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Existence and uniqueness of solutions for a hyperbolic Keller–Segel equation

In this work we describe a hyperbolic model with cell-cell repulsion with a dynamics in the population of cells. More precisely, we consider a population of cells producing a field (which we call “pressure”) which induces a motion of the cells following the opposite of the gradient. The field indicates the local density of population and we assume that cells try to avoid crowded areas and prefer locally empty spaces which are far away from the carrying capacity. We analyze the well-posed property of the associated Cauchy problem on the real line. Moreover we obtain a convergence result for bounded initial distributions which are positive and stay away from zero uniformly on the real line.


Introduction
In this article we are concerned with the following diffusion equation with logistic source: where χ > 0 is a sensing coefficient and p(t, x) is an external pressure. Model (1) describes the behavior of a population of cells u(t, x) living in a one-dimensional habitat x ∈ R, which undergo a logistic birth and death population dynamics, and in which individual cells follow the gradient of a field p. The constant χ characterizes the response of the cells to the effective gradient p x . In this work we will consider the case where p is itself determined by the state of the population u(t, x) as − σ 2 ∂ xx p(t, x) + p(t, x) = u(t, x), t > 0, x ∈ R.
This corresponds to a scenario in which the field p(t, x) is produced by the cells, diffuses to the whole space with diffusivity σ 2 (for σ > 0), and vanishes at rate one. As a result cells are pushed away from crowded area to emptier region. A similar model has been successfully used in our recent work [21] to describe the motion of cancer cells in a Petri dish in the context of cell co-culture experiments of Pasquier et al. [33]. Pasquier et al. [33] cultivated two types of breast cancer cells to study the transfer of proteins between them in a study of multi-drug resistance. It was observed that the two types of cancer cells form segregated clusters of cells of each kind after a 7-day co-culture experiment. In [21], the authors studied the segregation property of a model similar to (1)- (2), set in a circular domain in two spatial dimensions x ∈ R 2 representing a Petri dish. The study aims at describing the cancer cells motion in a Petri dish [21,33] in the context of a batch culture. The cell population should be regarded as a mono-layer attached to the bottom of the Petri dish covered a large quantity of nutritional liquid (used in the cell culture), which is constantly renewed.
Our model can be included in the family of non-local advection models for cell-cell adhesion and repulsion. As pointed out by many biologists, cell-cell interactions do not only exist in a local scope, but a long-range interaction should be taken into account to guide the mathematical modeling. Armstrong, Painter and Sherratt [1] in their early work proposed a model (APS model) in which a local diffusion is added to the non-local attraction driven by the adhesion forces to describe the phenomenon of cell mixing, full/partial engulfment and complete sorting in the cell sorting problem. Based on the APS model, Murakawa and Togashi [32] thought that the population pressure should come from the cell volume size instead of the linear diffusion. Therefore, the linear diffusion was changed into a nonlinear diffusion in order to capture the sharp fronts and the segregation in cell co-culture. Carrillo et al. [11] recently proposed a new assumption on the adhesion velocity field and their model showed a good agreement in the experiments in the work of Katsunuma et al. [25]. The idea of the long-range attraction and short-range repulsion can also be seen in the work of Leverentz, Topaz and Bernoff [28]. They considered a non-local advection model to study the asymptotic behavior of the solution. By choosing a Morse-type kernel which follows the attractive-repulsive interactions, they found that the solution can asymptotically spread, contract (blow-up), or reach a steady-state. Burger, Fetecau and Huang [8] considered a similar non-local adhesion model with nonlinear diffusion, for which they investigated the well-posedness and proved the existence of a compactly supported, non-constant steady state. Dyson et al. [19] established the local existence of a classical solution for a non-local cell-cell adhesion model in spaces of uniformly continuous functions. For Turing and Turing-Hopf bifurcation due to the non-local effect, we refer to Ducrot et al. [16] and Song et al. [37]. We also refer to Mogliner et al. [30], Eftimie et al. [20], Ducrot and Magal [17], Ducrot and Manceau [18] for more topics on non-local advection equations. For the derivation of such models, we refer to the work of Bellomo et al. [5] and Morale, Capasso and Oelschläger [31].
It can be noticed that, in the limit of slow diffusivity σ → 0 (and under the simplifying assumption that χ = 1), we get u(t, x) ≡ p(t, x) and (1) is equivalent to an equation with porous medium-type diffusion and logistic reaction The propagation dynamics for this kind of equation was first studied, to the extent of our knowledge, by Aronson [2], Atkinson, Reuter and Ridler-Rowe [3], and later by de Pablo and Vázquez [15], in the more general context of nonlinear diffusion u t = (u m ) xx + u(1 − u), with m > 1.
We refer to the monograph of Vázquez [38] for a detailed study of solutions to porous medium equations. The particular relation between the pressure p(t, x) and the density u(t, x) in (2) strongly reminds the celebrated model of chemotaxis studied by Patlak (1953) and Keller and Segel (1970) [34,26,27] (parabolicparabolic Keller-Segel model) and, more specifically, the parabolic-elliptic Keller-Segel model which is derived from the former by a quasi-stationary assumption on the diffusion of the chemical [24]. Indeed Equation (2) can be formally obtained as the quasistatic approximation of the following parabolic equation ε∂ t p(t, x) = χp xx (t, x) + u(t, x) − p(t, x), when ε → 0.
A rigorous derivation of the limit has been achieved in the case of the Keller-Segel model by Carrapatoso and Mischler [10]. We refer to [9,23,35] and the references therein for a mathematical introduction and biological applications. In these models, the field p(t, x) is interpreted as the concentration of a chemical produced by the cells rather than a physical pressure. One of the difficulties in attractive chemotaxis models is that two opposite forces compete to drive the behavior of the equations: the diffusion due to the random motion of cells, on the one hand, and on the other hand the non-local advection due to the attractive chemotaxis; the former tends to regularize and homogenize the solution, while the latter promotes cell aggregation and may lead to the blow-up of the solution in finite time [14,24].
Since the pressure p(t, x) is a non-local function of the density u(t, x) in (2), the spatial derivative appears as a non-local advection term in (1). In fact, our problem (1)-(2) can be rewritten as a transport equation in which the speed of particles is non-local in the density, where Traveling waves for a similar diffusive equation with logistic reaction have been investigated for quite general non-local kernels by Hamel and Henderson [22], who considered the model where K ∈ L p (R) is odd and p ∈ [1, ∞]. Notice that the attractive parabolic-elliptic Keller-Segel model is included in this framework by the particular choice They proved a spreading result for this equation (initially compactly supported solutions to the Cauchy problem propagate to the whole space with constant speed) and explicit bounds on the speed of propagation. Diffusive non-local advection also appears in the context of swarm formation [29]. Pattern formation for a model similar to (7) by Ducrot, Fu and Magal [16]. Let us mention that the inviscid equation (5) has been studied in a periodic cell by Ducrot and Magal [17]. A substantial literature has been produced for conservative systems of interacting particles and their kinetic limit (Balagué et al. [4], Carrillo et al. [12], Bernoff and Topaz [6], Bertozzi, Laurent and Rosado [7], among others). This paper is a part of a set of two papers. Here we study the well-posed character of the Cauchy problem (1)-(2). In a forthcoming paper, we will build on these results to study the propagation dynamics of compactly supported initial conditions and the existence of sharp discontinuous traveling waves for the model (1)- (2).
In this paper we focus on the particular case of (1)-(2) with σ > 0 and χ > 0. The paper is organized as follows. In Section 2, we present our main results. Section 3 is devoted to the well-posedness of the Cauchy problem for system (1)-(2).

Main results
We begin by defining our notion of solution to equation (1).
has a classical solution h(t, x) (i.e. for each x ∈ R fixed, the function t → h(t, x) is in C 1 ([0, T ], R) and satisfies (8)), and for a.e.
whereχ := χ σ 2 . We define weighted space L 1 η (R) as follows Our first result concerns the existence of integrated solutions to (1).
Next we show that the semiflow preserves some properties satisfied by the initial condition, namely the monotony, continuity and continuous differentiability. In the case of a C 1 (R) initial condition, we show that the solution integrated along the characteristics is actually a classical pointwise solution to the original problem (1)-(2).
Proposition 1 (Regularity of solutions). Let u(t, x) be an integrated solution to (1).
1. if u 0 (x) is continuous, then u(t, x) is continuous for each t > 0.
2. if u 0 (x) is monotone, then u(t, x) has the same monotony for each t > 0.
and u is then a classical solution to (1)-(2).
Next we show the long-time behavior of the solutions to (1). Theorem 2.3 (Long-time behavior). Let 0 ≤ u 0 (x) ≤ 1 be a nontrivial non-negative initial condition and u(t, x) be the corresponding integrated solution. Then 0 ≤ u(t, x) ≤ 1 for all t > 0 and x ∈ R. If moreover there exists δ > 0 such that δ ≤ u 0 (x) ≤ 1 then u(t, x) → 1, as t → ∞ and the convergence holds uniformly in x ∈ R.
The case of bounded initial conditions which are not positively bounded from below is more complex. In the case of initial conditions which are compactly supported, we expect that the support will expand to the whole space with constant speed and that the profile of the solution reaches an asymptotic shape (traveling wave). This situation will be investigated in a forthcoming paper.

Well-posedness of the Cauchy problem
In this section we investigate the existence and uniqueness of solutions for the system (8)- (9). The idea to construct a fixed point problem is to consider the two variables w(t, x) = u(t, h(t, x)) and p(t, x) = (ρ u)(t, x).
Before we state the theorem, let us introduce some functional spaces and definitions. We introduce the following weighted L 1 space for any η > 0, as endowed with the norm f L 1 η := η 2 R |f (y)|e −η|y| dy. Then for any η > 0 the space L 1 η (R) is a Banach space and for any 0 < η < η < +∞ we have We will say that a measurable set U ⊂ R is conull if |R\U| = 0, where |A| is the Lebesgue measure of the set A. In what follows we need to work in the space of regular bounded functions on a measurable set U ⊂ R. Let us recall that the space Finally we introduce the fixed point problem which is the key element of our proof of Theorem 2.2. Let τ > 0 and U ⊂ R be a conull set, we introduce the function spaces: and sup Clearly,X τ U is closed in the Banach space C 0 [0, τ ], L ∞ (U)).Ỹ τ is not closed in C 0 [0, τ ], W 1,∞ (R) , however for each K > 0, the setỸ . Moreover for each t ∈ [0, τ ] there exists a measurable set E t ⊂ R such that R\Et 1dx = 0, p n x (t, x) and p x (t, x) are well-defined for any x ∈ E t and lim n→+∞ p n x (t, x) = p x (t, x) for each x ∈ E. Let x, y ∈ E t , we have: Taking the limit n → ∞, we obtain hence p xx L ∞ ≤ K and p ∈Ỹ τ K . Given p ∈Ỹ τ , let h be the solution of the following equation ∂ ∂t h(t, s; x) = −χp x (t, h(t, s; x)), h(s, s; x) = x.
The existence of the solution h is ensured by p ∈Ỹ τ . Moreover, (i) for any x, the mapping t → p x (t, x) is continuous; (ii) the vector field p x (t, x) is Lipschitz continuous with respect to x and the Lipschitz coefficient is uniform with respect to t on [0, τ ]. In particular the image of U by h(t, s; ·) is still conull for any t, s ∈ [0, τ ].
We are now in the position to define the mapping T τ U [u 0 ] to which we aim at applying a fixed-point theorem: Remark 1. In formula (13), the function h must be understood as the solution of (12) where p the argument of the function T τ U [u 0 ](w, p).

Remark 2.
Since we only impose u 0 to be in L ∞ the time of local existence will depend on each value u 0 (x). That is why we are not considering the class of functions L ∞ for w(t, ·). Instead we work in the space L ∞ (U) for w(t, ·).
Our first result is the well-definition of T τ U [u 0 ]. We start with a series technical Lemma.

Therefore, we have
The result follows from Grönwall's inequality and the definition ofỸ τ K . Lemma 3.3 (Continuity properties). Let (w, p) ∈Z τ U be given. Then, the function u(t, x) := w(t, h(0, t; x)), defined for each t ∈ [0, τ ] and a.e. x ∈ R, is a continuous function of time for the L 1 η (R) topology (i.e., the map t → u(t, ·) is continuous in L 1 η (R)). The maps t → (ρ u)(t, ·) and t → (ρ x u)(t, ·) are continuous for the C 0 b (R) topology and moreover (ρ u)(t, ·) ∈ W 2,∞ (R) for all t ∈ [0, τ ]. Proof. Let (w, p) ∈Z τ U be given. We first remark that, since p x is Lipschitz continuous, the function h(t, s; ·) is locally Lipschitz continuous for all t, s ∈ [0, τ ] and therefore h(t, 0; U) is conull. In particular, We divide the rest of the proof in two steps.
Step 1. We show the continuity of t → u(t, ·). Let t ∈ [0, τ ] and ε > 0 be given. For s ∈ [0, τ ], we have: Next we select R > 0 sufficiently large, so that By the density of compactly supported smooth function in Then, we have: Next we estimate (16) and (17) (remark that (15) is a particular case of (17), for s = t), starting with (17). We have Moreover, changing the variable in the integral, we have where we recall that |h x | ≤ e Kχ|t−s| by (14) and s ≤ t + δ 0 . We have shown that for each s ∈ (t − δ 0 , t + δ 0 ), which is our desired estimate for (17) (and therefore for (15)). Next we estimate (16). Let , it is uniformly continuous on this set and there exists δ 1 > 0 such that whenever |t − s| ≤ δ 1 . This finishes our estimate of (16).
Summarizing, we have found δ 1 > 0 such that for all s ∈ [t − δ 1 , t + δ 1 ], the inequality holds. This finishes the proof of the continuity of u(t, ·) in L 1 η (R).

y)dy in the scope of this
Step. We first show that for Next we remark that for x ≤ y we have We deduce that In particular p x (t, ·) is globally Lipschitz continuous and thus p(t, ·) ∈ W 2,∞ (R).
Theorem 3.4 (Local existence and uniqueness of solutions). Let U be conull and u 0 ∈ L ∞ (U) be given. There exists τ > 0 such that T τ U [u 0 ] has a unique fixed point inZ τ . Moreover τ can be chosen as a continuous function τ u 0 L ∞ (U ) of u 0 L ∞ (U ) and the mapping u 0 ∈ L ∞ (U) → (w(t, x), p(t, x)) ∈Z τ is continuous in a neighborhood of u 0 .
Proof. We divide the proof in three steps. Step Let us denote Θ[u] := |u|e |u| , u ∈ R and recall the inequality e u − 1 ≤ |u|e |u| = Θ[u] for all u ∈ R. We have This implies Since χ[u] → 0 as u → 0, the continuity of w 1 is proved.
Step 2. Local stability of a vicinity. We show the stability of the set for any r > 0 and τ > 0 sufficiently small, where K : Let (w, p) ∈ B r , and define κ := (u 0 , ρ u 0 ) Z τ + r. By definition, we have On the one hand by (19) where Θ[u] = |u|e |u| . On the other hand, by (29) (with s = 0), for all t ∈ [0, τ ], Finally by (30), We conclude that for any r > 0 there is τ > 0 sufficiently small so that the inclusion T τ U [u 0 ](B r ) ⊂ B r holds.
Step 3. T τ U [u 0 ] is a contraction. More precisely, we show that T τ U [u 0 ] is contracting for τ sufficiently small.
Let r > 0 be given and τ > 0 be sufficiently small so that B r is left stable by T τ U [u 0 ], and define κ := (u 0 , ρ u 0 ) Z τ + r as in Step 2. Let (w, p) ∈ B r and (w,p) ∈ B r be given, we observe that for any t, s ∈ [0, τ ] and x ∈ U, where we have used the inequality |e u − 1| ≤ |u|e |u| =: Θ[u], ∀u ∈ R. Moreover, we have which yields sup This implies On the other hand, we have In order to estimate the term e , we write where we have used (22). Next we notice that thatp ∈Ỹ τ implies the inequality p xx L ∞ ((0,τ )×R) ≤ K, thus we obtain by a change of variable (recall the Lipschitz continuity ofh by Lemma 3.1) Finally we have We have shown the following estimate on p: Next we estimate the gradient of p. We have: For the need of this computation, let us introduce h − := min h (0, t; x), h(0, t; x) and h + := max h (0, t; x), h(0, t; x) . We have: According to Lemma 3.2 we have then Combining (32), (33) and (34), there exists a mapping τ → L(τ ) with L(τ ) → 0 as τ → 0 such that Thus for τ > 0 sufficiently small we have L(τ ) < 1 in which case T τ U [u 0 ] is a contraction on the complete metric space B r equipped with the topology induced by Z τ . By the Banach contraction principle, there exists then a unique fixed point to T τ U [u 0 ]. Moreover τ can be chosen as a continuous function of u 0 L ∞ (U ) . Finally, the continuous dependency of (w, p) with respect to u 0 is a direct application of the continuous dependency of the fixed point with respect to a parameter [39, Proposition 1.2].
In order to show the semigroup property satisfied by (w, p) and to make the link with the integrated solutions to (1), we need the following technical Lemma.
Proof. We divide the proof in three steps.
Step 1. We prove item (i). Let x ≤ y and t, s ∈ [0, τ ] be given, we first remark that  Next we remark that, with those functions f and g, we have For a given x ∈ R, we have uniformly in t, because of Lebesgue's dominated convergence theorem. Next we remark that, given t ∈ [0, τ ], if x is a Lebesgue point of the function z → u 0 (z)e t 0 1−w(l,z)dl ∈ C 0 ([0, τ ], L ∞ (U)), then g(t;x,y) y−x has a limit as y → x and  The formula (36) can be deduced from the remark h(t, s; x) = h(t, 0; h(0, s; x)), where the right-hand side is differentiable for all x ∈ h(s, 0; U ).
The formula for p x is proven similarly. Item (ii) is proved.
Step 3. We show item (iii). Using the formula for p x established in item (ii), we have therefore p x (t, ·) is differentiable each time x is a Lebesgue point of z → w(t, h(0, t; z)) and we have To finish our statement, we show that there exists U ⊂ U (see the definition of U given in item (i)) such that every x = h(t, 0; x 0 ) with x 0 ∈ U is a Lebesgue point of z → w(t, h(0, t; z)). Indeed, let U be the set given by Lemma A.1 applied to the function w ∈ C 0 ([0, τ ], L ∞ (U )). If x = h(t, 0; x 0 ) we have: Since h(0, t; x) is differentiable for each x ∈ h(t, 0; U ) ⊃ h(t, 0; U ), the right-hand side converges to 0 as y → x when x 0 ∈ U is a Lebesgue point of w(t, ·). Lemma 3.5 is proved.
Unfortunately, the solution (w, p) constructed in Theorem 3.4 does not satisfy a semigroup property. The reason is that, for a semigroup property to hold, the property p(t, x) = R ρ(x − y)w(t, y)dy would have to hold so that the vector (w(t, ·), p(t, ·)) can be taken as an initial condition; however, this is very unlikely in view of Lemma 3.5. In order to continue our construction of the integrated solutions, we first show that the solution can be defined in L ∞ with little modification.
Given u 0 ∈ L ∞ (R), we define the operator induced by the family T τ where T τ R [u 0 ] is obtained by (13) with an initial condition equal to u 0 a.e. and Z τ : The fact that T τ [u 0 ] is well-defined is shown in the following Corollary. Corollary 1 (Well-posedness in L ∞ (R)). Let u 0 ∈ L ∞ (R) be given. Let U and U be two conull set and u U 0 ∈ L ∞ (U) and u U 0 ∈ L ∞ (U ) be such that u 0 = u U 0 = u U 0 almost everywhere. There exists τ = τ ( u 0 L ∞ (R) ) > 0 and a conull set U ⊂ U ∩ U such that the solutions w U ∈ C 0 ([0, τ U ], L ∞ (U)) and w U ∈ C 0 ([0, τ U ], L ∞ (U )) given by Theorem 3.4 coincide for all t ∈ [0, τ U ] ∩ [0, τ U ] and x ∈ U . Moreover we have τ ≥ max(τ U , τ U ).
Proof. Most of the arguments involved in the proof of Corollary 1 are very classical therefore we concentrate on the most important point which is the well-definition of w in L ∞ . The set U ⊂ U ∩ U mentioned in the corollary can be defined as Since the existence time given by Theorem 3.4 depends only on u U 0 L ∞ (U ) , we have τ U ≥ max(τ U , τ U ). Moreover since U ⊂ U it follows from the uniqueness of the fixed point of T τ U [u 0 ] that w U and w U coincide on U , and similarly w U = w U on U . The remaining statements are classical.
We are now equipped with a family of operators T t defined for u ∈ L ∞ (R) and t ∈ [0, τ ( u 0 L ∞ )] as where w and τ ( u 0 L ∞ ) are given by Corollary 1. Next we show that the family T t satisfies a semigroup property. We deduce the existence of a maximal solution for each u 0 ∈ L ∞ (R).
Proof. The positiveness of τ * (u 0 ) is a consequence of Corollary 1. We show the existence of U as defined in the Theorem. Let U 0 := R and letũ 0 ∈ L ∞ (R) be a bounded measurable function on R such that ũ 0 L ∞ (R) = u 0 L ∞ (R) . In the rest of the proof we identify u 0 andũ 0 and consequently drop the tilde. We recursively construct a sequence of conull sets U n , n ∈ R, such that U n+1 ⊂ U n , and a sequence of functions u n 0 ∈ L ∞ (U n ), such that: (i) u n+1 0 (x) := w n (τ n , h n (0, τ n ; x)) where τ n := τ ( u n 0 L ∞ ), (w n , p n ) is the unique fixed point of the operator T τn U n (given by Theorem 3.4) with initial condition u n 0 and h n is the solution of (12) corresponding to p n .
We let U := n∈N U n . Remark that, since each U n is conull, the set U is still conull. Next we show that T τ [u 0 ] has a unique fixed point for each τ ∈ [0, n∈N τ n ).
Let T 0 = 0 and T n := n−1 k=0 τ n+1 , for all t ∈ [T n , T n + 1) we define We show that (w, p) is the unique fixed point of T τ U [u 0 ] for all τ ∈ [0, T ∞ ) by induction. Indeed, the property is a consequence of Theorem 3.4 for all τ ≤ T 1 . Suppose that (w, p) is the unique fixed point of T τ U [u 0 ] for all τ ≤ T n , n ≥ 1. The formula − (1 +χ)w n (l − T n , h(T n , 0; x))dl . (39) Next we remark that, by Lemma 3.5, the formula hold, therefore p(t, x) can be extended to a function p ∈ C 0 ([0, T n+1 ], W 1,∞ (R)) by defining p(t, x) = p n (t−T n , x) when t ≥ T n , and moreover the extended function h(t, s; x) defined on [0, if T n ≤ t, s solves (12). Therefore (39) can be rewritten as: where we have used the function w ∈ C 0 ([0, T n+1 ], L ∞ (U)) defined by the equality w(t, x) := w n (t−T n , h(0, T n ; x)) when t ≥ T n . Finally We have shown that (w, p) is a fixed point of T t U [u 0 ], for all t ≤ T n+1 . Uniqueness follows from the remark: let . Then w andw coincide in [0, T n ] (by uniqueness of the fixed point) therefore w(T n , x) =w(T n , x), w(T n , h(0, T n ; x)) =w(T n , h(0, T n ; x)) and by the uniqueness of the fixed point in the interval [T n , T n+1 ] we conclude w(t, ·) =w(t, ·). The uniqueness is proved. We have shown by induction that T τ U [u 0 ] has a unique fixed point for all τ ∈ [0, T ∞ ]. As a by-product, this is also true for T τ [u 0 ] and therefore T ∞ ≤ τ * (u 0 ).
and take a compact set K ⊂ E δ which is contained in a open set O with finite Lebesgue measure. Then K can be covered by a finite union of the interval in the family Ω µ of intervals I x,ε,µ := (x − ε, x + ε) such that x is a Lebesgue point of w(t, h(t, 0; ·)), I ⊂ O and lim sup n→+∞ Ix,ε,µ |w(t, h(t, 0; z)) − w(t, h n (t, 0; z))|dz ≤ 2µε.
Since O is independent of µ we take the limit µ → 0 to find |K| = 0 and therefore Since δ > 0 arbitrary we have shown that the set of where w(t, h n (t, 0; x)) does not converge to w(t, h(t, 0; x)) is included in n≥0 E 1/n , which is still negligible for the Lebesgue measure. We have shown the convergence of w(t, h n (t, 0; ·)) to w(t, h(t, 0; ·)) almost everywhere in R. The convergence of u n (t, ·) to u(t, ·) in L 1 η (R) is then a consequence of Lebesgue's dominated convergence Theorem.
We are now in the position to link the constructed maximal solution with the integrated solutions to (1).
Next we show that p(t, ·) ∈ W 2,∞ (R) for all t ∈ [0, τ ] and that the inequality sup t∈[0,τ ] p xx (t, ·) L ∞ < +∞ holds. Indeed, take x ≤ y, we have therefore p x is differentiable at each Lebesgue point of u and we have Next, define the solution h to (12). According to Definition 2.1, there exists a conull set U on which t → u(t, h(t, 0; x)) is a classical solution to (9). Therefore, by a direct integration, we have where w(t, x) := u(t, h(t, 0; x)). In particular, w(t, x) ∈ C 0 ([0, τ ], L ∞ (U)). By Lemma A.1, there exists a subset U ⊂ U such that for each x ∈ U and all t ∈ [0, τ ], x is a Lebesgue point of w(t, x). Since h(t, s; ·) is Lipschitz continuous for all t, s ∈ [0, τ ], we have where K is the Lipschitz constant of h(t, 0; ·). Therefore x is a Lebesgue point of u whenever h(0, t; x) is a Lebesgue point of w. In particular, for x ∈ U , p xx (t, h(t, 0; x)) is the derivative of p x and we have In particular, writing we find that the formula h x (t, 0; x) = eχ t 0 w(l,x)−p(l,h(l,0;x))dl holds for all x ∈ U . Therefore τ ] then by definition u is a fixed point of T τ [u 0 ] and we have see in Theorem 3.6 that there exists U ⊂ R conull such that T τ U [u 0 ](w, p) = (w, p) for a p ∈Ỹ τ , with w(t, x) = u(t, h(t, 0; x)). It then follows from Lemma 3.5 that p = ρ u and elementary computation then show that u is indeed a classical solution to (9) for all x ∈ U. This proves Item (ii).
This finishes the proof of Proposition 2.
In particular, the solution constructed in Step 1 and 2 can be extended up to T = +∞. We are now in the position to prove Theorem 2.2.
Existence and uniqueness. The existence and uniqueness of the integrated solution follows directly from Theorem 3.6 (existence and uniqueness of a fixed-point problem) and Proposition 2 (consistency between the fixed-point problem and the integrated solutions).
Continuity. The continuity in the space L 1 η (R) and the continuity of u 0 ∈ L ∞ (R) → T t u 0 ∈ L 1 η (R) have been shown in Theorem 3.6.
Other properties. The semigroup property follows directly from the form of the operator has been shown in Theorem 3.6. The uniform bound when 0 ≤ u 0 (x) ≤ 1 has been shown in Lemma 3.7 and the fact that τ * (u 0 ) = +∞ from the fact that the L ∞ norm of u(t, ·) cannot blow-up in finite time.
This ends the proof of Theorem 2.2.
Next we show that our model preserves certain properties of the initial condition.
Proposition 3 (Properties of the solutions). Let u(t, x) be an integrated solution to (1) and suppose u 0 ∈ L ∞ (R) with 0 ≤ u 0 ≤ 1. Then and u is then a classical solution to (1).
(iii) if u 0 (x) is monotone, then u(t, x) has the same monotony for each t > 0.
Since (t, x) → h(t, s; x) is continuous, the right-hand side is a continuous function. This shows (i).
Let us show (ii). By (i) we have u ∈ C 0 ([0, T ] × R). Thus, the spatial derivative of the vector field of (8) Therefore, the characteristic flow (t, s, x) → h(t, s; Finally we show (iii). We will assume that u 0 (x) is decreasing (the increasing case can be treated with a similar argument). We let w(t, x) := u(t, h(t, x)) where u is the solution to (1) starting from u(t = 0, x) ≡ u 0 (x), and h(t, s; x) be the corresponding characteristic flow, i.e. the solution to (12) with p(t, x) := R ρ(x − z)w(t, h(0, t; z))dz. Our aim is to show that w is a fixed point of the map we stress that h is the characteristic flow corresponding to the "real" solution to (1) and is independent ofw.
As the proof is more involved, we subdivide it in four steps.
Step one: Let r > 0, we show that there exists τ 0 such that the ball is left stable byT τ for 0 < τ ≤ τ 0 . Let w 0 ∈ B r . We compute: where C depends on u 0 L ∞ (R) , r, andχ. The existence of τ 0 is proved.
Step four: We conclude. Let τ := min(τ 0 , τ 1 ) where τ 0 , τ 1 are as in Step 1 and 2. By a direct application of the Banach contraction principle,T τ has a unique fixed point in B r , which is w (since w happens to be a fixed point). Moreover w can be obtained as the limit of the iteration scheme: w 0 (t, x) := u 0 (x), w n+1 (t, x) :=T τ (w n )(t, x).
Since u 0 is nonincreasing andT τ preserves the monotony, it follows that w is nonincreasing (D is closed for the considered topology).
Since τ does not depend on u 0 , the monotony of u(t, ·) for all t > 0 follows from an induction argument.
In particular It is not difficult to see that for each α ∈ (0, 1) there exists T α such that, for all t ≥ T α , we have inf x∈R w(t, x) ≥ αθ.
It is classical that for each q the set F q is negligible for the Lebesgue measure λ, i.e. λ(F q ) = 0. Since the family (F q ) q∈Q∩[0,τ ] is countable, we have λ   q∈Q∩[0,τ ] F q   = 0 therefore the set U := U\ q∈Q∩[0,τ ] F q is conull. Let us show that U is composed of Lebesgue points of w(t, ·). Let x ∈ U and t ∈ [0, τ ], then there exists a sequence of rational numbers t n ∈ Q such that t n → t. By definition of U , x is not in any F tn and therefore x is a Lebesgue point of the functions w(t n , ·) for all n ∈ N. We have:  |w(t n , x + εy) − w(t n , x)|dy + 2 w(t, ·) − w(t n , ·) L ∞ (U ) , therefore the right-hand side is arbitrarily small when ε → 0. We conclude that x is a Lebesgue point of w(t, ·). Lemma A.1 is proved.