii) for any jJ we have

    δ1,j(gj()gj(j))=A1,jc1,j(g1(+kjA1,j)g1(1)),(j,+j), (7.5)

    where gh is given in (7.2), c1,j in (7.4), A1,j in (4.10)2 and kj in (4.10)3, with ±j being solutions to

    ±jln(±j)=α1,jv1,jL±1,jln(L±1,j). (7.6)

    Proof. The proof consists in showing that, in the present case, (T) of Theorem 4.12 is equivalent to (Tl). The first two items in (T) and (Tl) are clearly equivalent. It remains to discuss the third one. Condition (4.17) is equivalent to

    δ1,j(gj(j(cjξ))gj(j))=A1,jc1,j(g1(φ1(c1ξ))g1(1)),ξR, (7.7)

    where φ1 is a solution to (3.5)-(7.3) and j(ξ)A1,jφ1(c1,jξ)kj for c1,j in (7.4), A1,j in (4.10)2 and kj in (4.10)3. By Theorem 3.2, φ1 is strictly increasing and so is the function j. Put j(cjξ). Hence (j,+j), by Lemma 4.10, and then (7.7) is equivalent to (7.5).

    In the following we focus on the case of (completely) non-stationary traveling waves with h=0 for some hH.

    Lemma 7.3. Assume that problem (2.1)-(2.2) admits a traveling wave. The following statements are equivalent:

    (i) 1=0;

    (ii) j=0 for all jJ;

    (iii) there exists jJ such that j=0.

    Proof. First, we prove that (i) implies (ii). Fix jJ. Since 1=0, then condition (7.6) implies that either j=0 or +j=1, for jJ. Assume by contradiction that +j=1. Since c1,j<0, condition (7.6) becomes

    jln(j)=α1,jv1,j+1ln(+1).

    Therefore, by (7.4), (4.10)2 and (4.10)3 we have that

    c1,j=v1,j +1ln(+1)jln(j) 1j+1=1jα1,j+1,Ai,j=1j+1,kj=1.

    Condition (7.5) can be written as

    ln()v1,j(1)(α1,j+1ln(+1)1j+1jα1,jδ1,j+1ln(11j))=0,

    for (j,1). By differentiating the above equation three times we obtain

    v1,j(1j)α1,jδ1,j+1(1)2=12,(j,1).

    This is a contradiction because the two sides have opposite sign. This proves (ii).

    Since the implication (ii) (iii) is obvious, it remains to show that (iii) (i). Let j=0 for some jJ. By (7.6) it follows that either 1=0 or +1=1. In the latter case by arguing as above it is easy to obtain a contradiction and then (iii) follows.

    At last, we give a result which is similar to the one given in Proposition 6.8. We denote

    Δj{α1,jδ1,j,δ1,j},jJ.

    By Lemma 7.3 we have either h=0, hH, or h0, hH. Below we consider the first case.

    Proposition 7.4. Problem (2.1)-(2.2) admits a (completely) non-stationary traveling wave with h=0, hH, if and only if either (6.11) holds true or

    0<v1,j<minΔj    or    v1,j>maxΔj,jJ,(α1,2δ1,2v1,2)δ1,2v21,2δ1,2==(α1,n+1δ1,n+1v1,n+1)δ1,n+1v21,n+1δ1,n+1. (7.8)

    In the first case, problem (2.1)-(2.2) has infinitely many of such waves; each of them satisfies (5.3) and (up to shifts) (2.5).

    In the second case, problem (2.1)-(2.2) has a unique (up to shifts) such wave and such wave, which does not satisfy (for no shifts) (2.5). Its end states are

    1=0=j,+1=(α1,jδ1,jv1,j)δ1,jv21,jδ1,j,+j=(α1,jδ1,jv1,j)v21,jv21,jδ1,j,jJ, (7.9)

    and do not satisfy (5.3).

    Proof. Fix jJ. Since c1,j>0, by (7.4) the formulas in (4.10) and (7.2) become

    c1,j=v1,jln(+1)ln(+j),A1,j=α1,jv1,jln(+1)ln(+j),kj=0=κj,gh()=vhln(+h),gh(0)=0.

    Hence (7.6) can be written as

    +jln(+j)=α1,jv1,j+1ln(+1) (7.10)

    and therefore (7.5) becomes

    (δ1,jv1,j+jα1,j+1)ln(+j)=0,(0,+j),

    namely

    +j=α1,jδ1,jv1,j+1. (7.11)

    System (7.10)-(7.11) admits a solution if and only if either (6.11) or (7.8) holds true. In the former case, (7.10)-(7.11) has infinitely many solutions and they satisfy (5.3); in the latter, the unique solution of (7.10)-(7.11) is (7.9)2,3. We examine separately these cases.

    Assume (6.11). In this case condition (Tl) of Proposition 7.2 with 1=0=j is equivalent to +1=+j(0,1),jJ, and then there are infinitely many traveling waves. They all satisfy (2.5) by Remark 6.3.

    Assume (7.8). In this case condition (Tl) of Proposition 7.2 with 1=0=j is equivalent to +h(0,1), hH, satisfying (7.10)-(7.11), namely to (7.8)-(7.9). In particular, (7.8)1, (7.9) imply that +j and +1 are distinct, namely they do not satisfy (5.3). Moreover, by Remark 6.3 the traveling wave does not satisfies (2.5).

    At last, the reverse implications are direct consequences of previous discussion about the solutions of (7.10)-(7.11) and then the proof is complete.

    Let ±h[0,1] with h+h. We introduce the change of variable

    rh+hρh+hh, (A.1)

    which implies ρh=+h(+hh)rh, ρh,t=(+hh)rh,t and ρh,x=(+hh)rh,x. Consequently, equation (2.1) can be written

    rh,t+Gh(rh)x=(Eh(rh)rh,x)x, (A.2)

    where

    Gh(rh)fh(+h(+hh)rh)fh(+h)+hh,Eh(rh)Dh(+h(+hh)rh).

    Furthermore, equation (A.2) has a wavefront solution ψh from 1 to 0 with wave speed θh if and only if equation (2.1) has a wavefront solution φh from h to +h with the same speed. Notice that ψh satisfies the equation

    (Eh(ψh)ψh)+(θhGh(ψh))ψh=0

    and φh is obtained by ψh by the change of variable (A.1), i.e.

    φh(ξ)=(h+h)ψh(ξ)++h,ξR. (A.3)

    We discuss now the existence of a wavefront solution rh(t,x)=ψh(xθht+σh)=ψh(ξ) of (A.2). In order to make use of [12,Theorem 9.1], we only need to show that

    Gh(rh)>rhGh(1),rh(0,1). (A.4)

    By the definition of Gh we have

    rhGh(1)=rhfh(+h)fh(h)+hh.

    Then, inequality (A.4) is equivalent to

    fh(+h)(fh(+h)f(h))rh<fh(+h(+hh)rh), for rh(0,1),

    if and only if h<+h. By the strict concavity of fh the last inequality is satisfied and then, by [12,Theorem 9.1], we deduce the existence of wavefront solutions ψh from 1 to 0 for (A.2). The wave speed, in this case, is θhGh(1). Furthermore, the profile ψh is unique up to shifts and, if ψh(0)ν for some 0<ν<1, then

    {ψh(ξ)=1for ξνh,νψh(ξ)Eh(s)Gh(s)+sGh(1)=ξfor νh<ξ<ν+h,ψh(ξ)=0for ξν+h, (A.5)

    where

    ν+hν0Eh(s)Gh(s)+sGh(1)ds,νh1νEh(s)Gh(s)+sGh(1)ds.

    Notice that, by differentiating (A.5) in the interval (νh,ν+h), we obtain that

    Eh(ψh(ξ))Gh(ψh(ξ))ψh(ξ)Gh(1)ψh(ξ)=1,ξ(νh,ν+h), (A.6)

    which implies ψh<0 in (νh,ν+h) because of (A.4).

    Consider now φh defined in (A.3); it satisfies (3.3) with Ih=(νh,ν+h) and φh>0 in Ih. Also condition (3.7) is true and φhC2(Ih,(h,+h) by the regularity of Dh and fh.

    Now it remains to consider the boundary conditions of φh at the extrema of Ih in the different cases. We have the following.

    (i) Assume h=0=Dh(0). We show that

    νh=1νEh(s)Gh(s)+sGh(1)ds>. (A.7)

    To prove (A.7), notice that Eh(1)=Dh(0)=0 and that Gh(s)+sGh(1)0 as s1. In addition, by means of the strict concavity of fh we obtain that

    lims1Eh(s)Gh(s)+Gh(1)=Eh(1)Gh(1)+Gh(1)=+hDh(0)fh(0)+fh(+h)+h0

    and then, by applying de l'Hospital Theorem we prove condition (A.7). Moreover, by condition (A.6), we get

    limξνhψh(ξ)={fh(0)fh(+h)+h+hDh(0)if Dh(0)>0,if Dh(0)=0.

    By applying (A.3) we conclude that φh(ξ)=h for ξνh and the estimates in (3.8) are satisfied. Furthermore, by the change of variables (A.1), we obtain that

    limξνhDh(φh(ξ))φh(ξ)=limξνh+hEh(ψh(ξ))ψh(ξ)

    and hence, by (A.6), we deduce (3.9).

    (ii) Assume 1+h=0=Dh(1). With a similar reasoning as in (i) we prove that ν+h>. In fact, Eh(0)=Dh(1)=0 and sGh(s)+sGh(1)0 as s0+. Moreover

    lims0+Eh(s)Gh(s)+Gh(1)=Eh(0)Gh(0)+Gh(1)=(1h)Dh(1)fh(1)+fh(h)1h0

    and again, by applying de l'Hospital Theorem we prove that ν+h>. Moreover, by the estimate (A.6), we have that

    limξνhψh(ξ)={fh(h)1h+fh(1)(h1)Dh(1)if Dh(1)<0,if Dh(1)=0.

    By applying (A.3) we conclude that φh(ξ)=+h for ξν+h and the estimates in (3.10) are satisfied; by (A.1) and (A.6) we derive (3.11).

    (iii) In all the other cases it is easy to show that Ih=R and again the slope condition (3.12) can be obtained by the estimate (A.6).

    The authors are members of INdAM-GNAMPA. They were supported by the Project Macroscopic models of traffic flows: qualitative analysis and implementation, sponsored by the University of Modena and Reggio Emilia. The first author was also supported by the INdAM -GNAMPA Project 2016 ''Balance Laws in the Modeling of Physical, Biological and Industrial Processes''. The last author was also supported by the INdAM -GNAMPA Project 2017 ''Nonlocal hyperbolic equations''.

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