BOUNDEDNESS IN PREY-TAXIS SYSTEM WITH ROTATIONAL FLUX TERMS

. This paper investigates prey-taxis system with rotational ﬂux terms (cid:40) u , , , under no-ﬂux boundary conditions in a bounded domain Ω ⊂ R n ( n ≥ 1) with smooth boundary. Here the matrix-valued function S ∈ C 2 (¯Ω × [0 , ∞ ) 2 ; R n × n ) fulﬁlls | S ( x,u,v ) | ≤ S 0 ( v ) (1+ u ) θ ( θ ≥ 0) for all ( x,u,v ) ∈ ¯Ω × [0 , ∞ ) 2 with some nondecreasing function S 0 . It is proved that for nonnegative initial data u 0 ∈ C 0 (Ω) and v 0 ∈ W 1 ,q (Ω) with some q > max { n, 2 } , if one of the following assumptions holds: (i) n = 1, (ii) n ≥ 2 ,θ = 0 and S 0 ( m ) m < 2 √ 3 n (11 n +2) , (iii) θ > 0, then the model possesses a global classical solution that is uniformly bounded. Where m := max {(cid:107) v 0 (cid:107) L ∞ (Ω) , 1 α } .

with some nondecreasing function S 0 . It is proved that for nonnegative initial data u 0 ∈ C 0 (Ω) and v 0 ∈ W 1,q (Ω) with some q > max{n, 2}, if one of the following assumptions holds: (i) n = 1, (ii) n ≥ 2, θ = 0 and S 0 (m)m < 1. Introduction. In this paper, we consider the global existence and boundedness of classical solutions to prey-taxis system with rotational flux terms u t = ∆u − ∇ · (uS(x, u, v)∇v) + uv − ρu, x ∈ Ω, t > 0, v t = ∆v − ξuv + µv(1 − αv), x ∈ Ω, t > 0, (∇u − uS(x, u, v)∇v) · ν = 0, ∇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 bounded domain Ω ⊂ R n (n ≥ 1) with smooth boundary, where ν is the outward unit normal vector field on ∂Ω, the scalar functions u = u(x, t) and v = v(x, t) denote the predator density and the prey population density, respectively. S(x, u, v) is a general chemotactic sensitivity function. When external forces are considered in prey-taxis system, the movement of predators may be affected. Comparing with bacterial chemotaxis, we assume that chemotactic sensitivity S is a tensor. We suppose that ρ, ξ, µ and α are positive constants, u 0 and v 0 are given nonnegative functions, and there exists nondecreasing function S 0 : [0,∞) → R such that S satisfies with some θ ≥ 0. Prey-taxis, the movement of predators towards the area with higher-density of prey population, plays an extremely important part in biological control and ecological balance such as maintaining the pest population below some economic threshold or leading to outbreaks of pest density [9,11]. Karevia and Odell first derived a PDE prey-taxis model to illustrate a population model of spatially heterogeneous predator-prey interactions [5]. The prototypical prey-taxis model can be written as where the term ∇ · (uφ(u, v)∇v) represents prey-taxis with a coefficient φ(u, v) and D is prey diffusion rate. uF (v) stands for the inter-specific interaction, uh(u) and f (v) accounts for the intra-specific interaction. F (v) accounts for the intake rate of predators, h(u) is the predator mortality rate function and f (v) is the prey growth function. The parameter γ > 0 denotes the intrinsic predation rate. Before introducing some results on (1.3), we interpret chemotaxis processes in which the cells are subject to external forces. Based on recent experiments, the movement of cells is not directed toward the concentration of chemical signal, but with a rotational motion. Consequently, the chemotactic sensitivity is a tensor, see [8] for more details. This can be formulated as where S is a matrix-valued function. Li et al. [8] proved that the initial-boundary value problem of (1.4) possesses a global bounded classical solution for n = 2 under the assumption that v 0 L ∞ (Ω) is sufficiently small. In [15], a global generalized solution was constructed for the initial-boundary value problem of (1.4) without any restriction on v 0 L ∞ (Ω) . For the case of n = 2, Winkler [17] showed that these solutions eventually becomes smooth and satisfies (u(·, t), v(·, t)) → (ū 0 , 0) as t → ∞ uniformly with respect to x ∈ Ω. Zhang [18] generalized the result of [8] to general bounded domain. It was shown in [18] that if one of the following assumptions holds: then the initial-boundary value problem of (1.4) admits a unique global classical solution that is uniformly bounded. (1.3) has also been studied by many authors, beginning the work of Lee et al. [6,7] and Ainseba et al. [1]. Recently, Jin and Wang [4] investigated the predatorprey system with prey-taxis in a two-dimensional bounded domain with Neumann boundary condition. They obtained the global boundedness of a unique classical solution. In [13], global classical solution was constructed for two-predator and one-prey models with nonlinear prey-taxis.
Winkler [16] considered a taxis system of the form x ∈ Ω, (1.5) in bounded convex domains Ω ⊂ R n (n ≥ 1). He showed that (1.5) possesses a global weak solution for n ≤ 5 and admits at least one global classical solution for n ≤ 2.
In [3], Jin et al. studied the following nutrient-taxis model with porous medium slow diffusion x ∈ Ω, (1.6) in a bounded domain Ω ⊂ R 3 . They obtained the global existence and uniform boundedness of weak solution to (1.6) for any m > 11 4 − √ 3. Moreover, they discussed the large time behavior of these global bounded solutions.
Inspired by Li et al. [8], we investigate the prey-taxis system with rotational flux terms (1.1). To the best of our knowledge, whether any kind of solution to (1.1) exists globally is an open problem. In this paper, we shall study the global existence and boundedness of classical solutions to (1.1).
Our main result reads as follows. Throughout the sequel, we shall suppose that S satisfies the conditions of Theorem 1.1. (1.2). Suppose that u 0 ∈ C 0 (Ω) and v 0 ∈ W 1,q (Ω) with some q > max{n, 2} are nonnegative functions. If one of the following assumptions is satisfied: then (1.1) possesses a global classical solution that is bounded in Ω × (0, ∞).
Remark 1.1. In contrast to (1.4), we obtain similar results to Zhang [18]. The difference between (1.1) and (1.4) is that (1.1) has uv − ρu, µv(1 − αv). uv indicates the intake of predators, −ρu is the linear degradation of the predator, µv(1 − αv) represents the growth of prey. For reader's convenience, we point out the problems induced by these terms. In contrast to [18], we need to deal with K 4 and K 5 in the proof of Lemma 4.1. According to standard parabolic regularity results ([10, Theorem 1.3]), whether uv + ρu satisfies the condition is crucial to the Hölder continuity of u. Remark 1.2. As already observed by Zhang [18], the important part (iii) strongly relies on the fact that unlike in the situation of classical Keller-Segel frameworks, the "signal" evolution in (1.1) is in its essence governed by an absorption process, rather than a production mechanism. The regularization effect of the signal absorption term is already clarified by Tao and Winkler [12], where they considered the oxygen consumption system The authors showed that for arbitrarily large initial data, this problem possesses a global weak solution for which there exists T > 0 such that (u, v) is bounded and smooth in Ω × (T, ∞).
The rest of this paper is organized as follows. In Section 2, we introduce a family of regularized problems and obtain the local existence of the regularized problems. In addition, we establish some preliminary estimates. In Section 3, A refined extensibility criterion is given in Lemma 3.2, it immediately implies the global existence of regularized problem for n = 1. In Section 4, we prove the global existence and boundedness of classical solutions to regularized problem in the highdimensional case. Finally, we give the proof of the Theorem 1.1 in Section 5.
2. Regularized problems. According to the idea from [8], we consider the regularized problems with ρ ε ⊂ C ∞ 0 (Ω) which satisfy 0 ≤ ρ ε ≤ 1 and ρ ε 1 as ε 0 in Ω. All the above approximate problems admit local-in-time smooth solutions. The proof is based on a well-established contraction mapping argument (see [2] for details).
Integrating the first and second equation in (2.1), we obtain Combining with (2.5), we have This yields (2.4).

3.
A refined extensibility criterion. This section is devoted to establishing an improved extensibility criterion which will be convenient for proving the global existence of classical solutions to (2.1). The proof of Lemma 3.1 and Lemma 3.2 is similar to [18], so we omit the details.
Next we can use the above lemma to derive an improved extensibility criterion.

4.
Global boundedness in the approximate problem. In this section, we turn to global boundedness of solutions to (2.1) in the dimension n ≥ 2. To this end, we need to establish a uniform bound for u ε in L p (Ω) with p ≥ 1. The approach pursued in Lemma 4.1 can go back to [14].
Combining Lemma 4.1 with Lemma 3.2, we have the following corollary.
Proof. Testing the second equation of (2.1) by v ε and integrating by parts, we have Integrating the above inequality over (0, T ) gives (5.2). Multiplying the first equation of (2.1) by u ε and integrating by parts, we obtain It follows from the Young's inequality and (2.5) that An integration over (0, T ) yields (5.1). (3.1) implies (5.3).
In view of (5.2) and (5.4), we find C(T ) > 0 such that Finally we can prove Theorem 1.1.