Global existence and asymptotic stability of solutions to a two-species chemotaxis system with any chemical diffusion

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Global existence and asymptotic stability of solutions to a two-species chemotaxis system with any chemical diffusion Masaaki Mizukami, Tomomi Yokota ∗,1 Department of Mathematics, Tokyo University of Science, Japan Received 28 December 2015; revised 5 April 2016

Abstract This paper deals with the two-species chemotaxis system ⎧ ⎪ ⎨ut = u − ∇ · (uχ1 (w)∇w) + μ1 u(1 − u) in  × (0, ∞), vt = v − ∇ · (vχ2 (w)∇w) + μ2 v(1 − v) in  × (0, ∞), ⎪ ⎩ wt = dw + h(u, v, w) in  × (0, ∞), where  is a bounded domain in Rn with smooth boundary ∂, n ∈ N; h, χi are functions satisfying some conditions. Negreanu–Tello (2014, 2015) [12,13] proved global existence and asymptotic behavior of solutions to the above system when 0 ≤ d < 1. The main purpose of the present paper is to remove the restriction of 0 ≤ d < 1. © 2016 Elsevier Inc. All rights reserved. MSC: primary 35M33; secondary 35A01, 35B35, 92C17 Keywords: Chemotaxis; Global existence; Asymptotic behavior; Stability

* Corresponding author.

E-mail addresses: [email protected] (M. Mizukami), [email protected] (T. Yokota). 1 Partially supported by Grant-in-Aid for Scientific Research (C), No. 25400119.

http://dx.doi.org/10.1016/j.jde.2016.05.008 0022-0396/© 2016 Elsevier Inc. All rights reserved.

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1. Introduction The chemotaxis system describes a part of the life cycle of cellular slime molds with chemotaxis. In more detail, slime molds move towards higher concentration of the chemical substance when they plunge into hunger. After the pioneering works of Keller–Segel [9], a number of variations of the chemotaxis system are proposed and investigated (see e.g., Bellomo–Bellouquid– Tao–Winkler [1], Hillen–Painter [5] and Horstmann [6]). Also, multi-species chemotaxis systems have been studied by e.g., Horstmann [8] and Wolansky [16]. In this paper we focus on a twospecies chemotaxis system which describes a situation in which multi-populations react on single chemoattractant. We especially center upon a model of the behavior of the chemoattractant substance described by a parabolic equation or by an ordinary differential equation. We consider the two-species chemotaxis system ⎧ ⎪ ut = u − ∇ · (uχ1 (w)∇w) + μ1 u(1 − u), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ v = v − ∇ · (vχ2 (w)∇w) + μ2 v(1 − v), ⎪ ⎨ t wt = dw + h(u, v, w), ⎪ ⎪ ⎪ ⎪ ∇u · ν = ∇v · ν = ∇w · ν = 0, ⎪ ⎪ ⎪ ⎪ ⎩u(x, 0) = u (x), v(x, 0) = v (x), w(x, 0) = w (x), 0 0 0

x ∈ , t > 0, x ∈ , t > 0, x ∈ , t > 0,

(1.1)

x ∈ ∂, t > 0, x ∈ ,

where  is a bounded domain in Rn (n ∈ N) with smooth boundary ∂ and ν is the outward normal vector to ∂. The initial data u0 , v0 and w0 are assumed to be nonnegative functions. The unknown functions u(x, t) and v(x, t) represent the population densities of two species and w(x, t) shows the concentration of the substance at place x and time t . In a mathematical view, global existence and behavior of solutions are fundamental theme. However, the problem (1.1) has some difficult points caused by the logistic term and by generalization of χi and h. For example, we cannot use the Lyapunov function. To overcome the difficulty, Negreanu–Tello [12,13] built a technical way to prove global existence and asymptotic behavior of solutions to (1.1). In [13] they dealt with (1.1) when d = 0, μi > 0 under the condition ∃ w ≥ w0 ; h(u, v, w) ≤ 0, where u, v satisfy some representations determined by w. In [12] they studied (1.1) when 0 < d < 1, μi = 0 under similar conditions as in [13] and χi +

1 χ2 ≤ 0 1−d i

(i = 1, 2).

(1.2)

They supposed in [12,13] that the functions h, χi for i = 1, 2 generalize of the prototypical case χ0,i χi (w) = (1+w) σi (χ0,i > 0, σi ≥ 1), h(u, v, w) = u + v − w. As to the special case that d = 1 and h(u, v, w) = u + v − w, Zhang–Li [17] proved global existence of solutions to (1.1) under χ0,i the assumption that μi is small and χi (w) ≤ (1+w) σi for σi > 1, χ0,i > 0 being small enough. The purpose of the present paper is to obtain global existence and asymptotic stability of solutions to (1.1) without the restriction of 0 ≤ d < 1. We shall suppose throughout this paper that h, χi (i = 1, 2) satisfy the following conditions:

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χi ∈ C 1+θ ([0, ∞)) ∩ L1 (0, ∞) (0 < ∃ θ < 1), h ∈ C 1 ([0, ∞) × [0, ∞) × [0, ∞)),

3

(i = 1, 2),

χi > 0

h(0, 0, 0) ≥ 0,

(1.4)

∂h ∂h ∂h (u, v, w) ≥ 0, (u, v, w) ≥ 0, (u, v, w) ≤ −γ , ∂u ∂v ∂w ∃ δ > 0, ∃ M > 0; |h(u, v, w) + δw| ≤ M(u + v + 1),

∃ γ > 0;

∃ ki > 0; −χi (w)h(0, 0, w) ≤ ki We also assume that ∃ p > n;

2dχi (w) +

(1.3)

(i = 1, 2).

(1.5) (1.6) (1.7)

   2 2 (d − 1)p + (d − 1) p + 4dp [χi (w)]2 ≤ 0

(i = 1, 2).

(1.8)

χ

0,i The above conditions cover the prototypical example χi (w) = (1+w) σi (χ0,i > 0, σi > 1), h(u, v, w) = u + v − w. Asymptotic behavior of solutions to (1.1) will be discussed under the following additional condition: There exists τ > 0 such that

1 max Cu (τ )χ1 (0)hu L∞ (Iτ ) , Cv (τ )χ2 (0)hv L∞ (Iτ ) < γ , 2 Cw (τ )2 χi (0)2 < 4μi

(i = 1, 2),

(1.9) (1.10)

where hu := ∂h/∂u, hv := ∂h/∂v, Iτ := (0, Cu (τ )) × (0, Cv (τ )) × (0, Cw (τ )) and Cu (τ ), Cv (τ ), Cw (τ ) will be defined in the proof of Theorem 1.1. We assume that the initial data u0 , v0 , w0 satisfy 0 ≤ u0 ∈ C() \ {0}, 0 ≤ v0 ∈ C() \ {0}, 0 ≤ w0 ∈ W 1,q () (∃ q > n).

(1.11)

Now the main results read as follows. The first theorem is concerned with global existence and boundedness in (1.1). Theorem 1.1. Let d ≥ 0, μi > 0 (i = 1, 2). Assume that h, χi satisfy (1.3)–(1.8). Then for any u0 , v0 , w0 satisfying (1.11) for some q > n, there exists an exactly one pair (u, v, w) of nonnegative functions u, v, w ∈ C( × [0, ∞)) ∩ C 2,1 ( × (0, ∞))

when d > 0,

u, v, w ∈ C([0, ∞); W 1,q ()) ∩ C 1 ((0, ∞); W 1,q ())

when d = 0,

which satisfy (1.1). Moreover, the solution (u, v, w) is uniformly bounded, i.e., there exists a constant C > 0 such that u(t)L∞ () + v(t)L∞ () + w(t)L∞ () ≤ C

for all t ≥ 0.

Remark 1.1. When 0 < d < 1, we note that the condition (1.8) in Theorem 1.1 relaxes (1.2) assumed in [12], because the following relation holds:  (d − 1)p + (d − 1)2 p 2 + 4dp 1 < . 2d 1−d

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Remark 1.2. We excluded the case μi = 0 (i = 1, 2) in Theorem 1.1. It seems to be indefinite whether our proof of boundedness is applicable to such case. The second one gives asymptotic behavior of solutions to (1.1). Theorem 1.2. Let d > 0, μi > 0 (i = 1, 2). Under the conditions (1.3)–(1.8) and (1.9), (1.10), the unique global solution (u, v, w) of (1.1) has the following asymptotic behavior: u(t) − 1L∞ () → 0,

v(t) − 1L∞ () → 0

w(t) − w L∞ () → 0

(t → ∞),

where w ≥ 0 such that h(1, 1, w ) = 0. Remark 1.3. From (1.4)–(1.6) there exists w such that h(1, 1, w ) = 0. Indeed, if we choose w ≥ 3M/δ, then (1.6) yields that h(1, 1, w) ≤ 3M − δw ≤ 0. On the other hand, (1.4) and (1.5) imply that h(1, 1, 0) ≥ h(0, 0, 0) ≥ 0. Hence, by the intermediate value theorem there exists w ≥ 0 such that h(1, 1, w ) = 0. Remark 1.4. Theorem 1.2 shows that the solutions u, v have the same behavior. This theorem is a new result not only in two-species chemotaxis systems but also in chemotaxis-growth systems (single-species case); note that there are not any result about asymptotic behavior of solutions to the fully parabolic system. The difference between one-species and two-species appears only in dealing with w because there are not competitive terms. However the difference is not so big because we can make an estimate for h(u, v, w) by using some conditions. By using a different method we will study the case that the two-species compete with each other, for example, μi u(1 − a1 u − a2 v) in a future work [11]. ´ ´ The strategy for the proof of Theorem 1.1 is to construct estimates for  up and  v p . One of the key for this strategy is to derive inequality d dt

ˆ

−r

u [f1 (w)] p



⎛

ˆ ≤a

−r

u [f1 (w)] p



−b⎝

⎞ p+1

ˆ

p

−r ⎠

u [f1 (w)] p

(1.12)



for some positive constants a, b, where f1 (w) := exp

⎧ w ⎨ˆ

⎫ ⎬ χ1 (s) ds . ⎭

⎩ 0

(p−1)p Negreanu–Tello [12,13] proved a similar inequality for “all” p ≥ 1 and r := p−d(p−1) . In this work we derive (1.12) for “some” p > n and some r = r(d, p) > 0 by modifying the proof in [12,13]. This enables us to improve the previous work and to remove the restriction of 0 ≤ d < 1. On the other hand, the strategy for the proof of Theorem 1.2 is ´to modify´ an argument in [13]. The key for this strategy is to obtain positive lower bounds for  u and  v by modifying an argument in [3]. This paper is organized as follows. In Section 2 we collect basic facts which will be used later. In Section 3 we prove global existence and boundedness (Theorem 1.1). Section 4 is devoted to the proof of asymptotic stability (Theorem 1.2).

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2. Preliminaries In this paper we need the following well-known facts concerning the Laplacian in  supplemented with homogeneous Neumann boundary conditions (for details, see [4,7]). Lemma 2.1. Suppose k > 0. Let  denote the realization of the Laplacian in Ls () with domain {z ∈ W 2,s () | ∇z · ν = 0 on ∂} for s ∈ (1, ∞). Then the operator − + k is sectorial and possesses closed fractional powers (− + k)η , η ∈ (0, 1), with dense domain D((− + k)η ). Moreover, the following holds. (i) If m ∈ {0, 1}, p ∈ [1, ∞] and q ∈ (1, ∞), then there exists a constant c1 > 0 such that for all z ∈ D((− + k)η ), zW m,p () ≤ c1 (− + k)η zLq () , provided that m < 2η and m − n/p < 2η − n/q. (ii) Suppose p ∈ [1, ∞). Then the associated heat semigroup (et )t≥0 maps Lp () into D((− + k)η ) in any of the space Lq (), q ≥ p, and there exist c2 > 0 and λ > 0 such that for all z ∈ Lp (), (− + k)η et (−k) zLq () ≤ c2 t

−η− n2 ( p1 − q1 ) −λt

e

zLp ()

(t > 0).

(iii) Let p ∈ (1, ∞). Then there exists λ > 0 such that for every ε > 0 there exists c3 > 0 such that for all Rn -valued ω ∈ C0∞ (),   (− + k)η et ∇ · ω

1

Lp ()

≤ c3 t −η−ε− 2 e−λt ωLp ()

(t > 0).

(2.1)

Accordingly, the operator (− + k)η et ∇· admits a unique extension to all of Lp () which, again denoted by (− + k)η et ∇·, satisfies (2.1) for all Rn -valued w ∈ Lp (). Lemma 2.2. Let d ≥ 0, μi ≥ 0 (i = 1, 2). Assume that h, χi satisfy (1.3), (1.4), (1.6). Then for any u0 , v0 , w0 satisfying (1.11) for some q > n, there exist Tmax ∈ (0, ∞] and an exactly one pair (u, v, w) of nonnegative functions u, v, w ∈ C( × [0, Tmax )) ∩ C 2,1 ( × (0, Tmax ))

when d > 0,

u, v, w ∈ C([0, Tmax ); W 1,q ()) ∩ C 1 ((0, Tmax ); W 1,q ())

when d = 0,

which satisfy (1.1). Moreover, either Tmax = ∞ or

lim

t→Tmax



 u(t)L∞ () + v(t)L∞ () + w(t)L∞ () = ∞.

Proof. We first consider the case d > 0. The proof of local existence of classical solutions to (1.1) is based on a standard contraction mapping argument, which can be found in [14,15]. The case d = 0 is show in [13]. Finally the maximum principle is applied to yield u > 0, v > 0, w ≥ 0 in  × (0, Tmax ). 2

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3. Global existence and boundedness Let (u, v, w) be the solution to (1.1) on [0, Tmax ) as in Lemma 2.2. We introduce the functions f1 = f1 (w) and f2 = f2 (w) by fi (w) := exp

⎧ w ⎨ˆ

⎫ ⎬ χi (s) ds

⎩

for i = 1, 2

⎭

(3.1)

0

to prove the following lemma. Lemma 3.1. Let d ≥ 0, μi ≥ 0 (i = 1, 2). Assume that χi satisfy (1.3) and (1.8) with some p > n. Then there exists r = r(d, p) > 0 such that d dt

ˆ p

u

f1−r

ˆ ≤ pμ1



d dt

p

u

f1−r (1 − u) − r



ˆ

v

p

f2−r



up f1−r χ1 (w)h(u, v, w),

(3.2)

v p f2−r χ2 (w)h(u, v, w).

(3.3)



ˆ ≤ pμ2

ˆ

v

p

f2−r (1 − v) − r



ˆ



Proof. We let p ≥ 1 be fixed later. From the first and third equations in (1.1) we have d dt

ˆ 

up f1−r = p

ˆ

up−1 f1−r ∇ · (∇u − uχ1 (w)∇w) + pμ1



ˆ

up f1−r χ1 (w)w − r

− rd 

ˆ

ˆ

up f1−r (1 − u)



up f1−r χ1 (w)h(u, v, w).



Denoting by I1 and I2 the first and third terms on the right-hand side as ˆ I1 := p 

up−1 f1−r ∇ · (∇u − uχ1 (w)∇w) , ˆ

I2 := −rd

up f1−r χ1 (w)w,



we can write as ˆ ˆ ˆ d up f1−r = I1 + I2 + pμ1 up f1−r (1 − u) − r up f1−r χ1 (w)h(u, v, w). dt 





We shall show that the following inequality: ∃ p > n, ∃ r > 0 ; I1 + I2 ≤ 0.

(3.4)

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Noting that  f1 ∇

u f1

 = ∇u − uχ1 (w)∇w,

we obtain ˆ

up−1 f1−r ∇

I1 = p 

ˆ 

=p 

p−1

u f1





· f1 ∇

u f1

−r+p−1 f1 ∇

ˆ 

= −p(p − 1) 

u f1

p−2



   u · f1 ∇ f1   2 u  ∇ f  1

−r+p 

f1

ˆ 

− p(−r + p − 1) 

u f1

p−1

−r+p

f1

 χ1 (w)∇

u f1

 · ∇w.

Similarly, we see that ˆ  I2 = −rd 

ˆ  = rdp 

u f1

u f1

ˆ 

+ rd 

p

−r+p

f1

p−1

u f1

χ1 (w)w

−r+p f1 χ1 (w)∇

p

−r+p

f1





u f1

 · ∇w

 (−r + p)[χ1 (w)]2 + χ1 (w) |∇w|2 .

Therefore it follows that I 1 + I2

ˆ 

= −p(p − 1) 

u f1

p−2

  2 u  ∇ f  1

−r+p 

f1

ˆ 

− (p(p − 1) − (1 + d)pr) ˆ  + 

u f1

p





p−1

−r+p

f1

 χ1 (w)∇

u f1

 · ∇w

  −r+p f1 dr(−r + p)[χ1 (w)]2 + drχ1 (w) |∇w|2

ˆ 

= −p(p − 1)

u f1

u f1

p−2

2       u p(p − 1) − (1 + d)pr u  (w) χ + ∇w ∇ 1   f1 2p(p − 1) f1

−r+p 

f1

7

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ˆ  + 

u f1

p

−r+p

f1



  (p(p − 1) − (1 + d)pr)2 + dr(−r + p) [χ1 (w)]2 + drχ1 (w) |∇w|2 . 4p(p − 1)

Here we write as   (p(p − 1) − (1 + d)pr)2 + dr(−r + p) [χ1 (w)]2 + drχ1 (w) 4p(p − 1) =

1 (a1 r 2 + 2a2 r + a3 ), 4p(p − 1)

where a1 , a2 , a3 are given by   a1 := (d − 1)2 p + 4d [χ1 (w)]2 ,   a2 := (p − 1) p(d − 1)[χ1 (w)]2 + 2dχ (w) , a3 := p(p − 1)2 [χ1 (w)]2 . Then there exists p > n such that the discriminant   Dr = 4(p − 1)2 (pχ12 (d − 1) + 2dχ1 )2 − pχ14 ((d − 1)2 + 4d) is nonnegative in view of (1.8). Therefore we have that there exists r > 0 such that I1 + I2 ≤ 0. Hence (3.4) implies ˆ ˆ ˆ d up f1−r ≤ pμ1 up f1−r (1 − u) − r up f1−r χ1 h(u, v, w). dt 





This means that (3.2) holds. In the same way, we obtain (3.3).

2

Lemma 3.2. Let d ≥ 0, μi > 0 (i = 1, 2). Assume that h, χi satisfy (1.3)–(1.5), (1.7), and (1.8) with some positive constants ki (i = 1, 2) and p > n, then    r/p pμ1 + rk1 χ  u(t)Lp () ≤ e 1 L1 (0,∞) max u0 Lp () , ||1/p , (3.5) pμ1     pμ2 + rk2 χ2 L1 (0,∞) r/p 1/p max v0 Lp () , || v(t)Lp () ≤ e . (3.6) pμ2 Proof. From the mean value theorem, the condition (1.5) and the fact that u, v > 0, it follows that for some ξ1 , ξ2 satisfying 0 ≤ ξ1 ≤ u and 0 ≤ ξ2 ≤ v, ∂h ∂h (ξ1 , v, w)u + (0, ξ2 , w)v + h(0, 0, w) ∂u ∂v ≥ h(0, 0, w).

h(u, v, w) =

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This together with the condition (1.7) leads to ˆ ˆ −r up f1−r χ1 (w)h(u, v, w) ≤ −r up f1−r χ1 (w)h(0, 0, w) 



ˆ

≤ k1 r

up f1−r .

(3.7)



Combining (3.2) with (3.7), we obtain ˆ ˆ ˆ d up f1−r ≤ (μ1 p + k1 r) up f1−r − μ1 p up+1 f1−r . dt 





Hence the Hölder inequality gives d dt

ˆ

up f1−r ≤ (μ1 p + k1 r)



ˆ

⎛ up f1−r − μ1 p||−1/p ⎝



ˆ

⎞(p+1)/p up f1−r ⎠

.



Solving this differential inequality, we infer ⎧⎛ ⎫ ⎛ ⎞1/p ⎞1/p ⎪ ⎪ ˆ ⎨ ˆ ⎬ ⎝ up f −r ⎠ ≤ max ⎝ up f −r ⎠ , pμ1 + rk1 ||1/p . 1 0 1 ⎪ ⎪ pμ1 ⎩ ⎭ 



Recalling the definition (3.1), we notice the relation 1 ≤ f1 (w) ≤ e In the same way, we obtain (3.6). 2

χ1 L1 (0,∞)

, which yields (3.5).

Remark 3.1. When d = 0, (3.2), (3.3), (3.5) and (3.6) still hold for all p ≥ 1. Indeed, we have only to choose r = p − 1 in the above proof. Proof of Theorem 1.1. First consider the case d > 0. We let τ ∈ (0, Tmax ). In view of Lemma 2.2 it is sufficient to make sure that u(t)L∞ () ≤ Cu (τ ),

v(t)L∞ () ≤ Cv (τ ),

w(t)L∞ () ≤ Cw (τ ),

holds with some Cu (τ ), Cv (τ ), Cw (τ ) > 0. We let ρ ∈ Writing as



p+n 2p , 1



t ∈ (τ, Tmax )

. This means 1 < 2ρ −

wt = d( − δ/d)w + h(u, v, w) + δw, and applying the variation of constants formula for w, we have ˆt w(t) = e

dt (−δ/d)

w0 +

ed(t−s)(−δ/d) (h(u(s), v(s), w(s)) + δw(s)) ds. 0

n p.

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From Lemma 2.1 and (1.6) we obtain that for all t ∈ (τ, Tmax ),   w(t)W 1,∞ () ≤ c1 (− + δ/d)ρ w(t)Lp () ≤ c1 c2 t −ρ e−λt w0 Lp () ˆt + c1 c 2

(t − s)−ρ e−λ(t−s) h(u(s), v(s), w(s)) + δw(s)Lp () ds

0

≤ c1 c 2 τ

−ρ −λτ

e

ˆt w0 Lp () + c1 c2 c4

(t − s)−ρ e−λ(t−s) ds,

0

where c4 :=



M(u(s)Lp () + v(s)Lp () + 1) (< ∞ by Lemma 3.2). Noting that

sup 0≤s
ˆt

−ρ −λ(t−s)

(t − s)

e

ˆ∞ ds ≤

0

r −ρ e−λr dr < ∞,

0

we deduce that ⎛ w(t)W 1,∞ () ≤ c1 c2 ⎝τ −ρ e−λτ w0 Lp () + c4

ˆ∞

⎞ r −ρ e−λr dr ⎠ =: Cw (τ ).

(3.8)

0

Since (1.8) implies χ1 < 0, it follows from (3.5) and (3.8) that for all t ∈ (τ/2, Tmax ), u(t)χ1 (w(t))∇w(t)Lp () ≤ χ1 (0)u(t)Lp () ∇w(t)L∞ () ≤ χ1 (0)

sup

u(t)Lp () Cw (τ/2) =: c5 .

(3.9)

0≤t
Employing the variation of constants formula for u yields u(t) = e(t−τ/2)(−1) u

τ  2

ˆt −

e(t−s)(−1) ∇ · (u(s)χ1 (w(s))∇w(s)) ds

τ/2

ˆt +

  e(t−s)(−1) (μ1 + 1)u(s) − μ1 u(s)2 ds

τ/2

=: J1 + J2 + J3 ,

t ∈ (τ, Tmax ).

    n 1 Let η ∈ 2p , 2 and ε ∈ 0, 12 − η . Then we observe that 0 < 2η − Lemmas 2.1 and 3.2 we see that for all t ∈ (τ, Tmax ),

n p

and η + ε +

1 2

< 1. By

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  τ    J1 L∞ () = e(t−τ/2)(−1) u  2 L∞ ()   τ    ≤ c1 (− + 1)η e(t−τ/2)(−1) u  2 Lp ()    τ −η −λt   τ  ≤ c1 c2 t − e u  2 2 Lp () ≤ 2η c1 c2 τ −η e−ητ

sup

u(t)Lp () .

0≤t
Using Lemma 2.1 and (3.9), we obtain ˆt J2 L∞ () ≤

e(t−s)(−1) ∇ · (u(s)χ1 (w(s))∇w(s))L∞ () ds

τ/2

ˆt ≤ c1

(− + 1)η e(t−s)(−1) ∇ · (u(s)χ1 (w(s))∇w(s))Lp () ds

τ/2

ˆt

(t − s)−η−ε−1/2 e−(ν+1)(t−s) u(s)χ1 (w(s))∇w(s)Lp () ds

≤ c 1 c3 τ/2

ˆ∞ ≤ c1 c 3 c 5

r −(η+ε+1/2) e−(ν+1)r dr.

0

Since the Neumann heat semigroup (et )t≥0 has the order preserving property, we infer ˆt J3 =

e

(t−s)(−1)

!   μ1 + 1 2 (μ1 + 1)2 + −μ1 u(s) − ds 2μ1 4μ1

τ/2

≤

(μ1 + 1)2 4μ1

ˆt

e(t−s) e−(t−s) ds,

τ/2

and moreover, by the maximum principle we have (μ1 + 1)2 J3 ≤ 4μ1

ˆt

e−(t−s) ds

τ/2

≤

 (μ1 + 1)2  1 − e−τ/2 . 4μ1

11

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Therefore we obtain that there exists Cu (τ ) > 0 such that u(t) ≤ J1 L∞ () + J2 L∞ () + J3 ≤ Cu (τ ),

t ∈ (τ, Tmax ).

The positivity of u yields that u(t)L∞ () ≤ Cu (τ ),

t ∈ (τ, Tmax ).

The same argument as for u gives the L∞ () bound for v. This completes the proof in the case d > 0. Next consider the case d = 0. From Remark 3.1 we have  

(p−1)/p pμ1 + (p − 1)k1 u(t)Lp () ≤ exp χ1 L1 (0,∞) max u0 Lp () , ||1/p pμ1 for all p ≥ 1. Taking the limits as p → ∞, we obtain the L∞ () bound for u, and similarly for v. The L∞ bound for w follows from w(t) = e

−δt

ˆt w0 +

e−δ(t−s) (h(u, v, w) + δw).

0

This completes the proof when d = 0. 2 4. Asymptotic behavior w∗

In this section we prove the asymptotic stability of solutions to (1.1). We denote by u∗ , v ∗ and the functions defined as u∗ (t) :=

1 ||

ˆ u(x, t) dx,

v ∗ (t) :=



1 ||

ˆ v(x, t) dx, 

w ∗ (t) :=

1 ||

ˆ w(x, t) dx. 

(4.1) We also put ∞ :=  × (0, ∞), Tτ :=  × (τ, T ) for τ ∈ [0, ∞), T ∈ (τ, ∞]. Lemma 4.1. Let (u, v, w) be a solution to (1.1). Under the condition (1.9), the following inequality holds: ¨

¨ |∇u|2 +

∞

¨ |∇v|2 +

∞

with some constant C > 0.

|∇w|2 + ∞

¨   u(u − 1)2 + v(v − 1)2 ≤ C ∞

(4.2)

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Proof. Let τ > 0 be a constant fixed later. We multiply the first equation in (1.1) by u − 1 and the second equation in (1.1) by v − 1, and integrate them over Tτ to obtain 1 2

ˆ 

(u(t) − 1)2

T τ

¨ =−



ˆ 1  2

Tτ

(v(t) − 1)2

T τ



uχ1 (w)∇u · ∇w − μ1 Tτ

¨

=−

¨

¨ |∇u|2 +

Tτ

¨

¨

|∇v|2 + Tτ

u(u − 1)2 ,

vχ1 (w)∇v · ∇w − μ2 Tτ

v(v − 1)2 . Tτ

Multiplying the third equation in (1.1) by −w and integrating it over Tτ , we have 1 2

ˆ ˆ ˆ  T 2 2 = −d |w| − h(u, v, w)w |∇w(t)| τ



Tτ

Tτ

ˆ

ˆ

= −d ˆ ≤

|w| +

Tτ

ˆ hu ∇u · ∇w +

2

Tτ

Tτ

hv ∇v · ∇w +

Tτ

ˆ

hu ∇u · ∇w +

ˆ Tτ

ˆ

hv ∇v · ∇w +

Tτ

hw |∇w|2

(−γ )|∇w|2 .

Tτ

Adding this to the above identities multiplied by positive constants λ1 and λ2 (fixed later), we see that ⎛ ⎜ λ1 ⎝

¨

¨ |∇u|2 + μ1

Tτ

⎛

⎜ + λ2 ⎝

⎞ ⎟ u(u − 1)2 ⎠

Tτ

¨

Tτ

¨ |∇v|2 + μ2 ¨

≤

⎞ ⎟ v(v − 1)2 ⎠ + γ

Tτ

¨ |∇w|2 Tτ

¨

(λ1 uχ1 (w) + hu )∇u · ∇w + Tτ

(λ2 vχ2 (w) + hv )∇v · ∇w + O(1)

(4.3)

Tτ

as T → ∞. Since (1.8) implies χ1 < 0, it follows that ˆ 

  (λ1 uχ1 (w) + hu )∇u · ∇w ≤ λ1 Cu (τ )χ1 (0) + hu L∞ (Iτ ) 

λ1 Cu (τ )χ1 (0) + hu L∞ (Iτ ) ≤ 2γ

ˆ |∇u · ∇w| 

2 ˆ

γ |∇u| + 2

ˆ |∇w|2 ,

2





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14

where Iτ := (0, Cu (τ )) × (0, Cv (τ )) × (0, Cw (τ )). In the same way as above, we obtain ˆ

 2 ˆ ˆ λ2 Cv (τ )χ2 (0) + hv L∞ (Iτ ) γ (λ2 vχ2 (w) + hv )∇v · ∇w ≤ |∇v|2 + |∇w|2 . 2γ 2





Therefore we have $



λ1 Cu (τ )χ1 (0) + hu L∞ (Iτ ) λ1 − 2γ $

2 % ¨ |∇u|2 Tτ





λ2 Cv (τ )χ2 (0) + hv L∞ (Iτ ) + λ2 − 2γ

2 % ¨ |∇v|2 ≤ O(1) Tτ

as T → ∞. Here we note that  2 λ1 Cu (τ )χ1 (0) + hu L∞ (Iτ ) λ1 − >0 2γ   ⇐⇒ (Cu (τ )χ1 (0))2 λ21 + 2 Cu (τ )χ1 (0)hu L∞ (Iτ ) − γ λ1 + hu 2L∞ (Iτ ) < 0. Since the discriminant   Dλ1 = 4γ γ − 2Cu (τ )χ1 (0)hu 2L∞ (Iτ ) is positive from the condition (1.9), we can find λ1 > 0 such that 

λ1 Cu (τ )χ1 (0) + hu L∞ (Iτ ) λ1 − 2γ

2 > 0.

The same argument yields that there exists λ2 > 0 such that  2 λ2 Cv (τ )χ2 (0) + hv L∞ (Iτ ) λ2 − > 0. 2γ Therefore (4.4) implies that ¨

¨ |∇u| +

|∇v|2 ≤ C

2

∞

∞

for some C > 0. Thanks to this estimate, we obtain from (4.3) that ¨ ¨ ¨ u(u − 1)2 + v(v − 1)2 + |∇w|2 ≤ C ∞

∞

for some C > 0. This completes the proof. 2

∞

(4.4)

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15

Lemma 4.2. Let (u, v, w) be a solution to (1.1). Under the condition (1.10), the following inequalities hold: ˆ ˆ u(t) ≥ C, v(t) ≥ C, t ∈ (0, ∞), 



with some constant C > 0. Proof. We shall prove that there exist 0 < β < 1, C > 0 and τ > 0 such that ˆ u(t)−β ≤ C (t > τ ).

(4.5)



Let 0 < β < 1 and τ > 0 be fixed later. In view of (3.8) we have that for all t > τ , ˆ ˆ ˆ d u−β = −β(β + 1) u−β−2 |∇u|2 + β(β + 1) u−β−1 χ1 (w)∇u · ∇w dt 

ˆ − μ1 β 

β(β + 1) ≤ 4  ≤β





u−β (1 − u) ˆ

u−β χ1 (w)2 |∇w|2 − μ1 β



β +1 Cw (τ )2 χ1 (0)2 − μ1 4

ˆ

ˆ

u−β (1 − u)



u−β + μ1 β



ˆ u1−β .

(4.6)



By the Hölder inequality we have ˆ

⎛ u(t)1−β ≤ ||1/β ⎝



ˆ

⎞1−β u(t)⎠

,

(4.7)



where the right-hand side is uniformly bounded from the estimate for u in L∞ norm (see Theorem 1.1) or in L1 norm (see [13, Lemma 2.4]). From (1.10) we can choose 0 < β < 1 such that β +1 Cw (τ )2 χ1 (0)2 − μ1 < 0. 4 Hence applying (4.7) to (4.6) gives d dt

ˆ 

u−β ≤ −b1

ˆ

u−β + b2



with some constants b1 , b2 > 0. Solving this differential inequality, we can obtain (4.5). Finally, noting by the Hölder inequality that

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16

⎛ ||(β+1)/β ⎝

ˆ

⎞−1/β u(t)−β ⎠

ˆ ≤



we have a positive lower bound for ´ for  v. 2

´

u

u(t), 

from (4.5). In the same way, we arrive at a lower bound

Lemma 4.3. Let (u, v, w) be a solution to (1.1). Under the conditions (1.9) and (1.10), it holds that u∗ (t) → 1,

v ∗ (t) → 1 as t → ∞.

Proof. Using the same argument as in [13, Lemma 3.3], we see that u∗ (t) → 0 or

u∗ (t) → 1 as t → ∞.

In light of Lemma 4.2 we conclude that u∗ (t) → 1 as t → ∞. Similarly, we have v ∗ (t) → 1 as t → ∞. 2 Lemma 4.4. Let (u, v, w) be a solution to (1.1). Under the conditions (1.9) and (1.10), it holds that u(t) − u∗ (t)L2 () → 0,

v(t) − v ∗ (t)L2 () → 0

as t → 0.

Proof. The proof is the same as in [13, Lemma 3.4]. 2 Proof of Theorem 1.2. Firstly boundedness of u, v, ∇w and standard parabolic regularity theory [10] yield that there exist α ∈ (0, 1) and C > 0 such that u

C

2+α,1+ α 2

(×[1,t])

+ v

C

2+α,1+ α 2

(×[1,t])

+ w

C

2+α,1+ α 2

(×[1,t])

≤C

for all t ≥ 1.

Therefore in view of the Arzelà–Ascoli theorem or the Gagliardo–Nirenberg inequality n

2

n+2 n+2 ϕL∞ () ≤ cϕW 1,∞ () ϕL2 () , it is sufficient to show that

u(t) − 1L2 () → 0,

v(t) − 1L2 () → 0,

w(t) − w L2 () → 0

as t → ∞.

Here the convergences of u and v come from Lemmas 4.3 and 4.4. As to w, we shall show that w(t) − w ∗ (t)L2 () → 0 as t → ∞

(4.8)

w ∗ (t) → w as t → ∞,

(4.9)

and

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17

where w ≥ 0 such that h(1, 1, w ) = 0. We denote by q the function defined as ˆ q(t) :=

|w(t) − w ∗ (t)|2 .



Then the Poincaré–Wirtinger inequality gives ˆ∞

¨ q(t) dt ≤ c

|∇w|2 < ∞.

∞

0

To use [2, Lemma 5.1] we prove that there exists C > 0 such that |q (t)| ≤ C. Indeed, using the third equation in (1.1), we have    ˆ     ∗ ∗   |q (t)| = 2  (w − w ) dw + h(u, v, w) − wt     ˆ ˆ   ≤ 2d |∇w|2 + |w − w ∗ | h(u, v, w) − wt∗  



≤C for some C > 0. From [2, Lemma 5.1] it follows that q(t) → 0 as t → ∞. This means (4.8). In the rest of the proof, we shall show (4.9). We denote by ε the function defined as ε(t) :=

1 ||

ˆ

h(u, v, w) − h(u∗ , v ∗ , w ∗ ).



By the mean value theorem we have |ε(t)| ≤

1 ||

C ≤ ||

ˆ

|h(u, v, w) − h(u∗ , v ∗ , w ∗ )|



ˆ





 |u − u∗ | + |v − v ∗ | + |w − w ∗ |

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18

for some C > 0. So we obtain from the Poincaré–Wirtinger inequality and Lemma 4.1 that ˆ∞ |ε(t)|2 dt < ∞.

(4.10)

0

Since h(1, 1, w ) = 0, it follows from the mean value theorem that ˆ d ∗ 1 h(u, v, w) w = dt || 

= h(u∗ , v ∗ , w ∗ ) − h(1, 1, w ) + ε = hu (u∗ − 1) + hv (v ∗ − 1) + hw (w ∗ − w ) + ε for some derivatives hu , hv , hw . Multiplying this equation by w ∗ − w , we have   1 d ∗ |2 = hw |w ∗ − w |2 + hu (u∗ − 1) + hv (v ∗ − 1) + ε (w ∗ − w ) |w − w 2 dt   |2 + C |u∗ − 1| + |v ∗ − 1| + |ε| |w ∗ − w | ≤ −γ |w ∗ − w for some C > 0. Putting ε˜ (t) := |u∗ (t) − 1| + |v ∗ (t) − 1| + |ε(t)| and solving the above differential inequality, we see that     ˆ ˆt   1 ∗ −γ t   |w (t) − w | ≤ e  + C e−γ (t−s) |˜ε (s)| ds.  || w0 − w   

(4.11)

0

It remains to show that ˆt

e−γ (t−s) |˜ε (s)| ds → 0 as t → ∞.

0

From [13, Lemma 3.3] we know that ˆ∞

∗

∗

ˆ∞

u (t)(u (t) − 1) dt < ∞, 2

0

v ∗ (t)(v ∗ (t) − 1)2 dt < ∞,

0

which together with Lemma 4.2 implies ˆ∞ (u∗ (t) − 1)2 dt < ∞,

ˆ∞ (v ∗ (t) − 1)2 dt < ∞.

0

0

(4.12)

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19

Collecting (4.10) and (4.12), we have ˆ∞ |˜ε (s)|2 ds < ∞. 0

Therefore it follows that ˆt e

−γ (t−s)

ˆt/2 ˆt −γ (t−s) |˜ε (s)| ds = e |˜ε (s)| ds + e−γ (t−s) |˜ε (s)| ds

0

0

t/2

⎛ ⎞1/2 ⎛ ⎞1/2 ˆt/2 ˆt ˆt 1 ⎜ ⎟ ⎜ ⎟ ≤ e− 2 γ t |˜ε (s)| ds + e−γ t ⎝ e2γ s ds ⎠ ⎝ |˜ε (s)|2 ds ⎠ 0

& ≤

t/2

t/2

⎛

⎞1/2 ⎞1/2 ' ⎛∞ ˆ ˆt t −1γt ⎝ 1 ⎜ ⎟ 2 |˜ε (s)|2 ds ⎠ + e 2 ⎝ |˜ε (s)| ds ⎠ → 0 2 2γ 0

t/2

as t → ∞. In view of (4.11) this means w ∗ (t) → w as t → ∞, the end of the proof. 2 Acknowledgments The authors would like to thank the referees for valuable comments improving the paper, e.g., the convergence is improved from L2 to L∞ in Theorem 1.2. References [1] N. Bellomo, A. Bellouquid, Y. Tao, M. Winkler, Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci. 25 (2015) 1663–1763. [2] A. Friedman, J.I. Tello, Stability of solutions of chemotaxis equations in reinforced random walks, J. Math. Anal. Appl. 272 (2002) 138–163. [3] K. Fujie, M. Winkler, T. Yokota, Blow-up prevention by logistic sources in a parabolic-elliptic Keller–Segel system with singular sensitivity, Nonlinear Anal. 109 (2014) 56–71. [4] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-Verlag, Berlin–New York, 1981. [5] T. Hillen, K.J. Painter, A user’s guide to PDE models for chemotaxis, J. Math. Biol. 58 (2009) 183–217. [6] D. Horstmann, From 1970 until present: the Keller–Segel model in chemotaxis and its consequences, Jahresber. Dtsch. Math.-Ver. 106 (2004) 51–69. [7] D. Horstmann, M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations 215 (2005) 52–107. [8] D. Horstmann, Generalizing the Keller–Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci. 21 (2011) 231–270. [9] E.F. Keller, L.A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol. 26 (1970) 399–415. [10] O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Ural’ceva, Linear and Quasi-Linear Equations of Parabolic Type, AMS, Providence, 1968. [11] M. Mizukami, in preparation.

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[12] M. Negreanu, J.I. Tello, On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal. 46 (2014) 3761–3781. [13] M. Negreanu, J.I. Tello, Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differential Equations 258 (2015) 1592–1617. [14] M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr. 283 (2010) 1664–1673. [15] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations 35 (2010) 1516–1537. [16] G. Wolansky, Multi-components chemotactic system in the absence of conflicts, European J. Appl. Math. 13 (2002) 641–661. [17] Q. Zhang, Y. Li, Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys. 66 (2015) 83–93.