
Citation: Caroline L. Marden, Ryan McDonald, Harold J. Schreier, Joy E.M. Watts. Investigation into the fungal diversity within different regions of the gastrointestinal tract of Panaque nigrolineatus, a wood-eating fish[J]. AIMS Microbiology, 2017, 3(4): 749-761. doi: 10.3934/microbiol.2017.4.749
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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
ρh,t+fh(ρh)x=(Dh(ρh)ρh,x)x,h=1,…,m+n, | (1.1) |
where
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
Following the above analogy, we understand the unknown functions
For each road we assign the functions
(f)
(D)
We emphasize that in (D) we can possibly have either
ρh,t+fh(ρh)x=(Dh(ρh)ρh,x)x,(t,x)∈R×Ωh, h∈H. | (2.1) |
Assumption (f) is standard when dealing with traffic flows [2]. More precisely, in that case
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+))=∑i∈Iαi,jFi(ρi(t,0−),ρi,x(t,0−)) for a.e. t∈R, j∈J, | (2.2) |
for given constant coefficients
∑j∈Jαi,j=1,i∈I. | (2.3) |
Conditions (2.2) and (2.3) imply
∑j∈JFj(ρj(t,0+),ρj,x(t,0+))=∑i∈IFi(ρi(t,0−),ρi,x(t,0−)) for a.e. t∈R, | (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+),t∈R, (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
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
Then, we are faced with the system of equations (2.1) that are coupled through (2.2), with the
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
Definition 3.1. A weak solution
This definition coincides with that given in [19,29] because we are considering non-constant profiles. The profile must satisfy the equation
(Fh(φh,φ′h)−chφh)′=0, | (3.2) |
namely,
(Dh(φh)φ′h)′−g′h(φh)φ′h=0, | (3.3) |
in the weak sense, where
gh(ρ)≐fh(ρ)−chρ | (3.4) |
is the reduced flux, see Figure 2.
This means that
∫R(Dh(φh(ξ))φ′h(ξ)−gh(φh(ξ)))ψ′(ξ)dξ=0, |
for every
φh(±∞)=ℓ±h, | (3.5) |
for
Ih≐{ξ∈R:ℓ−h<φh(ξ)<ℓ+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
0≤ℓ−h<ℓ+h≤1andch=fh(ℓ+h)−fh(ℓ−h)ℓ+h−ℓ−h. | (3.7) |
We have that
(i)
limξ↓ν−hφ′h(ξ)={ℓ+hf′h(0)−fh(ℓ+h)ℓ+hD′h(0) if D′h(0)>0,∞ if D′h(0)=0, | (3.8) |
limξ↓ν−hDh(φh(ξ))φ′h(ξ)=0. | (3.9) |
(ii)
limξ↑ν+hφ′h(ξ)={(1−ℓ−h)f′h(1)+fh(ℓ−h)(1−ℓ−h)D′h(1) if D′h(1)<0,∞ if D′h(1)=0, | (3.10) |
limξ↑ν+hDh(φh(ξ))φ′h(ξ)=0. | (3.11) |
(iii) In all the other cases
limξ→±∞φ′h(ξ)=0. | (3.12) |
We observe that for
gh(ℓ+h)=gh(ℓ−h)=−fh(ℓ+h)ℓ−h−fh(ℓ−h)ℓ+hℓ+h−ℓ−h | (3.13) |
and no
Theorem 3.2 motivates the following definition.
Definition 3.3. A traveling-wave solution
Remark 3.4. A consequence of assumption (f) is that if
In case (i) (or (ii)) of Theorem 3.2 does not hold we define
Ih=(ν−h,ν+h). |
The interval
In the case of non-stationary traveling-wave solutions
ωh≐min{c−1hν−h,c−1hν+h}. | (3.14) |
Lemma 3.5. Let
(a) If
(b) If
• either
•
In this case the function
Proof. We recall that
Now, we prove the second part of (b). Since
Finally, the converse is straightforward. In fact, if
Because of the smoothness properties of the profile proved in Theorem 3.2, we can integrate equation (3.2) in
chφh(ξ)−Fh(φh(ξ),φ′h(ξ))=chφh(ξ−)−Fh(φh(ξ−),φ′h(ξ−)). |
If
Fh(φh(ξ),φ′h(ξ))=chφh(ξ)+gh(ℓ±h),ξ∈Ih. | (3.15) |
We observe that (3.15) is trivially satisfied in case (i) when
Dh(φh(ξ))φ′h(ξ)=gh(φh(ξ))−gh(ℓ±h),ξ∈R. | (3.16) |
In this section we consider the traveling-wave solutions of problem (2.1)-(2.2) in the network
Definition 4.1. For any
•
•
•
•
Finally, we say that
For brevity, from now on we simply write "traveling wave" for "traveling-wave solution". In analogy to the notation above, we say that
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
Proposition 4.2. The function
cjφj(cjt)+gj(ℓ±j)=∑i∈Iαi,j(ciφi(cit)+gi(ℓ±i)),t∈R, j∈J. | (4.1) |
In (4.1) any combination of the signs
Proof. By plugging
Fj(φj(−cjt),φ′j(−cjt))=∑i∈Iαi,jFi(φi(−cit),φ′i(−cit)),t∈R, j∈J, |
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 "
Lemma 4.3. Assume that problem (2.1)-(2.2) admits a traveling wave. Then for any
max{fj(ℓ−j),fj(ℓ+j)}=∑i∈Iαi,jmax{fi(ℓ−i),fi(ℓ+i)}, | (4.2) |
min{fj(ℓ−j),fj(ℓ+j)}=∑i∈Iαi,jmin{fi(ℓ−i),fi(ℓ+i)}. | (4.3) |
Proof. Fix
Υj(t)=∑i∈Iαi,jΥi(t),t∈R, j∈J, |
where the map
limt→∞Υh(t)=max{fh(ℓ−h),fh(ℓ+h)},limt→−∞Υh(t)=min{fh(ℓ−h),fh(ℓ+h)}. |
Hence, by passing to the limit for
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
(ii)
(iii)
Proof. By subtracting (4.3) to (4.2) we obtain
|fj(ℓ+j)−fj(ℓ−j)|=∑i∈Iαi,j|fi(ℓ+i)−fi(ℓ−i)|. |
Since
Lemma 4.4 shows that either a traveling wave is stationary, and then
there exists i∈I such that ci≠0 and cj≠0 for every j∈J. | (4.4) |
Of course, by Lemma 4.4,
Proposition 4.5. Fix
{max[0,1]fj>∑i∈Iαi,jmax{fi(ℓ−i),fi(ℓ+i)} if c1=…=cm=0,max[0,1]fj≥∑i∈Iαi,jmax{fi(ℓ−i),fi(ℓ+i)} otherwise. | (4.5) |
In this case, the end states
Proof. Assume that there exist
Conversely, assume (4.5). If
By Proposition 4.5 and Lemma 4.4 we deduce that the end states
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≐{ℓ±i if cicj≥0,ℓ∓i if cicj<0. | (4.6) |
Proposition 4.6. Assume that problem (2.1)-(2.2) admits a traveling wave. Then for any
fj(ℓ±j)=∑i∈Iαi,jfi(L±i,j). | (4.7) |
Moreover, (4.7) is equivalent to (4.2)-(4.3).
Proof. Fix
max{fi(ℓ−i),fi(ℓ+i)}={fi(ℓ+i)if ci≥0fi(ℓ−i)if ci<0=fi(L+i,j),min{fi(ℓ−i),fi(ℓ+i)}={fi(ℓ−i)if ci≥0fi(ℓ+i)if ci<0=fi(L−i,j), |
and therefore (4.7) is equivalent to (4.2)-(4.3). The case
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(ℓ+h)=fh(ℓ−h),fj(ℓ−j)=∑i∈Iαi,jfi(ℓ−i) for h∈H, j∈J. | (4.8) |
Proof. Clearly, (4.8) is trivially satisfied if
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
∑j∈Jfj(ℓ−j)=∑i∈Ifi(ℓ−i). | (4.9) |
Conversely, consider the stationary case and assume conditions (4.8), (4.9); if moreover
∑i∈Ifi(ℓ−i)=∑j∈Jfj(ℓ−j)=∑(i,j)∈I×Jαi,jfi(ℓ−i), |
namely
Clearly, if both
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
ci,j≐cicj,Ai,j≐αi,jci,j,kj≐∑i∈Ic0(Ai,jL±i,j)−ℓ±j,κj≐cjkj, | (4.10) |
where
I0≐{i∈I:ci=0}={i∈I:fi(ℓ−i)=fi(ℓ+i)},Ic0≐I∖I0. |
We notice that
∑i∈Ic0Ai,j(L+i,j−L−i,j)=∑i∈Ic0αi,jc−1j(fi(L+i,j)−fi(L−i,j))=c−1j(fj(ℓ+j)−fj(ℓ−j))=ℓ+j−ℓ−j. |
Finally, by (f) we deduce that
for no j∈J we have both ℓ−j=0=1−ℓ+j. |
Proposition 4.9. The function
φj(ξ)=∑i∈Ic0(Ai,jφi(ci,jξ))−kj,ξ∈R, j∈J. | (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
cjφj(ξ)=−gj(ℓ±j)+∑i∈Iαi,j(ciφi(ci,jξ)+gi(L±i,j))=∑i∈Iαi,jciφi(ci,jξ)−κj, |
that is equivalent to (4.11).
We observe that
kj=∑i∈Ic0(Ai,jℓ−i+ℓ+i2)−ℓ−j+ℓ+j2,φj(ξ)=ℓ−j+ℓ+j2+∑i∈Ic0Ai,j(φi(ci,jξ)−ℓ−i+ℓ+i2). | (4.12) |
Proposition 4.9 shows how each outgoing profile
Lemma 4.10. Let
Proof. Since by Theorem 3.2 we know that
l+j−l−j=c−1j(fj(l+j)−fj(l−j))=∑i∈Iαi,jc−1j(fi(L+i,j)−fi(L−i,j))=∑i∈Iαi,jci,j(L+i,j−L−i,j)=∑i∈Ic0αi,j|ci,j|(ℓ+i−ℓ−i)>0. |
By definition of
cjℓj(±∞)=∑i∈Ic0(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
We notice that if both
The following result is similar to Lemma 4.4.
Lemma 4.11. Problem (2.1)-(2.2) admits a degenerate non-stationary traveling wave
(A) for some
(B) for every
ωi=ωj≐ω,i∈Ic0, j∈J. | (4.13) |
Proof. Let us introduce the following conditions:
Clearly
(Ⅰ)
(Ⅱ) for some
(Ⅲ) for some
where
φ′j(cjξ)=∑i∈Ic0αi,jc2i,jφ′i(ciξ) for a.e. ξ∈R, j∈J. | (4.14) |
More precisely, by Lemma 3.5, formula (4.14) holds for
As for Lemma 4.4, we notice that Lemma 4.11 implies that a non-stationary traveling wave
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)
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
(A) for some
(B) for every
The case when
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
γh(ℓ)≐{gh(ℓ)−gh(ℓ−h)Dh(ℓ) if Dh(ℓ)≠0,0 if Dh(ℓ)=0. | (4.15) |
In fact, when
φ′h(ξ)=gh(φh(ξ))−gh(ℓ−h)Dh(φh(ξ)),ξ∈R∖{ν−h,ν+h}, h∈H. | (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.
(
(i)
(ii) for any
(iii) for any
gj(ℓj(cjξ))−gj(ℓ−j)Dj(ℓj(cjξ))=∑i∈Ic0Ai,jci,jgi(φi(ciξ))−gi(ℓ−i)Di(φi(ciξ)) for a.e. ξ∈R, | (4.17) |
where
ℓj(ξ)≐∑i∈Ic0(Ai,jφi(ci,jξ))−kj,ξ∈R. | (4.18) |
Proof. First, assume that problem (2.1)-(2.2) admits a non-stationary traveling wave
φ′j(cjξ)=gj(φj(cjξ))−gj(ℓ−j)Dj(φj(cjξ)) | (4.19) |
for
φ′j(ξ)=∑i∈Ic0Ai,jci,jφ′i(ci,jξ)=∑i∈Ic0Ai,jci,jgi(φi(ci,jξ))−gi(ℓ−i)Di(φi(ci,jξ)) | (4.20) |
for
Conversely, assume that condition (
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
∑j∈Jmax{fj(ℓ−j),fj(ℓ+j)}=∑i∈Imax{fi(ℓ−i),fi(ℓ+i)}. | (4.21) |
Conversely, consider the non-stationary case and assume conditions (4.2), (4.21); if moreover
∑i∈Imax{fi(ℓ−i),fi(ℓ+i)}=∑j∈Jmax{fj(ℓ−j),fj(ℓ+j)}=∑(i,j)∈I×Jαi,jmax{fi(ℓ−i),fi(ℓ+i)}, |
namely
Remark 4.14. Fix
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
(i)
φj(cjt)=φi(cit)≐Φ(t),t∈R. | (5.1) |
(ii) If
(c−1jIj)=(c−1iIi)≐I, | (5.2) |
ℓ±j=ℓ±i=L±i,j≐ℓ±, | (5.3) |
cjgj(ℓ)−gj(ℓ±)Dj(ℓ)=cigi(ℓ)−gi(ℓ±)Di(ℓ), ℓ∈(ℓ−,ℓ+). | (5.4) |
(iii) If
cj=∑i∈Iαi,jci, ∑j∈Jcj=∑i∈Ici,κj=0,∑i∈IAi,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
cjφ′j(cjt)=ciφ′i(cit) for a.e. t∈R. | (5.6) |
Then (5.6) implies that either
(iii) To deduce
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
Proof. By (5.1) condition (2.5) holds in the stationary case if and only if
We point out that condition
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.
(
fh(ℓ−)≠fh(ℓ+), | (5.7) |
fj(ℓ±)=∑i∈Iαi,jfi(ℓ±), | (5.8) |
cjgj(φj(cjt))−gj(ℓ−)Dj(φj(cjt))=cigi(φi(cit))−gi(ℓ−)Di(φi(cit)) for a.e. t∈R, | (5.9) |
where
Proof. Assume that condition (
∑i∈IAi,j=∑i∈Iαi,jfi(ℓ+)−fi(ℓ−)fj(ℓ+)−fj(ℓ−)=1. |
By (5.7) we have that
ddt(φj(cjt)−φi(cit))=cjgj(φj(cjt))−gj(ℓ−)Dj(φj(cjt))−cigi(φi(cit))−gi(ℓ−)Di(φi(cit))=0. |
Therefore we conclude that (5.1) holds. Finally, (4.11) follows immediately from (5.1),
Consider in particular the case when the functions
fh(ℓ)≐vhf(ℓ),Dh(ℓ)≐δhD(ℓ),ℓ∈[0,1], | (5.10) |
for some constants
vi,j≐vivj,δi,j≐δiδj. | (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
v2i,j=δi,jand∑i∈Iαi,jvi,j=1. | (5.13) |
Proof. We only need to translate condition
Remark that by (5.12) condition
In this section we assume (5.10) for some constants
f(ρ)≐ρ(1−ρ), |
with no further mention. The case when only
For simplicity, in the whole section we focus on the case
In this case, condition (3.7) becomes
0≤ℓ−h<ℓ+h≤1 and ch=vh[1−ℓ+h−ℓ−h]. | (6.1) |
In particular, by
ρh is stationary ⟺ ℓ+h+ℓ−h=1. | (6.2) |
Moreover,
gh(ℓ)−gh(ℓ±h)=vh(ℓ+h−ℓ)(ℓ−ℓ−h), | (6.3) |
and therefore (3.16) becomes
δhD(φh(ξ))φ′h(ξ)=vh(ℓ+h−φh(ξ))(φh(ξ)−ℓ−h),ξ∈R. | (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,j≤1,(0,1−√1−α−11,jv−11,j2)if α1,jv1,j>1,j∈J. |
Proposition 6.1. Problem (2.1)-(2.2) admits infinitely many stationary traveling waves; their end states are characterized by the conditions
ℓ−1∈⋂j∈JL0j,ℓ+1+ℓ−1=1,ℓ±j=12(1±√1−4α1,jv1,jℓ+1ℓ−1),j∈J. |
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),
ℓ−h∈[0,1/2),ℓ+h+ℓ−h=1,ℓ−j(1−ℓ−j)=α1,jv1,jℓ−1(1−ℓ−1); |
then it is sufficient to compute
The latter part of the proposition is deduced by Proposition 5.2 because
In the following we treat the existence of non-stationary traveling waves. Since
c1,j=v1,j1−ℓ+1−ℓ−11−ℓ+j−ℓ−j,A1,j=α1,jv1,j1−ℓ+1−ℓ−11−ℓ+j−ℓ−j,kj=A1,jL±1,j−ℓ±j,κj=vjℓ−jℓ+j−α1,jv1ℓ−1ℓ+1. | (6.5) |
The following result translates Theorem 4.12 to the present case. We define the intervals
Lcj≐{[0,1]if α1,jv1,j≤1,[0,1]∖(1−√1−α−11,jv−11,j2,1+√1−α−11,jv−11,j2)if α1,jv1,j>1,j∈J. |
Proposition 6.2. Problem (2.1)-(2.2) admits a (completely) non-stationary traveling wave if and only if the following condition holds.
(
(i)
(ii)
(iii) for any
D(ℓ)=α1,jδ1,jv1,jD(ℓ+kjA1,j),ℓ∈(ℓ−j,ℓ+j), | (6.6) |
where
ℓ±j(1−ℓ±j)=α1,jv1,jL±1,j(1−L±1,j). | (6.7) |
Proof. The proof consists in showing that, in the present case, condition (
''
If we denote
{ℓ−j=12(1−√1−z−1,j),ℓ+j∈{12(1±√1−z+1,j)}, if cj>0, | (6.8) |
{ℓ−j∈{12(1±√1−z−1,j)},ℓ+j=12(1+√1−z+1,j), if cj<0. | (6.9) |
The square roots in (6.8)-(6.9) are real numbers if and only if
ℓ±1(1−ℓ±1)≤(4α1,jv1,j)−1. |
It is easy to see that the above estimate is equivalent to require
''
(ℓ+j−ℓj(cjξ))(ℓj(cjξ)−ℓ−j)D(ℓj(cjξ))=A1,jc1,jv1,jδ1,j(ℓ+1−φ1(c1ξ))(φ1(c1ξ)−ℓ−1)D(φ1(c1ξ)) | (6.10) |
for a.e.
ℓj(ξ)≐A1,j(φ1(c1,jξ)−L±1,j)+ℓ±j,ξ∈R. |
We point out that the above expression of
(ℓ+j−ℓj(cjξ))(ℓj(cjξ)−ℓ−j)=A21,j(L+1,j−φ1(c1ξ))(φ1(c1ξ)−L−1,j),(ℓ+1−φ1(c1ξ))(φ1(c1ξ)−ℓ−1)=(L+1,j−φ1(c1ξ))(φ1(c1ξ)−L−1,j), |
we have that (6.10) is equivalent to
D(ℓj(cjξ))=α1,jδ1,jv1,jD(φ1(c1ξ))for a.e. ξ∈R. |
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
We notice that if
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,j∈J. | (6.11) |
The special cases of constant or linear diffusivities are treated in the following subsections.
In this subsection we assume
D≐1, | (6.12) |
and in this case problem (3.5)-(6.4) reduces to
{δhφ′h(ξ)=vh(ℓ+h−φh(ξ))(φh(ξ)−ℓ−h),ξ∈R,φh(±∞)=ℓ±h. | (6.13) |
For any
ψh(ξ)≐ℓ+h1+e−vhδh(ℓ+h−ℓ−h)ξ+ℓ−h1+evhδh(ℓ+h−ℓ−h)ξ | (6.14) |
solves (6.13) because
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
φ(ξ)=(ψ1(ξ+σ1),…,ψn+1(ξ+σn+1)),ξ∈R, | (6.16) |
with
cjσ1=c1σj,j∈J. | (6.17) |
Proof. By Theorem 3.2, any solution to (6.13) has the form (6.16) with
fj(ℓ+j)ζj(t)+fj(ℓ−j)1+ζj(t)=α1,jf1(ℓ+1)ζ1(t)+f1(ℓ−1)1+ζ1(t),t∈R, j∈J, | (6.18) |
where
either fj(ℓ±j)=α1,jf1(ℓ±1), or fj(ℓ±j)=α1,jf1(ℓ∓1). |
(fj(ℓ+j)−fj(ℓ−j))(ζj(t)−ζ1(t))=0,t∈R, j∈J. |
Since by assumption
{vjδj(ℓ+j−ℓ−j)cj=v1δ1(ℓ+1−ℓ−1)c1,vjδj(ℓ+j−ℓ−j)σj=v1δ1(ℓ+1−ℓ−1)σ1,⇔{v1,jδ1,j=fj(ℓ+j)−fj(ℓ−j)f1(ℓ+1)−f1(ℓ−1)=α1,j,σjcj=σ1c1. |
(fj(ℓ+j)−fj(ℓ−j))(ζj(t)ζ1(t)−1)=0,t∈R, j∈J. |
Since by assumption
{vjδj(ℓ+j−ℓ−j)cj=−v1δ1(ℓ+1−ℓ−1)c1,vjδj(ℓ+j−ℓ−j)σj=−v1δ1(ℓ+1−ℓ−1)σ1,⇔{v1,jδ1,j=−fj(ℓ+j)−fj(ℓ−j)f1(ℓ+1)−f1(ℓ−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
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
The second part of the statement characterizes the end states. If a non-stationary traveling wave
In this subsection we assume
D(ρ)≐ρ. | (6.19) |
We notice that
{δhφhφ′h=vh(ℓ+h−φh)(φh−ℓ−h),ξ∈R,φh(±∞)=ℓ±h. | (6.20) |
If
ψh(ξ)≐{ℓ+h2(2−e−vhδhξ) if ξ≥−δhvhln2,0 if ξ<−δhvhln2, | (6.21) |
solves (6.20) because
(2exp(vhδhξ)ψh(ξ)−ℓ−hℓ+h−ℓ−h)ℓ−h=(2exp(vhδhξ)ℓ+h−ψh(ξ)ℓ+h−ℓ−h)ℓ+h | (6.22) |
solves (6.20) because
φ(ξ)=(ψ1(ξ+σ1),…,ψn+1(ξ+σn+1)),ξ∈R. | (6.23) |
In the sequel we prove that the shifts
v1,jσ1=δ1,jσj,j∈J. | (6.24) |
Lemma 6.7. Assume (6.19). If
v21,jδ1,j=1−ℓ+j−ℓ−j1−ℓ+1−ℓ−1 and ℓ−jℓ+j=α1,jv1,jℓ−1ℓ+1. | (6.25) |
Proof. In the present case, condition (6.6) becomes
We observe that (6.25)
ℓ±j(1−ℓ±j)=α1,jv1,jℓ±1(1−ℓ±1),j∈J. | (6.26) |
As a consequence
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,3√α1,jδ21,j},j∈J. |
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, j∈J,v1,2(δ1,2−v21,2)α1,2δ21,2−v31,2=…=v1,n+1(δ1,n+1−v21,n+1)α1,n+1δ21,n+1−v31,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
ℓ−1=0=ℓ−j,ℓ+1=v1,j(δ1,j−v21,j)α1,jδ21,j−v31,j,ℓ+j=α1,jδ1,j(δ1,j−v21,j)α1,jδ21,j−v31,j,j∈J. | (6.28) |
In both cases, any degenerate non-stationary traveling wave
Proof. We claim that the existence of a degenerate non-stationary traveling wave is equivalent to the existence of
ℓ+j=α1,jδ1,jv1,jℓ+1 and (α1,jδ21,j−v31,j)ℓ+1+v1,j(v21,j−δ1,j)=0,j∈J. | (6.29) |
In fact, by Proposition 6.2 the existence of a non-stationary traveling wave is equivalent to condition (
v21,jδ1,j=1−ℓ+j1−ℓ+1,ℓ+j(1−ℓ+j)=α1,jv1,jℓ+1(1−ℓ+1),j∈J. | (6.30) |
By (6.30) we obtain
Assume there is a degenerate non-stationary traveling wave; then
0<v1,j(δ1,j−v21,j)α1,jδ21,j−v31,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
Conversely, assume condition (6.11). In this case
Assume now condition (6.27). In this case the values for
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
φ′j(cjξ)=α1,jc21,jφ′1(c1ξ) for a.e. ξ∈R, j∈J, |
and the regularity of
1cj(δjvjln2+σj)=1c1(δ1v1ln2+σ1), |
which is equivalent to (6.17) because
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
Proof. Assume that there is a non-degenerate non-stationary traveling wave; then
(1−ℓ+j)ℓ−1=(1−ℓ+1)ℓ−j and (1−ℓ−j)ℓ+1=(1−ℓ−1)ℓ+j,j∈J. |
By adding the above relations we have
0=ℓ+1−ℓ+j+ℓ−1−ℓ−j=ℓ+1−1+(1−ℓ+1)ℓ−jℓ−1+ℓ−1−ℓ−j=1−ℓ+1−ℓ−1ℓ−1(ℓ−j−ℓ−1). |
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
ℓ+j+ℓ−j=ℓ+1+ℓ−1,ℓ−jℓ+j=ℓ−1ℓ+1,ℓ±j(1−ℓ±j)=ℓ±1(1−ℓ±1),j∈J. |
The same computations as before give that if we impose
At last, by Theorem 3.2, any solution to (6.20) has the form (6.23). Fix
ψj(cjt+σj)=ψ1(c1t+σ1),t∈R. |
This identity together with (6.22) and (5.3) imply
(2exp(vjδj(cjt+σj))ψ1(c1t+σ1)−ℓ−ℓ+−ℓ−)ℓ−=(2exp(vjδj(cjt+σj))ℓ+−ψ1(c1t+σ1)ℓ+−ℓ−)ℓ+,(2exp(v1δ1(c1t+σ1))ψ1(c1t+σ1)−ℓ−ℓ+−ℓ−)ℓ−=(2exp(v1δ1(c1t+σ1))ℓ+−ψ1(c1t+σ1)ℓ+−ℓ−)ℓ+. |
By dividing the above equalities and taking the logarithm we get
(vjδj(cjt+σj)−v1δ1(c1t+σ1))ℓ−=(vjδj(cjt+σj)−v1δ1(c1t+σ1))ℓ+,t∈R. |
Since
In this section we assume (5.10) for some constants
f(ρ)≐−ρln(ρ) |
for
Condition (3.7) becomes
0≤ℓ−h<ℓ+h≤1 and ch=−vhℓ+hln(ℓ+h)−ℓ−hln(ℓ−h)ℓ+h−ℓ−h. | (7.1) |
Moreover we have, for
gh(ℓ)=vhℓ(ℓ+hln(ℓ+h)−ℓ−hln(ℓ−h)ℓ+h−ℓ−h−ln(ℓ)),gh(ℓ)−gh(ℓ±h)=vh((ℓ−ℓ−h)ℓ+hln(ℓ+h)+(ℓ+h−ℓ)ℓ−hln(ℓ−h)ℓ+h−ℓ−h−ℓln(ℓ)). | (7.2) |
Therefore (3.16) becomes
φ′h(ξ)=vhδh[[φh(ξ)−ℓ−h]ℓ+hln(ℓ+h)+[ℓ+h−φh(ξ)]ℓ−hln(ℓ−h)ℓ+h−ℓ−h−φh(ξ)ln(φh(ξ))], | (7.3) |
for
We first consider the case of stationary waves. We define the intervals
L0j≐{[0,e−1)if α1,jv1,j≤1,(0,f−1ℓ(e−1α−11,jv−11,j))if α1,jv1,j>1,j∈J. |
Proposition 7.1. Problem (2.1)-(2.2) admits infinitely many stationary traveling waves; their end states are characterized by the conditions
ℓ−1∈⋂j∈JL0j,ℓ+1=f−1r(−ℓ−1ln(ℓ−1)),ℓ−j=f−1ℓ(−α1,jv1,jℓ−1ln(ℓ−1)),ℓ+j=f−1r(−α1,jv1,jℓ−1ln(ℓ−1)),j∈J. |
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)
ℓ−h∈[0,e−1),ℓ−hln(ℓ−h)=ℓ+hln(ℓ+h),ℓ−jln(ℓ−j)=α1,jv1,jℓ−1ln(ℓ−1). |
Hence
In the following we discuss the existence of non-stationary traveling waves. Since
c1,j=v1,j ℓ+1ln(ℓ+1)−ℓ−1ln(ℓ−1)ℓ+jln(ℓ+j)−ℓ−jln(ℓ−j) ℓ+j−ℓ−jℓ+1−ℓ−1. | (7.4) |
The following result translates Theorem 4.12 to the current framework. We define the intervals
Lcj≐{[0,1]if α1,jv1,j≤1,[0,1]∖(f−1ℓ(e−1α−11,jv−11,j),f−1r(e−1α−11,jv−11,j))if α1,jv1,j>1,j∈J. |
Proposition 7.2. Problem (2.1)-(2.2) admits a (completely) non-stationary traveling wave if and only if the following condition holds.
(
(i)
(ii)
(iii) for any
δ1,j(gj(ℓ)−gj(ℓ−j))=A1,jc1,j(g1(ℓ+kjA1,j)−g1(ℓ−1)),ℓ∈(ℓ−j,ℓ+j), | (7.5) |
where
ℓ±jln(ℓ±j)=α1,jv1,jL±1,jln(L±1,j). | (7.6) |
Proof. The proof consists in showing that, in the present case, (
δ1,j(gj(ℓj(cjξ))−gj(ℓ−j))=A1,jc1,j(g1(φ1(c1ξ))−g1(ℓ−1)),ξ∈R, | (7.7) |
where
In the following we focus on the case of (completely) non-stationary traveling waves with
Lemma 7.3. Assume that problem (2.1)-(2.2) admits a traveling wave. The following statements are equivalent:
(i)
(ii)
(iii) there exists
Proof. First, we prove that (i) implies (ii). Fix
ℓ−jln(ℓ−j)=α1,jv1,jℓ+1ln(ℓ+1). |
Therefore, by (7.4), (4.10)
c1,j=−v1,j ℓ+1ln(ℓ+1)ℓ−jln(ℓ−j) 1−ℓ−jℓ+1=−1−ℓ−jα1,jℓ+1,Ai,j=−1−ℓ−jℓ+1,kj=−1. |
Condition (7.5) can be written as
ℓln(ℓ)−v1,j(1−ℓ)(α1,jℓ+1ln(ℓ+1)1−ℓ−j+1−ℓ−jα1,jδ1,jℓ+1ln(1−ℓ1−ℓ−j))=0, |
for
−v1,j(1−ℓ−j)α1,jδ1,jℓ+1(1−ℓ)2=1ℓ2,ℓ∈(ℓ−j,1). |
This is a contradiction because the two sides have opposite sign. This proves (ii).
Since the implication (ii)
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},j∈J. |
By Lemma 7.3 we have either
Proposition 7.4. Problem (2.1)-(2.2) admits a (completely) non-stationary traveling wave with
0<v1,j<minΔj or v1,j>maxΔj,j∈J,(α1,2δ1,2v1,2)δ1,2v21,2−δ1,2=…=(α1,n+1δ1,n+1v1,n+1)δ1,n+1v21,n+1−δ1,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
ℓ−1=0=ℓ−j,ℓ+1=(α1,jδ1,jv1,j)δ1,jv21,j−δ1,j,ℓ+j=(α1,jδ1,jv1,j)v21,jv21,j−δ1,j,j∈J, | (7.9) |
and do not satisfy (5.3).
Proof. Fix
c1,j=v1,jln(ℓ+1)ln(ℓ+j),A1,j=α1,jv1,jln(ℓ+1)ln(ℓ+j),kj=0=κj,gh(ℓ)=vhℓln(ℓ+hℓ),gh(0)=0. |
Hence (7.6) can be written as
ℓ+jln(ℓ+j)=α1,jv1,jℓ+1ln(ℓ+1) | (7.10) |
and therefore (7.5) becomes
(δ1,j−v1,jℓ+jα1,jℓ+1)ln(ℓ+jℓ)=0,ℓ∈(0,ℓ+j), |
namely
ℓ+j=α1,jδ1,jv1,jℓ+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)
Assume (6.11). In this case condition (
Assume (7.8). In this case condition (
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
rh≐ℓ+h−ρhℓ+h−ℓ−h, | (A.1) |
which implies
rh,t+Gh(rh)x=(Eh(rh)rh,x)x, | (A.2) |
where
Gh(rh)≐−fh(ℓ+h−(ℓ+h−ℓ−h)rh)−fh(ℓ+h)ℓ+h−ℓ−h,Eh(rh)≐Dh(ℓ+h−(ℓ+h−ℓ−h)rh). |
Furthermore, equation (A.2) has a wavefront solution
(Eh(ψh)ψ′h)′+(θh−G′h(ψh))ψ′h=0 |
and
φh(ξ)=(ℓ−h−ℓ+h)ψh(ξ)+ℓ+h,ξ∈R. | (A.3) |
We discuss now the existence of a wavefront solution
−Gh(rh)>−rhGh(1),rh∈(0,1). | (A.4) |
By the definition of
−rhGh(1)=−rhfh(ℓ+h)−fh(ℓ−h)ℓ+h−ℓ−h. |
Then, inequality (A.4) is equivalent to
fh(ℓ+h)−(fh(ℓ+h)−f(ℓ−h))rh<fh(ℓ+h−(ℓ+h−ℓ−h)rh), for rh∈(0,1), |
if and only if
{ψh(ξ)=1for ξ≤ν−h,∫νψh(ξ)Eh(s)−Gh(s)+sGh(1)=ξfor ν−h<ξ<ν+h,ψh(ξ)=0for ξ≥ν+h, | (A.5) |
where
ν+h≐∫ν0Eh(s)−Gh(s)+sGh(1)ds,ν−h≐−∫1νEh(s)−Gh(s)+sGh(1)ds. |
Notice that, by differentiating (A.5) in the interval
Eh(ψh(ξ))Gh(ψh(ξ))−ψh(ξ)Gh(1)ψ′h(ξ)=1,ξ∈(ν−h,ν+h), | (A.6) |
which implies
Consider now
Now it remains to consider the boundary conditions of
(i) Assume
ν−h=−∫1νEh(s)−Gh(s)+sGh(1)ds>−∞. | (A.7) |
To prove (A.7), notice that
lims→1−E′h(s)−G′h(s)+Gh(1)=E′h(1)−G′h(1)+Gh(1)=−ℓ+hD′h(0)−f′h(0)+fh(ℓ+h)ℓ+h≥0 |
and then, by applying de l'Hospital Theorem we prove condition (A.7). Moreover, by condition (A.6), we get
limξ↓ν−hψ′h(ξ)={−f′h(0)−fh(ℓ+h)ℓ+hℓ+hD′h(0)if D′h(0)>0,−∞if D′h(0)=0. |
By applying (A.3) we conclude that
limξ↓ν−hDh(φh(ξ))φ′h(ξ)=limξ↓ν−h−ℓ+hEh(ψh(ξ))ψ′h(ξ) |
and hence, by (A.6), we deduce (3.9).
(ii) Assume
lims→0+E′h(s)−G′h(s)+Gh(1)=E′h(0)−G′h(0)+Gh(1)=(1−ℓ−h)D′h(1)f′h(1)+fh(ℓ−h)1−ℓ−h≥0 |
and again, by applying de l'Hospital Theorem we prove that
limξ↑νhψ′h(ξ)={fh(ℓ−h)1−ℓ−h+f′h(1)(ℓ−h−1)D′h(1)if D′h(1)<0,−∞if D′h(1)=0. |
By applying (A.3) we conclude that
(iii) In all the other cases it is easy to show that
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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