ASYMPTOTIC BEHAVIOR OF THE STOCHASTIC KELLER-SEGEL EQUATIONS.

This paper deals with the asymptotic behavior of the solutions of the non-autonomous one-dimensional stochastic Keller-Segel equations defined in a bounded interval with Neumann boundary conditions. We prove the existence and uniqueness of tempered pullback random attractors under certain conditions. We also establish the convergence of the solutions as well as the pullback random attractors of the stochastic equations as the intensity of noise approaches zero.

1. Introduction. In this paper, we investigate the long term dynamics of the nonautonomous stochastic Keller-Segel equations defined in a bounded interval I for t > τ with τ ∈ R: which are supplemented with homogeneous Neumann boundary conditions and appropriate initial conditions. The unknown functions in system (1.1)-(1.2) are u = u(x, t) and ρ = ρ(x, t), a, b and d are fixed positive constants, c : R → R + is a given function, f : R → R is a given nonlinearity. W is a two-sided real-valued Wiener process defined on a probability space and λ > 0 is the intensity of noise. The symbol • in (2.2) indicates that the equation is understood in the sense of Stratonovich's integration. The deterministic version (i.e., λ = 0) of system (1.1)-(1.2) was proposed by Keller and Segel in [26] to model the aggregation process of cellular slime mold by chemical attraction. From biological point of view, u and ρ represent the population density of biological individuals and the concentration of chemical substance, respectively, a is the diffusion rate of u, b is the diffusion rate of ρ, c and d are the degradation and production rates of ρ, respectively. The nonlinear function f in (1.1) is called a sensitivity function that is used to model the response of of cells to chemicals. The term − ∂ ∂x u ∂ ∂x f (ρ) is called a chemotactic term that is used to model the fact that cells are attracted by chemical stimulus. Several interesting nonlinear functions f are extensively investigated in the literature (see, e.g., [28,32,34]) including f (s) = s, s 2 , ln(1 + s), s 1 + s , and s 2 1 + s 2 (1. 3) for s ≥ 0. The deterministic Keller-Segel equations have been studied by many experts, see, e.g., [16,23,24,28,30,31,32,33,34,41]. In particular, the existence of solutions of the equations were investigated in [16,30,41], the blow-up of solutions were examined in [23,24,31], and the global attractors were discussed in [32,33]. However, as far as the authors are aware, there is no result available in the literature regarding the long term dynamics of the stochastic Keller-Segel system given by (1.1)- (1.2). The goal of the present paper is to investigate this problem and establish the existence of tempered pullback random attractors for the stochastic system in an invariant subset of L 2 (I) × H 1 (I). We will also examine the limiting behavior of the solutions of system (1.1)-(1.2) as λ → 0, and prove the convergence of the solutions as well as the pullback random attractors as λ → 0. The main difficulty of the paper lies in how to derive pullback uniform estimates for the solutions. Since system (1.1)-(1.2) is a quasilinear system for the unknown functions u and ρ, it is hard to derive such estimates. We will combine the semigroup method and the energy method to establish the desired a priori uniform estimates for the stochastic system.
Notice that (1.1) is a deterministic equation which is not perturbed by noise. When system (1.1)-(1.2) is supplemented with homogeneous Neumann boundary conditions, by (1.1) we find that where τ is the initial time. This means that the total population of biological individuals is conserved for all t ≥ τ , a fact of significance in both biology and mathematics. If (1.1) is perturbed by white noise, then the solutions of the system do not satisfy (1.4) anymore, which is not consistent with the deterministic system from biological point of view, and also introduces difficulty to derive uniform estimates of the solutions. That is why we do not perturb (1.1) by white noise in this paper. This paper is organized as follows. In the next section, we define a continuous cocycle for the non-autonomous stochastic system (1.1)-(1.2) in an invariant subset of L 2 (I) × H 1 (I). In Section 3, we derive pullback uniform estimates for the solutions which are needed for constructing pullback random absorbing sets. We then prove the existence of pullback random attractors in Section 4, and establish the convergence of the solutions as well as the pullback random attractors as λ → 0 in Section 5.
Hereafter, we use C and C i (i = 1, 2, · · · ,) to denote generic positive constants whose values may change from line to line.
For later purpose, we recall the following Gagliardo-Nirenberg interpolation inequality: for all u : I → R provided the right-hand side of (1.5) is finite.
Note that the space H s (I) is continuously embedded into C(Ī) for s > 1 2 , that is, there exists a positive constant C = C(s, I) such that (1.6) The following Agmon's inequality will also be used in this paper: for some C > 0.

2.
Cocycles for the stochastic Keller-Segel system. In this section, we prove the global existence of solutions for the non-autonomous stochastic Keller-Segel system under certain conditions, and define a continuous cocycle in an invariant subset of L 2 (I) × H 1 (I).
Given τ ∈ R, consider the following one-dimensional stochastic Keller-Segel equations defined in a bounded interval I = (a 1 , b 1 ) for t > τ : and initial conditions where a, b, d and λ are all positive constants.
Throughout this paper, we will assume that f : [0, ∞) → R is a smooth function such that there exist constants α 1 ≥ 0, α 2 ≥ 0 and α > 0 such that for all s ≥ 0, Note that all functions given by (1.3) satisfy condition (2.5).
We will also assume c : R → R + is continuous and bounded, (2.6) where c is the function in (2.2).
To describe the Wiener process W , we introduce the standard Wiener space (Ω, F, P ) where Ω = {ω ∈ C(R, R) : ω(0) = 0}, F is the Borel σ-algebra induced by the compact-open topology of Ω, and P is the Wiener measure on (Ω, F). Then the Wiener process W on (Ω, F, P ) takes the form: W (t, ω) = ω(t) for all t ∈ R and ω ∈ Ω. Denote by θ t : Ω → Ω the transformation Then by [2], (Ω, F, P, {θ t } t∈R ) is a metric dynamical system, and there exists a θ t -invariant set of full measure (which is still denoted by Ω) such that for every ω in that set, Next, we establish the existence and uniqueness of solutions of system (2.1)-(2.4) under (2.5). To that end, we need to transform the stochastic equation (2.2) into a deterministic one parametrized by the sample paths. Let v(x, t) = e −λω(t) ρ(x, t). Then by (2.1)-(2.2) we find that u and v satisfy and initial conditions with v 0 (x) = e −λω(τ ) ρ 0 (x). By the Galerkin method, one can verify that if f satisfies (2.5), then problem (2.7)-(2.10) has a unique local solution for every (u 0 , v 0 ) ∈ L 2 (I) × H 1 (I). More precisely, we have the following lemma.
Proof. The existence of local solutions follows from a standard process by applying the Galerkin method, see, e.g., [32]. The uniqueness and nonnegativity of solutions with nonnegative initial data can be obtained by the arguments of [32]. Since the local solution of problem (2.7)-(2.10) is given by the limit of the measurable solutions in ω of a family of finite-dimensional Galerkin systems, we infer that this local solution of system (2.7)-(2.10) is also measurable in ω ∈ Ω.
In what follows, we prove the local solution of problem (2.7)-(2.10) obtained in Lemma 2.1 is actually defined for all t ≥ τ when initial data are nonnegative. For that purpose, we only need to derive uniform estimates of the solutions on a finite time interval [τ, τ + T ] where the solution is defined. First, by integrating equation which together with the nonnegativity of solutions implies  Proof. For convenience, we write A = −b∂ xx + d with domain D(A) = {v ∈ H 2 (I) : v satisfies (2.9)}. Given θ ≥ 0, let A θ be the fractional power of A. It follows from [32] that D(A θ ) ⊆ H 2θ (I) for θ ≥ 0, which along with (1.6) implies that for every v 1 ∈ L 1 (I) and v 2 ∈ D(A θ ) with θ > 1 4 , (2.12) Note that Note that equation (2.8) can be reformulated as By (2.11) and (2.16) we obtain, for t ∈ [τ, τ + T ], from which the desired estimates follows.
Next, we derive uniform estimates on the component u of the solution (u, v) in L 2 (I).
Proof. By (2.7) we get We now estimate the last term on the right-hand side of (2.19). By (2.5) we get 1 2a e 2λω(t) To estimate the right-hand side of (2.20), we use the following interpolation inequalities from (1.5): L 2 (I) . (2.21) By (1.6) and (2.21), for the second term on the right-hand side of (2.20) we have For convenience, we write Then by Young's inequality, (2.11) and (2.22) we get (2.24) By the process to derive (2.24), we also obtain By (2.19) and (2) we obtain d dt Note that by (1.5), which along with (2.11) and (2.26) yields d dt On the other hand, by (2.8) we get By (2.27) we obtain It follows from (2) and (2.31) that By (2.23) and (2) we find that By (2) and Lemma 2.2 we infer that there exists from which (2.18) follows.
As an immediate consequence of Lemmas 2.1, 2.2 and 2.3 we obtain the global existence of solutions for problem (2.7)-(2.10).
In addition, (u(t), v(t)) is continuous with respect to initial data (u 0 , v 0 ) in ∈ L 2 (I)× H 1 (I) and is measurable with respect to ω ∈ Ω for every t ≥ τ .
As a consequence of Lemma 2.5 and the compactness of Sobolev embedding H 1 (I)×H 2 (I) → L 2 (I)×H 1 (I), we obtain the compactness of the solution operator of problem (2.7)-(2.10).
Let D = {D(τ, ω) ⊆ H : τ ∈ R, ω ∈ Ω} be a family of bounded nonempty subsets of H. Such a family D is called tempered if for every C > 0, τ ∈ R and ω ∈ Ω, We will use D to denote the collection of all tempered families of bounded nonempty subsets of H: 3. Uniform estimates. In this section, we derive uniform estimates for the cocycle Φ defined by (2). These estimates will be used to construct tempered pullback absorbing sets for system (2.1)-(2.4). We start with the uniform estimates on the component v of problem (2.7)-(2.10) in H 1 (I).
This completes the proof.
In other words, K ∈ D. This completes the proof.
Next, we prove the D-pullback asymptotic compactness of Φ in H.
We are now ready to present the main result of this section as given below.  , ω) : τ ∈ R, ω ∈ Ω} ∈ D in H. If, in addition, the function c : R → R + is periodic with period T > 0, then the attractor A is also periodic with period T , i.e., A(τ + T, ω) = A(τ, ω) for all τ ∈ R and ω ∈ Ω.

5.
Convergence of tempered random attractors. In this section, we investigate the limiting behavior of the solutions of the stochastic system (2.1)-(2.4) as the intensity λ of noise approaches zero. We will show that the D-pullback random attractors of the stochastic system converge to that of a deterministic system in terms of the Hausdorff semi-distance in L 2 (I) × H 1 (I) as λ → 0.