EXISTENCE OF SOLUTIONS TO CHEMOTAXIS DYNAMICS WITH LOGISTIC SOURCE

. This paper is concerned with a chemotaxis system with nonlinear diﬀusion and logistic growth term f ( b ) = κb − µ | b | α − 1 b with κ > 0, µ > 0 and α > 1 under the no-ﬂux boundary condition. It is shown that there exists a local solution to this system for any L 2 -initial data and that under a stronger assumption on the chemotactic sensitivity there exists a global solution for any L 2 -initial data. The proof is based on the method built by Marinoschi [8].


Introduction and main results.
1.1. Introduction. This paper is a continuation of our previous work in [11] for a chemotaxis model with Lipschitz growth f (b, c) or superlinear growth |b| α−1 b with α ≤ 4 (n = 1), α < 1 + 4 n (n = 2, 3). We shall study the case of logistic growth; more precisely, we consider existence of solutions to the chemotaxis model with logistic source where b 0 ∈ L 2 (Ω), Ω is a bounded domain in R n , n ≤ 3, ∂Ω is C 2 -class, D and K satisfy some conditions which will be given in Section 1.2. We denote by ν and ∂c ∂ν the unit outward normal vector and the normal derivative of c to ∂Ω, respectively. We emphasize that there is no restriction on an upper bound of α. It is also possible to consider the Dirichlet boundary condition for b instead of the no-flux boundary condition. The system (KS) was introduced by Keller and Segel [6] in 1970. The system describes a part of the life cycle of cellular slime molds with chemotaxis. In more detail, slime molds move towards higher concentrations of a chemical substance when they plunge into hunger. Here b(t, x) represents the density of the cell population and c(t, x) shows the concentration of the signal substance at time t and place x. A number of variations of the original Keller-Segel system with logistic source have been studied. Specially in the case that logistic source is added to the original model, i.e., D(b) = b and K(b, c) is constant, global existence has been proved (see, e.g., [1,9,10]). However, to our knowledge there are few results via operator theory when D and K are general forms in (KS) with logistic source. Employing the theory for nonlinear m-accretive operators (see Barbu [3]), Marinoschi [8] succeeded in showing local existence of solutions to (KS) for sufficiently small initial data in the case that f is Lipschitz continuous. The result in [8] was improved in our previous paper [11]. More precisely, we proved in [11] that there exists a local solution for any L 2 -initial data when f is Lipschitz continuous or superlinear. Ardeleanu and Marinoschi also dealt with the superlinear case in [2]. The purpose of this paper is to establish existence of solutions to (KS) with logistic source by the theory for nonlinear m-accretive operators.
1.2. Main results. In this paper we make the following assumption on D and K: To state our results we define the Hilbert spaces H, V 1 and the Banach space V 2 as Moreover, we set the Banach space V as Note that the dual space of V is given by and we have the continuous injections where the first one is compact. Now we define weak solutions to (KS).
In particular, if T > 0 can be taken arbitrarily, then (b, c) is called a global weak solution to (KS).
Remark 1. The definition of weak solutions is different from the one in our previous paper [11]. However, if there is some restriction on the size of α, the definition coincides with the previous one.
The first main theorem asserts local existence in (KS) and reads as follows.
Moreover, the following estimates hold : where The second theorem is concerned with global existence in (KS).
Assume the hypotheses of Theorem 1.2, and suppose that for some M > 0. Then for any b 0 ∈ H = L 2 (Ω), (KS) possesses a global weak solution (b, c). Moreover, for every T > 0 the following estimates hold : where M 5 , M 6 , M 7 , M 8 are positive constants which depend on b 0 H , n, Ω, T .

Remark 2.
In [8] and [11] there is the assumption on D as D(0) = 0. More precisely, we used the following inequality derived by the above assumption: To prove the theorems it suffices to use the inequality which is derived by (3) and D(0) ≥ 0 instead of the above inequality.
This paper is organized as follows. Section 2 gives an abstract formulation for (KS), introduces approximate problems, and provides some basic inequalities. We will prove key lemmas and our main results in Section 3.

Approach and preliminaries.
2.1. Problem approach. As introduced in [11] we rewrite (KS) as an abstract Cauchy problem in V 1 . We define the inner product on V 1 as where the maximal monotone operator A ∆ : D(A ∆ ) ⊂ H → H is defined as It is known that by a simple argument

TOMOMI YOKOTA AND NORIAKI YOSHINO
Then we rewrite (KS) as To show that (15) has a solution we proceed to construct an approximate problem as in [11]. For each N > 0 we define F N : H → H as and consider the Yosida approximation g λ of the nonlinear part: g(b) := |b| α−1 b in the logistic term. More precisely, by defining g : x)), J λ and g λ are given by J λ g := (I + λg) −1 , λ > 0, Then we consider the following approximate problem with N > 0, ε > 0, λ > 0: where A N,ε,λ : Since the mapping b → b − g λ (b) is Lipschitz continuous on H due to the property of the Yosida approximation, we obtain the following existence lemma as in [11].
2.2. Basic inequalities. We next state some known lemmas to prove the theorems. The following two lemmas are stated in [11].
Let Ω be a bounded domain in R n (n ∈ N) with C 2 -boundary. Let b ∈ L 2 (Ω) and c b = (I +A ∆ ) −1 b. Denote by J ε ∆ the resolvent of A ∆ as in (19), i.e., J ε ∆ := (I +εA ∆ ) −1 for each ε > 0. Then c b and J ε ∆ c b belong to H 2 (Ω) with the following estimates: where C R is a positive constant independent of ε.
The following inequality will be used in the estimate for approximate solutions.

Lemma 2.3.
Let Ω be a bounded domain in R n (n ≤ 3) with C 2 -boundary. Then there exist a ∈ (0, 1) and C GN > 0 such that for all u, v ∈ H 1 (Ω), 3. Proof of the main results. We will prove Theorems 1.2 and 1.3. Since the starting points of the proofs are different, we divide this section into two subsections.
3.1. Proof of Theorem 1.2. The following lemma is concerned with time-in-local estimates for approximate solutions and is the main part of this paper.
db N,ε,λ dt where C 1 , C 2 , and C 3 are positive constants which do not depend on N , ε, λ but depend on b 0 L 2 (Ω) .
Proof. We use the notation b instead of b N,ε,λ for simplicity. We first show (20). Testing the first equation in (17) by b and using the assumption D (r) ≥ D 0 give Denoting by K L the Lipschitz constant of (r 1 , r 2 ) → K(r 1 , r 2 )r 1 (see (4)), we observe that We estimate I 1 in the same way as in the proof of [11,Proposition 3.2]. For convenience we will present the detail of the proof. It follows from Lemma 2.3 that for some a ∈ (0, 1), Noting by (16) and Lemma 2.2 that we have Now employing Young's inequality, we consequently obtain 1−a . Next we consider the estimate for I 2 . The definition of the Yosida approximation implies and thus . Combining the estimates for I 1 and I 2 with (24), we infer that where c 2 := max{c 1 , D 0 + κ}. Applying the Gronwall type inequality (see e.g., [5, Theorem 21]) gives > 0. Now fix T 1 ∈ (0, T 0 ). Then (20) holds with We next prove (21) and (22). Integrating (25) and using (20), we infer that for t ∈ [0, T 1 ], Thus we arrive at (21) and (22) with Finally we estimate db dt L (α+1) (0,T1; V ) . Now we rewrite db dt as From (20)-(22), the terms ∆D(b) and κb − µg λ (b) are bounded in V 1 and V 2 , respectively. Moreover, noting that due to the Sobolev inequality, Lemma 2.2 and (20), we see by the general theory for Sobolev spaces (see e.g., [4, p. 291]) that where C S is a Sobolev constant. Thus in view of (5) we infer that db dt is bounded in L (α+1) (0, T 1 ; V ) and hence (23) holds with a positive constant C 3 .
We are now in a position to complete the proof of Theorem 1.2.
Proof of Theorem 1.2. Estimates (21) and (23) enable us to use the Lions-Aubin theorem (see Lions [7, p. 57]). As a consequence, there exist a subnet of (b N,ε,λ ) ε,λ>0 (still denoted by (b N,ε,λ ) ε,λ>0 ) and a function b N ∈ L 2 (0, The m-accretivity of g in L (α+1) (0, T ; V 2 ) entails that and thus by using a similar way to that in [11] we conclude that b N satisfies the equation Recalling the definition (16) and choosing N large enough, we see from (20) db N,ε,λ dt where C 4 , C 5 and C 6 are positive constants which do not depend on N , ε, λ but depend on b 0 L 2 (Ω) .
Proof. We first show (26). Testing the first equation in (17) by b and using the assumption D (r) ≥ D 0 , we obtain It follows from the assumption (10) that . From Lemma 2.2 we have On the other hand, as in the proof of Theorem 1.2, we have (Ω) . The estimates for I 3 and I 4 imply 1 2 where c 4 := 1 4 C 2 D0 + κ. Hence we infer from Gronwall's inequality that b(t) L 2 (Ω) ≤ 2e 2c4t , t ∈ (0, T ).
Thus (26) holds with C 4 := 2e 2c4T . Estimates (27) and (29) can be shown by a similar way as in the proof of Lemma 3.2. We note that the time for which estimates (27) and (29) are valid can be taken one in (26).
We are now in a position to complete the proof of Theorem 1.3.
Proof of Theorem 1.3. In the same way as in the proof of Theorem 1.2, we have the desired weak solution to (KS) on some interval. In view of the proof of Theorem 1.2, since the estimates for the approximate solutions are valid on (0, T ), the approximate solutions converge on (0, T ). Thus we conclude that the weak solution exists globally.