Kinks and solitons in linear and nonlinear-diffusion Keller-Segel type models with logarithmic sensitivity
aa r X i v : . [ m a t h . A P ] F e b KINKS AND SOLITONS IN LINEAR AND NONLINEAR–DIFFUSIONKELLER–SEGEL TYPE MODELS WITH LOGARITHMIC SENSITIVITY
JUAN CAMPOS, CLAUDIA GARC´IA, AND JUAN SOLER
Abstract.
This paper deals with the existence of traveling waves type patterns in the case ofthe Keller–Segel model with logarithmic sensitivity. The cases in which the diffusion is linearand nonlinear with flux-saturated (of the relativistic heat equation-type) are fully analyzed bycomparing the difference between both cases. Moreover, special attention is paid to travel-ing waves with compact support or with support in the semi-straight line. The existence ofthese patterns is rigorously proved and the differences between both cases (linear or nonlineardiffusion) are analyzed.
Contents
1. Introduction 12. The equations for traveling waves solutions 53. Linear diffusion 63.1. Analysis of equilibrium points 63.2. Phase space diagram 83.3. Unbounded solutions of the ( w, v ) system 83.4. Behavior of the ( w, v ) system 113.5. Going back to the solutions ( u, S ) of the original system 153.6. Solitons for both density and chemoattractant 194. Relativistic flux saturated operators with logarithmic sensitivity 204.1. Moving solutions with flux saturated mechanisms 204.2. Consequences on solutions in the system ( u, S ) 23References 241.
Introduction
The aim of this paper is to study traveling wave patterns of the Keller–Segel model withlogarithmic sensitivity, both in the case of linear diffusion and in the case of flux–saturated non–linear diffusion, focusing our analysis on the so-called relativistic heat equation. Our objectiveis to prove the existence of traveling waves of soliton type with compact support in both casesof linear and non-linear diffusion.Chemotaxis refers to the motion of the species up or down a chemical concentration gradient.Examples of this biological process are the propagation of traveling bands of bacterial towardthe oxygen [1, 2] or the outward propagation of concentric ring waves by the E. Coli [16, 17].The prototypical chemotaxis model was proposed by Keller and Segel [31] and in its general
Mathematics Subject Classification.
Key words and phrases.
Traveling waves, Keller-Segel, Flux-saturated, Patterns in nonlinear parabolic systems,Solitons, Logarithmic sensitivity, Cross-diffusion.This work has been partially supported by the MINECO-Feder (Spain) research grant number RTI2018-098850-B-I00, the Junta de Andaluc´ıa (Spain) Project PY18-RT-2422 & A-FQM-311-UGR18.) form reads as ∂ t u ( t, x ) = ∂ x (cid:26) u ( t, x )Φ (cid:18) ∂ x u ( t, x ) u ( t, x ) (cid:19) − au ( t, x ) ∂ x f ( S ) (cid:27) , x ∈ R , t > ,δ∂ t S ( t, x ) = γ∂ xx S ( t, x ) + k ( u, S ) , x ∈ R , t > ,u (0 , x ) = u ( x ) , x ∈ R . (1)The function u = u ( t, x ) refers to the cell density at position x and time t , whereas S ( t, x ) meansthe density of the chemoattractant. Then, the above system consists in two coupled equationsin terms of u and S . The parameter a ≥ δ and γ positive numbers where γ is thechemical diffusion coefficient. In the classical Keller–Segel model, the function Φ is taken tobe the identity map in order to have a classical diffusion in the first term of (1). Moreover, f refers the chemosentivity function describing the signal mechanism and k ( u, S ) characterizes thechemical growth and degradation.The chemosensitivity function f can be chosen in different ways. The linear law agreeswith f ( S ) = S , the logarithmic law is f ( S ) = log( S ) or the receptor law refers to f ( S ) = S m / (1 + S m ) for m ∈ N . The system with linear law and k ( u, S ) = S − u is called as theminimum chemotaxis model (see [26, 30]). The second one referring to the logarithmic lawfollows from the Weber–Frechner law, see [3, 8, 27, 31] for some applications. Although initiallythe Keller-Segel model was motivated by chemotaxis processes, the field of application of thesemodels is increasingly wide and covers fields of population dynamics, biopolymers or cross-diffusion in quantum mechanics, among others. We refer to [39] for a survey concerning thelogarithmic law.Then, here we will focus on the case of logarithmic sensitivity, meaning f ( S ) = log( S ) andwhere k ( u, S ) = u − λS with λ ≥
0. In this way, (1) agrees with ∂ t u ( t, x ) = ∂ x (cid:26) u ( t, x )Φ (cid:18) ∂ x u ( t, x ) u ( t, x ) (cid:19) − a ∂ x S ( t, x ) S ( t, x ) u ( t, x ) (cid:27) , x ∈ R , t > ,δ∂ t S ( t, x ) = γ∂ xx S ( t, x ) − λS ( t, x ) + u ( t, x ) , x ∈ R , t > ,u (0 , x ) = u ( x ) , x ∈ R . (2)We will assume that Φ verifies (H1) Φ ∈ C ( R ) , Φ( − s ) = − Φ( s ) , Φ ′ ( s ) > , ∀ s ∈ R . We will initially assume that Φ( s ) = s that corresponds to a linear diffusion (of the Laplaciantype) with respect to mass density (which is the classical Keller–Segel application), or morespecifically Φ( s ) = µs , where µ represents the viscosity coefficient.As we have mentioned before, we will give a comparative between the Keller–Segel withlinear diffusion and the flux–limited diffusion. More specifically, we will deal with the so-calledrelativistic heat equation, which corresponds to the choiceΦ( s ) = µ s q (cid:0) µc (cid:1) s , where µ and c are positive parameters.Let us remark that functions Φ satisfyinglim s →∞ Φ( s ) = ∞ , (which is verified for the linear diffusion) does not seem to have new phenomena except for acasuistic complication. On the other hand, if (H2) lim s →∞ Φ( s ) < ∞ , ELLER–SEGEL TYPE WITH LOGARITHMIC SENSITIVITY 3 we speak of a flux–saturated process on which we will analyze the qualitative differences withlinear diffusion, although the casuistry is similar. In the relativistic heat equation the limit in (H2) is exactly the parameter c , which expresses the growth rate of the support of the solutionfor such specific flux–saturated equation. We will refer to both hypothesis (H1) - (H2) as (H) . Inaddition to the relativistic heat equation, we should mention that the case of Larson operators,which are defined by Φ( s ) = µ s p q (cid:0) µc (cid:1) p | s | p , can be analyzed with exactly the same techniques used in this paper, as in general limitersverifying Φ( s ) − c = O ( s − p ) as s → ∞ , with p ∈ (1 , ∞ ).In addition to the complexity in the analysis introduced by the flux-saturated operatorsthat we study in this paper, there are many other operators of this type that would requirean adaptation or extension of the techniques used here, among them we mention the Wilsonoperator, which corresponds to p = ∞ in the Larson case, and is defined by φ ( s ) = µ s (cid:0) µc (cid:1) | s | . These are just some classical examples of diffusion by flux–saturated mechanisms, an extensivereview can be seen in [19]. Note that the above flux-saturated operators verify a sublineargrowth property, that is, there exist a, b ∈ R + such that | Φ( s ) | ≤ a | s | + b . Models with flux–saturated have been studied in various contexts and from diverse perspectives, from analyticalstudies, the appropriate concept of entropy solutions, hydrodynamic limits, etc., we refer to[4, 5, 6, 9, 10, 11, 12, 13, 14, 18, 21, 22, 23, 24, 25, 28, 33, 34, 35, 37, 38] for some references.In this work, we will focus on the study of traveling waves solutions, that is u ( t, x ) = u ( x − σt ) , and S ( t, x ) = S ( x − σt ) , (3)with σ >
0, for some profiles u, S : R → [0 , ∞ ) . The search for traveling waves solutions iscrucial to understand the mechanisms behind various propagating wave patterns. The analysisof traveling waves solutions for Keller-Segel type systems has been widely carried out fromvarious perspectives and techniques (variational or dynamic systems), see [29, 31, 32, 36, 15]and the references therein.We will show that the Keller–Segel system together with a flux–saturated mechanism exhibitsdiverse properties with respect to the classical one. Here, we will study both cases and showtheir differences. We refer to Figure 1 for the shapes of traveling waves solutions in the cases ofclassical diffusion and flux–saturated mechanisms. Note that the difference between the patternswith compact support in Figure 1 is mainly due to the associated flux–saturated mechanisms,where there are jumps in the connection with zero and these jumps have infinite slopes at bothends of the support, regardless of the parameter values. As we will see, there are other types ofmore classical traveling waves solutions with support in the straight or in the semi-straight line,although we believe that those that have compact support have a special interest in physics orbiology problems.Let us briefly explain the idea of the work. Assume that we have a solution of type (3), hence(2) agrees with − σu ′ = (cid:18) u Φ (cid:18) u ′ u (cid:19) − a S ′ S u (cid:19) ′ , (4) − σδS ′ = γS ′′ − λS + u, (5)where u ′ , S ′ and S ′′ represent the derivative with respect to the new variable s = x − σt .Although we have expressed the previous system taking into account all its terms, our studywill focus on the case where δ = 0. Under a suitable change of variables that is presented inProposition 2.1 in the case δ = 0, the above system of ordinary differential equations is relatedto w ′ = w Φ − ( av − σ ) − wv, (6) J. CAMPOS, C. GARC´IA, AND J. SOLER ss − s + A ss − s + B ss − s + C ss − s + D ss − s + E Figure 1.
Figures A, B and C correspond to the linear diffusion, in the cases0 < a < a = 1 and a > γv ′ = − γv − σδv − w + λ, , (7)with some initial conditions. Throughout this paper we will analyze the coupled system (6)–(7)and later come back to the original variables u and S in order to transfer and interpret the resultsobtained there. We will separate in two cases: first we will assume that Φ = Id giving rise toa linear diffusion and later we will deal with the nonlinear flux–saturated case for Φ satisfyingthe hypothesis (H) . The shape of the profiles u and S strongly depends on the previous cases.More specifically, in the case of a linear diffusion, i.e. Φ = Id, hence the system (6)–(7) is notsingular and classical theory for ODEs gives us existence and uniqueness of solution. Moreover,analyzing the phase diagram and coming back to u and ˜ S we are able to find different profileswith and without compact support. We refer to Figure 5 which illustrates the shapes of theprofiles. The existence of the diverse types of solutions strongly depends on the parameters a and σ . This is the main goal of Section 3.On the other hand, by virtue of the hypothesis (H) for Φ, one has that system (6)–(7) issingular at the boundary. Indeed, note that Φ − is defined only in ( − c, c ) as a consequenceof (H) and this gives us a boundary for the solutions of (6)–(7). Moreover, Φ − ( ± c ) = ±∞ which implies an infinite derivative of w on the boundary. That amounts to have the solutionsdescribed in Figure 8.The main goal of Section 4 is to analyze the existence of the different types of traveling wavesolutions in the case of the so-called relativistic heat equation (see Figure 1). The existence oftraveling waves were analyzed for the case of flux–saturated mechanisms for the first time in[20], while in the case of flux–saturated Keller–Segel in [7].Finally, the results obtained in this work can be summarized in the following (formal) theorem,which will be developed and specified in the following sections. Theorem 1.1.
There are traveling wave type profiles with compact support or with support inthe semi-straight line for the Keller-Segel model with logarithmic sensitivity both in the case oflinear diffusion and nonlinear flux–saturated mechanishms. In the latter case, the solutions mustbe understood in an entropic sense and present a jump with an infinite slope when extended byzero.
This work is organized as follows. Section 2 aims to give the equivalent equations for travelingwaves solutions in terms of ( w, v ). In Section 3 we analyze the classical Keller–Segel equationsby getting traveling waves solutions with the shapes described in the upper part of Figure 1.
ELLER–SEGEL TYPE WITH LOGARITHMIC SENSITIVITY 5
Finally, Section 4 deals with the Keller–Segel system with a flux–limited diffusion getting thesolutions presented in at the bottom of Figure 1.Let us establish the notation that will be used throughout the paper. We will denote u ( s ) = lim s → s u ( s ) , and u ( s ± ) = lim s → s ± u ( s ) . even when the values are not finite, we still use the right hand notation when the left hand onewas not clear. 2. The equations for traveling waves solutions
The idea of this section is to explore the existence of solutions of traveling waves type of thesystem (2), by analyzing (4)–(5).To make lighter the writing, we assume δ = 0, searching for solutions of the system (cid:18) u Φ (cid:18) u ′ u (cid:19) − a S ′ S u + σu (cid:19) ′ = 0 , (8) γS ′′ − λS + u = 0 . (9)We will look for solutions u ( s ) that are positive and bounded, and then S ( s ) is defined by (9).Let us consider solutions of (8) in a distributional framework in the sense of Z R (cid:18) u Φ (cid:18) u ′ u (cid:19) − a S ′ S u + σu (cid:19) ψ ′ ds = 0 , (10)for any test function ψ ∈ C ∞ ( R ). We observe that if u is positive, then from (9) we deduce that S >
0. Furthermore, in the case that u is bounded, then S ∈ C , since we are working in theone–dimensional case. Therefore, S ′ S ∈ L loc ( R ). The term u Φ (cid:16) u ′ u (cid:17) is the product of a boundedand integrable function u ∈ BV loc ( R ), by a semilinear function Φ, where u ′ makes sense as theRadon–Nikodym derivative of u . Then, the function s ∈ supp u u ( s )Φ (cid:18) u ′ ( s ) u ( s ) (cid:19) − a S ′ ( s ) S ( s ) u ( s ) + σu ( s ) , is a L loc ( R ) function. Therefore, the fact that u is a distributional (10) solution of (8) impliesthe existence of a constant k such that u ( s )Φ (cid:18) u ′ ( s ) u ( s ) (cid:19) − a S ′ ( s ) S ( s ) u ( s ) + σu ( s ) = k. In the case that supp u = R , we can deduce that k = 0, since (10) is defined for every testfunction ψ ∈ C ∞ ( R ). If supp u = R , we look for traveling wave profiles for which u ( s ) →
0, as s → + ∞ or s → −∞ , which again provides us that k = 0.Consequently, we look for functions u > S > s − , s + ), with −∞ ≤ s − < s + ≤ + ∞ , such thatΦ (cid:18) u ′ u (cid:19) − a S ′ S + σ = 0 , (11) γS ′′ − λS + u = 0 . (12)In the following proposition, we arrive at an equivalent system to (11)–(12) which will help uson the study of the existence of solutions. Proposition 2.1.
Let g : ( − c, c ) → R given by Φ( g ( y )) = y with y ∈ R , that is g = Φ − inthe sense of the composition of applications. Then, the solutions of (8)–(9) can be obtained bysolving the system w ′ = wg ( av − σ ) − wv, (13) J. CAMPOS, C. GARC´IA, AND J. SOLER γv ′ = − γv − w + λ, (14) where w ( s ) = u ( s ) S ( s ) , and v ( s ) = S ′ ( s ) S ( s ) . (15) Proof.
The calculations made previously allow us to see that w, v defined by (15) give rise to(13) and (14).To go back and recover the solutions of (8)–(9), we take s ∈ R , u > S > S ′ ∈ R andsolve the initial values problem consisting of (13)–(14) with initial data w ( s ) = u S , v ( s ) = S ′ S ,where S ( s ) = S exp( R ss v ( δ ) dδ ) and u ( s ) = w ( s ) S ( s ). (cid:3) Linear diffusion
This section aims to analyze the existence of traveling waves solutions for the classical Keller–Segel model with logarithmic sensitivity interacting with linear diffusion. Recall that throughoutthe paper we assume δ = 0, for the sake of simplicity. Therefore, the system under study in thissection is ∂ t u = ∂ x (cid:18) µ∂ x u − a ∂ x SS u (cid:19) , γ∂ xx S − λS + u. (16)Moreover, we assume that σ (the velocity of the traveling wave), a , γ and λ are positive realnumbers. Then, after the change of variable to traveling waves coordinates ( t, x ) → x − σt ,system (16) becomes: w ′ = w n ( a − v − σ o , (17) v ′ = λγ − v − γ w. (18)The main difference between a < a ≥ w, v ) ′ . In what follows, we will analyze the system for each value of a . Aftershowing the phase diagram for every case, we shall focus on the continuation of the solutionstogether with an asymptotic analysis. Finally, we need to come back to the original variables u and S via Proposition 2.1.3.1. Analysis of equilibrium points.
In this section, we explore a local analysis of the solu-tions around the fixed points by studying the linearized equation around them.From now on define v ⋆ as v ⋆ := s λγ , (19)and σ ⋆ := | − a | s λγ = | − a | v ⋆ . (20)The value of σ ⋆ will determine the different scenarios for the solutions. This will be analyzed inthe following proposition. Proposition 3.1.
Define ( w , v ) = (0 , v ⋆ ) , ( w , v ) = (0 , − v ⋆ ) , and ( w , v ) = (cid:18) λ − γσ ( a − , σa − (cid:19) , where v ⋆ is defined in (19) . Hence, (1) If a > and σ ⋆ < σ , then (17) – (18) has two fixed points given by ( w i , v i ) , with i = 1 , . The point ( w , v ) is a stable point, whereas ( w , v ) is a saddle point. The stable manifoldassociated to the saddle point is generated by ( γ ((1 + a ) v ⋆ + σ ) , , and the unstable oneis generated by (0 , . ELLER–SEGEL TYPE WITH LOGARITHMIC SENSITIVITY 7 (2) If a ∈ (0 , and σ < σ ⋆ , then (17) – (18) has three fixed points given by ( w i , v i ) , with i = 1 , , . The point ( w , v ) is a stable point, ( w , v ) is an unstable point and ( w , v ) is a saddle point. Both the stable and unstable manifold associated to the saddle pointare one–dimensional. (3) If a > and σ < σ ⋆ , then (17) – (18) has three fixed points given by ( w i , v i ) , with i = 1 , , . The points ( w , v ) and ( w , v ) are saddle points, whereas ( w , v ) is astable point or a stable focus. The stable manifold associated to ( w , v ) is generated by (0 , and the unstable one is generated by ( γ ( − (1 + a ) v ⋆ + σ ) , . Moreover, the stablemanifold associated to ( w , v ) is generated by ( γ ((1 + a ) v ⋆ + σ ) , , and the unstable oneis generated by (0 , .Proof. The linearized problem associated to such system is given by( w, v ) ′ = A ( w i , v i )( w, v ) , for ( w i , v i ) a fixed point, where A ( w, v ) = (cid:18) ( a − v − σ ( a − w − γ − v (cid:19) . The eigenvalues are given by the solutions of(( a − v − σ − x )( − v − x ) + ( a − wγ = 0 . In the case of ( w , v ), we have that the eigenvalues are x = ( a − v ⋆ − σ, and , x = − v ⋆ , where v ⋆ >
0. Then, since x < a − v ⋆ − σ < w , v ) isstable. In the case that ( a − v ⋆ − σ > w , v ). The associated eigenvalues read as x = − ( a − v ⋆ − σ, and , x = 2 v ⋆ . In this case x is positive. Hence if (1 − a ) v ⋆ < σ , ( w , v ) is a saddle point. Otherwise, it is anunstable point.Finally, in the case that a = 1, there exists ( w , v ) such that (17)–(18) holds. The associatedeigenvalues to this point are described by x = − σa − − s σ ( a − − ( a − γ (cid:18) λ − γσ ( a − (cid:19) ,x = − σa − s σ ( a − − ( a − γ (cid:18) λ − γσ ( a − (cid:19) . Note that since we are studying solutions with w >
0, we consider such fixed point only in thecase when w >
0. That correspond to λ − γσ ( a − > , which agrees with σ ⋆ > σ . In the case that a > x < x <
0. Moreover, if σ ( a − − ( a − γ (cid:18) λ − γσ ( a − (cid:19) > , we obtain a stable point. Otherwise, it is a stable focus. On the other hand, if a < x > x <
0, finding a saddle point. Finally, the generators of the stable and unstablemanifold associated to the saddle points are given through the associated eigenfunctions. (cid:3)
J. CAMPOS, C. GARC´IA, AND J. SOLER
Remark 3.2.
From the previous proposition, note that if a < and σ = σ ⋆ , then ( w , v ) =( w , v ) . Moreover, one gets that ( w , v ) is a stable point. However, for ( w , v ) = ( w , v ) onegets a zero eigenvalue and another positive one.In the case that a > and σ = σ ⋆ , then ( w , v ) = ( w , v ) . In such a case, ( w , v ) is asaddle point. However, the point ( w , v ) = ( w , v ) is degenerate as it has a zero eigenvalueand a negative one. Phase space diagram.
Here, we will give a formal discussion about the monotonicity ofthe solutions and show the phase diagram depending also on the choice of the parameters.The isocline map depends on the position (in case it exists) of the vertical line v = σa − andthe parabola w = λ − γv , with w ≥
0. This can be described via v ⋆ and σ ⋆ defined in (19)–(20).First, consider small chemotactic sensitivity agreeing with a <
1. That corresponds to theupper part of Figure 2, and, following its notation, we consider Case A, for 0 < σ < σ ⋆ , and CaseB, for σ > σ ⋆ . In both cases, ( w , v ) and ( w , v ) are fixed points. Moreover, if 0 < σ < σ ⋆ (CaseA), we have an extra fixed point given by the intersection between the parabola and the verticalline: this is ( w , v ) defined in Proposition 3.1. Note also that limit cycles are not allowedhere. Indeed, for Case B we have that the region under the parabola is positively invariant.Therefore, either the solution enters the area under the parabola and the component of v ofthe system changes from decreasing to increasing, or the solution never touches the parabolaand v is always decreasing. In Case A, the region under the parabola can be divided in twosmall regions described in Figure 2-(A) . There, the region to the left-hand side of the verticalline is positively invariant and a limit cycle can not touch it. On the other hand, the region tothe right-hand side of the vertical line is negative invariant. Finally, putting together all theprevious arguments we find the phase diagrams described in the upper part of Figure 2.The case a = 1 is very special since we do not have any vertical line: it corresponds toProposition 3.1–1. Indeed, the system reduces to w ′ = − σw,v ′ = λγ − v − wγ . As in the previous case, the area under the parabola remains positively invariant. We refer thereader to Figure 2–C for its phase diagram.Finally, if we have a large chemotactic sensitivity parameter, i.e., a >
1, we get again twopossibilities: either the vertical line intersects the parabola. This is described at the bottom ofFigure 2. We denote then Cases D when 0 < σ < σ ⋆ and E when σ > σ ⋆ . Here, we also havethat ( w , v ) and ( w , v ) are fixed points for both cases, and ( w , v ) only for Case D. The nonexistence of limit cycles in these cases is not clear here. For Case E, we can work as for a < w , v ) may appear.3.3. Unbounded solutions of the ( w, v ) system. In this section, we study the unboundedsolutions (the upper blue lines) in Figure 3. To do so, first we shall assume that the initialcondition ( w , v ) satisfies v > v ⋆ and w >
0, where v ⋆ is defined in (19). That will bepresented in the following proposition: Proposition 3.3.
Let a > , v > v ⋆ and w > , where v ⋆ is defined in (19) . Consider ( w, v ) the maximal solution to (17) – (18) , with initial data ( w ( s ) , v ( s )) = ( w , v ) , defined in ( s − , s + ) .Then, we have s − ∈ R , v ( s − ) = + ∞ , and w ( s − ) = + ∞ , for a < ,w ( s − ) ∈ R + , for a = 1 ,w ( s − ) = 0 , for a > . Proof.
Notice that since v > v ⋆ the solution in ( s − , s ) lies outside the parabola, which impliesthat v decreases. On the other hand, the sign of w depends on a and σ . We refer to Figure 2. ELLER–SEGEL TYPE WITH LOGARITHMIC SENSITIVITY 9 vw A − v ∗ v ∗− σ − a vw B - v ∗ v ∗− σ − a vw C − v ∗ v ∗ vw D − v ∗ v ∗ σa − vw E − v ∗ v ∗ σa − Figure 2.
Top (A and B): a <
1. Center (C): a = 1. Bottom (D and E): a > σ < σ ⋆ . Right (B and E): σ > σ ⋆ . The red curves determine thechanges of direction of ( v, w ) and the black arrows refer to such directions.First, let us prove that s − ∈ R by a reductio ad absurdum argument . In this way, assumethat s − = −∞ and we will arrive to a contradiction. By the monotonicity of v , we have twopossibilities: lim s →−∞ v ( s ) = L > v ⋆ , or lim s →−∞ v ( s ) = + ∞ . Consider the first one. Hence, there exists a sequence s n → −∞ such thatlim n → + ∞ v ′ ( s n ) = 0 , and using the equation for v we arrive atlim n → + ∞ v ′ ( s n ) = λγ − L < , getting a contradiction. Hence, now assumelim s →−∞ v ( s ) = + ∞ . In this case, one has that v ( s ) tends to 0 as s → −∞ . As a consequence, there exists s n → −∞ such that lim n → + ∞ (cid:18) v ( s n ) (cid:19) ′ = 0 , which agrees with0 = − lim n → + ∞ v ′ ( s n ) v ( s n ) = lim n → + ∞ (cid:26) w ( s n ) γv ( s n ) − λγv ( s n ) (cid:27) = lim n → + ∞ (cid:26) w ( s n ) γv ( s n ) (cid:27) > , getting again a contradiction. Then, we can conclude that s − ∈ R .In the next step we will prove that if lim s → s − w ( s ) = + ∞ , (21)hence lim s → s − v ( s ) = + ∞ . (22)As a consequence and using that s − ∈ R , we will have that (22) always happens. We work againwith a reductio ad absurdum argument . Assume thatlim s → s − v ( s ) = L > v ⋆ , occurs. Here, we will use the graph system associated to (17)–(18). Denote V ( x ) = v ( w − ( x )).Hence lim x → + ∞ V ( x ) = L , and we deduce that there exists a sequence x n → + ∞ such thatlim n → + ∞ V ′ ( x n ) = 0 . By using the equations for ( w, v ), we have that V ′ ( x ) = λ − γV ( x ) − xγx { ( a − V ( x ) − σ } , and then lim x → + ∞ V ′ ( x ) = lim x → + ∞ a − V ( x ) − σ (cid:26) − γ + λ − γV ( x ) γx (cid:27) = 0 , getting a contradiction. Then, we can conclude that if (21) happens, (22) also does. Since s − ∈ R , we achieve that in any case (22) occurs. That concludes the first part of the lemma.In order to have the behavior of w at s − , we need to use the graph system for W ( x ) = w ( v − ( x )), that satisfies W ′ ( x ) = γW ( x ) { ( a − x − σ } λ − W ( x ) − γx . (23)Here, we will use a comparison argument. Consider first the case a <
1, where we have that w ′ < v ′ <
0, for s ∈ ( s − , s ). Take x such that x ≤ v ( s ) for any s ∈ ( s − , s ). Our goalwill be to prove that lim s → s − w ( s ) = + ∞ . (24)Note that the graph system is well–defined in x ∈ [ v , + ∞ ), since v is monotone. From (23) onehas that W ′ >
0, and then W ( x ) ≥ W ( x ), for any x ∈ [ v , + ∞ ). Moreover W ′ ( x ) ≥ α W ( x ) x , for some α >
0. Hence, considering y the solution to y ′ ( x ) = α y ( x ) x , with y ( v ) = W ( x ), we find that W ( x ) ≥ y ( x ). Consequently, one has that W ( x ) ≥ W ( x ) x α , achieving (24).The case a = 1 is very special since the equation for w can be integrated: note that it doesnot depend on v . Hence, one has w ( s ) = w e − σ ( s − s ) .The last case a > s < s such that v ( s ) > σa − . Then, for any s ∈ ( s − , s ) one has that v decreases and w increases. ELLER–SEGEL TYPE WITH LOGARITHMIC SENSITIVITY 11
Hence, we can set x such that x ≤ v ( s ), for any s ∈ ( s − , s ). We can define the graph system(23) for x ∈ [ v ( s ) , + ∞ ), and we can check W ′ ( x ) W ( x ) ≤ − αx , for some α >
0. By using again the comparison principle one finds W ( x ) ≤ Cx − α , concludingthe proof. (cid:3) The analysis for solutions with initial data v < − v ⋆ is completely symmetric. In fact, it isenough to make an investment in the path of s . Then, we achieve the following result. Proposition 3.4.
Let a > , v < − v ⋆ and w > , where v ⋆ is defined in (19) . Consider ( w, v ) the maximal solution to (17) – (18) , with initial data ( w , v ) , defined in ( s − , s + ) . Then, we have s + ∈ R , v ( s + ) = −∞ , and w ( s + ) = + ∞ , for a < ,w ( s + ) ∈ R + , for a = 1 ,w ( s + ) = 0 , for a > . Behavior of the ( w, v ) system. Here, we shall prove that there exists each of the solutions(the blue curve) of Figure 3. There, the initial condition is taking to be in the right hand sideof the parabola. The proof will have three parts. First, we will check the existence of w = w ⋆ such that we have a unique solution ending in ( w , v ) for Cases B, C, D and E; and ending in( w , v ) for Case A. The idea of the proof is the uniqueness of the stable manifold associated toeach saddle point. Later, we shall analyze the solutions with either w > w ⋆ or w < w ⋆ . At theend of this section, we will study the special case in which the starting point is a saddle fixedpoint by studying solutions with initial data v < − v ⋆ . More specifically, we will focus on a < σ < σ ⋆ corresponding to Figure 3-A.By using the uniqueness of the stable manifold associated to a saddle point, we are able toprove the existence of w ⋆ in Figure 2. Proposition 3.5.
Let a > and v > v ⋆ , where v ⋆ is defined in (19) . There exists w ⋆ > anda maximal solution to (17) – (18) , with initial data ( w ⋆ , v ) , defined in ( s − , + ∞ ) with s − ∈ R ,satisfying the following. • Asymptotic behavior at s − : We have that lim s → s − v ( s ) = + ∞ , for any a > , and lim s → s − w ( s ) = + ∞ , for a < , lim s → s − w ( s ) ∈ R + , for a = 1 , lim s → s − w ( s ) = 0 , for a > . • Asymptotic behavior at + ∞ : We find that lim s → + ∞ ( w ( s ) , v ( s )) = ( w , v ) , for a < and σ < σ ⋆ . Otherwise, lim s → + ∞ ( w ( s ) , v ( s )) = ( w , v ) . Remark 3.6.
Here we are considering σ = σ ⋆ . By virtue of Remark 3.2, we can prove also theexistence of such w ⋆ , but not the uniqueness: we can not ensure that there is a unique curveending at ( w , v ) . A vww ∗ v ∗ vw B w ∗ v ∗ vw C w ∗ v ∗ vw D w ∗ v ∗ vw E w ∗ v ∗ Figure 3.
Top (A and B): a <
1. Center (C): a = 1. Bottom (D and E): a > σ < σ ⋆ . Right (B and E): σ > σ ⋆ . The blue curves refer to thesolutions with initial data ( v , w ) with v > v ⋆ and where w depends on theposition with respect to w ⋆ . Proof.
Let us explain the existence of w ⋆ . It appears since the fixed point ( w , v ), for the case a < σ < σ ⋆ , or ( w , v ) = (0 , − v ⋆ ), for the other cases described in Proposition 3.1, aresaddle points.Let us focus on the case of having the saddle point ( w , v ), which is the case of Figure 2:B, C, D and E. There, the stable manifold is generated by the vector ( γ ((1 + a ) v ⋆ + σ ) , v = − v ⋆ of the parabola w = − γv + λ is 2 γv ⋆ . Sincethe vector ( γ (2 v ⋆ + σ ) ,
1) is steepest than the parabola at that point (2 γv ⋆ , , − v ⋆ ) as the starting point of the time reversed system and it will not enter in the parabola.The same argument of Lemma 3.3 can be applied here to prove that s − ∈ R andlim s → s − v ( s ) = + ∞ . Hence, we have that such solution must intersect the line v = v . In this way, we find theexistence of such w ⋆ , see Figure 3. Moreover, from Lemma 3.3 one finds the behavior at s − .Note that by the continuation argument, one has that s + = + ∞ . (cid:3) As a consequence of the uniqueness of solutions and Propositions 3.3 and 3.4, we get thefollowing asymptotic behavior for the solutions with w > w ⋆ , where w ⋆ is defined in Proposition3.5. Proposition 3.7.
Let a > , v > v ⋆ and w > w ⋆ , where v ⋆ in defined in (19) and w ⋆ isdefined in Proposition 3.5. Then, any maximal solution (17) – (18) , with initial data ( w , v ) ,defined in ( s − , s + ) satisfies the following: (1) We have that s − , s + ∈ R . (2) Moreover, we find lim s → s − v ( s ) = + ∞ , and lim s → s + v ( s ) = −∞ , ELLER–SEGEL TYPE WITH LOGARITHMIC SENSITIVITY 13 for any a > , and lim s → s − w ( s ) = lim s → s + w ( s ) = + ∞ , for a < , lim s → s − w ( s ) , lim s → s + w ( s ) ∈ R + , for a = 1 , lim s → s − w ( s ) = lim s → s + w ( s ) = 0 , for a > . The solutions starting from w < w ⋆ will enter in the parabola and, hence, we can check thatthey are bounded close to s + . Indeed, for cases A, B, C and E of Figure 2 we can prove thatthey converge to a fixed point. Since for case D we can not ensure the non existence of limitcycles around ( w , v ), we can not ensure the convergence to such point. However, the solutionwill be bounded. Proposition 3.8.
Let a > , v > v ⋆ and w < w ⋆ , where v ⋆ is defined in (19) and w ⋆ is definedin Proposition 3.5. Then, any maximal solution (17) – (18) , with initial data ( w , v ) , defined in ( s − , s + ) satisfies the following: (1) We have that s − ∈ R , and s + = + ∞ . (2) Moreover, the following assertions hold true: • Asymptotic behavior at s − : We have lim s → s − v ( s ) = + ∞ , for any a > , and lim s → s − w ( s ) = + ∞ , for a < , lim s → s − w ( s ) ∈ R + , for a = 1 , lim s → s − w ( s ) = 0 , for a > . • Asymptotic behavior at + ∞ : We find that the solution is bounded and satisfies − v ⋆ < lim inf s → + ∞ v ( s ) ≤ lim sup s → + ∞ < v ⋆ , and lim inf s → + ∞ w ( s ) > , for a > and σ < σ ⋆ . Otherwise lim s → s + ( w ( s ) , v ( s )) = ( w , v ) . Proof.
From Lemma 3.3 one finds that s − ∈ R and the mentioned behavior at s − .By the uniqueness of solution, we have that such solution must enter in the parabola as s approaches s + . Once it enters, one has that v ′ > a > σ < σ ⋆ , which correspond to Case D of Figure 2, we can not ensure thenon existence of limit cycles. What we know is that the solution is bounded as it approaches s + and then we can continue it having s + = + ∞ . Moreover, we get that − v ⋆ < lim inf s → + ∞ v ( s ) ≤ lim sup s → + ∞ < v ⋆ , and lim inf s → + ∞ w ( s ) > . On the other hand, from the phase portrait one has that the solution must converges to a fixedpoint: ( w , v ). By the continuation principle one finally achieves that s + = + ∞ , concludingthe proof. (cid:3) Hence, the previous Propositions 3.5, 3.7 and 3.8 yield the following theorem.
Theorem 3.9.
Let a > and v > v ⋆ , where v ⋆ is defined in (19) . Consider any maximalsolution (17) – (18) , with initial data ( w , v ) , defined in ( s − , s + ) . Then, there exists w ⋆ > suchthat the following is satisfied: • If w > w ⋆ , then s − , s + ∈ R . Moreover, we find lim s → s − v ( s ) = + ∞ , and lim s → s + v ( s ) = −∞ , for any a > , and lim s → s − w ( s ) = lim s → s + w ( s ) = + ∞ , for a < , lim s → s − w ( s ) , lim s → s + w ( s ) ∈ R + , for a = 1 , lim s → s − w ( s ) = lim s → s + w ( s ) = 0 , for a > . • If w = w ⋆ , then s − ∈ R and s + = + ∞ . We have lim s → s − v ( s ) = + ∞ , for any a > , and lim s → s − w ( s ) = + ∞ , for a < , lim s → s − w ( s ) ∈ R + , for a = 1 , lim s → s − w ( s ) = 0 , for a > . Moreover, we find lim s → + ∞ ( w ( s ) , v ( s )) = ( w , v ) , for a < and σ < σ ⋆ , and otherwise one gets lim s → + ∞ ( w ( s ) , v ( s )) = ( w , v ) . • If w < w ⋆ , then s − ∈ R and s + = + ∞ . We have lim s → s − v ( s ) = + ∞ , for any a > , and lim s → s − w ( s ) = + ∞ , for a < , lim s → s − w ( s ) ∈ R + , for a = 1 , lim s → s − w ( s ) = 0 , for a > . Moreover, we find that the solution is bounded and satisfies − v ⋆ < lim inf s → + ∞ v ( s ) ≤ lim sup s → + ∞ < v ⋆ , and lim inf s → + ∞ w ( s ) > , for a > and σ < σ ⋆ , and otherwise lim s → s + ( w ( s ) , v ( s )) = ( w , v ) . Finally, let us focus on a < σ < σ ⋆ . From the phase diagram in Figure 2, we findinteresting solutions when considering v < − v ⋆ in such a case. Those solutions are describedin Figure 4. Theorem 3.10.
Let a < , σ < σ ⋆ , w > and v < − v ⋆ , where v ⋆ and σ ⋆ are defined in (19) – (20) . Consider the maximal solution ( w, v ) to (17) – (18) , with initial data ( w ( s ) , v ( s )) =( w , v ) , defined in ( s − , s + ) . Then, there exists w ⋆ satisfying the following. (1) Asymptotic behavior at s + : We have that s + ∈ R and lim s → s + v ( s ) = −∞ , and lim s → s + w ( s ) = + ∞ . (2) Asymptotic behavior at s − : • If w > w ⋆ then s − ∈ R and lim s → s − v ( s ) = + ∞ , and lim s → s − w ( s ) = + ∞ . ELLER–SEGEL TYPE WITH LOGARITHMIC SENSITIVITY 15 − v ⋆ v ⋆ w ⋆ v v w Figure 4.
Case a ∈ (0 ,
1) and σ < σ ⋆ . The blue curves refer to the solutionswith initial data ( v , w ) with v < − v ⋆ and where w depends on the positionwith respect to w ⋆ . • If w = w ⋆ then s − = −∞ and lim s →−∞ ( w ( s ) , v ( s )) = ( w , v ) . • If w < w ⋆ then s − = −∞ and lim s →−∞ ( w ( s ) , v ( s )) = ( w , v ) . Proof.
The asymptotic behavior at s + is achieved as a consequence of Lemma 3.4.The existence of w ⋆ comes from the uniqueness of the stable manifold associated to the saddlepoint ( w , v ). This is similar to the proof of the existence of w ⋆ in Proposition 3.5. By thecontinuation of the solutions one gets that s − = −∞ in such a case.In the case that w > w ⋆ , then the solution can not enter in the parabola by the uniquenessof solution. Then, a similar scenario to Proposition 3.7 occurs here obtaining the announcedresult.In the latter case w < w ⋆ , the solution will converge to the unstable point ( w , v ) (whichis a stable point of the time reversed system). By the continuation principle one gets again s − = −∞ . (cid:3) Going back to the solutions ( u, S ) of the original system. Different values of a willexhibit diverse scenarios. In particular, we will have the distinct types of solutions A1, A2, A3and A4, for the original variables ( u, S ), defined as follows • Type A1:
A function f : ( s − , s + ) → R is of Type A1 if s − , s + ∈ R and f ( s − ) = f ( s + ) = 0. • Type A2:
A function f : ( s − , s + ) → R is of Type A2 if s − ∈ R and s + = + ∞ .Moreover, f satisfies f ( s − ) = 0, andlim s → + ∞ f ( s ) = 0 . • Type A3:
A function f : ( s − , s + ) → R is of Type A3 if s − ∈ R and s + = + ∞ .Moreover, f satisfies f ( s − ) = 0, andlim s → + ∞ f ( s ) = + ∞ . • Type A4:
A function f : ( s − , s + ) → R is of Type A4 if s − = −∞ and s + ∈ R .Moreover, f satisfies f ( s + ) = 0, andlim s →−∞ f ( s ) = + ∞ . See Figure 5 referring to the different type of solutions.Finally, we will recover the solution u ( t, x ) = u ( s ) and S ( t, x ) = S ( s ) to (2) via Proposition2.1. First, we need to introduce two preliminary results. s − s + s − s − s + Figure 5.
Types of solutions: A1, A2, A3, A4 in the case of linear diffusionKeller–Segel system with logarithmic sensitivity.
Lemma 3.11.
Let ( w, v ) be solutions to (17) – (18) defined in ( s , s + ) . If s + < + ∞ , then wehave lim s → s + S ( s ) = 0 , lim s → s + u ( s ) = 0 . Proof.
Analyzing the different possibilities in terms of a , we can deduce that in each of the caseslim s → s + v ( s ) = −∞ . From here and from equality v = S ′ /S we deduce S ( s ) = C e R ss v ( τ ) dτ , for some s , C associated to the initial data. Let us see that v is not integrable and, therefore, R s + s v ( τ ) dτ = −∞ . To demonstrate this fact, we distinguish two cases based on the value of theparameter a .In the case a ≥
1, just make the limit by using L’Hˆopital rule (note that v ′ ( s ) < s → s + s + − s − v ( s ) = lim s → s + v ( s ) v ′ ( s ) = lim s → s +
11 + w ( s ) γv ( s ) − λγv ( s ) = 1 , where we have used that w ( s ) v ( s ) → w + ∞ = 0 , since for a ≥ w ( s ) → w + , as s → s + , where w + ∈ [0 , + ∞ ).In the case 0 < a <
1, the limit w + = ∞ and we need to analyze the behaviour of w ( s ) v ( s ) . Takenow z = − v , which implies z ( s ) → ∞ , as s → s + . We can build a solution of w ′ = w ((1 − a ) z − σ ) , (25) z ′ = z + wγ − λγ (26)using the associated graph system. That is, define a function W : ( z , + ∞ ) → R , such that W ( z ( s )) = w ( s ). This takes the form W ′ = W (1 − a ) z − σz + Wγ − λγ . ELLER–SEGEL TYPE WITH LOGARITHMIC SENSITIVITY 17
Let W ( z ) = z R ( z ). Then, R ( s ) > z R ′ R = (1 − a ) − σz Rγ − λγz − ≤ (1 − a ) − σz − λγz − . Then, for z large enough and z ≥ z we have z R ′ R ≤ . Therefore, we find a bound for
R R ( z ) ≤ R ( z ) z z → , or, equivalently, W ( z ) z → , which concludes the proof for the case 0 < a < (cid:3) In a symmetric way, we find also the following result for the case s − > −∞ . Lemma 3.12.
Let ( w, v ) be solutions to (17) – (18) defined in ( s − , s + ) . If s − > −∞ , then wehave lim s → s − S ( s ) = 0 , lim s → s − u ( s ) = 0 . By using Lemmas 3.11 and 3.12, we achieve the following result concerning the traveling wavessolutions coming from Propositions 3.5, 3.7 and 3.8.
Theorem 3.13.
Let a > , v > v ⋆ , where v ⋆ is defined in (19) , and w ⋆ defined in Proposition3.5. For any σ > , there exists u ( t, x ) = u ( s ) and S ( t, x ) = S ( s ) , with s = x − σt , travelingwaves solution to (2) , with initial data ( u , S ) , verifying the following. (1) If u S > w ⋆ , then u and S are of Type A1. (2) If u S = w ⋆ , then u and S are of Type A2. (3) If u S < w ⋆ and av ⋆ < σ then S is of Type A3 and u is of Type A2. (4) If u S < w ⋆ and av ⋆ > σ , then S and u are of Type A3.Proof. From (15), we recover the value of u and S as u ( s ) = C w ( s ) e R ss v ( τ ) dτ , and S ( s ) = C e R ss v ( τ ) dτ , (27)for some C > w, v ) definedin ( s − , s + ). If u S ≤ w ⋆ , then s + = + ∞ , and in the other case s + ∈ R . Moreover, s − ∈ R .Hence, u and S are defined in ( s − , s + ).By using Lemma 3.12, one achieves lim s → s − u ( s ) = 0 , and lim s → s − S ( s ) = 0 , for any a > • Case u S > w ⋆ . The behavior at s + in this case can be obtaining using Lemma 3.11. • Case u S = w ⋆ . From Proposition 3.1 one has that s + = + ∞ andlim s → + ∞ ( w ( s ) , v ( s )) = ( w , v ) , if a < σ < | − a | v ⋆ , or lim s → + ∞ ( w ( s ) , v ( s )) = ( w , v ) , in other case. Hence, since v and v are negative numbers one findslim s → + ∞ S ( s ) = lim s → + ∞ C e R ss v ( τ ) dτ = 0 , and then lim s → + ∞ u ( s ) = 0 . • Case u S < w ⋆ . From Proposition 3.8 one finds that s + = + ∞ andlim s → + ∞ ( w ( s ) , v ( s )) = ( w , v ) , if a < a > σ < σ ⋆ . Since v is a positive number one achieveslim s → + ∞ S ( s ) = lim s → + ∞ C e R ss v ( τ ) dτ = + ∞ . Moreover, since w = 0, there is a competition between w and S . Note thatlim s → + ∞ R ss v ( τ ) dτ ln( w ( s )) = lim s → + ∞ v ( s )( a − v − σ = v ⋆ ( a − v ⋆ − σ < . Then lim s → + ∞ (cid:26) ln( w ( s )) + Z ss v ( τ ) dτ (cid:27) = lim s → s + ln( w ( s )) (cid:26) v ⋆ ( a − v ⋆ − σ (cid:27) . Hence, if 1 + v ⋆ ( a − v ⋆ − σ < , we find lim s → + ∞ u ( s ) = + ∞ . In the case that 1 + v ⋆ ( a − v ⋆ − σ > , we get lim s → + ∞ u ( s ) = 0 . (cid:3) The solutions constructed in Theorem 3.10 gives us solutions of Types A1 and A4.
Theorem 3.14.
Let a < , σ < σ ⋆ , v < − v ⋆ , where v ⋆ and σ ⋆ are defined in (19) – (20) , and w ⋆ defined in Proposition 3.10. There exists u ( t, x ) = u ( s ) and S ( t, x ) = S ( s ) , with s = x − σt ,for any σ > , which is a t raveling waves solutions to (2) , with initial data ( u , S ) , verifyingthe following: (1) If u S > w ⋆ , then u and S are of Type A1. (2) If u S ≤ w ⋆ , then u and S are of Type A4. ELLER–SEGEL TYPE WITH LOGARITHMIC SENSITIVITY 19
Solitons for both density and chemoattractant.
Note that the first part of Theorem3.13 corresponds to the existence of soliton-type patterns for both the density and the chemoat-tractant. Here, we will concrete the study of such solutions and, in particular, focus on thebehavior of the derivative of u .Consider a > v > v ⋆ and w > w ⋆ , where v ⋆ is defined in (19) and w ⋆ is defined in Propo-sition 3.5. Note that, as it was pointed out in Proposition 2.1, we can recover the informationon the original variables ( u, S ) solutions of (8)–(9), by taking s ∈ R , u > S > S ′ ∈ R and solving (13)–(14) with initial data w ( s ) = u S , v ( s ) = S ′ S , where S ( s ) = S exp( R ss v ( δ ) dδ )and u ( s ) = w ( s ) S ( s ).Then, as a consequence of Theorem 3.13 we deduce the existence of( u, S ) : ( s − , s + ) → (0 , + ∞ )with −∞ < s − < s + < + ∞ and such that u ( s − ) = S ( s − ) = 0 = u ( s + ) = S ( s + ) . Thus, by using that S is a solution of a boundary problem, the strong maximum principle assuresus that S ′ ( s − ) > , S ′ ( s + ) < . On the other hand, we can use (11) with Φ( s ) = s to write u ′ = (cid:18) a S ′ S − σ (cid:19) u. Integrating this equation we find u ( s ) = u (cid:18) S ( s ) S (cid:19) a e − σ ( s − s ) , (28)with u = u ( s ). Note that the term e − σ ( s − s ) causes a lateral displacement of u with respectto S , as can be seen in Figure 6.Hence, thanks to (28), we deduce the following assertions: • If 0 < a <
1, then u ′ ( s − ) = ∞ and u ′ ( s + ) = −∞ ; • If a = 1, then u ′ ( s − ) > u ′ ( s + ) < • If a >
1, then u ′ ( s − ) = u ′ ( s + ) = 0 , which is reproduced in the following Figure 6. ss − s + Case 0 < a < ss − s + Case a = 1. ss − s + Case a > uS Figure 6.
This figure represents the different patterns, depending on the pa-rameter a , of the soliton type for both the density and the chemoattractant. Relativistic flux saturated operators with logarithmic sensitivity
In this section we are going to obtain soliton-type traveling waves for the flux–saturatedequation ∂ t u = ∂ x µ u∂ x u q u + µ c ( ∂ x u ) − a ∂ x SS u , γ∂ xx S − λS + u. (29)Here, we shall focus on this system, which is the logarithmic Keller–Segel model associated withthe relativistic heat equation. However, most of the results and techniques used in this case canbe extended to other flux saturated mechanisms under certain conditions that will be pointedout through this section. The advantage of this particular case is that most of the calculationscan be done explicitly.System (29) fits the case of the general equation (2), whereΦ( s ) = µ s q µ c s . (30)Note that the flux is bounded and then it is in particular a bounded sublinear function, that is,it satisfies (H2) . By virtue of Propostion 2.1, the existence of traveling waves can be achievedby analyzing the following system (cid:26) w ′ = w ( g ( av − σ ) − v ) ,v ′ = λγ − v − γ w. (31)where g = Φ − . In our case g : ( − c, c ) → R is defined as g ( y ) = cyµ √ c − y . Thus, (31) is definedfor w > σ − ca < v < σ + ca . Note that in the search of the fixed points for (31), one obtains that the equation λ − γv − w = 0could have one, two or three different solutions. That gives rise to a very varied casuistry.4.1. Moving solutions with flux saturated mechanisms.
This section aims to study theexistence of some special patterns for (29) as well as its dynamic properties. We shall focus onthe most relevant shapes for us that are the solutions of soliton types.In the following theorem, we show the existence of ( w, v ) solutions of (31) joining the boundarypoints σ − ca and σ + ca . Indeed, we achieve singular solutions due to the behavior of w ′ in thosepoints. Theorem 4.1.
For any v ∈ ( σ − ca , σ + ca ) , there exists w ∗ such that every maximal solution of(31) with initial data ( v , w ) is defined on an interval ( s − , s + ) , with −∞ < s − < s + < + ∞ ,for any w > w ∗ . This maximal solution verifies i) v ( s − ) = σ + ca , v ( s + ) = σ − ca , ii) v ∈ C [ s − , s + ] and v ′ ( s ) < , iii) w ( s − ) , and w ( s + ) ∈ (0 , + ∞ ) , iv) w ′ ( s − ) = + ∞ , w ′ ( s + ) = −∞ . The type of solutions defined in the statement of Theorem 4.1 can be synthesized in Figure7. To prove Theorem 4.1 we need to introduce a previous result on the properties of the graphsystem associated with (31). Let ( v, w ) be a solution of (31) defined on an interval I . We define w ( s ) = W ( v ( s )), for each s ∈ I . In this way, W verifies W ′ = W g ( av − σ ) − v − v − Wγ + λγ . (32)Let us introduce the following result about the solution W of (32). ELLER–SEGEL TYPE WITH LOGARITHMIC SENSITIVITY 21 sw s - s + w ( s -) w ( s + ) sv s - s + σ + caσ - ca w vw ( s -) w ( s + ) σ − ca σ + ca v w Figure 7.
Graph of the solution described by Theorem 4.1. First, we representthe v component by placing w below. The graph on the right hand side reflectsthe trace in the ( v, w ) plane. Note that the solution has infinite slopes at theends of the interval support. Lemma 4.2.
Let v ∈ ( σ − ca , σ + ca ) . There exists w ∗ for which the solution of (32) satisfying W ( v ) = w is defined in ( σ − ca , σ + ca ) , for any w > w ∗ , and λ < inf (cid:26) W ( v ) : v ∈ (cid:18) σ − ca , σ + ca (cid:19)(cid:27) . (33) Furthermore, the solution of (32) verifies lim v → σ ± ca W ( v ) ∈ (0 , ∞ ) , (34)lim v → σ ± ca W ′ ( v ) = ±∞ . (35) Proof.
Setting Y ( v ) = W ( v ) in (32), we find Y ′ = Y g ( av − σ ) − v γ − Y ( v − λγ ) . (36)We look for ǫ > Y ( v ) of (36) with Y ( v ) = Y , for Y small enough, isdefined in ( σ − ca , σ + ca ) and, furthermore, Y ( v ) < ǫ .Let us check that for each ǫ > δ > Y < ǫ , then Y ( v ) < δ . Setting ε >
0, in a neighborhood of v , which we denote by ˜ I , we have Y ( v ) < ǫ and, therefore, we canobtain (cid:12)(cid:12)(cid:12)(cid:12) Y ′ ( v ) Y ( v ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Y ( v ) f ( v ) γ − ǫα , (37)where f ( v ) = | g ( av − σ ) | + | v | , and α = sup (cid:26) v ( s ) + λγ , v ∈ (cid:18) σ − ca , σ + ca (cid:19)(cid:27) . We choose ǫ such that γ − ǫα >
0. Using that Y ( v ) < ǫ , Gronwall lemma applied to (37)leads to Y ( v ) ≤ Y e ǫ γ − ǫα (cid:12)(cid:12)(cid:12)(cid:12)Z vv f ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12)! ≤ Y e ǫ γ − ǫα k f k L ! , for any v ∈ ˜ I . Let δ < ǫ such that δe ǫ γ − ǫα k f k L < ǫ and Y < δ . Then we have Y ( v ) < ǫ , for v ∈ ˜ I .Finally, a standard argument for the prolongation of solutions in differential equations allowsus to ensure that ˜ I = ( σ − ca , σ + ca ), since δ < ε . As a consequence we obtain that Y ( v ) < δ , forany v ∈ ( σ − ca , σ + ca ).Hence, we can come back to the original function W ( v ) = Y ( v ) , and setting ε < λ , it is enoughto take w ∗ = δ to deduce the statement of the theorem.It remains to check the behavior at the boundary of definition of W , that is, (34) and (35). Byusing (33), we get that W is uniformly positive and the denominator is uniformly negative. Asa consequence, we find (35), taking into account (32). From this it follows that W is monotonenear the extremes and therefore it reaches a global maximum in the interval ( σ − ca , σ + ca ), whichis bounded, obtaining then that (34) holds. (cid:3) With the help of the preceding result we can assemble the proof of Theorem 4.1.
Proof of Theorem 4.1.
Let v ∈ ( σ − ca , σ + ca ), we take w ∗ achieved in Lemma 4.2. Then, weconsider w > w ∗ . The solution of the initial value problem is obtained by integrating thedifferential equation v ′ = − v − W ( v ) γ + λγ , v (0) = v , (38)which is defined in a maximal interval ( s , s ). Now, let us choose w : ( s , s ) → (0 , ∞ ) definedby w ( s ) = W ( v ( s )), where W : ( σ − ca , σ + ca ) → (0 , ∞ ) is given by Lemma 4.2. Using that thisfunction is bounded together with (33), we obtain the existence of two constants 0 < ε < ε such that − ε < v ′ ( s ) < − ε , s ∈ ( s , s ) . (39)From this, we deduce that s and s are finite, and that v is decreasing, having limits at theends of the interval. A prolongation argument provides the first statement i) of the theorem.To prove ii), we just have to check that the limits lim s → s ± v ′ ( s ) exist, which can be deducedfrom (38) and from the asymptotic behavior of W near the extremes of the interval, which havebeen shown in Lemma 4.2.To deduce iii) we just need to keep in mind that w ( s ) = W ( v ( s )) and (34).Finally, using the chain rule and taking into account ii) and (35), we can prove that iv)holds. (cid:3) Remark 4.3.
In this result it has been crucial that f is integrable, which is equivalent to that g is integrable in ( − c, c ) . Therefore, it is possible to use other types of limiters that verify thiscondition of integrability without a modification of the arguments of Theorem 4.1. The following result focuses on the existence of singular solutions joining σ − ca with σ + ca locatedinside the parabola, see Figure 8. Theorem 4.4.
Assume [ σ − ca , σ + ca ] ⊂ ( − v ∗ , v ∗ ) . There exist w ∗ such that any maximal solutionof (29), with initial data ( v , w ) , is defined on an interval ( s − , s + ) , −∞ < s − < s + < + ∞ , forany v ∈ ( σ − ca , σ + ca ) and < w < w ∗ . This solution verifies i) v ( s − ) = σ − ca , v ( s + ) = σ + ca , ii) v ∈ C [ s − , s + ] and v ′ ( s ) > , iii) w ( s − ) and w ( s + ) ∈ (0 , + ∞ ) , iv) w ′ ( s − ) = −∞ , w ′ ( s + ) = ∞ . From the hypotheses of Theorem 4.4 we can deduce that w ( s ) < λ − γv ( s ). Hence, similarlyto Theorema 4.1, we can synthesize the characteristics of the solutions in Figure 8.The proof of Theorem 4.4 is similar to that of Theorem 4.1. Indded, his proof is based on thefollowing result that it is the equivalent of Lemma 4.2. Lemma 4.5.
Assume [ σ − ca , σ + ca ] ⊂ ( − v ∗ , v ∗ ) . There exist w ∗ > such that every solution of(32) that satisfies W ( v ) = w is defined in ( σ − ca , σ + ca ) , for any v ∈ ( σ − ca , σ + ca ) and < w < w ∗ . ELLER–SEGEL TYPE WITH LOGARITHMIC SENSITIVITY 23 sw s - s + sv s - s + σ - caσ + ca w v - v ∗ v ∗ σ - ca σ + ca v w ∗ Figure 8.
Graph of the solution described by Theorem 4.4. In the upper leftfigure we show the v component, and in the one below the corresponding w component. The graph on the right reflects the trace in the ( v, w ) plane. Notethat the solution has infinite slopes at the ends of the interval support. Furthermore W ( v ) < λ − γv , for all v ∈ (cid:18) σ − ca , σ + ca (cid:19) , (40) and lim v → σ ± ca W ( v ) ∈ (0 , ∞ ) , (41)lim v → σ ± ca W ′ ( v ) = ∓∞ . (42) Proof.
The proof of this lemma is very similar to Lemma 4.2, here one must work directly with(32). Let ε >
M > < W < ε , then
M < λγ − Wγ . Setting 0 < δ < ε and0 < w < δ , then the solution of (32) with W ( v ) = w verifies W ( v ) < ε over an interval ˜ I containing v . Using the Gronwall lemma, we obtain W ( v ) < δe (cid:18) M Z vv | f ( s ) | ds (cid:19) ≤ δe (cid:18) k f k M (cid:19) . Assuming δe (cid:18) k f k M (cid:19) < ε = w ∗ , we can deduce that ˜ I = I , and we conclude with the samearguments as those of the Lemma 4.2 proof. Similarly, (41) and (42) are proved in the sameway as their equivalents in Lemma 4.2. (cid:3) Consequences on solutions in the system ( u, S ) . In this subsection, we revert ourresults to the original context of problem (29), which adapts to the case (2). The search fortraveling waves u ( t, x ) = u ( x − σt ) and S ( t, x ) = S ( x − σt ) , leads to Φ (cid:18) u ′ u (cid:19) − a S ′ S + σ = 0 , (43) γS ′′ − λS + u = 0 . (44)where Φ is the map (30). Now we choose s ∈ [ s − , s + ] and S ∈ (0 , ∞ ), then we need to solve (cid:26) S ′ ( s ) = v ( s ) S ( s ) , S ( s ) = S ,u ( s ) = S ( s ) w ( s ) . (45)Recall that we are using v = S ′ S and w = uS .Let us obtain a representation of the solution of (29) in Figure 9. Note that in Figure 9 thecomponent S is logarithmically concave in [ s − , s + ] and has two discontinuities in the second ss - s + uS Figure 9.
Graphs of the solution ( u, S ) of (29) obtained by applying (45) tothe solution of Theorem 4.1. The function u must be continued by zero outside[ s − , s + ] while S has to be continued in a C sense by solutions of the equation γS ′′ − λS = 0.derivative ( S is not a function C ) at s − and s + . About the u component of the solution, weonly know its behavior around the extremes, which presents a saturation front.On the other hand, if c > a q λγ holds, we can use Theorem 4.4, provided that σ < c a q λγ ,obtaining a solution qualitatively similar to Figure 10. ss - s + uS Figure 10.
Graphs of the solution ( u, S ) of (29) obtained by applying (45) tothe solution of Theorem 4.4.Note that here the S component is logarithmically convex in [ s − , s + ] and also has two sin-gularities in its second derivative ( S is not C ) at s − and s + . Again, we only know from the u component of the solution its behavior around the extremes of the interval. In this case, theshape of the flux saturation is the reverse of the previous one.Finally, let us mention that the previous arguments can be extended to more general fluxsaturated mechanics which are presented in the introduction by assuming that Φ − is integrable.However, there the calculations are less explicit than in this particular case. We refer to Remark4.3 for more details. References [1] J. Adler,
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Discrete Cont Dyn–B , 601–641. Departamento de Matem´atica Aplicada and Research Unit “Modeling Nature” (MNat), Facul-tad de Ciencias. Universidad de Granada, 18071-Granada, Spain
Email address : [email protected] Departamento de Matem´atica Aplicada and Research Unit “Modeling Nature” (MNat), Facul-tad de Ciencias. Universidad de Granada, 18071-Granada, Spain.
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