ASYMPTOTIC BEHAVIOR IN A CHEMOTAXIS-GROWTH SYSTEM WITH NONLINEAR PRODUCTION OF SIGNALS

. We consider the chemotaxis-growth system (cid:40) u t = ∆ u − χ ∇ · ( u ∇ v ) + µu (1 − u ) , x ∈ Ω , t > 0 , v t = ∆ v − v + h ( u ) , x ∈ Ω , t > 0 , under no-ﬂux boundary conditions, in a convex bounded domain Ω ⊂ R 3 with smooth boundary, where χ > 0 and µ > 0 are given parameters, and h ( s ) is a prescribed function on [0 , ∞ ). It is shown that under the assumption that 4 | h (cid:48) | < (cid:112) 2 µ − 7 χ 2 , for any given nonnegative u 0 ∈ C 0 (¯Ω) and v 0 ∈ W 1 , ∞ (Ω) the system possesses a global classical solution which is bounded in Ω × (0 , ∞ ). Moreover, whenever χ | h (cid:48) | < (cid:112) 8 µ, any bounded classical solution constructed above stabilizes to the constant stationary solution (1 ,h (1)) as the time goes to inﬁnity.

1. Introduction. We consider the initial-boundary value problem x ∈ Ω, t > 0, ∂u ∂ν = ∂v ∂ν = 0, x ∈ ∂Ω, t > 0, u(x, 0) = u 0 (x), v(x, 0) = v 0 (x), x ∈ Ω, (1.1) in a convex bounded domain Ω ⊂ R 3 with smooth boundary, where χ > 0 and µ > 0 are given parameters, and h(s) is a prescribed function on [0, ∞). In (1.1), u denotes the density of a cell population, v stands for the concentration of a chemical substance, and h represents the production of the chemical substance by the cells. The model (1.1) was proposed in [12] for modeling the first steps of tumor-related 466 YUANYUAN LIU AND YOUSHAN TAO angiogenesis, and it was also established in [8] and [6] for describing the pattern formation of bacteria. Depending on various biological processes, the function h takes the following three typical forms: • A linear function h(u) = u, see [4].
Very recently, when the second parabolic equation in (1.1) is replaced by the elliptic counterpart 0 = ∆v−v+h(u), Chaplain and Tello [1] studied the asymptotic behavior of solutions to the corresponding simplified parabolic-elliptic chemotaxis system under the assumption that 2χ|h | < µ.
The approach used in [1] is the sup-and sub-solutions method, which seems not employable for the fully parabolic chemotaxis system (1.1). Therefore, the main purpose of this paper is to extend the result in [1] to a new one for the parabolicparabolic system (1.1) based on a Lyapunov functional technique in [16].
Throughout this paper we assume that the signal production function h is C 1regular fulfilling h(s) ≥ 0, h(0) = 0, and 0 ≤ h (s) ≤ L (1.2) where L > 0 is a constant.
Our first result on the global existence and boundedness reads as follows.
Theorem 1.1. Let Ω ⊂ R 3 be a convex bounded domain with smooth boundary, and suppose that the parameters χ and µ are positive, and assume that h is a prescribed function from C 1 ([0, ∞)) satisfying (1.2). Then whenever for any given nonnegative u 0 ∈ C 0 (Ω) and v 0 ∈ W 1,∞ (Ω) the problem (1.1) possesses a global classical solution (u, v) which is bounded in Ω × (0, ∞) in the sense that there exists C > 0 fulfilling Here we note that the convexity assumption on the domain Ω in Theorem 1.1 can actually be removed (cf. [7] and [3], for instance); however, to shorten our presentation, we refrain us from addressing this issue here.
If µ > 0 is suitably large, then the global classical solution constructed above stabilizes to the homogeneous steady state (1, h(1)). More precisely, we have the following: Let Ω ⊂ R 3 be a convex bounded domain with smooth boundary, and assume that h satisfies (1.2). Then whenever

5)
for any bounded classical solution (u, v) of (1.1) with u 0 ≡ 0 constructed in Theorem 1.1 has the property that We should mention that one can further claim the exponential convergence rate of the solution in Theorem 1.2 by establishing some higher regularity estimates and making full use of the dissipation relation (4.6) below, as done in [16]. However, we refrain from repeating the details here.
Since a strong chemotaxis or high production of chemotactic signal may give rise to the formation of singularity of solutions to (1.1), the condition (1.3) or (1.5) biologically implies that a suitably considerable logistic damping can balance the above-said effects caused by the chemotactic term or the signal production term.
Finally, we remark that all generic constants C and c i (i = 1, 2, · · · ) throughout this paper may depend on |Ω|, u 0 C 0 (Ω) , v 0 W 1,∞ (Ω) , χ, µ and L, but they are independent of t or T max given in Lemma 2.1 below.
2. Preliminaries. The following local existence and extensibility result can be found in the literature ( [19]).
Let Ω ⊂ R 3 be a bounded domain with smooth boundary, let χ and µ be positive, suppose that h ∈ C 1 ([0, ∞)) and assume that u 0 and v 0 are nonnegative functions from C 0 (Ω) and W 1,∞ (Ω) respectively. Then there exist T max ∈ (0, ∞] and a classical solution (u, v) of (1.1) in Ω × (0, T max ) such that and that u and v are nonnegative in Ω × (0, T max ), and such that Some basic properties of any such solution are readily checked.

2)
and in particular we have and t+τ t Ω where τ := min 1, Proof. From and integration of the first equation in (1.1) we immediately obtain (2.2). Since u is nonnegative, and since Ω u 2 ≥ 1 |Ω| ( Ω u) 2 for all t ∈ (0, T max ) by the Cauchy-Schwarz inequality, this implies that On an ODE comparison, this yields (2.3), whereas (2.4) results from a time integration of (2.2).
Later on, we shall need the following auxiliary lemma (cf. [15,Lemma 3.4]). Then With the help of Lemma 2.3, we can derive an elementary estimate on ∇v.
3. Global existence. Proof of Theorem 1.1. In order to prove Theorem 1.1, we first estimate Ω u 2 and Ω |∇v| 4 . We begin with the following.
Proof. Testing the first equation in (1.1) against u we obtain for all t ∈ (0, T max ), which yields (3.1) because for all t ∈ (0, T max ).
To absorb the first integral on the right sides of (3.1) and (3.2), we shall make use of the logistic dampening effects in the first equation in (1.1).
Proof. By a straightforward computation using the first two equations in (1.1), integrating by parts and employing the identity ∇v · ∇∆v = 1 for all t ∈ (0, T max ). Here by (3.4) and nonnegativity of u, and by Young's inequality as well as for all t ∈ (0, T max ). In light of (3.7)-(3.10) the identity (3.6) immediately implies (3.5).
Proof of Theorem 1.1. The first equation in (1.1) can be rewritten in the form where G := ∇v fulfills thanks to Lemma 3.5. Relying on this, noting that µs − µs 2 ≤ µ 4 for all s ∈ R, invoking the known smoothing properties of the Neumann heat semigroup (cf. e.g. [20, Lemma 1.3 (iv)]) and using the maximum principle we can obtain ≤ C for all t > 1 (4.1) with some θ ∈ (0, 1) and C > 0 being independent of t.
Then, the claim for u readily results from [14,Theorem 1.3], whereas the assertion for v can easily be obtained from (4.2) and the Sobolev embedding.
We then focus on constructing a Lyapunov functional which implies some weak convergence information for the solution. As a preparation, we begin with the following differential inequalities.