STOCHASTIC CHEMOTAXIS MODEL WITH FRACTIONAL DERIVATIVE DRIVEN BY MULTIPLICATIVE NOISE

We introduce stochastic model of chemotaxis by fractional Derivative generalizing the deterministic Keller Segel model. These models include fluctuations which are important in systems with small particle numbers or close to a critical point. In this work, we study of nonlinear stochastic chemotaxis model with Dirichlet boundary conditions, fractional Derivative and disturbed by multiplicative noise. The required results prove the existence and uniqueness of mild solution to time and space-fractional, for this we use analysis techniques and fractional calculus and semigroup theory, also studying the regularity properties of mild solution for this model.


Introduction
In this study, we consider on the following generalized SKSM with time-space fractional derivative on a  (see [1]). We denote by c D β t the Caputo derivative of order β, which is defined by (see [17]) The rest of the paper is organized as follows. In Section 2, we will introduce some notations and preliminaries, which play a crucial role in our theorem analysis. In Section 3, the existence and uniqueness of mild solution to the problem of time-space fractional (2.1) and in Section 4, the spatial and temporal regularity properties of mild solution to this time-space fractional (2.1) are proved. In Section 5, the existence and uniqueness of mild solution to the problem of time-space fractional (2.6). Finally, the spatial and temporal regularity properties of mild solution to this time-space fractional (2.6) are proved. We use stochastic analysis techniques, fractional calculus and semigroup theory.
Next, we mention some Notations and preliminaries the task at work.

Notations and preliminaries
Denote the basic functional space L p (D), 1 ≤ p < ∞ and H s (D) by the usual Lebesgue and Sobolev spaces, respectively. We assume that A is the negative Laplacian −∆ in a bounded domain D with zero Since the operator A is self-adjoint on H with discrete spectral, i.e., there exists the eigenvectors e n with corresponding eigenvalues λ n such that Ae n = λ n e n , e n = √ 2 sin(nπ), λ n = π 2 n 2 , n ∈ N + .
For any s > 0, letḢ s be the domain of the fractional power A where {W (t)} t≥0 is a Q− Wiener process with linear bounded covariance operator Q such that T r(Q) < ∞.
Further, there exists the eigenvalues λ n and corresponding eigenfunctions e n satisfy Q n = λ n e n , n = 1, 2, ..., then the Wiener process is given by in which {β n } n≥1 is a sequence of real-valued standard Brownian motions. Let L 2 0 = L 2 (Q for any p ≥ 2. We shall also need the following result with respect to the fractional operator A α (see Ref. [18]).
Lemma 2.1. For any α > 0, an analytic semigroup S α (t) = e −tAα , t ≥ 0 is generated by the operator −A α on L p , and for any ν ≥ 0, there exists a constant C αν dependent on α and ν such that in which £(B) denotes the Banach space of all linear bounded operators from B to itself.
Next, we will introduce the following lemma to estimate the stochastic integrals, which contains the Burkholder-Davis-Gundy's inequality.
For any 0 ≤ t 1 < t 2 ≤ T and p ≥ 2, and for any predictable stochastic process Now, we give the following definition of mild solution for our time-space fractional stochastic Keller-Segel model.
C [0, T ];Ḣ ν P -a. e, and it holds, and (2.6) respectefily for a. s. ω ∈ Ω, where the generalized Mittag-Leffler operators E β (t) and E ββ (t) are defined as which contain the Mainardi's Wright-type function with β ∈ (0, 1) given by in which the Mainardi function M β (θ) act as a bridge between the classical integral-order and fractional derivatives of differential equations, for more details see [19,20]. Here, the derivation of mild solution (2.5) and (2.6) can be found in Appendix (7) and Appendix (8) (respectively).

Existence and uniqueness of mild solution
Our main purpose of this section is to prove the existence and uniqueness of mild solution to the problem (2.1). To do this, the following assumptions are imposed.
3.1. Assumption . The measurable function g : Ω × H → L 2 0 satisfies the following globalLipschitz and growth conditions: for all u, v ∈ H.

3.2.
Assumption . Let C, C 1 are a positive real number, then the bounded bilinear operator satisfies the following properties: where C 1 depend a norm the c in L 2 0 (D), and for all u, v, c ∈ L 2 0 (D).
Proof. We fix an ω ∈ Ω and use the standard Picard's iteration argument to prove the existence of mild solution. To begin with, the sequence of stochastic process {u(t)} n≥0 is constructed as The proof will be split into three steps.
Step1 For each n ≥ 0, we show that The application of the Lemma (2.1) gives Applying the following Hölder inequality to the second term of the right-hand side of (3.6) Making use of the Hölder inequality and Lemma (2.2) to the third term of the right-hand side of (3.6), we get Using the above estimates (3.6) and (3.9), we have By means of the extension of Gronwall's lemma, it holds that Step2 Show that the sequence {u n (t)} n≥0 is a Cauchy sequence in the space L p (Ω;Ḣ ν ). For any n ≥ m ≥ 1, applying the similar arguments employed to obtain (3.8) and (3.9), we get A direct application of Gronwall's lemma yields for all T > 0. Taking limits to the stochastic sequence {u n (t)} n≥0 in (3.4) as n → ∞, we finish the proof of the existence of mild solution to (2.1).
Step3 We show the uniqueness of mild solution. Assume u and v are two mild solutions of the problem (2.1), using the similar calculations as in Step 2, we can obtain for all T > 0, which implies that u = v, it follows that the uniqueness of mild solution. Obviously, when ν = 0, the above three steps still work. Thus the proof of Theorem 3.1 is completed.

Regularity of mild solution
In this section, we will prove the spatial and temporal regularity properties of mild solution to time-space fractional SKSM based on the analytic semigroup.
Proof. For any 0 ≤ t ≤ T and 1 ≤ ν < α < 2, we have Using Theorem (2.1), the first term can be estimated by It is easy to know that The application of Theorem (2.1) and Assumptions (3.2), we get By means of Theorem (2.1), Assumptions (3.1) and Lemma (2.2), we can deduce Thus, we conclude the proof of Theorem (4.2) by combining with the estimates (4.2)-(4.6).
Next, we will devote to the temporal regularity of the mild solution.
Taking expectation on the both side of (4.8), and in view of the estimates (4.11)-(4.16), we conclude that }. This completes the proof of Theorem (4.2)

Existence and uniqueness of mild solution
Our main purpose of this section is to prove the existence and uniqueness of mild solution to the problem (2.6). To do this, the following assumptions are imposed.

Assumption.
The measurable function f : Ω × H → L 2 0 satisfies the following global Lipschitz and growth conditions: for all u, v ∈ H.

5.2.
Assumption. Let C, is a positive real number, then the bounded bilinear operator L : L 2 0 (D) → H −1 (D) satisfies the following properties: and for all v, c ∈ L 2 0 (D).

Assumption.
Assume that the initial value c 0 : Ω →Ḣ ν is a F 0 -measurable random variable, it holds that for any 0 ≤ ν < α < 2. Proof. We fix an ω ∈ Ω and use the standard Picard's iteration argument to prove the existence of mild solution. To begin with, the sequence of stochastic process {c n (t)} n≥0 is constructed as The proof will be split into three steps.
Step1 For each n ≥ 0, we show that The application of the Lemma (2.1) gives Applying the following Hölder inequality to the second term of the right-hand side of (5.7) . Making use of the Hölder inequality and Lemma (2.2) to the third term of the right-hand side of (5.7), we get (5.10) Using the above estimates (5.7)-(5.10), we have Step1: Show that the sequence {c n (t)} n≥0 is a Cauchy sequence in the space L p (Ω;Ḣ ν ). For any n ≥ m ≥ 1, applying the similar arguments employed to obtain (5.9) and (5.10), we get in wich (5.12)

}.
A direct application of Gronwall's lemma yields Taking limits to the stochastic sequence {c n (t)} n≥0 in (5.5) as n → ∞, we finish the proof of the existence of mild solution to (2.6).
Step 3: We show the uniqueness of mild solution. Assume c and v are two mild solutions of the problem (2.6), using the similar calculations as in Step 2, we can obtain for all T > 0, which implies that c = v, it follows that the uniqueness of mild solution. Obviously, when ν = 0, the above three steps still work. Thus the proof of Theorem (6.1) is completed.

Regularity of mild solution
In this section, we will prove the spatial and temporal regularity properties of mild solution to time-space fractional SKSM based on the analytic semigroup.
Proof. For any 0 ≤ t ≤ T and 1 ≤ ν < α < 2, we have Using Theorem (2.1), the first term can be estimated by It is easy to know that The application of Theorem (2.1) and Assumptions (5.2), we get (6.5) By means of Theorem (2.1), Assumptions (5.1) and Lemma (2.2), we can deduce Thus, we conclude the proof of Theorem (6.1) by combining with the estimates (6.2)-(6.6).
Next, we will devote to the temporal regularity of the mild solution.

Appendix A
Considering the following abstract formulation of time-space fractional stochastic of equation (2.1) We derive the mild solution to (7.1) by means of Laplace transform, which denoted by . Let λ > 0, and we define thatû Upon Laplace transform, using the formula cD β t u(λ) = λ βû − λ β−1 u 0 . Then applying the Laplace transform to (7.1), we obtain in which I is the identity operator, and S α (t) = e −tAα is an analytic semigroup generated by the operator −A α . We introduce the following one-sided stable probability density function: whose Laplace transform is given by Making use of above expression (7.4), then the terms on the right-hand side of (7.2) can be written as Together with (7.2) and (7.5)-(7.7) helps us to get Now, by means of inverse Laplace transform to (7.8), we have achieved that (7.9) Here, we also introduce the Mainardi's Wright-type function where 0 < β < 1 and θ ∈ (0, ∞). Further, the relationships between the probability density function W β (θ) and Mainardi's Wright-type function M β (θ) are shown that We denote the generalized Mittag-Leffler operators E α (t) and E ββ (t) as Therefore, the equation (7.9) can be written as (7.10) Up to now, we have deduced the mild solution (7.10) to the time-space fractional stochastic equation (2.1).
Making use of above expression (8.4), then the terms on the right-hand side of (8.2) can be written as Here, we also introduce the Mainardi's Wright-type function where 0 < β < 1 and θ ∈ (0, ∞). Further, the relationships between the probability density function W β (θ) and Mainardi's Wright-type function M β (θ) are shown that We denote the generalized Mittag-Leffler operators E α (t) and E ββ (t) as E α (t) = Up to now, we have deduced the mild solution (8.10) to the time-space fractional stochastic equation (2.6).

Conflicts of Interest:
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.