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Solvability of the Fractional Hyperbolic Keller-Segel System

We study a new nonlocal approach to the mathematical modelling of the Chemotaxis problem, which describes the random motion of a certain population due a substance concentration. Considering the initial-boundary value problem for the fractional hyperbolic Keller-Segel model, we prove the solvability of the problem. The solvability result relies mostly on fractional calculus and kinetic formulation of scalar conservation laws.


Introduction
We introduce and study in this paper the Fractional Hyperbolic Keller-Segel (FHKS for short) model for chemotaxis described by the following system in Ω, ∇K s c · ν = 0, on Γ, (1.1) where u(t, x) is the density of cells and c(t, x) is the chemoattractant concentration, which is responsible for the cell aggregation. The problem is posed in a bounded open subset Ω ⊂ R n , (n = 1, 2, or 3), with C 2 −boundary denoted by Γ, and as usual we denote by ν(r) the outward normal to Ω at r ∈ Γ. The given measurable bounded function u 0 is the initial condition of the cells, and we assume 0 ≤ u 0 (x) ≤ 1, for a.e. x ∈ Ω. (1.2) Moreover, since the normal fractional flux in the equation on u vanishes on Γ, that is, we assume that the boundary is characteristic, it is not necessary to prescribe boundary conditions for u. This assumption besides the natural one, it prevents some specific difficulties related to the trace problem, see [18] for instance.
We also consider the non-local operator K s given by definition in (2.5), which can not be interpreted as the inverse of (−∆ N ) s . Indeed, we are dealing with the spectral fractional Laplacian (−∆ N ) s , which is defined from the eigenvalues of the Laplacian with Neumann boundary conditions, where the first eigenvalue is zero and the corespondent eigenfunction is a constant. Hence the NSFL is not an injective operator in its domain, actually it becomes injective when restrict to the set f ∈ D (−∆ N ) s ; which implies that c satisfies condition (1.3) if, and only if, u satisfies it. Last but not least, we consider (conveniently) the function g(u) = u (1 − u), which prevents blow up. Indeed, from a maximum principle established result, and the assumption (1.2) we are going to show that, 0 ≤ u(t, x) ≤ 1 almost everywhere. Consequently, we could not impose that the chemoattractant concentration c(t, x) satisfies condition (1.3), otherwise u = 0 a.e.. This implies an inherent fractional characterization of the problem studied here.
The theory of chemotaxis modeling goes back to E. F. Keller and L. A. Segel [11,12,13], where a detailed description of the movement of cells oriented by chemical cues can be found. In fact, a nonlocal version of the Keler-Segel model has been proposed by Caffarelli, Vazquez in [3] (Section 8. Comments and Extensions). Clearly, the study of a nonlocal version of the Keler-Segel model becomes interesting as a proposed open problem in that seminal cited paper [3]. But not just because of that, it seems natural to assume that the random motion of a certain population due a substance concentration can be described by Lévy process instead of the Brownian motion, which is in fact a special case of the continuous Lévy process. The fractional model proposed here in (1.1) is a fractional generalization of the model considered in Perthame-Dalibard [19]. Indeed, in that paper they studied the following system in Ω, ∇S · ν = 0, on Γ, (1.6) which follows from the system (1.1), at least formally passing to the limit as s → 0 + . In particular, it is allowed to have jumps in the chemoattractant concentration c(t, x) in the proposed model (1.1), which is not the case for the original model (1.6). This is very important for practical applications.
The mathematical analysis developed in this paper combine two different aspects (fractional calculus and kinetic formulation of scalar conservation laws). The first one is quite new, where well-posedness and several useful estimates are obtained for a fractional parabolic-elliptic system, see Section 3. Then, the rigidity result of Perthame-Dalibard [19] is well adapted in Section 4 to tackle the limiting problem to obtain the proof of the Theorem 1.1 (Main Theorem). Indeed, the non-local description of the chemotaxis model studied here brings new difficulties to the entropy structure of the first equation in (1.1), see (1.7). In particular, because of the non-local space dependence of the entropy flux, which leads to specific difficulties. More precisely, due to the lack of strong compactness or contraction properties, we apply the kinetic formulation associated to the propagation of oscillations of the transport equations (Theorem 4.1). To this end, it is essential to use the renormalization procedure in the kinetic equation, and we are allowed to do that for any s ∈ (0, 1/2]. We would like to remark that the uniqueness property is not addressed in this paper, neither the asymptotic behavior. First, let us stress that no uniqueness result is expected even for the standard system (1.6). On the other hand, longtime behavior was establish to (1.6) in [19], and similarly the main ingredient require here is Ω g(u(t))∇K s c(t) · ∇c(t, x) dx ≥ 0 for each t > 0, which does not seen that such inequality holds. Indeed, we may write where ξ ∈ (min{u, c}, max{u, c}). Formally, the first integral in the right hand side is positive (see Caffarelli, Soria, Vázquez [4] p.1706), although we do not have sign control in the second one. We leave this question for future studies.
Finally, we discuss in Section 5 some related problems that can be handled similarly. In particular, one of the systems considered does not satisfy exactly the condition (1.5), compare it with (5.3).

Statement of the FHKS system
The aim of this section is to formulate the mathematical problem for the FHKS system. We begin observing that, the first equation in (1.1) is a hyperbolic scalar conservation law, thus the density of cells function u may admit shocks. Therefore, in order to select the more correct physical solution, we need an admissible criteria, which is given by the entropy condition.
is called an entropy pair for the first equation in (1.1), if there exists η : R → R a Lipschitz continuous and also convex function and the function q : R → R, which satisfies for almost all u ∈ R. Also, we call η(u) an entropy and q(u)∇K s c the associated entropy flux for the first equation in (1.1).
Analogously to the most important example of the Kružkov's entropies, we consider Another two examples of parameterized family of entropy pairs for (1.1), which will be conveniently used for the kinetic formulation, are given by for each v ∈ R, where |v| ± := max{±v, 0}, and In order to formulate the mathematical problem for the system (1.1), we formally assume enough regularity to the pair (u(t, x), c(t, x)). Then for any entropy η ∈ C 2 , we multiply by η ′ (u) the fist equation in (1.1) to obtain where we have used that, −div∇K s c = u − c. One recalls that, any smooth entropy pair F(u) = (η(u), q(u)) for (1.1) can be recovered by the family of Kružkov's entropies. Therefore, the following definition tells us in which sense a pair of functions (u, c) is a weak solution of FHKS system: (1.1)-(1.2).
Definition 1.2. Given an initial data u 0 ∈ L ∞ (Ω) satisfying (1.2) and any s ∈ (0, 1), a pair of functions is called a weak solution to the FHKS system, if for almost all t > 0, the pair (u, c) satisfies the condition (1.5), and the integral inequality for any fixed v ∈ R, and each nonnegative function φ ∈ C ∞ c (R × R d ) and also the following integral identity holds for any ψ ∈ H 1 (Ω). Now, we are able to state plainly the main result of this paper. Then, we have the following Theorem 1.1 (Main Theorem). Let u 0 ∈ L ∞ (Ω) be an initial data satisfying (1.2) and 0 < s ≤ 1/2. Then, there exists a pair of functions which is a weak solution to the FHKS system, and it satisfies for almost all t > 0 and x ∈ Ω. Remark 1.1. In fact, the condition (1.5) follows from (1.9) with an integration by parts, due to the regularity of the function c(t, ·) ∈ D((−∆ N ) 1−s ) for almost all t > 0. Indeed, the second equation in (1.1) is satisfied almost everywhere, and (1.5) follows integrating it on Ω.

Notation and Functional Spaces
Let Ω be a bounded open set in R n . We denote by dx (or dξ, etc.) the Lebesgue measure, and by H θ the θ−dimensional Hausdorff measure. By L p (Ω) we denote the set of real p−summable functions with respect to the Lebesgue measure, and the vector counterparts of these spaces are denoted by L p (Ω) = L p (Ω) n .
• The space W s,p (Ω) The fractional Sobolev space is denoted by W s,p (Ω), where a real s 0 is the smoothness index, and a real p 1 is the integrability index. More precisely, for s ∈ (0, 1), p ∈ [1, +∞), the fractional Sobolev space of order s with Lebesgue exponent p is defined by Moreover, for s > 1 we write s = m + σ, where m is an integer and σ ∈ (0, 1). In this case, the space W s,p (Ω) consists of those equivalence classes of functions u ∈ W m,p (Ω) whose distributional derivatives D α u, with |α| = m, belong to W σ,p (Ω), that is which is a Banach space with respect to the norm If s = m is an integer, then the space W s,p (Ω) coincides with the Sobolev space W m,p (Ω). Also, it is very interesting the case when p = 2, i.e. W s,2 (Ω). In this case, the (fractional) Sobolev space is also a Hilbert space, and we can consider the inner product where ·, · is the inner product in L 2 (Ω).
• The space H s (Ω) Following Lions, Magenes [15], we can define for s ∈ (0, 1), the spaces H s (Ω) by interpolation between H 1 (Ω) and L 2 (Ω), i.e. According to this definition, this space is a Hilbert space with the natural norm given by the interpolation. We recall that, when Ω has Lipschitz boundary regularity, then the spaces W s,2 (Ω) and H s (Ω) are equivalent.

The NSFL Operator
Here and subsequently Ω ⊂ R n is a bounded open set with C 2 −boundary Γ. Following [9,10], we are interested in fractional powers of a strictly positive selfadjoint operators defined in a domain, which is dense in a (separable) Hilbert space. More precisely, let us denote by (−∆ N ) the operator (−∆) subject to Neumann boundary conditions. One observes that (−∆ N ) is a nonnegative and self-adjoint operator defined in By the spectral theory, there exists a complete orthonormal basis {ϕ k } ∞ k=0 of L 2 (Ω), where ϕ k satisfies the following eigenvalue problem in Ω, Therefore, we have that ϕ k is the eigenfunction corresponding to eigenvalue λ k for each k ≥ 0, where one repeats each eigenvalue λ k according to its (finite) multiplicity: Moreover, it is not difficult to show that, ϕ 0 = 1/ |Ω|, that is a constant value, and Then, we may write Now, applying the functional calculus, we define for each s ∈ (0, 1), the Neumann spectral fractional Laplacian operator, that is given by Moreover, D (−∆ N ) s is a Hilbert space, with the inner product In particular, the norm | · | s is defined by The next proposition give us the main properties of the (−∆ N ) s , and K s operators defined above.
Let Ω ⊂ R n be a bounded open set, s ∈ (0, 1), and consider (−∆ N ) s , and K s the operators defined respectively by (2.3) and (2.5). Then, we have: (2) The operator (−∆ N ) s and K s are self-adjoint.
(4) For any λ > 0, and s ∈ [0, 1], the operator Proof. The proofs of items (1) − (3) follow analogously to Proposition 2.1 in [9], hence we omit them. Let us show item (4), first we note that for any which implies for any λ > 0 Therefore, the linear operator and check that, v ∈ D((−∆ N ) s ) and satisfies the above equation. Therefore, Remark 2.1. 1. One remarks that, for each λ > 0 the operator is invertible. For the extremal case (λ = 0), this assertion is false which is due to the fact that, 2. Thanks to the above observation, we have that (−∆ N ) s is not invertible in its domain. Then, the operator K s could not be seen as the inverse of (−∆ N ) s . However, if we restrict the domain of the fractional Laplacian to a specific subset of D (−∆ N ) s , we obtain the existence of the inverse (−∆ N ) −s .
Let us mention an important result, which help us to show the existence of solutions for the parabolic regularization of the system (1.1).

Proposition 2.2.
Given v ∈ L 2 (Ω), then for all s ∈ (0, 1) Proof. It follows from Proposition 2.1 together with the definition of the K s operator.

The inverse of the restricted NSFL operator
Here, we consider a subset of the domain D (−∆ N ) s , such that there exists the inverse of the NSFL operator when restricted to this set. To this end, let us define for each s ∈ [0, 1) the following set Hence we have the following Proposition 2.3. Under the conditions of Proposition 2.1, it follows that: (1) For all u ∈ H 2s N , there exists α > 0 such that where α is the coercivity constant of (−∆ N ).
Remark 2.2. Applying Proposition 2.3, it follows that for each u ∈ H 0 N the inverse of the NSFL is given by From now own, we write (−∆ N ) −s to denote the inverse of the NSFL operator, whenever this makes sense.
Proposition 2.4. Let (−∆ N ) −s be the operator defined by (2.12) for any fixed s ∈ (0, 1). Then, we have: Proof. The proof proceeds analogously to the Proposition 2.1 in [9], and hence we omit it.

Some auxiliary results
First, we recall that using the language of semigroups, as introduced in [20] (see also [21]), one can check that (−∆ N ) s is indeed a nonlocal operator. In fact, the NSFL is also given by where e t∆N u(x) is the heat diffusion semigroup generated by the Neumann Laplacian acting on u. Now, the aim is to characterize the space D((−∆ N ) s ). To begin, we study D((−∆ N ) 1/2 ), indeed, by using the L 2 normalization and the weak formulation of the equation (2.1), we see that ϕ k 2 H 1 (Ω) = 1 + λ k . It is easy to check that {ϕ k } k∈N0 is also an orthogonal basis of H 1 (Ω). Hence, we find for all u ∈ H 1 (Ω). Similarly, we have the following Let Ω ⊂ R n be a bounded open set. Then for any s ∈ (0, 1), In particular, for each u ∈ H s N , there exist C 1 , C 2 > 0 such that Proof. See Theorem 2.5 in [22], and Lemma 7.1 in [2].
Here and subsequently, we denote for each s ∈ (0, 1) the operator H s = K 1/2 s . Then, we consider the following Proof. The proof follows directly from Proposition 2.1, item (3). Proposition 2.6. Let Ω ⊂ R n be a bounded open set and 0 < s < 1.
2. Now, we show item 2. Since u ∈ H 1 (Ω), it is enough to consider that u ∈ D( −∆ N ) and thus apply a standard density argument. First, we integrate by parts to obtain where we have used the definition of K s u and ∇K s u · ν = 0 on Γ. Due to the fractional Laplacian being self-adjoint, it follows that Therefore, using the equivalence norm (2.13) together with the definition of H s u, we have

On a Perturbed Problem
The aim of this section is to introduce and study the properties of a perturbed system associated to (1.1). More precisely, given ε > 0 we consider the following fractional parabolic-elliptic system in Ω, ∇K s c ε · ν = 0 and ∇u ε · ν = 0, on Γ, where u 0,ε is a regularized initial data, such that Then, we show existence and uniqueness of (u ε , c ε ). To this end, we are going to apply the Banach Fixed Point Theorem to prove the local in time existence of solution to (3.1), and thus applying a contradiction argument we extend it to be global in time. Since (3.1) is a fractional non-standard parabolic-elliptic system we present the proof in details. To begin, we consider the following . Then for each u 0,ε ∈ L 2 (Ω) the fractional parabolic-elliptic system admits a unique weak solution Proof. 1. Givenũ ∈ L ∞ ((0, ∞) × Ω) from (2.8) and Remark 2.1, it is possible to write, for almost all t > 0, the chemoattractant density as follows In particular c ε ∈ L ∞ (0, ∞); D (−∆ N ) 1−s and, for almost all t > 0 and almost everywhere x ∈ Ω, satisfies the equation Moreover, due to Proposition 2.2 we have that where (3.6) follows by the definition of K s together with (3.4). Observe that, K s c ε is an operator which depends onũ, so we define where the last assertion is obtained by (3.5). Then the first equation in (3.3) together with the initial-boundary condition can be written as follows in Ω, Therefore, it is enough to show the existence of a unique solution for (3.7). We claim that, for anyũ ∈ L ∞ ((0, ∞) × Ω), div g(ũ) ∇Lũ ∈ L ∞ ((0, ∞); H −1 (Ω)).
From the proof of the previous lemma, one observes that the fractional parabolic-elliptic system (3.1) can be decoupled. Therefore, equivalently to show existence and uniqueness of solution (u ε , c ε ) for (3.1), we study the following initial-boundary Neumann value problem (3.8)

Local existence of solution
For convenience, let us denote for any T > 0, and also Ω T = (0, T ) × Ω. Then, we consider the following Proof. 1. Hereupon, we denote by B T R the following set where T > 0 and R > 0 are chosen a posteriori. Also we define the mapping T : B T R → W (T ),ũ → u ε = T (ũ) where u ε is the unique solution of (3.7) (for each ε > 0 fixed), that is to say, for eachũ ∈ L ∞ (Ω T ) and t ∈ [0, T ), we may write g(ũ(τ, y)) ∇ y K(t − τ, x, y) · ∇Lũ(τ, y) dy dτ, (3.9) where K(t, x, y) is the heat kernel of (−∆) with Neumann boundary condition, namely (see [7], Theorem 2.1.4) 2. Now, we show that T (ũ) ∈ B T R for eachũ ∈ B T R . Indeed, from (3.9) and applying Hölder inequality, we obtain (3.10) where C, C 1 are positive constants and we have used estimates of heat kernels with Neumann boundary conditions, (see [25], Lemma 3.3).
To follow, from (3.6) let G s (x, y) be the kernel of L, that is to say, for each s ∈ (0, 1), we define Consequently, ∇Lũ could be written as follows for almost everywhere t ∈ (0, T ), x ∈ Ω. Therefore, applying Theorem 6.4 in [1] (see also [2], Proposition 5.2), we have where C 2 is a positive constant. From the above estimate and (3.10), we obtain Therefore, taking R = 6 5 C u 0 L ∞ (Ω) and T = 1 (12C 1 C 2 (R + R 2 )) 2 , it follows that T (ũ) ∈ B T R . 3. Finally, we prove that T is a contraction on B T R . Indeed, we consider u i ∈ B T R , (i = 1, 2), and similarly to item (2) we have Therefore, the mapping T is a contraction, and hence we can apply the Banach Fixed Point Theorem. Thus T has a fixed point, which is by construction the unique solution of (3.8).

Global existence of solution
The main issue of this section is to show that, under the condition (1.2) for the initial data u 0 , we obtain (by contradiction) global in time existence of solution of the problem (3.1). To this end, we show first the uniform boundedness of (u ε , c ε ). Proof. 1. First, we observe that, div (g(u ε )∇Lu ε ) ∈ L 2 (0, T M ); L 2 (Ω) . Therefore, from equation (3.8) and standard parabolic regularity theory, we obtain u ε ∈ C [0, T M ); H 1 (Ω) ∩ L 2 (0, T M ); H 2 (Ω) and ∂ t u ε ∈ L 2 (Ω TM ) .
Consequently, the pair (u ε , c ε ) satisfies the partial differential equations in (3.1) in the strong sense, that is, for almost all (t, x) ∈ Ω TM .

To show (3.15), we multiply the first equation in (3.1) by ϕ
Then, we obtain for each t ∈ (0, T M ) where we have used that 0 ≤ u 0 ≤ 1, and the boundary condition in (3.1). On the other hand, one observes that and passing to the limit as δ → 0 + , we have Thus for a.e. (t, x) ∈ Ω TM , |u ε (t, x) − 1| + = 0 and similarly we show that |u ε (t, x)| − = 0, from which follows (3.15).
3. Now, we recall that Ω ⊂ R n (with n = 1, 2, 3), then by the regularity of c ε together with Morrey's Inequality, we have that c ε ∈ L ∞ 0, T M ; C(Ω) , thus the inf c ε is finite, where the infimum is taking over (0, T M ) ×Ω.
of the problem (3.1), and it satisfies the following uniform bounds Proof of Claim: Indeed, let us suppose that T M < +∞. Therefore, applying Remark 3.1 there exists t j → T − M as j → ∞ such that lim j→∞ u ε (t j , ·) L ∞ (Ω) = +∞, which is a contradiction, thus T M = +∞.

Perturbed problem estimates
The aim of this section is to investigate some important properties of the global solution (u ε , c ε ) of the problem (3.1) as given by Theorem 3.2. Henceforth, we consider that (u ε , c ε ) satisfies the partial differential equations of the problem (3.1) in the strong sense, (see item 1 in the proof of Proposition 3.1).
Lemma 3.2. Let (u ε , c ε ) be the unique solution of the problem (3.1). Then for any entropy η ∈ C 2 , Proof. First, multiplying the first equation in (3.1) by η ′ (u ε ) we obtain On the other hand, one observes that where we have used q ′ (u) = η ′ (u)g ′ (u). Then from the second equation in (3.1), we have which substituting in (3.18) concludes the proof.

Lemma 3.3 (Mass conservation). For almost all
Proof. It is enough to integrate (3.1) (first and second equations) over Ω. where C = C(T, |Ω|, u 0 L ∞ ) is a positive constant.
Proof. Let us multiply (3.1) (first equation) by u ε and integrating by parts over Ω T , we obtain , and integrating by parts we have This, together with (3.20) implies that which proves the assertion.

Proof of Main Theorem
In this section we prove Theorem 1.1, and to this end we are mostly concerned to pass to the limit in (3.17) as ε → 0. More precisely, we write (3.17) in weak sense (using the entropy pair F(u, v)), and jointly with (3.1) (second equation) we obtain (1.8), (1.9) respectively, after pass to the limit as ε → 0. Since (3.17) has non-linear terms, the uniform estimates on the sequence {u ε } are not sufficient to take the limit transition on ε as it goes to 0. In fact, we need strong convergence, and as usual for scalar conservation laws we apply (following closer Chemetov, Neves [6] and also Perthame, Dalibard [19]) the Kinetic Theory.
Therefore, in view of the above results (passing to subsequences still denoted by ε), there exist functions ∇K s c ε → ∇K s c, strongly in L 2 loc (0, ∞); L 2 (Ω) .
The above convergences are enough to pass to the limit (as ε > 0 goes to zero) in the second equation of the system (3.1), that is to say also in equations (4.3) and (4.7), the only exception is the term g ′ (k)∇K s c ε ·∇f ε , which yields an extra effort. First, we can write Moreover, we have in the sense of distributions as ε → 0, Although, from the moment, we cannot assert that the weak limit of u ε f ε is uf . However, we know that {u ε f ε } ε>0 is uniformly bounded in L ∞ ((0, ∞) × Ω × R).
Finally, we take φ = ∂ k ψ in (4.14) and (4.15), with ψ being a nonnegative function in C ∞ 0 (Ω T × R). Then integrating by parts on k, we obtain that f satisfies, respectively, the following transport equations (in distribution sense) and Here ∇ (k,x) = ∂ ∂k , ∇ x and the vector field b : where R := g ′ (k)(ρ − uf ).
Proof. Under the conditions of the vector field b given by (4.21), we can apply the renormalization procedure which means that, the equations (4.18) and (4.19) are regularized on a parameter θ, and respectively multiplied by (1 − f θ ) and f θ , (f θ being the regularization of f ). Then, the obtained equations are added and, taking the limit as θ → 0 we obtain (4.22), where the inequality follows from the following relation in view of (4.16), (4.17). Actually, we omit the details as now it is a standard procedure in the renormalization theory for transport equations. Now, let us study the trace concept of f at time t = 0.
Proposition 4.2. The function f (t, x, k) has the trace f 0 (x, k) at time t = 0, such that Proof. 1. First, let E be a countable dense subset of C 1 0 (Ω). Then, for each ζ ∈ E and k ∈ Q ∩ [0, 1], we define the following set of full measure in (0, T ), where the intersection is taken over E × (Q ∩ [0, 1]). Also E is a set of full measure in (0, T ).
2. To show the existence of the essential limit of f (t, x, k), as t goes to zero, we use the inequalities (4.14), (4.15). Indeed, we consider the test function for any t 0 ∈ E (fixed), and j ≥ 1, where H j is a standard regularization of the Heaviside function, and ψ is a non-negative function which belongs to E. Then, we have from (4.14) Passing to the limit in the above equation as j → ∞, and taking into account that t 0 is Lebesque point of I(t), we obtain where we have used the Dominated Convergence Theorem. Since t 0 ∈ E is arbitrary, and in view of the density of E in L 1 (Ω), it follows from (4.23) that ess lim for all non-negative ψ ∈ L 1 (Ω), which implies for almost everywhere x in Ω, Similarly, we obtain from (4.15) ess lim t→0 for all non-negative ψ ∈ L 1 (Ω), which implies for almost everywhere x in Ω, Therefore, the ess lim t→0 f (t, x, k) exists, and in particular we have almost everywhere in Ω × R, which concludes the proof.
Proof. Follows from Lemma 3.1 in [19]. Now, we are ready to show the strong convergence of the family {u ε }, which is the main issue to prove Theorem 1.1 (Main Theorem). Proof. First, let us recall that R and F are identically zero for k ≤ 0 and k ≥ 1. Now let us define for δ > 0 sufficiently small, the function ζ δ as follows and consider Ψ δ (t, where d(x) = min y∈Γ |x − y| is the distance function from x ∈ Ω to Γ, and also t 0 is a Lebesgue point of the function f 0 − (f 0 ) 2 dk dx = 0. (4.28) For the term I δ 2 one observes that, since Γ is a C 2 −boundary, there exists a sufficiently small δ > 0 such that, each point x ∈ Ω δ := {x ∈ Ω : d(x) < δ} has a unique projection x b = x b (x) on the boundary Γ. For every x ∈ Ω δ , we have where τ = d(x). Therefore, we obtain where we have used that ∇K s c ∈ L 2 (0, T ); H 1 (Ω) , ∇K s c · ν = 0 on Γ, and (x b , τ ) forms an orthogonal coordinate system in a neighborhood of τ = 0.
The above theorem implies that, the kinetic function f takes only the values 0 and 1 almost everywhere in (0, ∞)×Ω×R, and since f is monotone decreasing on k, there exists a function w = w(t, x), such that f (t, x, k) = sgn + (w(t, x) − k).
Therefore for any G ∈ C 1 ([0, 1]), with G(0) = 0, it follows that weakly star in L ∞ (Ω T ), which implies w = u almost everywhere, and the strong convergence of the family {u ε } to u in L p (Ω T ) for any p < ∞. Then, we write (3.17) in weak sense using the entropy pair F(u, v), and jointly with the second equation in (3.1), also written in the weak sense, we derive passing to the limit as ε → 0 that, the pair (u, c) satisfies (1.8) and (1.9), which ends the proof of Theorem 1.1.

Comments and Extensions
One remarks that, the results established in this paper apply to some interesting correlated versions of the system (1.1).
Therefore, we have that ϕ k is the eigenfunction corresponding to eigenvalue µ k , which is given by µ k = λ k + 1 for each k ≥ 0, where the pair (ϕ k , λ k ) is the solution of (2.1). Thus, applying the functional calculus we can define (−∆ N + I d ) s u := ∞ k=0 (λ k + 1) s u, ϕ k ϕ k . Now, we are alloyed to take K s = (−∆ N +I d ) s −1 = (−∆ N +I d ) −s . Similar to the system (1.1), it is not difficult to show that the condition (1.5) is satisfied, that is to say Ω c(t, x) dx = Ω u(t, x) dx.