Traveling waves for degenerate diffusive equations on networks

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.


Introduction
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 [8,10,11,16,23,29]. 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 h = 1, . . . , m + n, (1.1) where t denotes time and x the position along the road. In this case ρ h is a vehicle density; about the diffusivity D h (ρ h ) ≥ 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 [17,24]. We refer to [3,15,17,19,21,26] 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 [10,11,25] 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 [20]; in the case of a second-order model without diffusion but including a relaxation term, we refer to [9,27]; 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 [29,30] for the semilinear diffusive case and to [18] 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 [1,10,11,25]; 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.

The model
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 d ∈ R m+n has components d i = 1 for i ∈ I . = {1, . . . , m} and d j = −1 for j ∈ J . = {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 x ∈ R − . = (−∞, 0] and numbered by the index i, outgoing roads by x ∈ R + . = [0, ∞) and j; the crossroad is located at x = 0 for both parameterizations. We denote the generic road by Ω h for h ∈ H; then Ω i . = R − for i ∈ I and Ω j . = R + for j ∈ J. The network is defined as N .
Following the above analogy, we understand the unknown functions ρ h as vehicular densities in the roads Ω h , h ∈ H; ρ h ranges in [0, ρ h ], where ρ h is the maximal density in the road Ω h . Without loss of generality we assume that ρ h = 1 for every h ∈ H; 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 . = (ρ 1 , . . . , ρ m+n ) : R × N → [0, 1] m+n understanding that ρ(t, x 1 , . . . , x m+n ) = (ρ 1 (t, x 1 ), . . . , ρ m+n (t, x m+n )).
For each road we assign the functions f h , the hyperbolic flux, and D h , the diffusivity; we assume for every h ∈ H We emphasize that in (D) we can possibly have either D h (0) = 0 or D h (1) = 0, or even both possibilities at the same time. The evolution of the flow is described by the equations Assumption (f) is standard when dealing with traffic flows [2]. More precisely, in that case is either linear or strictly concave, decreasing and satisfying v h (1) = 0, see [17,24]. The prototype of such a velocity satisfying which was introduced in [14]; another example is given in [22]. The simplest model for the diffusivity is then where δ h is an anticipation length [3,20].
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 for given constant coefficients α i,j ∈ (0, 1] satisfying 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 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 α i,j > 0 for every i and j. The case when α i,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 v h are positive. Then, we are faced with the system of equations (2.1) that are coupled through (2.2), with the α i,j satisfying (2.3). Solutions to (2.1)-(2.2) are meant in the weak sense, namely ρ h ∈ C 1 (R × Ω h ; [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.

Traveling waves for a single equation
In this section we briefly remind some definitions and results about traveling waves [12] for the single equation where we keep for future reference the index h. Equation (3.1) has no source terms (differently from [29,30]) and then any constant is a solution; for simplicity we discard constant solutions in the following analysis.
This definition coincides with that given in [18,28] because we are considering nonconstant profiles. The profile must satisfy the equation namely, is the reduced flux, see Figure 2.
PSfrag replacements Figure 2: A flux f h satisfying (f), solid curve, and the corresponding reduced flux g h defined in (3.4), dashed curve, in the case c h < 0, left, and in the case c h > 0, right.

This means that
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. (3.7) We have that ϕ h ∈ C 2 I h ; (ℓ − h , ℓ + h ) is unique (up to shifts) and ϕ ′ h (ξ) > 0 for ξ ∈ I h ; moreover, the following holds true. We observe that for c h given by (3.7), we deduce by (f) that g h (ρ) ≥ 0 for all ρ ∈ [ℓ − h , ℓ + h ], see Figure 2. Moreover, we have 3.13) and no ρ = ℓ ± h makes g h (ρ) equal to that value. Theorem 3.2 motivates the following definition. Remark 3.4. A consequence of assumption (f) is that if ρ h is degenerate, then the profile ϕ h is singular either at ν − h in case (i) or at ν + h in case (ii), in the sense that ϕ ′ h 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 ν − h . = −∞ (respectively, ν + h . = ∞). In this way the interval (ν − h , ν + h ) is always defined and coincides with the interval I h defined in (3.6): . The interval I h is bounded if and only if both (i) and (ii) hold; in this case ρ h is both degenerate and stationary. As a consequence, if ρ h is non-stationary then I h is unbounded and coincides either with a half line (if ρ h is degenerate) or with R (if ρ h is non-degenerate). At last, ρ h is degenerate if and only if either ν − h or ν + h is finite. In the case of non-stationary traveling-wave solutions ρ h we use the notation (3.14) Lemma 3.5. Let ρ h be a traveling-wave solution of (3.1); then we have the following.
h . This means that at least one of the end states must be 0 or 1, say 0; but then c h = 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 c h = 0, exactly one between (i) and (ii) of Theorem 3.2 occurs, namely, exactly one between ν − h and ν + h is finite. If ν − h is finite and The statement about the smoothness of ξ → ϕ ′ h (c h ξ) is proved as above. Finally, the converse is straightforward. In fact, if ω h is finite, then either ω h = c −1 h ν − h and ν − h is finite, or ω h = c −1 h ν + h and ν + h is finite; in both cases ρ h is degenerate. Because of the smoothness properties of the profile proved in Theorem 3.2, we can integrate equation (3.2) in (ξ − , ξ) ⊂ I h and we obtain h in the previous expression, by applying (3.9) or (3.12) we deduce We observe that (3.15) is trivially satisfied in case (i) when ξ < ν − h 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 = ν − h and in case (ii) at ξ = ν + h , respectively. In conclusion, we have that (3.15) holds in the whole R, namely

Traveling waves in a network
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 h ∈ H, let ρ h be a traveling-wave solution of (2.1) h in the sense of Definition 3.1 and set ρ . = (ρ 1 , . . . , ρ m+n ). With reference to Definition 3.3, we say that: • ρ is stationary if each component ρ h is stationary; • ρ is completely non-stationary if none of its components is stationary; • ρ is degenerate if at least one component ρ h 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 . = (ϕ 1 , . . . , ϕ m+n ) is a profile for ρ if ϕ h is a profile corresponding to ρ h for every h ∈ H.
For clarity of exposition, we collect our general results for stationary and non-stationary traveling waves in the following subsections.

General results
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 h ∈ H; both conditions in (3.7) are tacitly assumed as well.
In (4.1) any combination of the signs ± is allowed.
2) and recalling that by Theorem 3.2 the profiles are continuous in R, we obtain 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.
Proof. Fix j ∈ J. We notice that (4.1) is equivalent to Hence, by passing to the limit for t → ±∞ in (4.1) we obtain (4.2) and (4.3), respectively. (ii) c i = 0 for all i ∈ I; (iii) c j = 0 for all j ∈ J.
Proof. By subtracting (4.3) to (4.2) we obtain , 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 c h = 0 for every h ∈ H, or it is non-stationary, and then there exists i ∈ I such that c i = 0 and c j = 0 for every j ∈ J.
In this case, the end states ℓ ± j are uniquely determined if and only if c i = 0 for every i ∈ I.
Conversely, assume (4.5). If c i = 0 for every i ∈ I, then ℓ − j < ℓ + j are uniquely determined because of the strict concavity of f j . Assume, on the contrary, that c i = 0 for some i ∈ I; then c j = 0 by Lemma 4.4, i.e., f j (ℓ − j ) = f j (ℓ + j ). Thus (4.2)-(4.3) determine exactly four possible choices of end states ℓ ± j with ℓ − j < ℓ + j , see Figure 3.
By Proposition 4.5 and Lemma 4.4 we deduce that the end states ℓ ± j are uniquely determined in terms of the end states ℓ ± 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 Proposition 4.6. Assume that problem (2.1)-(2.2) admits a traveling wave. Then for any j ∈ J we have Moreover, (4.7) is equivalent to (4.2)-(4.3).

The stationary case
In this short subsection we briefly consider stationary traveling waves.
Proof. Clearly, (4.8) is trivially satisfied if ℓ − h = 0 and ℓ + h = 1 for all h ∈ H. We claim that there exist infinitely many choices of ℓ ± 1 , . . . , ℓ ± m+n satisfying (4.8). To prove the claim, we are sufficiently small to satisfy the first condition in (4.5) for all j ∈ J. Then, by a continuity argument, we can choose . 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).

The non-stationary case
In this subsection we consider non-stationary traveling waves. By Lemma 4.4 this is equivalent to consider the scenario in (4.4): We can therefore introduce the following notation: where L i,j is defined in (4.6) and We notice that I c 0 = ∅ by (4.4) and that both I 0 and I c 0 depend on the end states ℓ ± i , i ∈ I, indeed. Moreover, k j is well defined because by (4.7) Finally, by (f) we deduce that Proof. By Proposition 4.2 it is sufficient to prove that by (4.4) condition (4.1) is equivalent to (4.10). By (3.13) we have that is equivalent to (4.10).
We observe that k j and (4.10) can be written in a little bit more explicit form by avoiding the use of L ± i,j as follows Proposition 4.8 shows how each outgoing profile ϕ j can be expressed by (4.10) in terms of the ingoing profiles ϕ i , i ∈ I. We know a priori that ϕ j 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 ϕ j are enjoined by the function defined by the right-hand side of (4.10).
Lemma 4.9. Let ϕ i , for i ∈ I, be the profiles provided by Theorem 3.2 and assume that I c 0 = ∅; fix j ∈ J and consider any l ± j ∈ [0, 1] satisfying (4.7) and such that, for the corresponding c j , it holds c j = 0. Then l − j < l + j . Moreover, denote by ℓ j (ξ) the right-hand side of (4.10); then ξ → ℓ j (ξ) is non-decreasing and ℓ j (±∞) = l ± j .
Proof. Since by Theorem 3.2 we know that ℓ − i < ℓ + i , then by (4.7) j because by the definitions of ℓ j and κ j we have We notice that Proposition 4.8 exploits condition (2.2) through its expression (4.1) for the profiles; the diffusivities D h are not involved in (4.10). Indeed, Proposition 4.8 imposes strong necessary conditions on the diffusivities as we discuss now as a preparation to (4.16).
We notice that if both but not both. In this case we have Proof. Let us introduce the following conditions: Clearly where ω h is defined in (3.14). Differentiating (4.10) gives More precisely, by Lemma 3.5, formula (4.13) holds for ξ ∈ R \ {ω j } ∪ i∈I c 0 ω i ; moreover, by the same lemma we know that ξ → ϕ ′ h (c h ξ) is singular at ξ = ω h and C 1 elsewhere, for h ∈ I c 0 ∪ J. Hence, (4.13) implies (4.12). By (4.12) 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 4.10 implies that a non-stationary traveling wave ρ is either non-degenerate, and then ρ h is non-degenerate for every h ∈ H, or ρ is degenerate, and then either there exists i ∈ I 0 such that ρ i is degenerate, or ρ h is degenerate for all h ∈ I c 0 ∪ J. In both cases a non-stationary traveling wave ρ satisfies (4.12). 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): The proof of the following result is an immediate consequence of Lemma 4.10 and, hence, omitted. 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: Since ϕ h satisfies equation (3.16), we are led to extend the quotient ℓ → to the whole of R by defining (4.14) In fact, when ℓ is replaced by We remark that condition D h (ℓ) = 0 occurs at most when either ℓ = 0 or ℓ = 1. To avoid the introduction of the new notation (4.14), in the following we simply keep on writing for γ h (ℓ). As a consequence, any non-stationary traveling wave of problem (2.1)-(2.2) satisfies for a.e. ξ ∈ R, (4.16) where ϕ 1 , . . . , ϕ m are solutions to (3.5)-(3.16) and, for k j as in (4.9), Proof. First, assume that problem (2.1)-(2.2) admits a non-stationary traveling wave ρ with profiles ϕ h , end states ℓ ± h and speeds c h , for h ∈ H. By Theorem 3.2 we have that ℓ ± h and c h satisfy (3.7). By Proposition 4.8 the profiles ϕ h satisfy (3.5)-(3.16) and (4.10). The end states ℓ ± j , j ∈ J, satisfy (4.7) by Proposition 4.6. Since ρ is non-stationary we are in the scenario given by (4.4): for ξ ∈ R in the non-degenerate case and for ξ ∈ R \ {ω} with ω given by (4.12) in the degenerate case. On the other hand, by differentiating (4.10) and applying (4.15) with h = i we deduce for ξ ∈ R in the non-degenerate case and for ξ ∈ R \ {ω} with ω given by (4.12) in the degenerate case. Identity (4.16) follows because ℓ j ≡ ϕ j by (4.10) and by comparing (4.18), (4.19). Conversely, assume that condition (T) holds. We remark that the existence of ϕ i , i ∈ I, is assured by Theorem 3.2. Fix j ∈ J. By defining ϕ j . = ℓ j we obtain (4.10). We know by assumption that I c 0 = ∅, ℓ ± j ∈ [0, 1] satisfy (4.7) and c j = 0; we can apply therefore Lemma 4.9 and deduce that ℓ − j < ℓ + j , ϕ j is non-decreasing and satisfies (3.5) with h = j. By Proposition 4.8, what remains to prove is that ϕ j satisfies (3.16). But by (4.10) we deduce (4.19) for a.e. ξ ∈ R, because ϕ 1 , . . . , ϕ m satisfy (3.16) and hence, recalling the extension (4.14), also (4.15); then by (4.16) we conclude that ϕ j satisfies (4.15) for a.e. ξ ∈ R and then (3.16) for a.e. ξ ∈ R. Finally, (3.16) holds by the regularity ensured by Theorem 3.2 for the profiles.
Remark 4.13. Fix ℓ ± i ∈ [0, 1], i ∈ I, so that ℓ − i < ℓ + i and (4.5) holds. We know by Proposition 4.5 that for every j ∈ J there exists (ℓ − j , ℓ + j ) ∈ [0, 1] 2 , with ℓ − j < ℓ + j , that satisfies (4.7), but it is not unique. If beside (4.7) we impose also (4.16), then we may have three possible scenarios: such (ℓ − j , ℓ + 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.

The continuity condition
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 [18,29,30] 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.
(ii) If ρ satisfies (2.5), then either it is stationary (hence (5.1) reduces to ϕ j (0) = ϕ i (0)), or it is completely non-stationary and the speeds c h have the same sign (hence c i,j > 0). In the latter case, ρ is either non-degenerate or completely degenerate; moreover Proof. We split the proof according to the items in the statement.
(ii) Since we are discarding constant profiles, by (5.1) we have that either c h = 0 for all h ∈ H or c h = 0 for all h ∈ H. The stationary case is trivial; therefore we consider below only the non-stationary case and assume that c h = 0 for all h ∈ H. By differentiating (5.1) with respect to t we deduce 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 ρ h is degenerate if and only if the map ξ → ϕ ′ h (c h ξ) is singular at ξ = ω h ∈ R and C 1 elsewhere. By taking t ∈ I in (5.6) we deduce that c i and c j have the same sign. As a consequence we have L ± i,j = ℓ ± i and then ℓ ± i = ℓ ± j by letting t → ±∞ in (5.1). By (5.1), (3.16) and (5.3) we have D h Φ(ξ) Φ ′ (ξ) = c h g h (Φ(ξ)) − g h (ℓ ± ) for all h ∈ H, whence (5.4) by the extension (4.14).
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 ℓ ± h satisfy (4.8) and are such that S . = h∈H (ℓ − h , ℓ + h ) = ∅. Proof. By (5.1) condition (2.5) holds in the stationary case if and only if ϕ i (0) = ϕ j (0) for (i, j) ∈ I × J. Recalling the proof of Theorem 4.7, it is sufficient to take ℓ ± h ∈ [0, 1] satisfying (4.8) and such that S = ∅, ℓ 0 ∈ S and the unique solution ϕ h to (3.5)-(3.16) such that ϕ h (0) = ℓ 0 . There are infinitely many of such profiles because of the arbitrariness of ℓ ± h . We point out that condition S = ∅ can occur if the functions f h 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. (T c ) There exist ℓ ± ∈ [0, 1] with ℓ − < ℓ + , such that for any h ∈ H, i ∈ I and j ∈ J

Consider in particular the case when the functions f and D satisfy (f) and (D), respectively, and assume that
We notice that now we have v i,j = c i,j . Remark that by (5.12) condition (5.13) 2 is equivalent to (5.5) 4 .
6 Application to the case of a quadratic flux, m = 1 In this section we assume (5.10) for some constants v h , δ h > 0, D satisfying (D) and the quadratic flux [14] f (ρ) 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.
and therefore (3.16) becomes We first consider stationary traveling waves and specify Theorem 4.7 and Proposition 5.2 in the current framework. We define the intervals Proposition 6.1. Problem (2.1)-(2.2) admits infinitely many stationary traveling waves; their end states are characterized by the conditions 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 h ∈ H and j ∈ J then it is sufficient to compute ℓ ± j and to observe that the definition of L 0 j guarantees that they are real numbers.
The latter part of the proposition is deduced by Proposition In the following we treat the existence of non-stationary traveling waves. Since m = 1, by Lemma 4.4 this is equivalent to assume c h = 0 for h ∈ H, namely, the traveling wave is completely non-stationary. By (4.7), (6.1) and (6.2), from (4.9) we deduce (6.5) The following result translates Theorem 4.12 to the present case. We define the intervals Proposition 6.2. Problem (2.1)-(2.2) admits a (completely) non-stationary traveling wave if and only if the following condition holds.
(T q ) There exist ℓ ± 1 ∈ [0, 1] with ℓ − 1 < ℓ + 1 such that: where k j is defined in (6.5) with ℓ ± j being solutions to Proof. The proof consists in showing that, in the present case, condition (T) of Theorem 4.12 is equivalent to (T q ).
• The first item of (T) is clearly equivalent to the first item of (T q ).
• We prove now that the second item of (T) is equivalent to the second item of (T q ).
• Finally, to prove that (T q ) 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 (6.14) and (6.24). 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 v 2 1,j = δ 1,j and α 1,j v 1,j = 1, j ∈ J. (6.10) The special cases of constant or linear diffusivities are treated in the following subsections.

The case of constant diffusivities
In this subsection we assume D . = 1, (6.11) and in this case problem (3.5)-(6.4) reduces to For any h ∈ H, the function 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 (6.13) to problem (6.12).
Proposition 6.4. Assume (6.11). Problem (2.1)-(2.2) admits a (completely) non-stationary traveling wave if and only if α 1,j δ 1,j = v 1,j . (6.14) In this case any non-stationary traveling wave ρ has a profile ϕ of the form with ℓ ± h satisfying (i), (ii) and (6.7) in Proposition 6.2 and σ h ∈ R, h ∈ H, such that Proof. By Theorem 3.2, any solution to (6.12) has the form (6.15) with σ h ∈ R, h ∈ H. Therefore, by Proposition 4.2 it only remains to prove that (4.1) is equivalent to (6.14)-(6.16). Straightforward computations show that in the present case (4.1) can be written as • In the former case, identity (6.17) is equivalent to Since by assumption • In the latter case, identity (6.17) is equivalent to Since by assumption f j (ℓ + j ) = f j (ℓ − j ), it must be ζ j ζ 1 ≡ 1, i.e. z j (t) = −z 1 (t), namely In both cases we proved that (4.1) is equivalent to (6.14)-(6.16); this concludes the proof.
Remark 6.5. Consider conditions (6.10) 1 , (6.10) 2 and (6.14). Any two of them implies the third one. Proof. The first part of the statement is just Remark 6.3. In this case, since (6.10) implies (6.14), by Proposition 6.4 any (completely) non-stationary traveling wave ρ has a profile of the form (6.15)-(6.16). 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 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 In both cases, any degenerate non-stationary traveling wave ρ has a profile ϕ of the form (6.22) with ψ h defined by (6.20) and σ h ∈ R, h ∈ H, satisfying (6.23).
At last, by Theorem 3.2, any solution to (6.19) has the form (6.22). By (4.13), that in the present case becomes for a.e. ξ ∈ R, j ∈ J, and the regularity of ψ h defined in (6.20), we have which is equivalent to (6.16) because c 1,j = δ 1,j /v 1,j .
The following result treats the non-degenerate case.
We use the notation introduced in (5.11); then, in the present case the diffusivity D h coincides with the anticipation length δ h 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 f h , D h and m. Condition (3.7) becomes (7.1) Moreover we have, for h ∈ H, We first consider the case of stationary waves. We define the intervals
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 h ∈ H and j ∈ J .
and it is sufficient to determine ℓ ± j . Observe that the definition of L 0 j guarantees that they can be uniquely computed. At last, the latter part of the proposition follows by the proof of Proposition 5.2 since e −1 ∈ S .
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 The following result translates Theorem 4.12 to the current framework. We define the intervals (T l ) There exist ℓ ± 1 ∈ [0, 1] with ℓ − 1 < ℓ + 1 such that: where g h is given in (7.2), c 1,j in (7.4), A 1,j in (4.9) 2 and k j in (4.9) 3 , with ℓ ± j being solutions to ℓ ± j ln(ℓ ± j ) = α 1,j v 1,j L ± 1,j ln(L ± 1,j ).
Proof. The proof consists in showing that, in the present case, (T) of Theorem 4.12 is equivalent to (T l ). The first two items in (T) and (T l ) are clearly equivalent. It remains to discuss the third one. Condition (4.16) is equivalent to where ϕ 1 is a solution to (3.5)-(7.3) and ℓ j (ξ) . = A 1,j ϕ 1 (c 1,j ξ) − k j for c 1,j in (7.4), A 1,j in (4.9) 2 and k j in (4.9) 3 . By Theorem 3.2, ϕ 1 is strictly increasing and so is the function ℓ j . Put ℓ . = ℓ j (c j ξ). Hence ℓ ∈ (ℓ − j , ℓ + j ), by Lemma 4.9, and then (7.7) is equivalent to (7.5).
In the following we focus on the case of (completely) non-stationary traveling waves with ℓ − h = 0 for some h ∈ H. Lemma 7.3. Assume that problem (2.1)-(2.2) admits a traveling wave. The following statements are equivalent: (ii) ℓ − j = 0 for all j ∈ J; (iii) there exists j ∈ J such that ℓ − j = 0.

Condition (7.5) can be written as
for ℓ ∈ (ℓ − j , 1). By differentiating the above equation three times we obtain 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 ℓ − j = 0 for some j ∈ J. By (7.6) it follows that either ℓ − 1 = 0 or ℓ + 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 ∆ j . = α 1,j δ 1,j , δ 1,j , j ∈ J.
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 and do not satisfy (5.3).
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.

A Proof of Theorem 3.2
Let ℓ ± h ∈ [0, 1] with ℓ − h = ℓ + h . We introduce the change of variable Furthermore, equation (A.2) has a wavefront solution ψ h from 1 to 0 with wave speed θ h if and only if equation (2.1) has a wavefront solution ϕ h from ℓ − h to ℓ + h with the same speed. Notice that ψ h satisfies the equation and ϕ h is obtained by ψ h by the change of variable (A.1), i.e.
We discuss now the existence of a wavefront solution r h (t, x) = ψ h (x − θ h t + σ h ) = ψ h (ξ) of (A.2). In order to make use of [12, Theorem 9.1], we only need to show that By the definition of G h we have

Then, inequality (A.4) is equivalent to
if and only if ℓ − h < ℓ + h . By the strict concavity of f h the last inequality is satisfied and then, by [12,Theorem 9.1], we deduce the existence of wavefront solutions ψ h from 1 to 0 for (A.2). The wave speed, in this case, is θ h . = G h (1). Furthermore, the profile ψ h is unique up to shifts and, if ψ h (0) . = ν for some 0 < ν < 1, then ds.
Notice that, by differentiating (A.5) in the interval (ν − h , ν + h ), we obtain that which implies ψ ′ h < 0 in (ν − h , ν + h ) because of (A.4). Consider now ϕ h defined in (A.3); it satisfies (3.3) with I h = (ν − h , ν + h ) and ϕ ′ h > 0 in I h . Also condition (3.7) is true and ϕ h ∈ C 2 (I h , (ℓ − h , ℓ + h ) by the regularity of D h and f h . Now it remains to consider the boundary conditions of ϕ ′ h at the extrema of I h in the different cases. We have the following.
(i) Assume ℓ − h = 0 = D h (0). We show that To prove (A.7), notice that E h (1) = D h (0) = 0 and that −G h (s) + sG h (1) → 0 as s → 1 − . In addition, by means of the strict concavity of f h we obtain that and then, by applying de l'Hospital Theorem we prove condition (A.7). Moreover, by condition (A.6), we get By applying (A.3) we conclude that ϕ h (ξ) = ℓ − h for ξ ≤ ν − h and the estimates in (3.8) are satisfied. Furthermore, by the change of variables (A.1), we obtain that and hence, by (A.6), we deduce (3.9).
(iii) In all the other cases it is easy to show that I h = R and again the slope condition (3.12) can be obtained by the estimate (A.6).