Uniqueness and nonuniqueness of fronts for degenerate diffusion-convection-reaction equations

We consider a scalar parabolic equation in one spatial dimension. The equation is constituted by a convective term, a reaction term with one or two equilibria, and a positive diffusivity which can however vanish. We prove the existence and several properties of traveling-wave solutions to such an equation. In particular, we provide a sharp estimate for the minimal speed of the profiles; for wavefronts, we improve previous results about their regularity; we discover a new family of semi-wavefronts.

In the above notation, the numbers suggest where it is mandatory that the corresponding function vanishes. Notice that (D1) leaves open the possibility for D to vanish or not at 0, and (D0) for D at 1. We refer to Figure 1 for a graphical illustration of these assumptions. Notice that the product Dg always vanishes at both 0 and 1 under both set of assumptions. We also require the following condition on the product of D and g: lim sup ϕ→0 + D(ϕ)g(ϕ) ϕ < +∞. (1.2) Condition (1.2) is equivalent to D(ϕ)g(ϕ) ≤ Lϕ, for some L > 0 and ϕ in a right neighborhood of 0, and it is satisfied by minimal regularity assumptions on D and/or g.
In (1.1), the notation ρ = ρ(t, x) suggests a density; this is indeed the case. In the last twenty years, the modeling of vehicular traffic flows or pedestrian dynamics has attracted the interest of several mathematicians, providing new and challenging problems [11,12,26]. This paper was partly motivated by such a research stream and carries on the analysis of a scalar parabolic model begun in [6,7,8]. Indeed, if f (ρ) = ρv(ρ), where the velocity v is an assigned function, then equation (1.1) can be understood as a simplified model for a crowd walking with velocity v along a straight path with side entries for other pedestrians, which are modeled by g; here ρ is understood as the crowd normalized density. Assumption (g01), for instance, means that pedestrians do not enter if the road is empty (g(0) = 0, modeling an aggregative behavior) or if it is fully occupied (g(1) = 0, because of lack of space). If the diffusivity is small, then the diffusion term accounts for some "chaotic" behavior, which is common in crowds movements. In this framework it is usual to assume that D degenerates at the extrema of the interval where it is defined [2,3,5,24]. For more details on this modeling we refer to [7] and references there, in particular for the dependence of D on ρ. On the other hand, the assumption (g0) is better motivated by population dynamics. In this case g is a growth term which, for instance, increases with the population density ρ. We refer to [23] for analogous modelings in biology. Anyhow, apart from the above possible applications, equation (1.1) is a quite general diffusion-convection-reaction equation that deserves to be full understood.
A traveling-wave solution is, roughly speaking, a solution to (1.1) of the form ρ(t, x) = ϕ(x − ct), for some profile ϕ = ϕ(ξ) and constant wave speed c, see [13] for general information. In this case the profile must satisfy, in some sense, the equation where ′ denotes the derivative with respect to ξ. A unique (up to shifts) solution to (1.3) is usually determined by imposing conditions on ϕ at ±∞, which must coincide with the equilibria of (1.1), i.e., with the zeros of g. We consider in this paper non-constant, monotone profiles, and focus on the case they are decreasing. This leads to require either ϕ(−∞) = 1, ϕ(+∞) = 0, (1.4) or simply ϕ(+∞) = 0, (1.5) according to we make assumption (g01) or (g0). The former profiles are called wavefronts, the latter are semi-wavefronts, see Figure 2; precise definitions are provided in Definition 2.1. Notice that in both cases the equilibria may be reached for a finite value of the variable ξ as a consequence of the degeneracy of D at those points. These solutions represent single-shape smooth transitions between the two constant densities 0 and 1. In the case of wavefronts, their interest lies in the fact that they are viscous approximations of shock waves to the inviscid version of equation (1.1), i.e., when D = 0. Semi-wavefronts lack of this motivation but are nevertheless meaningful for applications [7]; moreover, wavefronts connecting "nonstandard" end states can be constructed by pasting semi-wavefronts [8], see also the end of this Introduction. At last, we point out that assumption (1.2) is usual in this framework, when looking for decreasing profiles, see e.g. [1], and implies that the dynamical system underlying (1.3) has a node at the origin. If D(ρ) ≥ 0, the existence of solutions to the initial-value problem for (1.1) is more or less classical [28]; however, the fine structure of traveling waves reveals a variety of different patterns. We refer to [19,20], respectively, for the cases where D is non degenerate, i.e., D > 0, and for the degenerate case, where D can vanish at either 0 or 1. The main results of those papers is that there is a critical threshold c * , depending on both f and the product Dg, such that traveling waves satisfying (1.4) exist if and only if c ≥ c * . The smoothness of the profiles depend on f , D and c but not on g. In both papers the source term satisfies (g01); see [6,7] for the case when g has only one zero.
The case when D changes sign, which is not studied in this paper, also has strong motivations: we quote [17,25] for biological models, [8] for applications to collective movements, and [9,14,15] for other models. Several results about traveling waves have been obtained in [8,10,16,17,18].
In this paper we study semi-wavefronts and wavefronts for equation (1.1); in particular we complete the analysis that began in [6,7]. We prove that in both cases there is a threshold c * , as above, such that profiles only exists for c ≥ c * ; we also study their regularity and strict monotonicity, namely whether they are classical (i.e., C 1 ) or sharp (and then reach an equilibrium at a finite ξ in a no more than continuous way). Several explicit examples are scattered throughout the paper to show that our assumptions are necessary in most cases.
This research has some important novelties. First, we give a refined estimate for c * , which allows to better understand the meaning of this threshold. Second, we improve a result obtained in [20] about the appearance of wavefronts with a sharp profile. Third, in the case of semi-wavefronts, we make the surprising discovery of a whole family of profiles which was previously unknown, to the best of our knowledge. This is a consequence of a detailed study of a singular first-order problem, as we briefly explain now.
The main tool to investigate (1.3) is the analysis of singular first-order problems as (1.6) Problem (1.6) is deduced by problem (1.3)-(1.5) by the singular change of variables z(ϕ) := D(ϕ)ϕ ′ , where the right-hand side is understood to be computed at ϕ −1 (ϕ), see e.g. [7,19]. Notice that ϕ −1 exists by the assumption of monotony of ϕ. The use of (1.6) to tackle problem (1.3)-(1.5) is not new, but in this paper the analysis is pushed to a detail that was never considered in previous papers and reveals unexpected solutions, which lead to the loss of uniqueness of semi-wavefronts to (1.3).
On the other hand, the analysis of problem (1.6) is fully exploited in the forthcoming paper [4], which deals with the case in which D changes sign once. In that paper we show that there still exist wavefronts joining 1 with 0, which travel across the region where D is negative; they are constructed by pasting two semi-wavefronts obtained in the current paper. Similar results in the case g = 0 are proved in [8].
Here follows an account of the content of the paper. In Section 2 we provide some basic definitions, state our main results and make several comments. The analysis of problem (1.6) and of other related singular problems occupies Sections 3 to 8. There, we study in great detail the existence, uniqueness and qualitative properties of the solutions to (1.6). Then, in Sections 9 and 10 we exploit such results to construct semi-wavefronts and wavefronts, respectively; there, we prove our main results, show some consequences and provide further comments.

Main results
We begin this section with some definitions on traveling waves and the related profiles, under assumptions somewhat weaker than those stated in the Introduction. We denote with I ⊆ R an open interval.
. Consider a function ϕ ∈ C(I) with values in [0, 1], which is differentiable a.e. and such that D(ϕ)ϕ ′ ∈ L 1 loc (I); let c be a real constant. The function ρ(x, t) := ϕ(x − ct), for (x, t) with x − ct ∈ I, is a traveling-wave solution of equation (1.1) with wave speed c and wave profile ϕ if, for every ψ ∈ C ∞ 0 (I), The previous definition can be made more precise as follows. Below, monotonic means that ξ 1 < ξ 2 implies ϕ(ξ 1 ) ≤ ϕ(ξ 2 ); in the fourth item we assume g(0) = g(1) = 0, while in the two last ones we only require that g vanishes at the point which is specified by the semi-wavefront. A traveling-wave solution is • global if I = R and strict if I = R and ϕ is not extendible to R; • classical if ϕ is differentiable, D(ϕ)ϕ ′ is absolutely continuous and (1.3) holds a.e.; • sharp at ℓ if there exists ξ ℓ ∈ I such that ϕ(ξ ℓ ) = ℓ, with ϕ classical in I \ {ξ ℓ } and not differentiable at ξ ℓ ; • a wavefront if it is global, with a monotonic, non-constant profile ϕ satisfying either (1.4) or the converse condition.
In the last two items we say that ϕ connects ϕ(a + ) (1 or 0) with 1 or 0 (resp., with ϕ(b − )). The smoothness of a profile is related to the degeneracy of D, see [8,13]. More precisely, assume (f), and either (D1), (g0) or (D0), (g01); let ρ be any traveling-wave solution of (1.1) with profile ϕ defined in I and speed c. Then ϕ is classical in every interval I ± ⊆ I where D ϕ(ξ) ≷ 0 for ξ ∈ I ± ; moreover, ϕ ∈ C 2 (I ± ). Notice that profiles always are determined up to a space shift.
Our first main result concerns semi-wavefronts.
For a fixed c > c * , the profiles of Theorem 2.1 are not unique. This lack of uniqueness is not due only to the action of space shifts but, more intimately, to the non-uniqueness of solutions to problem (1.6) that is proved in Proposition 5.1 below. Roughly speaking, these profiles depend on a parameter b ranging in the interval [β(c), 0], for a suitable threshold β(c) ≤ 0. As a conclusion, the family of profiles can be precisely written as Moreover, β(c) < 0 if c > c * and β(c) → −∞ as c → +∞. The threshold β(c) essentially corresponds to the minimum value that the quantity D(ϕ b )ϕ ′ b may achieve when ϕ b reaches 1, for b ∈ [β(c), 0]. This loss of uniqueness is a novelty if we compare Theorem 2.1 with analogous results in [6,7]. In particular, in [7,Theorem 2.7] the assumptions on the functions D and g are reversed: both of them are positive in (0, 1) with D(0) = 0 < g(0), D(1) > 0 = g(1); in [6, Theorem 2.3] D and g are still positive in (0, 1) but the vanishing conditions are D(1) = 0 = g (1). In both cases the profiles exist for every c ∈ R and are unique. Once more, the different results are due to the different nature of the equilibria of the dynamical systems of (1.3).
The estimates (2.2) deserve some comments. First, the left estimate improves analogous bounds (see [22] for a rather comprehensive list) by including the term sup ϕ∈(0,1] f (ϕ)/ϕ ≥ h(0) on the left-hand side. This improvement looks more significative if we also assumė (Dg)(0) = 0, as we do in the following Theorem 2.2; the latter condition holds, for instance, if D(0) = 0 orġ(0) = 0. In this case (2.2) reduces to which can be written with obvious notation as where the indexes label velocities related to the convection (we use the letter a to suggest advection, in order to avoid the awkward notation c c for convection) or diffusion-reaction components. In spite of several different bounds provided for c * in the literature [22], in (2.5) the same term, accounting for the dependence on f , occurs in both the lower and upper bound. This symmetry, which shows the shift of the critical threshold as a consequence of the convective term f , occurs in none of the previous estimates. The interpretation of c dr is well known since [1]: by a shooting argument, only profiles with a sufficiently high speed can reach 1 (a saddle for the dynamical system) starting from 0 (which consequently turns out to be a node instead of a center).
We now comment on c a . In the diffusion-convection case (i.e., when g = 0), there exist profiles connecting ℓ ∈ (0, 1] to 0 if and only i.e., if the line joining the points (0, 0) and ℓ, f (ℓ) lies strictly above the graph of f for ϕ ∈ (0, ℓ), see [13,Theorem 9.1]. In this case, then, the quantity c a represents the maximal speed that can be reached by the profiles connecting ℓ to 0, for ℓ ranging in (0, 1]. Notice that condition (2.6) is also necessary and sufficient in the purely hyperbolic case (i.e., when also D = 0) in order that the equation u t + f (u) x = 0 admits a shock wave of speed f (ℓ)/ℓ with ℓ as left state and 0 as right state. This is not surprising since the viscous profiles approximate the shock wave and converge to it in the vanishing viscosity limit. Indeed, condition (2.6) does not depend on D.
The presence of the positive reaction term g satisfying (g01) (notice that when (g0) holds, and then g(1) > 0, we only have semi-wavefronts, but nevertheless the same bounds still hold) does not allow profile speeds to be less than c a : assuming that z satisfies Equation (1.6) 1 , by the positivity of both D and g we deduce The following corollary investigates the qualitative properties of the profiles when they reach the equilibrium 0; the classification is complete, apart from some possibilities corresponding to c * = h(0); in these minor sub-cases, further assumptions are needed, see e.g. Remark 10.1. Notice that below the existence of the lim ξ→a + D ϕ(ξ) ϕ ′ (ξ) is a consequence of the definition (9.20) and Lemma 3.1.   imply that ϕ is classical; moreover, ϕ reaches 0 at some ξ 0 > a, and then it is not strictly decreasing, if imply that ϕ is sharp at 0 (reached at some ξ 0 > a) with We shall prove in Proposition 8.2 thatβ(c) = β(c) under suitable assumptions, but in general it is still open whether the two thresholds differ. Notice that β is related to the existence of the semi-wavefronts whileβ deals with their smoothness (see Figure 3).
We now present our result on wavefronts; we assume that D and g satisfiy (D0) and (g01). The goal is to extend results contained in [20, Theorems 2.1 and 6.1] regarding the existence and, more importantly, the regularity of wavefronts of Equation (1.1). In particular, the next theorem has the merit to derive the classification of wavefronts under (D0), merely, without additional assumptions (which were instead required in [20, Theorems 2.1 and 6.1]). Notice that in the following result we require that D vanishes at 0; this assumption leads to improve not only the left-hand bound (2.2) on c * by (2.5), but also the right-hand bound, by means of a recent integral estimate provided in [22]. (ii) if c = c * and c * > h(0), then ϕ is sharp at 0. Furthermore, ϕ reaches 0 at some ξ 0 ∈ R and it holds that As in analogous cases [7], Theorem 2.2 provides no information about the smoothness of the profiles when c = c * = h(0). We show in Remark 10.1 that in such a case profiles may be either sharp or classical. Further assumptions on the joint behavior of f , D and g are needed to establish whether which of the possibility occurs.

Singular first-order problems
In this section we begin the analysis of the auxiliary problem (1.6), which shall be concluded with Section 8. First, we consider, for c ∈ R, the problem where we assume q ∈ C 0 [0, 1] and q > 0 in (0, 1).
We summarize here below [7, Lemmas 4.1 and 4.3] in a version ad hoc for our purposes, by also exploiting Lemma 3.1. These technical tools were obtain in [7] under a bit more specific assumptions on q. Nonetheless it is easy to verify that they also apply in the current case, in virtue of (3.2).
Because of Lemmas 3.1 and 3.3, in the following we always look for (and understand) solutions z to problem (3.1), and analogous ones, in the class C[0, 1] ∩ C 1 (0, 1), without any further mention. Motivated by Lemma 3.1, in the next sections we focus the following problem, where the boundary condition is given on the left extremum of the interval of definition: This problem is exploited in the case of semi-wavefronts. Notice that the value of z at ϕ = 1 is not prescribed; obviously, from (3.11) 2 , we have z(1) ≤ 0. Later on, we shall also briefly deal with an analogous problem, see (6.1), where however the boundary condition is given on the right extremum of the interval of definition.
The extremal case, i.e., z(1) = 0, has a peculiar role in what follows. It is then worth displaying explicitly that occurrence: Such a system is needed in the study of wavefronts.

The singular problem with two boundary conditions
Problems (3.11) and (3.12) have solutions only when c is larger than a critical threshold c * . In this section we first give a new estimate to c * under mild conditions on q; then, we obtain a result of existence and uniqueness of solutions to (3.12) if c ≥ c * .
Recalling (D1), (g0) and (1.2) and (D0)-(g01), throughout the next sections (until Section 8) we need to strengthen the very weak assumptions (3.2) of the previous section; for commodity we gather them all here below. We assume We improve, in the same spirit of [22, Theorem 3.1], a well-known result [1,13,19]. More precisely, in the case that q is differentiable at 0, in [22,Theorem 3.1] it is proved that Problem (3.11) has a solution if The last assumption in (q) is a bit weaker than the differentiability of q at 0 and, as a consequence, our result below is less stronger than the one in [22]. It is an open problem whether the existence of solutions to Problem (3.12) under (4.1) can be achieved by only assuming lim sup ϕ→0 + q(ϕ)/ϕ < +∞.
Lemma 4.1. Assume (q) and suppose that c > sup Then Problem (3.12) admits a solution.
We now give a result about existence and uniqueness of solutions to (3.12). We point out that Proposition 4.1, apart from the estimate due to Lemma 4.1, was already given in [21,Proposition 1]. For the reader's convenience, we give its complete proof.
such that there exists a unique z satisfying (3.12) if and only if c ≥ c * .
Proof. Consider the following problem Since (q) holds, we observe that (4.4) was studied in [19]. In fact, (4 As a consequence, by applying [19, Lemma 2.2], we have that also

The singular problem with left boundary condition
In this section we face problem (3.11), where the value at 1 of the solution is not constrained.
We always put ourselves under the assumptions of Proposition 4.1 and then assume (q). Moreover, we always refer to the speed threshold c * introduced in that proposition and denote by z * the corresponding unique solution to (3.12). We refer to Figure 5 for an illustration of Propositions 4.1 and 5.1. In the above proposition, the threshold case c = c * is a bit more technical; we shall prove in Proposition 6.1 that β(c * ) = 0 under some further assumptions. We show that A c = [β, 0), for some β = β(c) < 0, by dividing the proof into four steps.
Step (iv): β ∈ A c . Let {b n } n ⊂ A c be a strictly decreasing sequence such that b n → β + . Since b n ∈ A c , each b n is associated with a solution z n of (3.11) and z n (1) = b n . From the uniqueness of the solution of Cauchy problem for (3.11) 1 , the sequence z n is decreasing.
A monotonicity of solutions of (3.11) follows. We omit its proof since it is quite standard, once that Lemma 3.
In particular, since z c 1 < 0 in (0, 1], then, for any 0 < δ < 1, there exists M > 0 such that Thus, for any ϕ ∈ (δ, 1), At last, we collect some important consequences of (5.4) and Lemma 4.1 (or [22, Theorem 3.1]), concerning a sharper estimate to c * . To the best of our knowledge these estimates are new, and we provide a comment to their meaning.
The bound from above in (2.12) is then proved. The bound from below in (2.12) is instead due directly to (5.5), because ofq(0) = 0.
Remark 5.2. We can now make precise the statement following formula (2.7) about the gap between c a and c * . Indeed, if the value c a is obtained at some ϕ ∈ (0, 1], then the sup in the right-hand side of (2.7) is strictly larger than c a because z < 0 in (0, 1). Then c * > c a . Otherwise, if sup ϕ∈(0,1] f (ϕ)(ϕ) = h(0), then c a = h(0) and by (5.5) we still deduce c * > c a .

Further existence and non-existence results
Propositions 4.1 and 5.1 completely treat the existence of solutions of (3.12) and (3.11), respectively, in the cases c ≥ c * and c > c * . In this section, we investigate the remaining cases and show that such propositions are somehow optimal. Preliminarily, we give a lemma. It deals with the following problem, where c ∈ R but, differently from (3.11), the boundary condition is imposed on the right extremum of the interval of definition: The differential equation in (3.11) and (6.1) is the same, and inherits the properties of the dynamical system underlying equation (1.3), which has a center or a node at (0, 0) and a saddle at (1, 0); no wonder then that the corresponding results, cf. Proposition 5.1 and Lemma 6.1, are different. This is well known, see e.g. Lemma 3.2 (1).
Moreover, while in problem (3.11) the threshold c * discriminated the existence of solutions, for problem (6.1) solutions will be proved to exist for every c ∈ R; instead, the threshold c * enters into the problem to discriminate whether solutions reach 0 or not (see Figure 9). A related behavior was pointed out in [7, Theorem 2.6]. On the contrary, the monotonicity properties stated in Corollary 5.1 and in Lemma 6.1 are the same. (ii) it holds that z * (ϕ) = lim c→c * ζ c (ϕ) for any ϕ ∈ [0, 1]. (6.2) Proof. The part regarding the existence and the uniqueness was proved in [7, Theorem 2.6], while the monotonicity as stated in (i) was given in [7, Lemma 5.1]. It remains to prove (ii). We show that, for any ϕ ∈ [0, 1], For any ϕ ∈ [0, 1], by (i) we have These inequalities and Lemma 3.3 imply that there exist two functions w, w ∈ C 0 [0, 1] ∩ C 1 (0, 1) such that and that both w and w satisfy (3.1) with c = c * . Since w(1) = w(1) = 0, both of them then solve (6.1). By the uniqueness of solutions of (6.1) it follows that w = w = z * .
Proof. Take c < c * and assume by contradiction that problem (3.11) has a solution z. Let ζ = ζ c be the solution of (6.1) given by Lemma 6.1. Note that necessarily such a ζ must satisfy ζ(0) < 0, by Proposition 4.1. Then it holds that ζ(ϕ 0 ) = z(ϕ 0 ) =: y 0 < 0, for some ϕ 0 ∈ (0, 1); see Figure 11. This contradicts the uniqueness of the Cauchy problem associated to (6.1) 1 . Thus, the proof is concluded.  In this section and in the next one we investigate the behavior of the solutions z to (3.11) at 1 and 0. We now deal with the former case. In the following proposition, under a bit more regularity on the term q at the right extremum, we prove thatż(1) exists and explicitly compute its value. We suppose thaṫ q(1) exists and is finite.
A direct check shows that the right-hand side of (7.2) corresponds exactly to r + . Thus, if we prove that µ = r + then we conclude the proof.
We now show that, under (6.4), the thresholdβ(c) defined in (8.4) and occurring in Proposition 8.1 coincides with the threshold β(c) introduced in Proposition 5.1.
Proof. Consider ε > 0 and let z ε be the solution of Lemma 3.2 (1.b) implies that z ε exists and it is defined in its maximal-existence interval [0, δ], for some 0 < δ ≤ 1. By the uniqueness of solutions of the Cauchy problem associated to (3.1) 1 , we have necessarily z ε < z β in [0, δ], where z β was defined in the statement of Lemma 8.1. Since z β (δ) < 0 then δ = 1.
We claim that z ε converges for ε → 0 + to bothẑ and z β , whereẑ is defined in (8.3), see Figure 12. From the uniqueness of the limit, it follows then thatẑ and z β must coincide and hence that β =β.

Strongly non-unique strict semi-wavefronts
In this section, we apply some of the results showed above, regarding Problems (3.11) and (3.12), to study semi-wavefronts of Equation (1.1) when D and g satisfy (D1), (g0) and (1.2); in particular, we prove Theorem 2.1. Indeed, all the results obtained in Sections 4 -8 apply when we set q := Dg, (9.1) since clearly such q fulfills (q) We highlight that throughout all the present section, when we refer to c * we always intend the threshold given by Proposition 4.1 for q given by (9.1), for which it holds (2.2), as observed in Remark 5.1.
We now prove Theorem 2.1.
Proof of Theorem 2.1. To begin with, we prove that there exists a semi-wavefront to 0 of (1.1) if c ≥ c * . To this end, for q = Dg, consider one of the solutions z = z(ϕ) of (3.11) (which is (1.6)), provided by Propositions 4.1 and 5.1. Consider the Cauchy problem (9.11) The right-hand side of (9.11) 1 is of class C 1 in a neighborhood of 1 2 , and then there exists a unique solution ϕ in its maximal-existence interval (a, ξ 0 ), for −∞ ≤ a < ξ 0 ≤ ∞. Since z(ϕ)/D(ϕ) < 0 for ϕ ∈ (0, 1), we deduce that ϕ is decreasing and then (see Figure 15) A direct consequence of (9.11) 1 is that ϕ satisfies (1.3) in (a, ξ 0 ). We show that, if ξ 0 ∈ R, we can extend ϕ and obtain a solution of (1.3), in the sense of Definition 2.1, defined in all the half-line (a, +∞).
Remark 9.2. In the proof of Theorem 2.1 we have showed en passant that there exists a bijection between solutions z of (1.6) and strict semi-wavefronts to 0 (modulo space shifts) ϕ of (1.1), connecting 1 to 0. In particular, this bijection is given by (9.11) and (9.20). We can now prove Corollary 2.1.

New regularity classification of wavefronts
In this section we assume that D and g satisfy (D0) and (g01) and prove Theorem 2.2. Analogously to Section 9, but now thanks to assumptions (D0) -(g01), we apply results of Sections 4 -8 to the case q = Dg. Proof of Theorem 2.2. We first show that wavefronts are allowed if and only if c ≥ c * for some c * which satisfies (2.12). The proof is quite standard and is mostly contained in the proof of Theorem 2.1. Then, we prove (i) and (ii), by making use of some of the arguments detailed in the proof of Corollary 2.1. Set q = Dg. Clearly, q satisfies (q), with in particularq(0) = 0. We apply Proposition 4.1. Problem (3.12) admits a unique solution z if and only if c ≥ c * where for c * the estimates in (4.3) hold. As already observed in Remark 5.1, since (D0) and (g01) hold true, in this case c * satisfies (2.12).
Direct computations show that ϕ 1 and ϕ 2 are two wave profiles defining two wavefronts, both of them associated with c = h(0). Plainly, ϕ 1 is sharp at ξ = log(2) while ϕ 2 is classical.