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Large time behavior of solutions to a quasilinear attraction–repulsion chemotaxis system with logistic source
Nonlinear Analysis: Real World Applications 54 (2020) 103095
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Nonlinear Analysis: Real World Applications www.elsevier.com/locate/nonrwa
Large time behavior of solutions to a quasilinear attraction–repulsion chemotaxis system with logistic source✩ Xiao He a,b , Miaoqing Tian c , Sining Zheng d ,∗ a
Department of Mathematics, Dalian Minzu University, Dalian 116600, PR China Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, PR China College of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou 450002, PR China d School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, PR China b c
article
info
Article history: Received 13 October 2019 Received in revised form 31 December 2019 Accepted 4 January 2020 Available online xxxx Keywords: Attraction–repulsion Chemotaxis Convergence rate Large time behavior Logistic source
abstract This paper studies the quasilinear attraction–repulsion chemotaxis system with a logistic source ut = ∇ · (D(u)∇u) − ∇ · (Φ(u)∇v) + ∇ · (Ψ (u)∇w) + f (u), τ1 vt = ∆v + αu − βv, τ2 wt = ∆w + γu − δw, under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ RN (N ≥ 1), where τ1 , τ2 ∈ {0, 1}, D, Φ, Ψ ∈ C 2 ([0, +∞)) nonnegative with D(s) ≥ (s + 1)p for s ≥ 0, Φ(s) ≤ χsq , ξsr ≤ Ψ (s) ≤ ζsr for s ≥ s0 > 0, f (s) ≤ µs(1 − sk ) for s > 0, f (0) ≥ 0. In a previous paper of the authors (Tian et al., 2016), the criteria for global boundedness of solutions were established for the case of τ2 = 0, depending on the interaction among the multi-nonlinear mechanisms (diffusion, attraction, repulsion and source) in the model. This paper continuously determines the global boundedness conditions for the case of τ2 = 1. In particular, we obtain the large time behavior of the globally bounded solutions for the situation of D(s) = (s+1)p , Φ(s) = χsq , Ψ (s) = ξsr , f (s) = µs(1 − s), s ≥ 0 with p = 2(q − 1) = 2(r − 1) ≥ 0, τ1 , τ2 ∈ {0, 1}. © 2020 Elsevier Ltd. All rights reserved.
1. Introduction In this paper, we study the quasilinear attraction–repulsion chemotaxis ( ) ( ) ⎧ ut = ∇ · (D(u)∇u) − ∇ · Φ(u)∇v + ∇ · Ψ (u)∇w + f (u), ⎪ ⎪ ⎪ ⎪ ⎪ τ1 vt = ∆v + αu − βv, ⎪ ⎪ ⎨ τ2 wt = ∆w + γu − δw, ⎪ ∂v ∂w ∂u ⎪ ⎪ ⎪ = = = 0, ⎪ ⎪ ∂n ∂n ⎪ ⎩ ∂n u(x, 0) = u0 (x), τ1 v(x, 0) = τ1 v0 (x), τ2 w(x, 0) = τ2 w0 (x),
system (x, t) ∈ Ω × (0, T ), (x, t) ∈ Ω × (0, T ), (x, t) ∈ Ω × (0, T ),
(1.1)
(x, t) ∈ ∂Ω × (0, T ), x ∈ Ω,
✩ Supported by the National Natural Science Foundation of China (11701067, 11171048), Natural Science Foundation of Liaoning, China (2019-ZD-0180) and Key Research Plan of Henan Province Colleges, China (20B110021) ∗ Corresponding author. E-mail address: [email protected] (S. Zheng).
https://doi.org/10.1016/j.nonrwa.2020.103095 1468-1218/© 2020 Elsevier Ltd. All rights reserved.
2
X. He, M. Tian and S. Zheng / Nonlinear Analysis: Real World Applications 54 (2020) 103095
where Ω ⊂ RN (N ≥ 1) is a bounded domain with smooth boundary, n is the unit outer normal vector on the boundary, τ1 , τ2 ∈ {0, 1}, α, β, γ, δ > 0. The nonlinear functions D, Φ, Ψ ∈ C 2 ([0, ∞)) satisfy D(s) ≥ (s + 1)p , s ≥ 0, q
(1.2)
Φ(s) ≤ χs , s ≥ s0 ,
(1.3)
ξsr ≤ Ψ (s) ≤ ζsr , s ≥ s0 ,
(1.4)
with χ, ξ, ζ, s0 > 0, p, q, r ∈ R. The logistic source f (s) is smooth and satisfies f (0) ≥ 0, f (s) ≤ µs(1 − sk ) for s > 0
(1.5)
with µ, k > 0. In the model (1.1), u represents the cell density, v and w denote the concentrations of attractive and repulsive signals produced by the cells respectively. The logistic source f (u) included in (1.1) means that the cell density cannot grow unlimitedly. The properties of solutions would be determined by the interactions among the four nonlinear mechanisms—diffusion, attraction, repulsion and logistic source. By taking D(u) ≡ 1, Φ(u) = χu and Ψ (u) = f (u) ≡ 0 the system (1.1) just is the classical Keller–Segel chemotaxis system, which has been studied extensively since 1970 (refer to e.g. [1–6] and the references therein). To describe the limited-growth of cells, the effects of logistic sources were introduced subsequently, e.g., f satisfies (1.5) with D(u) ≡ 1, Φ(u) = χu, Ψ (u) ≡ 0 and k = 1 in (1.1). For the parabolic–elliptic system with logistic source, Tello and Winkler [7] showed that the solutions are globally bounded if µ > NN−2 χ. The same result was obtained for the parabolic–parabolic case with logistic source for either N = 2 [8], or N ≥ 3 with µ sufficiently large [9]. From then on, general quasilinear chemotaxis systems with logistic source were studied extensively. It has been obtained in [10,11] that if max{p + N2 , 1} > q, then the solution is global; if q = 1 (the balance case), the global boundedness of solutions can be ensured by µ > αχ(1 − 2/N (1 − p)+ ) for the parabolic–elliptic system [12], and µ > µ0 with some µ0 > 0 for the parabolic–parabolic system [13,14]. For the fully parabolic system with D(u) ≡ 1, Φ(u) = χu and Ψ (u) = ξu, the large time behavior was studied in [15]. Recently, people pay attention to the Keller–Segel systems with multi-species and multi-stimulus. Refer to [16–23] for details. This paper is a continuous work of our previous paper [24], where the criteria for global boundedness of solutions were established for the cases of τ2 = 0 with τ1 = 0, 1, depending on various dominations of the four nonlinear mechanisms (diffusion, attraction, repulsion and source). It has been shown that if τ1 = τ2 = 0, then the global boundedness of solutions would be ensured by one of the following conditions (i) q < max{r, k}; (ii) q − p < N2 ; (iii) q = max{r, k}, q − p ≥ N2 with (a) q = r = k, 2 µ > (αχ − γξ)(1 − N (q−p) )/(1 + N2(q−1) (q−p) ), or (b) q = r > k, αχ − γξ < 0, or (c) q = k > r, 2 µ > αχ(1 − N (q−p) )/(1 + N2(q−1) (q−p) ) ([24, Theorem 1]). Also, for τ2 = 0, τ1 = 1, the solutions would be globally bounded if one of the following is true (i) q < max{r, k}; (ii) q − p < N2 ; (iii) q = max{r, k}, and q − p ≥ N2 with (a) q = r = k, with µ and γξ large that µ + γξθ0 > µ0 , or (b) q = r > k, with γξ large that γξ > µ1 , or (c) q = k > r, with µ large that µ > µ0 ([24, Theorem 2]). In this paper, we at first establish the conditions for the other two cases of τ2 = 1 with τ1 = 0, 1, and then determine the large time behavior of solutions for all cases of τ1 , τ2 ∈ {0, 1}. These are stated in the following theorems.
X. He, M. Tian and S. Zheng / Nonlinear Analysis: Real World Applications 54 (2020) 103095
3
Theorem 1. Let τ1 = 0, τ2 = 1 or τ1 = τ2 = 1 in (1.1), Ω ⊂ RN (N ≥ 1) be a bounded domain with smooth boundary, D, Φ, Ψ and f satisfy (1.2)–(1.5), nonnegative initial data (u0 (x), v0 (x), w0 (x)) ∈ ¯ ) × W 1,σ (Ω ) × W 1,σ (Ω ) (σ > N ). C(Ω (i) If max{q, r} < max{k, p + N2 }, then Eq. (1.1) admits a globally bounded solution. (ii) Assume max{q, r} = k ≥ p + N2 . There exists µ0 > 0 such that if µ > µ0 , then the solution of Eq. (1.1) is globally bounded. Next theorem is on the large time behavior of solutions of Eq. (1.1) for the situation of D(s) = (s + 1)p , Φ(s) = χsq , Ψ (s) = ξsr , f (s) = µs(1 − s), s ≥ 0
(1.6)
p = 2(q − 1) = 2(r − 1) ≥ 0.
(1.7)
with χ, ξ > 0, and
Theorem 2. Let (u, v, w) be a globally bounded solution of (1.1) with τ1 , τ2 ∈ {0, 1}, Ω ⊂ RN (N ≥ 1) be a bounded domain with smooth boundary, D, Φ, Ψ , f satisfy (1.6). There exist µ1 , λ, C > 0 such that ∥u(·, t) − 1∥L∞ (Ω) + ∥v(·, t) −
α γ ∥L∞ (Ω) + ∥w(·, t) − ∥L∞ (Ω) ≤ Ce−λt β δ
(1.8)
provided µ > µ1 . The rest of the paper is arranged as follows. In Section 2 we introduce the local existence of solutions to (1.1) as preliminaries, and then prove Theorems 1 and 2 in Sections 3 and 4, respectively. 2. Preliminaries In this section, we give the known result on the local existence of solutions to Eq. (1.1) with necessary estimates as preliminaries. Lemma 2.1. Suppose Ω ⊂ RN (N ≥ 1) is a bounded domain with smooth boundary, D, Φ, Ψ and f satisfy ( ) ¯ ) as τ1 = τ2 = 0, or nonnegative pairs u0 (x), v0 (x) ∈ (1.2)–(1.5). Then for nonnegative u0 (x) ∈ C(Ω ( ) ¯ ) × W 1,σ (Ω ) (σ > N ) as τ1 = 1, τ2 = 0, u0 (x), w0 (x) ∈ C(Ω ¯ ) × W 1,σ (Ω ) (σ > N ) as τ1 = 0, τ2 = 1, C(Ω ( ) 1,σ 2 ¯ or nonnegative u0 (x), v0 (x), w0 (x) ∈ C(Ω ) × (W (Ω )) (σ > N ) as τ1 = τ2 = 1, there exist nonnegative ¯ × [0, Tmax )) ∩ C 2,1 (Ω ¯ × (0, Tmax )) with Tmax ∈ (0, ∞] classically solving (1.1). functions u, v, w ∈ C 0 (Ω Moreover, if Tmax < ∞, then lim ∥u(·, t)∥L∞ (Ω) = ∞. t→Tmax
The proof of Lemma 2.1 is standard under a suitable framework of the fixed point theory. Refer to [3,10,12,25] for the details. The following estimates were obtained in [24] with an idea from [26]. Lemma 2.2 (Lemma 2.2 [24]). Let (u, v, w) be a solution of (1.1) ensured by Lemma 2.1. Then ∫ {∫ } u ≤ max u0 , |Ω | := M, t ∈ (0, Tmax ). Ω
(2.1)
Ω
Moreover, for any η > 0, θ > 1, there is c0 = c0 (η, θ) > 0 such that ∫ ∫ vθ ≤ η uθ + c0 , t ∈ (0, Tmax ) Ω
Ω
(2.2)
X. He, M. Tian and S. Zheng / Nonlinear Analysis: Real World Applications 54 (2020) 103095
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with τ1 = 0, and
∫
wθ ≤ η
∫
Ω
with τ2 = 0.
uθ + c0 , t ∈ (0, Tmax )
(2.3)
Ω
□
3. Proof of Theorem 1 In this section, we deal with the global boundedness of solutions to the parabolic–parabolic–parabolic case with τ1 = τ2 = 1 and the parabolic–elliptic–parabolic case with τ1 = 0, τ2 = 1 in (1.1). The following lemma as a variation of Maximal Sobolev Regularity [27] is crucial for our proof. Lemma 3.1 (Lemma 2.2 [14]). Let σ > 1, a, b > 0. Consider the following evolution equation ⎧ ⎪ ⎪ Zt = ∆Z + au − bZ, (x, t) ∈ Ω × (0, T ), ⎨ ∂Z = 0, (x, t) ∈ ∂Ω × (0, T ), ⎪ ⎪ ⎩ ∂n Z(x, 0) = Z0 (x), x ∈ Ω. ( ) σ σ 0 For each Z0 ∈ W 2,σ (Ω ) (σ > N ) with ∂Z ∂n = 0 on ∂Ω and any u ∈ L (0, T ); L (Ω ) , there exists a unique solution ( ) ( ) Z ∈ W 1,σ (0, T ); Lσ (Ω ) ∩ Lσ (0, T ); W 2,σ (Ω ) . 0) Moreover, there exists Cσ > 0, such that if t0 ∈ [0, T ), Z(·, t0 ) ∈ W 2,σ (Ω ) (σ > N ) with ∂Z(·,t = 0, then ∂n ∫ T∫ ∫ T∫ ( ) b b b σ e 2 στ uσ + Cσ e 2 σt0 ∥Z(·, t0 )∥σLσ (Ω) + ∥∆Z(·, t0 )∥σLσ (Ω) . e 2 στ |∆Z| ≤ Cσ aσ
t0
t0
Ω
Ω
¯ ) with Given t0 ∈ (0, Tmax ) with t0 ≤ 1, we know from Lemma 2.1 that u(·, t0 ), v(·, t0 ), w(·, t0 ) ∈ C 2 (Ω ∂w(·,t0 ) ¯ > 0 such that = ∂n = 0 on ∂Ω . Pick M
∂v(·,t0 ) ∂n
¯, sup ∥u(·, τ )∥L∞ (Ω) , sup ∥v(·, τ )∥L∞ (Ω) , sup ∥w(·, τ )∥L∞ (Ω) ≤ M
0≤τ ≤t0
0≤τ ≤t0
0≤τ ≤t0
∥∆v(·, t0 )∥L∞ (Ω) , ∥∆w(·, t0 )∥L∞ (Ω)
(3.1)
¯. ≤M
Proof of Theorem 1. The key step of the proof is to show that for any l > 1, there exists c = c(l) > 0 such that ∫ ul ≤ c, t ∈ (0, Tmax ). (3.2) Ω
Without loss of generality, assume l > max{3 + 2|q|, 3 + 2|r|, 3 + |p|}. Suppose τ1 = τ2 = 1. Multiply (1.1)1 by (u + 1)l−1 and then integrate by parts on Ω , ∫ ∫ ∫ 1 d 2 (u + 1)l ≤ −(l − 1) (u + 1)l−2 D(u)|∇u| + (l − 1) (u + 1)l−2 Φ(u)∇u · ∇v l dt Ω ∫Ω ∫ Ω l−2 − (l − 1) (u + 1) Ψ (u)∇u · ∇w + µ u(1 − uk )(u + 1)l−1 Ω Ω ∫ ∫ l+p 2 4(l − 1) 2 | + (l − 1) ≤− |∇(u + 1) ∇h1 (u) · ∇v (l + p)2 Ω Ω ∫ ∫ ∫ l−1 − (l − 1) ∇h2 (u) · ∇w + µ u(u + 1) −µ uk+1 (u + 1)l−1 Ω Ω Ω ∫ ∫ ∫ l+p 2 4(l − 1) 2 | − (l − 1) =− |∇(u + 1) h (u)∆v + (l − 1) h2 (u)∆w 1 (l + p)2 Ω Ω Ω ∫ ∫ +µ u(u + 1)l−1 − µ uk+1 (u + 1)l−1 , t ∈ (0, Tmax ) Ω
Ω
(3.3)
X. He, M. Tian and S. Zheng / Nonlinear Analysis: Real World Applications 54 (2020) 103095
5
with ∫ h1 (u) :=
u
(s + 1)l−2 Φ(s)ds, h2 (u) :=
∫
u
(s + 1)l−2 Ψ (s)ds.
0
0
Meanwhile, by (1.3) and (1.4), there exist b1 = b1 (l), b2 = b2 (l) > 0 such that s0
∫
∫
u
χ (u + 1)l+q−1 , l+q−1 0 s0 ∫ s0 ∫ u ζ l−2 h2 (u) = (u + 1)l+r−1 (s + 1) Ψ (s)ds + (s + 1)l−2 Ψ (s)ds ≤ b2 + l+r−1 0 s0
h1 (u) =
(s + 1)l−2 Φ(s)ds +
(s + 1)l−2 Φ(s)ds ≤ b1 +
(3.4) (3.5)
with t ∈ (0, Tmax ). Thus by Young’s inequality with (3.3)–(3.5), we obtain 1 d l dt
4(l − 1) (u + 1) ≤ − (l + p)2 Ω
∫
l
∫ χ(l − 1) |∇(u + 1) | + (u + 1)l+q−1 |∆v| l+q−1 Ω Ω ∫ ∫ ∫ ζ(l − 1) (u + 1)l+r−1 |∆w| + b2 (l − 1) |∆w| + b1 (l − 1) |∆v| + l+r−1 Ω Ω Ω ∫ ∫ +µ u(u + 1)l−1 − µ uk+1 (u + 1)l−1 Ω Ω ∫ ∫ l+p 2 χ(l − 1) 4(l − 1) 2 | + ≤− |∇(u + 1) (u + 1)l+q (l + p)2 Ω l+q Ω ∫ ∫ ζ(l − 1) 2χ(l − 1) l+q |∆v| + (u + 1)l+r + (l + q − 1)(l + q) Ω l+r Ω ∫ ∫ ∫ 2ζ(l − 1) l+r l−1 |∆w| +µ u(u + 1) −µ uk+1 (u + 1)l−1 + (l + r − 1)(l + r) Ω Ω Ω + c1 , t ∈ (0, Tmax ) l+p 2 2
∫
(3.6)
with c1 = c1 (l) > 0. Multiplying (1.1)1 by ul−1 and then integrating by parts on Ω , similarly to the procedure for (3.6), we have ∫ ∫ ∫ ∫ χ(l − 1) 2χ(l − 1) ζ(l − 1) 1 d l+q ul ≤ ul+q + |∆v| + ul+r l dt Ω l+q (l + q − 1)(l + q) l + r Ω Ω ∫ ∫ Ω ∫ 2ζ(l − 1) l+r l l+k + |∆w| +µ u −µ u + c2 (l), t ∈ (0, Tmax ). (3.7) (l + r − 1)(l + r) Ω Ω Ω By Lemma 3.1, we know from (1.1)2 that for any given σ > 1, ∫ t∫ t0
β
σ
Ω
∫ t∫
( β β e 2 στ uσ + Cσ e 2 σt0 ∥v(·, t0 )∥σLσ (Ω) t0 Ω ) + ∥∆v(·, t0 )∥σLσ (Ω) , t ∈ (t0 , Tmax ).
e 2 στ |∆v| ≤ Cσ ασ
(3.8)
Notice that we cannot do the same thing as (3.8) directly for (1.1)3 , where the cases of δ = β and δ ̸= β will be considered separately. If δ = β, we know by Lemma 3.1 that ∫ t∫ t0
Ω
β
σ
∫ t∫
( β β e 2 στ uσ + Cσ e 2 σt0 ∥w(·, t0 )∥σLσ (Ω) t0 Ω ) + ∥∆w(·, t0 )∥σLσ (Ω) , t ∈ (t0 , Tmax ).
e 2 στ |∆w| ≤ Cσ γ σ
(3.9)
X. He, M. Tian and S. Zheng / Nonlinear Analysis: Real World Applications 54 (2020) 103095
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For δ ̸= β, rewrite (1.1)3 as wt = ∆w + γu − βw + (β − δ)w. By Lemma 3.1, ∫ t∫ e t0
β 2 στ
σ
∫ t∫
( β β σ e 2 στ |γu + (β − δ)w| + Cσ e 2 σt0 ∥w(·, t0 )∥σLσ (Ω) t0 Ω ) + ∥∆w(·, t0 )∥σLσ (Ω) ∫ t∫ ∫ t∫ β β ≤ Cσ (2γ)σ e 2 στ uσ + Cσ (2|β − δ|)σ e 2 στ wσ
|∆w| ≤ Cσ
Ω
t0
+ Cσ e
β 2 σt0
(
Ω
t0
∥w(·, t0 )∥σLσ (Ω)
+
Ω
∥∆w(·, t0 )∥σLσ (Ω)
)
, t ∈ (t0 , Tmax ).
(3.10)
From the Gagliardo–Nirenberg inequality and Lemma 2.2, we have 1−a a ∥w∥Lσ (Ω) ≤ cσ (∥∆w∥aLσ (Ω) ∥w∥L 1 (Ω) + ∥w∥L1 (Ω) ) ≤ cσ (∥∆w∥Lσ (Ω) + 1)
with cσ > 0 and a =
σ−1 σ−1+ 2σ N
∈ (0, 1). And then for any ϵ > 0, ∫
σ wσ ≤ cσ (∥∆w∥aσ Lσ (Ω) + 1) ≤ ϵ∥∆w∥Lσ (Ω) + cϵ,σ
Ω 1 2Cσ (2|β−δ|)σ .
with cϵ,σ > 0. Choose ϵ = ∫ t∫ e t0
β 2 στ
Together with (3.10), we know
∫ t∫
t β ( β u + Cσ e 2 στ + Cσ e 2 σt0 ∥w(·, t0 )∥σLσ (Ω) t0 Ω t0 ) + ∥∆w(·, t0 )∥σLσ (Ω) , t ∈ (t0 , Tmax ).
σ
|∆w| ≤ Cσ
Ω
e
β 2 στ
∫
σ
(3.11)
l−1 At first consider the item (i) with max{q, r} < max{k, p + N2 }. Notice that l−1 l+q , l+r < 2 and l−1 l−1 (l+q−1)(l+q) , (l+r−1)(l+r) < 1. If max{q, r} =: K < k, we get from (3.7) with Young’s inequality that ∫ ∫ ∫ ∫ ∫ ∫ β(l + K) 1 d l+K l+K ul + ul ≤ 2χ ul+K + 2χ |∆v| + 2ζ ul+K + 2ζ |∆w| l dt Ω 2l Ω Ω Ω Ω Ω ∫ ∫ ( β(l + K) ) l + µ+ u −µ ul+k + c3 2l Ω Ω ∫ ∫ ∫ µ l+K l+K ≤ 2χ |∆v| ul+k + c4 , t ∈ (t0 , Tmax ) + 2ζ |∆w| − 4 Ω Ω Ω
with c3 = c3 (l), c4 = c4 (l) > 0. Then for t ∈ (t0 , Tmax ), 1 l
∫
β
ul ≤ e− 2 (l+K)(t−t0 )
Ω
∫
t
+ 2ζ
1 l
∫
ul (·, t0 ) + 2χ
∫
Ω
β
β
e− 2 (l+K)(t−τ )
t0
e− 2 (l+K)(t−τ )
t0
t
∫ |∆w|
l+K
Ω
−
∫
l+K
|∆v| Ω
µ 4
∫
t
β
e− 2 (l+K)(t−τ )
t0
∫
ul+k +
Ω
2c4 . β(l + K)
Combine (3.8) and (3.1) to get ∫
t
e t0
β
− 2 (l+K)(t−τ )
∫
l+K
|∆v| Ω
∫
t
β
− 2 (l+K)(t−τ )
∫
β
≤ Cl e ul+K + Cl e− 2 (l+K)(t−t0 ) t0 Ω ( ) l+K × ∥v(·, t0 )∥l+K + ∥∆v(·, t )∥ 0 l+K (l+K) L (Ω) L (Ω) ∫ t ∫ β µ ≤ e− 2 (l+K)(t−τ ) ul+k + cl , t ∈ (t0 , Tmax ). 16χ t0 Ω
X. He, M. Tian and S. Zheng / Nonlinear Analysis: Real World Applications 54 (2020) 103095
7
The inequalities (3.9), (3.11) and (3.1) yield ∫ ∫ ∫ t ∫ t β β β l+K e− 2 (l+K)(t−τ ) |∆w| ≤ Cl e− 2 (l+K)(t−τ ) ul+K + Cl + Cl e− 2 (l+K)(t−t0 ) t0 Ω t Ω ( 0 ) l+K × ∥w(·, t0 )∥L(l+K) (Ω) + ∥∆w(·, t0 )∥l+K Ll+K (Ω) ∫ t ∫ β µ e− 2 (l+K)(t−τ ) ≤ ul+k + cl , t ∈ (t0 , Tmax ). 16ζ t0 Ω This concludes (3.2) that 1 l
∫
ul ≤
Ω
1 l
∫
ul (·, t0 ) + cl ≤ c5 , t ∈ (t0 , Tmax )
Ω
with c5 = c5 (l) > 0. For max{q, r} =: K < p + N2 , without loss of generality, assume K ≥ k. By Young’s inequality, it follows from (3.6) that ∫ ∫ ∫ l+p 2 1 d 4(l − 1) 2 | + 2(χ + ζ) (u + 1)l+K (u + 1)l ≤ − |∇(u + 1) l dt Ω (l + p)2 Ω ∫ ∫ ∫ Ω l+K l+K + 2χ |∆v| + 2ζ |∆w| + µ (u + 1)l + c6 (l), t ∈ (t0 , Tmax ). (3.12) Ω
Ω
Ω
We know from (2.1) with the Gagliardo–Nirenberg inequality, ∫ l+K l+p 2 (u + 1)l+K = ∥(u + 1) 2 ∥ l+p l+K 2 L l+p (Ω)
Ω
≤ c7 ∥∇(u + 1) ≤ c8 ∥∇(u + 1)
l+p 2
l+p 2
2a l+K
l+p ∥L2 (Ω) ∥(u + 1)
l+p 2
2(1−a) l+K l+p
∥
2 L l+p (Ω)
+ c7 ∥(u + 1)
l+p 2
2 l+K l+p
∥
2
L l+p (Ω)
2a l+K
l+p ∥L2 (Ω) + c8 , t ∈ (t0 , Tmax )
( (l+p) ) ( N (l+p) ) N with c7 = c7 (l), c8 = c8 (l) > 0 and a = N (l+p) −N ∈ (0, 1). Since K < p + N2 2 2(l+K) / 1 − 2 + 2 l+K implies a l+p < 1, we have by Young’s inequality with ε > 0 that ∫ l+p (u + 1)l+K ≤ ε∥∇(u + 1) 2 ∥2L2 (Ω) + c9 (3.13) Ω
for c9 = c9 (l, ε) > 0. So (3.12) and (3.13) yield ∫ ∫ 1 d β(l + K) l (u + 1) + (u + 1)l l dt Ω 2l Ω ∫ ∫ ∫ ∫ 4(l − 1) l+K l+K l+K l+K ≤− (u + 1) + 2(χ + ζ) (u + 1) + 2χ |∆v| + 2ζ |∆w| (l + p)2 ε Ω Ω Ω Ω ∫ ( β(l + K) ) l + µ+ (u + 1) + c10 2l Ω ∫ ∫ ∫ ( 4(l − 1) ) l+K l+K l+K ≤ − + 2(χ + ζ) + 1 (u + 1) + 2χ |∆v| + 2ζ |∆w| + c11 (l + p)2 ε Ω Ω Ω with c10 = c10 (l, ε), c11 = c11 (l, ε) > 0 for t ∈ (t0 , Tmax ). Combining (3.8), (3.9), (3.11) and (3.1), we obtain ∫ ∫ β 1 1 (u + 1)l ≤ e− 2 (l+K)(t−t0 ) (u(·, t0 ) + 1)l l Ω l Ω ∫ ∫ ( 4(l − 1) ) t − β (l+K)(t−τ ) 2 + 2(χ + ζ) + 1 e (u + 1)l+K + − (l + p)2 ε t0 Ω
8
X. He, M. Tian and S. Zheng / Nonlinear Analysis: Real World Applications 54 (2020) 103095
∫
t
+ 2χ
β
− 2 (l+K)(t−τ )
e t0
+
∫
∫
l+K
|∆v|
t
+ 2ζ
Ω
β
− 2 (l+K)(t−τ )
e
∫
t0
l+K
|∆w| Ω
2c11 (l, ε) β(l + K) β
≤ e− 2 (l+K)(t−t0 )
∫
1 l
(u(·, t0 ) + 1)l ∫ ∫ ( 4(l − 1) ) t − β (l+K)(t−τ ) 2 e + − + 2(χ + ζ) + 1 (u + 1)l+K (l + p)2 ε t0 Ω ∫ t ∫ β e− 2 (l+K)(t−τ ) + 2χCl+K αl+K ul+K + 2ζCl+K γ
l+K
Ω
∫
t0 t
Ω β
− 2 (l+K)(t−τ )
e t0
∫
ul+K + c12 , t ∈ (t0 , Tmax )
(3.14)
Ω
with c12 = c12 (l, ε) > 0. For l > max{3 + 2|q|, 3 + 2|r|, 3 + |p|}, choose ε > 0 small such that −
4(l − 1) + 2(χ + ζ) + 1 + 2χCl+K αl+K + 2ζCl+K γ l+K ≤ 0. (l + p)2 ε
This proves the desired (3.2) by (3.14). Next treat the item (ii) with k = max{q, r} ≥ p + N2 . Suppose k = q > r. Then by (3.7) and Young’s inequality with ε1 > 0, ∫ ∫ ∫ ∫ χ(l − 1) 2χ(l − 1) 1 d l+q l l+q u ≤ u + |∆v| + ε1 ul+q l dt Ω l+q (l + q − 1)(l + q) Ω Ω Ω ∫ ∫ ∫ l+q + ε1 |∆w| +µ ul − µ ul+q + c13 , t ∈ (t0 , Tmax ) Ω
Ω
Ω
with c13 = c13 (l, ε1 ) > 0, and hence, ∫ ∫ ∫ ∫ ∫ β(l + q) χ(l − 1) 2χ(l − 1) 1 d l+q ul + ul ≤ ul+q + |∆v| + ε1 ul+q l dt Ω 2l l+q (l + q − 1)(l + q) Ω Ω Ω Ω ∫ ∫ ∫ ( β(l + q) ) l+q l + ε1 |∆w| + µ+ u −µ ul+q + c14 2l Ω Ω Ω ∫ ∫ ∫ χ(l − 1) 2χ(l − 1) l+q ≤ ul+q + |∆v| + ε1 ul+q l+q (l + q − 1)(l + q) Ω Ω Ω ∫ ∫ ∫ l+q l+q + ε1 |∆w| + ε1 u −µ ul+q + c14 (l, ε1 ) Ω Ω Ω ∫ ∫ ( ) χ(l − 1) 2χ(l − 1) l+q =− µ− − 2ε1 ul+q + |∆v| l+q (l + q − 1)(l + q) Ω Ω ∫ l+q + ε1 |∆w| + c14 , t ∈ (t0 , Tmax ) Ω
with c14 = c14 (l, ε1 ) > 0. Similarly to (3.14), we have by the variation-of-constants formula with (3.8), (3.9), (3.11) and c15 = c15 (l, ε1 ) > 0 that ∫ ( ) 1 χ(l − 1) 2χ(l − 1) ul ≤ − µ − − 2ε1 − Cl+q αl+q − ε1 Cl+q γ l+q l Ω l+q (l + q − 1)(l + q) ∫ t ∫ −β(l+q)(t−τ ) 2 × e ul+q + c15 , t ∈ (t0 , Tmax ). (3.15) t0
Ω
Let µ > µ0,1 with µ0,1 :=
χ(l − 1) 2χ(l − 1) + Cl+q αl+q l+q (l + q − 1)(l + q)
(3.16)
and then choose ε1 small enough such that µ − µ0,1 − 2ε1 − ε1 Cl+q γ l+q > 0. This concludes (3.2) by (3.15).
X. He, M. Tian and S. Zheng / Nonlinear Analysis: Real World Applications 54 (2020) 103095
If k = q = r, we have by (3.7) that ∫ ∫ ∫ ∫ ζ(l − 1) 1 d χ(l − 1) 2χ(l − 1) l+q ul ≤ ul+q + |∆v| + ul+q l dt Ω l+q (l + q − 1)(l + q) l + q Ω Ω ∫ ∫ Ω ∫ 2ζ(l − 1) l+q l l+q + |∆w| +µ u −µ u + c2 , t ∈ (t0 , Tmax ). (l + q − 1)(l + q) Ω Ω Ω
9
(3.17)
By the variation-of-constants formula with (3.8), (3.9), (3.11) and (3.17), similarly to (3.15), we have ∫ ( 1 χ(l − 1) ζ(l − 1) 2χ(l − 1) ul ≤ − µ − − − ε1 − Cl+q αl+q l Ω l+q l+q (l + q − 1)(l + q) ∫ )∫ t β 2ζ(l − 1) l+q − 2 (l+q)(t−τ ) − Cl+q γ e ul+q + c16 (l, ε1 ), t ∈ (t0 , Tmax ). (3.18) (l + q − 1)(l + q) t0 Ω Let µ > µ0,2 with µ0,2 :=
2χ(l − 1) 2ζ(l − 1) χ(l − 1) ζ(l − 1) + + Cl+q αl+q + Cl+q γ l+q l+q l+q (l + q − 1)(l + q) (l + q − 1)(l + q)
and choose ε1 small enough such that µ − µ0,2 − ε1 > 0. Thus (3.2) is true by (3.18). If k = r > q, by (3.7) and Young’s inequality with ε2 > 0, we know ∫ ∫ ∫ ∫ ζ(l − 1) 1 d l+k ul ≤ ε2 ul+k + ε2 |∆v| + ul+k l dt Ω l+k Ω Ω Ω ∫ ∫ ∫ 2ζ(l − 1) l+k l+k |∆w| + ε2 u −µ ul+k + c17 , t ∈ (t0 , Tmax ) + (l + k − 1)(l + k) Ω Ω Ω
(3.19)
(3.20)
with c17 = c17 (l, ε2 ) > 0. Similarly to (3.18), we have by the variation-of-constants formula with (3.8), (3.9), (3.11) and (3.20) that ∫ ( ) 2ζ(l − 1) 1 ζ(l − 1) ul ≤ − µ − − 2ε2 − Cl+k γ l+k − ε2 Cl+k αl+k l Ω l+k (l + k − 1)(l + k) ∫ t ∫ β ul+k + c18 (l, ε2 ), t ∈ (t0 , Tmax ). (3.21) × e− 2 (l+k)(t−τ ) Ω
t0
Let µ > µ0,3 with µ0,3 :=
2ζ(l − 1) ζ(l − 1) + Cl+k γ l+k l+k (l + k − 1)(l + k)
(3.22)
and choose ε2 small enough such that µ − µ0,3 − 2ε2 − ε2 Cl+k αl+k > 0. We conclude (3.2) by (3.21). Define µ0 := max{µ0,1 , µ0,2 , µ0,3 } in the item (ii) of the theorem. This obtains the desired preliminary estimate (3.2). For the case of τ1 = 0, τ2 = 1, similarly to the proof of [24, Theorem 2] with τ1 = 1, τ2 = 0, we can get (3.2) as well. We omit the details here. Furthermore, by the framework of Moser iteration [13, Lemma A.1], there is l1 (N, p) > N > 0 such that the known ∥u(·, t)∥Ll1 (Ω) < ∞, t ∈ (τ0 , Tmax ) implies ∥u(·, t)∥L∞ (Ω) ≤ C, t ∈ (τ0 , Tmax ) for some C > 0. Now, take l = l1 in (3.16), (3.19) and (3.22). Together with (3.1) and (3.2), we know that ∥u(·, t)∥L∞ (Ω) ≤ C, t ∈ (0, Tmax ). By Lemma 2.1, this concludes Tmax = ∞. The boundedness of v and w is obtained by the standard elliptic and parabolic regularity theory directly. □
10
X. He, M. Tian and S. Zheng / Nonlinear Analysis: Real World Applications 54 (2020) 103095
Remark 1. Observe that the results of Theorem 1 with τ1 = 0, τ2 = 1 are not as accurate as those in [24, Theorem 2] with τ1 = 1, τ2 = 0. The possible positive contribution of the repulsion component to the global boundedness of solutions has not been shown yet in the theorem. This is because of the current technical difficulty for treating the case when the repulsion equation is parabolic in chemotaxis systems, corresponding to the cases with τ2 = 1 here. 4. Proof of Theorem 2 In this section, motivated by [28], we study the large time behavior of the globally bounded solutions. Proof of Theorem 2. The parabolic regularity [25] with the global boundedness of (u, v, w) indicates that there exist θ ∈ (0, 1) and K > 0 such that ∥u∥
C
2+θ,1+ θ 2
¯ (Ω×[t,t+1])
, ∥v∥
C
2+θ,1+ θ 2
¯ (Ω×[t,t+1])
, ∥w∥
C
2+θ,1+ θ 2
¯ (Ω×[t,t+1])
≤ K, t ≥ t0 > 0.
Define functionals A(t), B(t) and C(t) as ∫ ∫ ∫ α γ A(t) = (u − 1 − ln u), B(t) = η1 (v − )2 and C(t) = η2 (w − )2 β δ Ω Ω Ω
(4.1)
(4.2)
with D, Φ, Ψ and f taking the case of (1.6), and η1 , η2 > 0 to be determined. For the case of τ1 = τ2 = 1, denote E(t) := A(t) + B(t) + C(t). Since p = 2(q − 1) = 2(r − 1) ≥ 0, we have for ϵ1 ∈ (0, 1) that ∫ u−1 d A(t) = ut dt u ∫Ω ) u − 1( = ∇ · ((u + 1)p ∇u) − χ∇ · (uq ∇v) + ξ∇ · (ur ∇w) + µu(1 − u) u Ω ∫ ∫ ∫ ∫ (u + 1)p 2 q−2 r−2 |∇u| + χ u ∇u · ∇v − ξ u ∇u · ∇w − µ (u − 1)2 =− 2 u ∫Ω ∫ Ω ∫ Ω ∫ Ω χ2 2 2 2 2 p−2 2(q−2) ≤− u |∇u| + ϵ1 |∇v| + (1 − ϵ1 ) u2(r−2) |∇u| u |∇u| + 4ϵ1 Ω Ω Ω Ω ∫ ∫ ξ2 2 + |∇w| − µ (u − 1)2 4(1 − ϵ1 ) Ω ∫ ∫ Ω ∫ χ2 ξ2 2 2 = |∇v| + |∇w| − µ (u − 1)2 , (4.3) 4ϵ1 Ω 4(1 − ϵ1 ) Ω Ω ∫ d α B(t) = 2η1 (v − )vt dt β ∫Ω α = 2η1 (v − )(∆v + αu − βv) β Ω ∫ ∫ ∫ α α 2 (4.4) = −2η1 |∇v| + 2η1 α (v − )(u − 1) − 2η1 β (v − )2 , β β Ω Ω ∫ Ω d γ C(t) = 2η2 (w − )wt dt δ ∫Ω γ = 2η2 (w − )(∆w + γu − δw) δ Ω ∫ ∫ ∫ γ γ 2 = −2η2 |∇w| + 2η2 γ (w − )(u − 1) − 2η2 δ (w − )2 . (4.5) δ δ Ω Ω Ω
X. He, M. Tian and S. Zheng / Nonlinear Analysis: Real World Applications 54 (2020) 103095
Let η1 =
χ2 8ϵ1
and η2 =
ξ2 8(1−ϵ1 ) .
11
Then for any ϵ2 > 0, (4.3)–(4.5) imply that
∫ ∫ ∫ d χ2 α α ξ2γ γ E(t) ≤ (v − )(u − 1) + (w − )(u − 1) − µ (u − 1)2 dt 4ϵ1 Ω β 4(1 − ϵ1 ) Ω δ Ω ∫ ∫ χ2 β α 2 γ 2 ξ2δ − (v − ) − (w − ) 4ϵ1 Ω β 4(1 − ϵ1 ) Ω δ ∫ ∫ ∫ 2 2 2 α 2 χ α ξ 2 δ(1 − ϵ2 ) γ χ β(1 − ϵ2 ) 2 (v − ) + (u − 1) + (w − )2 ≤ 4ϵ1 β 16ϵ β(1 − ϵ ) 4(1 − ϵ ) δ 1 2 1 Ω Ω ∫ ∫ Ω 2 2 2 ∫ ξ γ α χ β + (u − 1)2 − µ (u − 1)2 − (v − )2 16(1 − ϵ1 )(1 − ϵ2 )δ Ω 4ϵ β 1 Ω Ω ∫ ξ2δ γ 2 − (w − ) 4(1 − ϵ1 ) Ω δ ∫ ∫ γ ξ 2 δϵ2 χ2 βϵ2 α 2 (w − )2 =− (v − ) − 4ϵ1 Ω β 4(1 − ϵ1 ) Ω δ ∫ ( ( χ2 α2 ξ 2 γ 2 )) 1 + (u − 1)2 . − µ− 16(1 − ϵ2 ) ϵ1 β (1 − ϵ1 )δ Ω Define g(ϵ) =
χ2 α2 ϵβ
+
ξ2 γ 2 (1−ϵ)δ .
We can take ϵ1 ∈ (0, 1) such that g(ϵ1 ) = inf g(ϵ).
(4.6)
0<ϵ<1
Let µ > µ1 with µ1 =
1 g(ϵ1 ), 16 2
2
2 2
2 2
µ1 ξ δϵ2 ξ γ 1 2 and choose ϵ2 > 0 such that µ − 1−ϵ > 0. By taking ϵ3 = min{ χ4ϵβϵ , 4(1−ϵ , µ − 16(1−ϵ ( χϵ1αβ + (1−ϵ )}, 2 1 1) 2) 1 )δ we have ∫ ( d α γ ) E(t) ≤ −ϵ3 (u − 1)2 + (v − )2 + (w − )2 . (4.7) dt β δ Ω
Integrating (4.7) from t0 to ∞, we get ∫ ∞∫ ( α γ ) E(t0 ) (u − 1)2 + (v − )2 + (w − )2 ≤ β δ ϵ3 Ω t0 by E(t) ≥ 0. Together with (4.1), this implies ∫ ( γ ) α lim (u − 1)2 + (v − )2 + (w − )2 = 0. t→∞ Ω β δ
(4.8)
The Gagliardo–Nirenberg inequality says N
2
N +2 N +2 ∥h∥L∞ (Ω) ≤ c∞ ∥h∥W 1,∞ (Ω) ∥h∥L2 (Ω)
(4.9)
γ for all h ∈ W 1,∞ (Ω ) ∩ L2 (Ω ). Choosing h as u − 1, v − α β and w − δ in (4.9) respectively, we have from (4.8) and (4.1) that
∥u(·, t) − 1∥L∞ (Ω) + ∥v(·, t) − By L’Hˆ opital’s rule,
α γ ∥L∞ (Ω) + ∥w(·, t) − ∥L∞ (Ω) → 0 as t → ∞. β δ
u − 1 − ln u 1 = , (u − 1)2 2 ∫ ∫ 1 (u − 1)2 ≤ A(t) ≤ (u − 1)2 , t > T1 4 Ω Ω lim
t→∞
and hence
X. He, M. Tian and S. Zheng / Nonlinear Analysis: Real World Applications 54 (2020) 103095
12
2
2
χ }. So, it follows from (4.7) that , ξ for some T1 > 0. Let d = max{1, 8ϵ 1 8(1−ϵ1 )
d E(t) ≤ −ϵ4 E(t), t > T1 dt with ϵ4 =
ϵ3 d ,
and thus E(t) ≤ E(T1 )e−ϵ4 (t−T1 ) , t > T1 .
Therefore 1 4
∫
(u − 1)2 +
Ω
χ2 8ϵ1
∫ (v − Ω
ξ2 α 2 ) + β 8(1 − ϵ1 )
∫ (w − Ω
γ 2 ) ≤ E(t) ≤ E(T1 )e−ϵ4 (t−T1 ) , t > T1 . δ
By using the Gagliardo–Nirenberg inequality (4.9) again together with (4.1), we have ϵ4
∥u(·, t) − 1∥L∞ (Ω) ≤ c0 e− N +2 (t−T1 ) , ϵ4 α ∥v(·, t) − ∥L∞ (Ω) ≤ c0 e− N +2 (t−T1 ) , β ϵ4 γ ∥w(·, t) − ∥L∞ (Ω) ≤ c0 e− N +2 (t−T1 ) , δ
(4.10) (4.11) (4.12)
with t > T1 and c0 > 0. For the case of τ1 = 1 and τ2 = 0, define E1 (t) = A(t) + B(t). Multiplying by w − γδ on (1.1)3 and integrating by parts, we get ∫ ∫ ∫ ∫ ∫ γ δ γ γ2 γ 2 (w − )2 + (u − 1)2 , |∇w| = γ (u − 1)(w − ) − δ (w − )2 ≤ − δ δ 2 δ 2δ Ω Ω Ω Ω Ω and thus
∫ (w − Ω
Taking η1 =
χ2 8ϵ1
γ 2 γ2 ) ≤ 2 δ δ
∫
(u − 1)2 .
Ω
in (4.4), together with (4.3), we have
∫ ∫ ∫ d ξ2γ γ ξ2δ γ E1 (t) ≤ (w − )(u − 1) − (w − )2 − µ (u − 1)2 dt 4(1 − ϵ1 ) Ω δ 4(1 − ϵ1 ) Ω δ Ω ∫ ∫ χ2 α α χ2 β α 2 + (v − )(u − 1) − (v − ) 4ϵ1 Ω β 4ϵ1 Ω β ∫ ∫ χ2 βϵ2 α 2 ( 1 ( χ2 α2 ξ 2 γ 2 )) ≤− (v − ) − µ − + (u − 1)2 4ϵ1 Ω β 16 ϵ1 β(1 − ϵ2 ) (1 − ϵ1 )δ Ω ∫ ∫ ( χ2 α2 α 2 ( 1 ξ 2 γ 2 )) χ2 βϵ2 ≤− (v − ) − µ − + (u − 1)2 . 4ϵ1 Ω β 16(1 − ϵ2 ) ϵ1 β (1 − ϵ1 )δ Ω Similar to the procedure of the case τ1 = τ2 = 1, we know with µ > µ1 that ∫ ∫ 1 χ2 α 2 (u − 1) + (v − )2 ≤ E1 (t) ≤ E1 (T1 )e−ϵ4 (t−T1 ) , t > T1 . 4 Ω 8ϵ1 Ω β It follows from (4.13) that ∫ (w − Ω
γ 2 4γ 2 ) ≤ 2 E1 (T1 )e−ϵ4 (t−T1 ) , t > T1 . δ δ
Then (4.10)–(4.12) can be derived by (4.9) and (4.1). For case of τ1 = 0 and τ2 = 1, the simple result from (1.1)2 that ∫ ∫ ∫ ∫ ∫ α α β α α2 2 |∇v| = α (u − 1)(v − ) − β (v − )2 ≤ − (v − )2 + (u − 1)2 β β 2 β 2β Ω Ω Ω Ω Ω
(4.13)
X. He, M. Tian and S. Zheng / Nonlinear Analysis: Real World Applications 54 (2020) 103095
implies
α2 α (v − )2 ≤ 2 β β Ω
∫
∫
(u − 1)2 .
13
(4.14)
Ω
This concludes (4.10)–(4.12) as well. If τ1 = τ2 = 0, (4.13) and (4.14) are obviously true. So we have ∫ ∫ ∫ d χ2 α α χ2 β α 2 γ ξ2γ A(t) ≤ (v − )(u − 1) − (v − ) + (w − )(u − 1) dt 4ϵ1 Ω β 4ϵ1 Ω β 4(1 − ϵ1 ) Ω δ ∫ ∫ 2 ξ δ γ − (w − )2 − µ (u − 1)2 4(1 − ϵ1 ) Ω δ Ω ∫ ( 2 2 ( 1 χ α ξ 2 γ 2 )) ≤− µ− + (u − 1)2 . 16 ϵ1 β (1 − ϵ1 )δ Ω By a similar procedure as that for the case of τ1 = τ2 = 1, we know with µ > µ1 that ∫ 1 (u − 1)2 ≤ A(t) ≤ A(T1 )e−ϵ4 (t−T1 ) , t > T1 . 4 Ω Together with (4.13) and (4.14), we obtain ∫ ∫ γ 2 4γ 2 α 4α2 −ϵ4 (t−T1 ) (w − ) ≤ 2 A(T1 )e , (v − )2 ≤ 2 A(T1 )e−ϵ4 (t−T1 ) , t > T1 , δ δ β β Ω Ω and hence the desired (4.10)–(4.12) by (4.9) and (4.1).
□
Acknowledgment The authors would like to express their sincere appreciation to the anonymous reviewer for his/her significant suggestions to complete the present version of the paper. References [1] M.A. Herrero, J.J.L. Vel´ azquez, A blow-up mechanism for a chemotaxis model, Ann. Sc. Norm. Super. 24 (1997) 633–683. [2] D. Horstmann, From 1970 until present: The Keller–Segel model in chemotaxis and its consequences I, Jber. DMV 105 (2003) 103–165. [3] D. Horstmann, M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations 215 (2005) 52–107. [4] W. J¨ ager, S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc. 329 (1992) 819–824. [5] M. Winkler, Aggregation. vs, Aggregation vs global diffusive behavior in the higher-dimensional Keller–Segel model, J. Differential Equations 248 (2010) 2889–2905. [6] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller–Segel system, J. Math. Pures Appl. 100 (2013) 748–767. [7] J.I. Tello, M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations 32 (2007) 849–877. [8] K. Osaki, A. Yagi, Global existence of a chemotaxis-growth system in R2 , Adv. Math. Sci. Appl. 12 (2002) 587–606. [9] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations 35 (2010) 1516–1537. [10] X.R. Cao, Boundedness in a quasilinear parabolic-parabolic Keller–Segel system with logistic source, J. Math. Anal. Appl. 412 (2014) 181–188. [11] M. Winkler, K.C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filing effect, Nonlinear Anal. 72 (2010) 1044–1064. [12] X.R. Cao, S.N. Zheng, Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source, Math. Methods Appl. Sci. 37 (2014) 2326–2330. [13] Y. Tao, M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller–Segel system with subcritical sensitivity, J. Differential Equations 252 (2012) 692–715. [14] C.B. Yang, X.R. Cao, Z.X. Jiang, S.N. Zheng, Boundedness in a quasilinear fully parabolic Keller–Segel system of higher dimension with logistic source, J. Math. Anal. Appl. 430 (2015) 585–591.
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[15] J. Li, Y.Y. Ke, Y.F. Wang, Large time behavior of solutions to a fully parabolic attraction–repulsion chemotaxis system with logistic source, Nonlinear Anal. RWA 39 (2018) 261–277. [16] 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. [17] H.Y. Jin, Boundedness of the attraction–repulsion Keller–Segel system, J. Math. Anal. Appl. 422 (2015) 1463–1478. [18] J. Liu, Z.A. Wang, Classical solutions and steady states of an attraction–repulsion chemotaxis in one dimension, J. Biol. Dyn. 6 (2012) 31–41. [19] K.J. Painter, T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q. 10 (2002) 501–543. [20] Y. Tao, Z.A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci. 23 (2013) 1–36. [21] L.C. Wang, C.L. Mu, P. Zheng, On a quasilinear parabolic–elliptic chemotaxis system with logistic source, J. Differential Equations 256 (2014) 1847–1872. [22] Q.S. Zhang, Y.X. Li, Global existence and asymptotic properties of the solution to a two-species chemotaxis system, J. Math. Anal. Appl. 418 (2014) 47–63. [23] Q.S. Zhang, Y.X. Li, An attraction–repulsion chemotaxis system with logistic source, Z. Angew. Math. Mech. (2015) 1–15. [24] M.Q. Tian, X. He, S.N. Zheng, Global boundedness in quasilinear attraction–repulsion chemotaxis system with logistic source, Nonlinear Anal. RWA 30 (2016) 1–15. [25] O.A. Ladyzenskaya, V.A. Solonnikov, N.N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, 1968. [26] M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities? J. Nonlinear Sci. 24 (2014) 809–855. [27] M. Hieber, J. Prss, Heat kernels and maximal Lp − Lq estimates for parabolic evolution equations, Comm. Partial Differential Equations 22 (1997) 1647–1669. [28] X.L. Bai, M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J. 65 (2016) 553–583.
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Nonlinear Analysis: Real World Applications www.elsevier.com/locate/nonrwa
Large time behavior of solutions to a quasilinear attraction–repulsion chemotaxis system with logistic source✩ Xiao He a,b , Miaoqing Tian c , Sining Zheng d ,∗ a
Department of Mathematics, Dalian Minzu University, Dalian 116600, PR China Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, PR China College of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou 450002, PR China d School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, PR China b c
article
info
Article history: Received 13 October 2019 Received in revised form 31 December 2019 Accepted 4 January 2020 Available online xxxx Keywords: Attraction–repulsion Chemotaxis Convergence rate Large time behavior Logistic source
abstract This paper studies the quasilinear attraction–repulsion chemotaxis system with a logistic source ut = ∇ · (D(u)∇u) − ∇ · (Φ(u)∇v) + ∇ · (Ψ (u)∇w) + f (u), τ1 vt = ∆v + αu − βv, τ2 wt = ∆w + γu − δw, under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ RN (N ≥ 1), where τ1 , τ2 ∈ {0, 1}, D, Φ, Ψ ∈ C 2 ([0, +∞)) nonnegative with D(s) ≥ (s + 1)p for s ≥ 0, Φ(s) ≤ χsq , ξsr ≤ Ψ (s) ≤ ζsr for s ≥ s0 > 0, f (s) ≤ µs(1 − sk ) for s > 0, f (0) ≥ 0. In a previous paper of the authors (Tian et al., 2016), the criteria for global boundedness of solutions were established for the case of τ2 = 0, depending on the interaction among the multi-nonlinear mechanisms (diffusion, attraction, repulsion and source) in the model. This paper continuously determines the global boundedness conditions for the case of τ2 = 1. In particular, we obtain the large time behavior of the globally bounded solutions for the situation of D(s) = (s+1)p , Φ(s) = χsq , Ψ (s) = ξsr , f (s) = µs(1 − s), s ≥ 0 with p = 2(q − 1) = 2(r − 1) ≥ 0, τ1 , τ2 ∈ {0, 1}. © 2020 Elsevier Ltd. All rights reserved.
1. Introduction In this paper, we study the quasilinear attraction–repulsion chemotaxis ( ) ( ) ⎧ ut = ∇ · (D(u)∇u) − ∇ · Φ(u)∇v + ∇ · Ψ (u)∇w + f (u), ⎪ ⎪ ⎪ ⎪ ⎪ τ1 vt = ∆v + αu − βv, ⎪ ⎪ ⎨ τ2 wt = ∆w + γu − δw, ⎪ ∂v ∂w ∂u ⎪ ⎪ ⎪ = = = 0, ⎪ ⎪ ∂n ∂n ⎪ ⎩ ∂n u(x, 0) = u0 (x), τ1 v(x, 0) = τ1 v0 (x), τ2 w(x, 0) = τ2 w0 (x),
system (x, t) ∈ Ω × (0, T ), (x, t) ∈ Ω × (0, T ), (x, t) ∈ Ω × (0, T ),
(1.1)
(x, t) ∈ ∂Ω × (0, T ), x ∈ Ω,
✩ Supported by the National Natural Science Foundation of China (11701067, 11171048), Natural Science Foundation of Liaoning, China (2019-ZD-0180) and Key Research Plan of Henan Province Colleges, China (20B110021) ∗ Corresponding author. E-mail address: [email protected] (S. Zheng).
https://doi.org/10.1016/j.nonrwa.2020.103095 1468-1218/© 2020 Elsevier Ltd. All rights reserved.
2
X. He, M. Tian and S. Zheng / Nonlinear Analysis: Real World Applications 54 (2020) 103095
where Ω ⊂ RN (N ≥ 1) is a bounded domain with smooth boundary, n is the unit outer normal vector on the boundary, τ1 , τ2 ∈ {0, 1}, α, β, γ, δ > 0. The nonlinear functions D, Φ, Ψ ∈ C 2 ([0, ∞)) satisfy D(s) ≥ (s + 1)p , s ≥ 0, q
(1.2)
Φ(s) ≤ χs , s ≥ s0 ,
(1.3)
ξsr ≤ Ψ (s) ≤ ζsr , s ≥ s0 ,
(1.4)
with χ, ξ, ζ, s0 > 0, p, q, r ∈ R. The logistic source f (s) is smooth and satisfies f (0) ≥ 0, f (s) ≤ µs(1 − sk ) for s > 0
(1.5)
with µ, k > 0. In the model (1.1), u represents the cell density, v and w denote the concentrations of attractive and repulsive signals produced by the cells respectively. The logistic source f (u) included in (1.1) means that the cell density cannot grow unlimitedly. The properties of solutions would be determined by the interactions among the four nonlinear mechanisms—diffusion, attraction, repulsion and logistic source. By taking D(u) ≡ 1, Φ(u) = χu and Ψ (u) = f (u) ≡ 0 the system (1.1) just is the classical Keller–Segel chemotaxis system, which has been studied extensively since 1970 (refer to e.g. [1–6] and the references therein). To describe the limited-growth of cells, the effects of logistic sources were introduced subsequently, e.g., f satisfies (1.5) with D(u) ≡ 1, Φ(u) = χu, Ψ (u) ≡ 0 and k = 1 in (1.1). For the parabolic–elliptic system with logistic source, Tello and Winkler [7] showed that the solutions are globally bounded if µ > NN−2 χ. The same result was obtained for the parabolic–parabolic case with logistic source for either N = 2 [8], or N ≥ 3 with µ sufficiently large [9]. From then on, general quasilinear chemotaxis systems with logistic source were studied extensively. It has been obtained in [10,11] that if max{p + N2 , 1} > q, then the solution is global; if q = 1 (the balance case), the global boundedness of solutions can be ensured by µ > αχ(1 − 2/N (1 − p)+ ) for the parabolic–elliptic system [12], and µ > µ0 with some µ0 > 0 for the parabolic–parabolic system [13,14]. For the fully parabolic system with D(u) ≡ 1, Φ(u) = χu and Ψ (u) = ξu, the large time behavior was studied in [15]. Recently, people pay attention to the Keller–Segel systems with multi-species and multi-stimulus. Refer to [16–23] for details. This paper is a continuous work of our previous paper [24], where the criteria for global boundedness of solutions were established for the cases of τ2 = 0 with τ1 = 0, 1, depending on various dominations of the four nonlinear mechanisms (diffusion, attraction, repulsion and source). It has been shown that if τ1 = τ2 = 0, then the global boundedness of solutions would be ensured by one of the following conditions (i) q < max{r, k}; (ii) q − p < N2 ; (iii) q = max{r, k}, q − p ≥ N2 with (a) q = r = k, 2 µ > (αχ − γξ)(1 − N (q−p) )/(1 + N2(q−1) (q−p) ), or (b) q = r > k, αχ − γξ < 0, or (c) q = k > r, 2 µ > αχ(1 − N (q−p) )/(1 + N2(q−1) (q−p) ) ([24, Theorem 1]). Also, for τ2 = 0, τ1 = 1, the solutions would be globally bounded if one of the following is true (i) q < max{r, k}; (ii) q − p < N2 ; (iii) q = max{r, k}, and q − p ≥ N2 with (a) q = r = k, with µ and γξ large that µ + γξθ0 > µ0 , or (b) q = r > k, with γξ large that γξ > µ1 , or (c) q = k > r, with µ large that µ > µ0 ([24, Theorem 2]). In this paper, we at first establish the conditions for the other two cases of τ2 = 1 with τ1 = 0, 1, and then determine the large time behavior of solutions for all cases of τ1 , τ2 ∈ {0, 1}. These are stated in the following theorems.
X. He, M. Tian and S. Zheng / Nonlinear Analysis: Real World Applications 54 (2020) 103095
3
Theorem 1. Let τ1 = 0, τ2 = 1 or τ1 = τ2 = 1 in (1.1), Ω ⊂ RN (N ≥ 1) be a bounded domain with smooth boundary, D, Φ, Ψ and f satisfy (1.2)–(1.5), nonnegative initial data (u0 (x), v0 (x), w0 (x)) ∈ ¯ ) × W 1,σ (Ω ) × W 1,σ (Ω ) (σ > N ). C(Ω (i) If max{q, r} < max{k, p + N2 }, then Eq. (1.1) admits a globally bounded solution. (ii) Assume max{q, r} = k ≥ p + N2 . There exists µ0 > 0 such that if µ > µ0 , then the solution of Eq. (1.1) is globally bounded. Next theorem is on the large time behavior of solutions of Eq. (1.1) for the situation of D(s) = (s + 1)p , Φ(s) = χsq , Ψ (s) = ξsr , f (s) = µs(1 − s), s ≥ 0
(1.6)
p = 2(q − 1) = 2(r − 1) ≥ 0.
(1.7)
with χ, ξ > 0, and
Theorem 2. Let (u, v, w) be a globally bounded solution of (1.1) with τ1 , τ2 ∈ {0, 1}, Ω ⊂ RN (N ≥ 1) be a bounded domain with smooth boundary, D, Φ, Ψ , f satisfy (1.6). There exist µ1 , λ, C > 0 such that ∥u(·, t) − 1∥L∞ (Ω) + ∥v(·, t) −
α γ ∥L∞ (Ω) + ∥w(·, t) − ∥L∞ (Ω) ≤ Ce−λt β δ
(1.8)
provided µ > µ1 . The rest of the paper is arranged as follows. In Section 2 we introduce the local existence of solutions to (1.1) as preliminaries, and then prove Theorems 1 and 2 in Sections 3 and 4, respectively. 2. Preliminaries In this section, we give the known result on the local existence of solutions to Eq. (1.1) with necessary estimates as preliminaries. Lemma 2.1. Suppose Ω ⊂ RN (N ≥ 1) is a bounded domain with smooth boundary, D, Φ, Ψ and f satisfy ( ) ¯ ) as τ1 = τ2 = 0, or nonnegative pairs u0 (x), v0 (x) ∈ (1.2)–(1.5). Then for nonnegative u0 (x) ∈ C(Ω ( ) ¯ ) × W 1,σ (Ω ) (σ > N ) as τ1 = 1, τ2 = 0, u0 (x), w0 (x) ∈ C(Ω ¯ ) × W 1,σ (Ω ) (σ > N ) as τ1 = 0, τ2 = 1, C(Ω ( ) 1,σ 2 ¯ or nonnegative u0 (x), v0 (x), w0 (x) ∈ C(Ω ) × (W (Ω )) (σ > N ) as τ1 = τ2 = 1, there exist nonnegative ¯ × [0, Tmax )) ∩ C 2,1 (Ω ¯ × (0, Tmax )) with Tmax ∈ (0, ∞] classically solving (1.1). functions u, v, w ∈ C 0 (Ω Moreover, if Tmax < ∞, then lim ∥u(·, t)∥L∞ (Ω) = ∞. t→Tmax
The proof of Lemma 2.1 is standard under a suitable framework of the fixed point theory. Refer to [3,10,12,25] for the details. The following estimates were obtained in [24] with an idea from [26]. Lemma 2.2 (Lemma 2.2 [24]). Let (u, v, w) be a solution of (1.1) ensured by Lemma 2.1. Then ∫ {∫ } u ≤ max u0 , |Ω | := M, t ∈ (0, Tmax ). Ω
(2.1)
Ω
Moreover, for any η > 0, θ > 1, there is c0 = c0 (η, θ) > 0 such that ∫ ∫ vθ ≤ η uθ + c0 , t ∈ (0, Tmax ) Ω
Ω
(2.2)
X. He, M. Tian and S. Zheng / Nonlinear Analysis: Real World Applications 54 (2020) 103095
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with τ1 = 0, and
∫
wθ ≤ η
∫
Ω
with τ2 = 0.
uθ + c0 , t ∈ (0, Tmax )
(2.3)
Ω
□
3. Proof of Theorem 1 In this section, we deal with the global boundedness of solutions to the parabolic–parabolic–parabolic case with τ1 = τ2 = 1 and the parabolic–elliptic–parabolic case with τ1 = 0, τ2 = 1 in (1.1). The following lemma as a variation of Maximal Sobolev Regularity [27] is crucial for our proof. Lemma 3.1 (Lemma 2.2 [14]). Let σ > 1, a, b > 0. Consider the following evolution equation ⎧ ⎪ ⎪ Zt = ∆Z + au − bZ, (x, t) ∈ Ω × (0, T ), ⎨ ∂Z = 0, (x, t) ∈ ∂Ω × (0, T ), ⎪ ⎪ ⎩ ∂n Z(x, 0) = Z0 (x), x ∈ Ω. ( ) σ σ 0 For each Z0 ∈ W 2,σ (Ω ) (σ > N ) with ∂Z ∂n = 0 on ∂Ω and any u ∈ L (0, T ); L (Ω ) , there exists a unique solution ( ) ( ) Z ∈ W 1,σ (0, T ); Lσ (Ω ) ∩ Lσ (0, T ); W 2,σ (Ω ) . 0) Moreover, there exists Cσ > 0, such that if t0 ∈ [0, T ), Z(·, t0 ) ∈ W 2,σ (Ω ) (σ > N ) with ∂Z(·,t = 0, then ∂n ∫ T∫ ∫ T∫ ( ) b b b σ e 2 στ uσ + Cσ e 2 σt0 ∥Z(·, t0 )∥σLσ (Ω) + ∥∆Z(·, t0 )∥σLσ (Ω) . e 2 στ |∆Z| ≤ Cσ aσ
t0
t0
Ω
Ω
¯ ) with Given t0 ∈ (0, Tmax ) with t0 ≤ 1, we know from Lemma 2.1 that u(·, t0 ), v(·, t0 ), w(·, t0 ) ∈ C 2 (Ω ∂w(·,t0 ) ¯ > 0 such that = ∂n = 0 on ∂Ω . Pick M
∂v(·,t0 ) ∂n
¯, sup ∥u(·, τ )∥L∞ (Ω) , sup ∥v(·, τ )∥L∞ (Ω) , sup ∥w(·, τ )∥L∞ (Ω) ≤ M
0≤τ ≤t0
0≤τ ≤t0
0≤τ ≤t0
∥∆v(·, t0 )∥L∞ (Ω) , ∥∆w(·, t0 )∥L∞ (Ω)
(3.1)
¯. ≤M
Proof of Theorem 1. The key step of the proof is to show that for any l > 1, there exists c = c(l) > 0 such that ∫ ul ≤ c, t ∈ (0, Tmax ). (3.2) Ω
Without loss of generality, assume l > max{3 + 2|q|, 3 + 2|r|, 3 + |p|}. Suppose τ1 = τ2 = 1. Multiply (1.1)1 by (u + 1)l−1 and then integrate by parts on Ω , ∫ ∫ ∫ 1 d 2 (u + 1)l ≤ −(l − 1) (u + 1)l−2 D(u)|∇u| + (l − 1) (u + 1)l−2 Φ(u)∇u · ∇v l dt Ω ∫Ω ∫ Ω l−2 − (l − 1) (u + 1) Ψ (u)∇u · ∇w + µ u(1 − uk )(u + 1)l−1 Ω Ω ∫ ∫ l+p 2 4(l − 1) 2 | + (l − 1) ≤− |∇(u + 1) ∇h1 (u) · ∇v (l + p)2 Ω Ω ∫ ∫ ∫ l−1 − (l − 1) ∇h2 (u) · ∇w + µ u(u + 1) −µ uk+1 (u + 1)l−1 Ω Ω Ω ∫ ∫ ∫ l+p 2 4(l − 1) 2 | − (l − 1) =− |∇(u + 1) h (u)∆v + (l − 1) h2 (u)∆w 1 (l + p)2 Ω Ω Ω ∫ ∫ +µ u(u + 1)l−1 − µ uk+1 (u + 1)l−1 , t ∈ (0, Tmax ) Ω
Ω
(3.3)
X. He, M. Tian and S. Zheng / Nonlinear Analysis: Real World Applications 54 (2020) 103095
5
with ∫ h1 (u) :=
u
(s + 1)l−2 Φ(s)ds, h2 (u) :=
∫
u
(s + 1)l−2 Ψ (s)ds.
0
0
Meanwhile, by (1.3) and (1.4), there exist b1 = b1 (l), b2 = b2 (l) > 0 such that s0
∫
∫
u
χ (u + 1)l+q−1 , l+q−1 0 s0 ∫ s0 ∫ u ζ l−2 h2 (u) = (u + 1)l+r−1 (s + 1) Ψ (s)ds + (s + 1)l−2 Ψ (s)ds ≤ b2 + l+r−1 0 s0
h1 (u) =
(s + 1)l−2 Φ(s)ds +
(s + 1)l−2 Φ(s)ds ≤ b1 +
(3.4) (3.5)
with t ∈ (0, Tmax ). Thus by Young’s inequality with (3.3)–(3.5), we obtain 1 d l dt
4(l − 1) (u + 1) ≤ − (l + p)2 Ω
∫
l
∫ χ(l − 1) |∇(u + 1) | + (u + 1)l+q−1 |∆v| l+q−1 Ω Ω ∫ ∫ ∫ ζ(l − 1) (u + 1)l+r−1 |∆w| + b2 (l − 1) |∆w| + b1 (l − 1) |∆v| + l+r−1 Ω Ω Ω ∫ ∫ +µ u(u + 1)l−1 − µ uk+1 (u + 1)l−1 Ω Ω ∫ ∫ l+p 2 χ(l − 1) 4(l − 1) 2 | + ≤− |∇(u + 1) (u + 1)l+q (l + p)2 Ω l+q Ω ∫ ∫ ζ(l − 1) 2χ(l − 1) l+q |∆v| + (u + 1)l+r + (l + q − 1)(l + q) Ω l+r Ω ∫ ∫ ∫ 2ζ(l − 1) l+r l−1 |∆w| +µ u(u + 1) −µ uk+1 (u + 1)l−1 + (l + r − 1)(l + r) Ω Ω Ω + c1 , t ∈ (0, Tmax ) l+p 2 2
∫
(3.6)
with c1 = c1 (l) > 0. Multiplying (1.1)1 by ul−1 and then integrating by parts on Ω , similarly to the procedure for (3.6), we have ∫ ∫ ∫ ∫ χ(l − 1) 2χ(l − 1) ζ(l − 1) 1 d l+q ul ≤ ul+q + |∆v| + ul+r l dt Ω l+q (l + q − 1)(l + q) l + r Ω Ω ∫ ∫ Ω ∫ 2ζ(l − 1) l+r l l+k + |∆w| +µ u −µ u + c2 (l), t ∈ (0, Tmax ). (3.7) (l + r − 1)(l + r) Ω Ω Ω By Lemma 3.1, we know from (1.1)2 that for any given σ > 1, ∫ t∫ t0
β
σ
Ω
∫ t∫
( β β e 2 στ uσ + Cσ e 2 σt0 ∥v(·, t0 )∥σLσ (Ω) t0 Ω ) + ∥∆v(·, t0 )∥σLσ (Ω) , t ∈ (t0 , Tmax ).
e 2 στ |∆v| ≤ Cσ ασ
(3.8)
Notice that we cannot do the same thing as (3.8) directly for (1.1)3 , where the cases of δ = β and δ ̸= β will be considered separately. If δ = β, we know by Lemma 3.1 that ∫ t∫ t0
Ω
β
σ
∫ t∫
( β β e 2 στ uσ + Cσ e 2 σt0 ∥w(·, t0 )∥σLσ (Ω) t0 Ω ) + ∥∆w(·, t0 )∥σLσ (Ω) , t ∈ (t0 , Tmax ).
e 2 στ |∆w| ≤ Cσ γ σ
(3.9)
X. He, M. Tian and S. Zheng / Nonlinear Analysis: Real World Applications 54 (2020) 103095
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For δ ̸= β, rewrite (1.1)3 as wt = ∆w + γu − βw + (β − δ)w. By Lemma 3.1, ∫ t∫ e t0
β 2 στ
σ
∫ t∫
( β β σ e 2 στ |γu + (β − δ)w| + Cσ e 2 σt0 ∥w(·, t0 )∥σLσ (Ω) t0 Ω ) + ∥∆w(·, t0 )∥σLσ (Ω) ∫ t∫ ∫ t∫ β β ≤ Cσ (2γ)σ e 2 στ uσ + Cσ (2|β − δ|)σ e 2 στ wσ
|∆w| ≤ Cσ
Ω
t0
+ Cσ e
β 2 σt0
(
Ω
t0
∥w(·, t0 )∥σLσ (Ω)
+
Ω
∥∆w(·, t0 )∥σLσ (Ω)
)
, t ∈ (t0 , Tmax ).
(3.10)
From the Gagliardo–Nirenberg inequality and Lemma 2.2, we have 1−a a ∥w∥Lσ (Ω) ≤ cσ (∥∆w∥aLσ (Ω) ∥w∥L 1 (Ω) + ∥w∥L1 (Ω) ) ≤ cσ (∥∆w∥Lσ (Ω) + 1)
with cσ > 0 and a =
σ−1 σ−1+ 2σ N
∈ (0, 1). And then for any ϵ > 0, ∫
σ wσ ≤ cσ (∥∆w∥aσ Lσ (Ω) + 1) ≤ ϵ∥∆w∥Lσ (Ω) + cϵ,σ
Ω 1 2Cσ (2|β−δ|)σ .
with cϵ,σ > 0. Choose ϵ = ∫ t∫ e t0
β 2 στ
Together with (3.10), we know
∫ t∫
t β ( β u + Cσ e 2 στ + Cσ e 2 σt0 ∥w(·, t0 )∥σLσ (Ω) t0 Ω t0 ) + ∥∆w(·, t0 )∥σLσ (Ω) , t ∈ (t0 , Tmax ).
σ
|∆w| ≤ Cσ
Ω
e
β 2 στ
∫
σ
(3.11)
l−1 At first consider the item (i) with max{q, r} < max{k, p + N2 }. Notice that l−1 l+q , l+r < 2 and l−1 l−1 (l+q−1)(l+q) , (l+r−1)(l+r) < 1. If max{q, r} =: K < k, we get from (3.7) with Young’s inequality that ∫ ∫ ∫ ∫ ∫ ∫ β(l + K) 1 d l+K l+K ul + ul ≤ 2χ ul+K + 2χ |∆v| + 2ζ ul+K + 2ζ |∆w| l dt Ω 2l Ω Ω Ω Ω Ω ∫ ∫ ( β(l + K) ) l + µ+ u −µ ul+k + c3 2l Ω Ω ∫ ∫ ∫ µ l+K l+K ≤ 2χ |∆v| ul+k + c4 , t ∈ (t0 , Tmax ) + 2ζ |∆w| − 4 Ω Ω Ω
with c3 = c3 (l), c4 = c4 (l) > 0. Then for t ∈ (t0 , Tmax ), 1 l
∫
β
ul ≤ e− 2 (l+K)(t−t0 )
Ω
∫
t
+ 2ζ
1 l
∫
ul (·, t0 ) + 2χ
∫
Ω
β
β
e− 2 (l+K)(t−τ )
t0
e− 2 (l+K)(t−τ )
t0
t
∫ |∆w|
l+K
Ω
−
∫
l+K
|∆v| Ω
µ 4
∫
t
β
e− 2 (l+K)(t−τ )
t0
∫
ul+k +
Ω
2c4 . β(l + K)
Combine (3.8) and (3.1) to get ∫
t
e t0
β
− 2 (l+K)(t−τ )
∫
l+K
|∆v| Ω
∫
t
β
− 2 (l+K)(t−τ )
∫
β
≤ Cl e ul+K + Cl e− 2 (l+K)(t−t0 ) t0 Ω ( ) l+K × ∥v(·, t0 )∥l+K + ∥∆v(·, t )∥ 0 l+K (l+K) L (Ω) L (Ω) ∫ t ∫ β µ ≤ e− 2 (l+K)(t−τ ) ul+k + cl , t ∈ (t0 , Tmax ). 16χ t0 Ω
X. He, M. Tian and S. Zheng / Nonlinear Analysis: Real World Applications 54 (2020) 103095
7
The inequalities (3.9), (3.11) and (3.1) yield ∫ ∫ ∫ t ∫ t β β β l+K e− 2 (l+K)(t−τ ) |∆w| ≤ Cl e− 2 (l+K)(t−τ ) ul+K + Cl + Cl e− 2 (l+K)(t−t0 ) t0 Ω t Ω ( 0 ) l+K × ∥w(·, t0 )∥L(l+K) (Ω) + ∥∆w(·, t0 )∥l+K Ll+K (Ω) ∫ t ∫ β µ e− 2 (l+K)(t−τ ) ≤ ul+k + cl , t ∈ (t0 , Tmax ). 16ζ t0 Ω This concludes (3.2) that 1 l
∫
ul ≤
Ω
1 l
∫
ul (·, t0 ) + cl ≤ c5 , t ∈ (t0 , Tmax )
Ω
with c5 = c5 (l) > 0. For max{q, r} =: K < p + N2 , without loss of generality, assume K ≥ k. By Young’s inequality, it follows from (3.6) that ∫ ∫ ∫ l+p 2 1 d 4(l − 1) 2 | + 2(χ + ζ) (u + 1)l+K (u + 1)l ≤ − |∇(u + 1) l dt Ω (l + p)2 Ω ∫ ∫ ∫ Ω l+K l+K + 2χ |∆v| + 2ζ |∆w| + µ (u + 1)l + c6 (l), t ∈ (t0 , Tmax ). (3.12) Ω
Ω
Ω
We know from (2.1) with the Gagliardo–Nirenberg inequality, ∫ l+K l+p 2 (u + 1)l+K = ∥(u + 1) 2 ∥ l+p l+K 2 L l+p (Ω)
Ω
≤ c7 ∥∇(u + 1) ≤ c8 ∥∇(u + 1)
l+p 2
l+p 2
2a l+K
l+p ∥L2 (Ω) ∥(u + 1)
l+p 2
2(1−a) l+K l+p
∥
2 L l+p (Ω)
+ c7 ∥(u + 1)
l+p 2
2 l+K l+p
∥
2
L l+p (Ω)
2a l+K
l+p ∥L2 (Ω) + c8 , t ∈ (t0 , Tmax )
( (l+p) ) ( N (l+p) ) N with c7 = c7 (l), c8 = c8 (l) > 0 and a = N (l+p) −N ∈ (0, 1). Since K < p + N2 2 2(l+K) / 1 − 2 + 2 l+K implies a l+p < 1, we have by Young’s inequality with ε > 0 that ∫ l+p (u + 1)l+K ≤ ε∥∇(u + 1) 2 ∥2L2 (Ω) + c9 (3.13) Ω
for c9 = c9 (l, ε) > 0. So (3.12) and (3.13) yield ∫ ∫ 1 d β(l + K) l (u + 1) + (u + 1)l l dt Ω 2l Ω ∫ ∫ ∫ ∫ 4(l − 1) l+K l+K l+K l+K ≤− (u + 1) + 2(χ + ζ) (u + 1) + 2χ |∆v| + 2ζ |∆w| (l + p)2 ε Ω Ω Ω Ω ∫ ( β(l + K) ) l + µ+ (u + 1) + c10 2l Ω ∫ ∫ ∫ ( 4(l − 1) ) l+K l+K l+K ≤ − + 2(χ + ζ) + 1 (u + 1) + 2χ |∆v| + 2ζ |∆w| + c11 (l + p)2 ε Ω Ω Ω with c10 = c10 (l, ε), c11 = c11 (l, ε) > 0 for t ∈ (t0 , Tmax ). Combining (3.8), (3.9), (3.11) and (3.1), we obtain ∫ ∫ β 1 1 (u + 1)l ≤ e− 2 (l+K)(t−t0 ) (u(·, t0 ) + 1)l l Ω l Ω ∫ ∫ ( 4(l − 1) ) t − β (l+K)(t−τ ) 2 + 2(χ + ζ) + 1 e (u + 1)l+K + − (l + p)2 ε t0 Ω
8
X. He, M. Tian and S. Zheng / Nonlinear Analysis: Real World Applications 54 (2020) 103095
∫
t
+ 2χ
β
− 2 (l+K)(t−τ )
e t0
+
∫
∫
l+K
|∆v|
t
+ 2ζ
Ω
β
− 2 (l+K)(t−τ )
e
∫
t0
l+K
|∆w| Ω
2c11 (l, ε) β(l + K) β
≤ e− 2 (l+K)(t−t0 )
∫
1 l
(u(·, t0 ) + 1)l ∫ ∫ ( 4(l − 1) ) t − β (l+K)(t−τ ) 2 e + − + 2(χ + ζ) + 1 (u + 1)l+K (l + p)2 ε t0 Ω ∫ t ∫ β e− 2 (l+K)(t−τ ) + 2χCl+K αl+K ul+K + 2ζCl+K γ
l+K
Ω
∫
t0 t
Ω β
− 2 (l+K)(t−τ )
e t0
∫
ul+K + c12 , t ∈ (t0 , Tmax )
(3.14)
Ω
with c12 = c12 (l, ε) > 0. For l > max{3 + 2|q|, 3 + 2|r|, 3 + |p|}, choose ε > 0 small such that −
4(l − 1) + 2(χ + ζ) + 1 + 2χCl+K αl+K + 2ζCl+K γ l+K ≤ 0. (l + p)2 ε
This proves the desired (3.2) by (3.14). Next treat the item (ii) with k = max{q, r} ≥ p + N2 . Suppose k = q > r. Then by (3.7) and Young’s inequality with ε1 > 0, ∫ ∫ ∫ ∫ χ(l − 1) 2χ(l − 1) 1 d l+q l l+q u ≤ u + |∆v| + ε1 ul+q l dt Ω l+q (l + q − 1)(l + q) Ω Ω Ω ∫ ∫ ∫ l+q + ε1 |∆w| +µ ul − µ ul+q + c13 , t ∈ (t0 , Tmax ) Ω
Ω
Ω
with c13 = c13 (l, ε1 ) > 0, and hence, ∫ ∫ ∫ ∫ ∫ β(l + q) χ(l − 1) 2χ(l − 1) 1 d l+q ul + ul ≤ ul+q + |∆v| + ε1 ul+q l dt Ω 2l l+q (l + q − 1)(l + q) Ω Ω Ω Ω ∫ ∫ ∫ ( β(l + q) ) l+q l + ε1 |∆w| + µ+ u −µ ul+q + c14 2l Ω Ω Ω ∫ ∫ ∫ χ(l − 1) 2χ(l − 1) l+q ≤ ul+q + |∆v| + ε1 ul+q l+q (l + q − 1)(l + q) Ω Ω Ω ∫ ∫ ∫ l+q l+q + ε1 |∆w| + ε1 u −µ ul+q + c14 (l, ε1 ) Ω Ω Ω ∫ ∫ ( ) χ(l − 1) 2χ(l − 1) l+q =− µ− − 2ε1 ul+q + |∆v| l+q (l + q − 1)(l + q) Ω Ω ∫ l+q + ε1 |∆w| + c14 , t ∈ (t0 , Tmax ) Ω
with c14 = c14 (l, ε1 ) > 0. Similarly to (3.14), we have by the variation-of-constants formula with (3.8), (3.9), (3.11) and c15 = c15 (l, ε1 ) > 0 that ∫ ( ) 1 χ(l − 1) 2χ(l − 1) ul ≤ − µ − − 2ε1 − Cl+q αl+q − ε1 Cl+q γ l+q l Ω l+q (l + q − 1)(l + q) ∫ t ∫ −β(l+q)(t−τ ) 2 × e ul+q + c15 , t ∈ (t0 , Tmax ). (3.15) t0
Ω
Let µ > µ0,1 with µ0,1 :=
χ(l − 1) 2χ(l − 1) + Cl+q αl+q l+q (l + q − 1)(l + q)
(3.16)
and then choose ε1 small enough such that µ − µ0,1 − 2ε1 − ε1 Cl+q γ l+q > 0. This concludes (3.2) by (3.15).
X. He, M. Tian and S. Zheng / Nonlinear Analysis: Real World Applications 54 (2020) 103095
If k = q = r, we have by (3.7) that ∫ ∫ ∫ ∫ ζ(l − 1) 1 d χ(l − 1) 2χ(l − 1) l+q ul ≤ ul+q + |∆v| + ul+q l dt Ω l+q (l + q − 1)(l + q) l + q Ω Ω ∫ ∫ Ω ∫ 2ζ(l − 1) l+q l l+q + |∆w| +µ u −µ u + c2 , t ∈ (t0 , Tmax ). (l + q − 1)(l + q) Ω Ω Ω
9
(3.17)
By the variation-of-constants formula with (3.8), (3.9), (3.11) and (3.17), similarly to (3.15), we have ∫ ( 1 χ(l − 1) ζ(l − 1) 2χ(l − 1) ul ≤ − µ − − − ε1 − Cl+q αl+q l Ω l+q l+q (l + q − 1)(l + q) ∫ )∫ t β 2ζ(l − 1) l+q − 2 (l+q)(t−τ ) − Cl+q γ e ul+q + c16 (l, ε1 ), t ∈ (t0 , Tmax ). (3.18) (l + q − 1)(l + q) t0 Ω Let µ > µ0,2 with µ0,2 :=
2χ(l − 1) 2ζ(l − 1) χ(l − 1) ζ(l − 1) + + Cl+q αl+q + Cl+q γ l+q l+q l+q (l + q − 1)(l + q) (l + q − 1)(l + q)
and choose ε1 small enough such that µ − µ0,2 − ε1 > 0. Thus (3.2) is true by (3.18). If k = r > q, by (3.7) and Young’s inequality with ε2 > 0, we know ∫ ∫ ∫ ∫ ζ(l − 1) 1 d l+k ul ≤ ε2 ul+k + ε2 |∆v| + ul+k l dt Ω l+k Ω Ω Ω ∫ ∫ ∫ 2ζ(l − 1) l+k l+k |∆w| + ε2 u −µ ul+k + c17 , t ∈ (t0 , Tmax ) + (l + k − 1)(l + k) Ω Ω Ω
(3.19)
(3.20)
with c17 = c17 (l, ε2 ) > 0. Similarly to (3.18), we have by the variation-of-constants formula with (3.8), (3.9), (3.11) and (3.20) that ∫ ( ) 2ζ(l − 1) 1 ζ(l − 1) ul ≤ − µ − − 2ε2 − Cl+k γ l+k − ε2 Cl+k αl+k l Ω l+k (l + k − 1)(l + k) ∫ t ∫ β ul+k + c18 (l, ε2 ), t ∈ (t0 , Tmax ). (3.21) × e− 2 (l+k)(t−τ ) Ω
t0
Let µ > µ0,3 with µ0,3 :=
2ζ(l − 1) ζ(l − 1) + Cl+k γ l+k l+k (l + k − 1)(l + k)
(3.22)
and choose ε2 small enough such that µ − µ0,3 − 2ε2 − ε2 Cl+k αl+k > 0. We conclude (3.2) by (3.21). Define µ0 := max{µ0,1 , µ0,2 , µ0,3 } in the item (ii) of the theorem. This obtains the desired preliminary estimate (3.2). For the case of τ1 = 0, τ2 = 1, similarly to the proof of [24, Theorem 2] with τ1 = 1, τ2 = 0, we can get (3.2) as well. We omit the details here. Furthermore, by the framework of Moser iteration [13, Lemma A.1], there is l1 (N, p) > N > 0 such that the known ∥u(·, t)∥Ll1 (Ω) < ∞, t ∈ (τ0 , Tmax ) implies ∥u(·, t)∥L∞ (Ω) ≤ C, t ∈ (τ0 , Tmax ) for some C > 0. Now, take l = l1 in (3.16), (3.19) and (3.22). Together with (3.1) and (3.2), we know that ∥u(·, t)∥L∞ (Ω) ≤ C, t ∈ (0, Tmax ). By Lemma 2.1, this concludes Tmax = ∞. The boundedness of v and w is obtained by the standard elliptic and parabolic regularity theory directly. □
10
X. He, M. Tian and S. Zheng / Nonlinear Analysis: Real World Applications 54 (2020) 103095
Remark 1. Observe that the results of Theorem 1 with τ1 = 0, τ2 = 1 are not as accurate as those in [24, Theorem 2] with τ1 = 1, τ2 = 0. The possible positive contribution of the repulsion component to the global boundedness of solutions has not been shown yet in the theorem. This is because of the current technical difficulty for treating the case when the repulsion equation is parabolic in chemotaxis systems, corresponding to the cases with τ2 = 1 here. 4. Proof of Theorem 2 In this section, motivated by [28], we study the large time behavior of the globally bounded solutions. Proof of Theorem 2. The parabolic regularity [25] with the global boundedness of (u, v, w) indicates that there exist θ ∈ (0, 1) and K > 0 such that ∥u∥
C
2+θ,1+ θ 2
¯ (Ω×[t,t+1])
, ∥v∥
C
2+θ,1+ θ 2
¯ (Ω×[t,t+1])
, ∥w∥
C
2+θ,1+ θ 2
¯ (Ω×[t,t+1])
≤ K, t ≥ t0 > 0.
Define functionals A(t), B(t) and C(t) as ∫ ∫ ∫ α γ A(t) = (u − 1 − ln u), B(t) = η1 (v − )2 and C(t) = η2 (w − )2 β δ Ω Ω Ω
(4.1)
(4.2)
with D, Φ, Ψ and f taking the case of (1.6), and η1 , η2 > 0 to be determined. For the case of τ1 = τ2 = 1, denote E(t) := A(t) + B(t) + C(t). Since p = 2(q − 1) = 2(r − 1) ≥ 0, we have for ϵ1 ∈ (0, 1) that ∫ u−1 d A(t) = ut dt u ∫Ω ) u − 1( = ∇ · ((u + 1)p ∇u) − χ∇ · (uq ∇v) + ξ∇ · (ur ∇w) + µu(1 − u) u Ω ∫ ∫ ∫ ∫ (u + 1)p 2 q−2 r−2 |∇u| + χ u ∇u · ∇v − ξ u ∇u · ∇w − µ (u − 1)2 =− 2 u ∫Ω ∫ Ω ∫ Ω ∫ Ω χ2 2 2 2 2 p−2 2(q−2) ≤− u |∇u| + ϵ1 |∇v| + (1 − ϵ1 ) u2(r−2) |∇u| u |∇u| + 4ϵ1 Ω Ω Ω Ω ∫ ∫ ξ2 2 + |∇w| − µ (u − 1)2 4(1 − ϵ1 ) Ω ∫ ∫ Ω ∫ χ2 ξ2 2 2 = |∇v| + |∇w| − µ (u − 1)2 , (4.3) 4ϵ1 Ω 4(1 − ϵ1 ) Ω Ω ∫ d α B(t) = 2η1 (v − )vt dt β ∫Ω α = 2η1 (v − )(∆v + αu − βv) β Ω ∫ ∫ ∫ α α 2 (4.4) = −2η1 |∇v| + 2η1 α (v − )(u − 1) − 2η1 β (v − )2 , β β Ω Ω ∫ Ω d γ C(t) = 2η2 (w − )wt dt δ ∫Ω γ = 2η2 (w − )(∆w + γu − δw) δ Ω ∫ ∫ ∫ γ γ 2 = −2η2 |∇w| + 2η2 γ (w − )(u − 1) − 2η2 δ (w − )2 . (4.5) δ δ Ω Ω Ω
X. He, M. Tian and S. Zheng / Nonlinear Analysis: Real World Applications 54 (2020) 103095
Let η1 =
χ2 8ϵ1
and η2 =
ξ2 8(1−ϵ1 ) .
11
Then for any ϵ2 > 0, (4.3)–(4.5) imply that
∫ ∫ ∫ d χ2 α α ξ2γ γ E(t) ≤ (v − )(u − 1) + (w − )(u − 1) − µ (u − 1)2 dt 4ϵ1 Ω β 4(1 − ϵ1 ) Ω δ Ω ∫ ∫ χ2 β α 2 γ 2 ξ2δ − (v − ) − (w − ) 4ϵ1 Ω β 4(1 − ϵ1 ) Ω δ ∫ ∫ ∫ 2 2 2 α 2 χ α ξ 2 δ(1 − ϵ2 ) γ χ β(1 − ϵ2 ) 2 (v − ) + (u − 1) + (w − )2 ≤ 4ϵ1 β 16ϵ β(1 − ϵ ) 4(1 − ϵ ) δ 1 2 1 Ω Ω ∫ ∫ Ω 2 2 2 ∫ ξ γ α χ β + (u − 1)2 − µ (u − 1)2 − (v − )2 16(1 − ϵ1 )(1 − ϵ2 )δ Ω 4ϵ β 1 Ω Ω ∫ ξ2δ γ 2 − (w − ) 4(1 − ϵ1 ) Ω δ ∫ ∫ γ ξ 2 δϵ2 χ2 βϵ2 α 2 (w − )2 =− (v − ) − 4ϵ1 Ω β 4(1 − ϵ1 ) Ω δ ∫ ( ( χ2 α2 ξ 2 γ 2 )) 1 + (u − 1)2 . − µ− 16(1 − ϵ2 ) ϵ1 β (1 − ϵ1 )δ Ω Define g(ϵ) =
χ2 α2 ϵβ
+
ξ2 γ 2 (1−ϵ)δ .
We can take ϵ1 ∈ (0, 1) such that g(ϵ1 ) = inf g(ϵ).
(4.6)
0<ϵ<1
Let µ > µ1 with µ1 =
1 g(ϵ1 ), 16 2
2
2 2
2 2
µ1 ξ δϵ2 ξ γ 1 2 and choose ϵ2 > 0 such that µ − 1−ϵ > 0. By taking ϵ3 = min{ χ4ϵβϵ , 4(1−ϵ , µ − 16(1−ϵ ( χϵ1αβ + (1−ϵ )}, 2 1 1) 2) 1 )δ we have ∫ ( d α γ ) E(t) ≤ −ϵ3 (u − 1)2 + (v − )2 + (w − )2 . (4.7) dt β δ Ω
Integrating (4.7) from t0 to ∞, we get ∫ ∞∫ ( α γ ) E(t0 ) (u − 1)2 + (v − )2 + (w − )2 ≤ β δ ϵ3 Ω t0 by E(t) ≥ 0. Together with (4.1), this implies ∫ ( γ ) α lim (u − 1)2 + (v − )2 + (w − )2 = 0. t→∞ Ω β δ
(4.8)
The Gagliardo–Nirenberg inequality says N
2
N +2 N +2 ∥h∥L∞ (Ω) ≤ c∞ ∥h∥W 1,∞ (Ω) ∥h∥L2 (Ω)
(4.9)
γ for all h ∈ W 1,∞ (Ω ) ∩ L2 (Ω ). Choosing h as u − 1, v − α β and w − δ in (4.9) respectively, we have from (4.8) and (4.1) that
∥u(·, t) − 1∥L∞ (Ω) + ∥v(·, t) − By L’Hˆ opital’s rule,
α γ ∥L∞ (Ω) + ∥w(·, t) − ∥L∞ (Ω) → 0 as t → ∞. β δ
u − 1 − ln u 1 = , (u − 1)2 2 ∫ ∫ 1 (u − 1)2 ≤ A(t) ≤ (u − 1)2 , t > T1 4 Ω Ω lim
t→∞
and hence
X. He, M. Tian and S. Zheng / Nonlinear Analysis: Real World Applications 54 (2020) 103095
12
2
2
χ }. So, it follows from (4.7) that , ξ for some T1 > 0. Let d = max{1, 8ϵ 1 8(1−ϵ1 )
d E(t) ≤ −ϵ4 E(t), t > T1 dt with ϵ4 =
ϵ3 d ,
and thus E(t) ≤ E(T1 )e−ϵ4 (t−T1 ) , t > T1 .
Therefore 1 4
∫
(u − 1)2 +
Ω
χ2 8ϵ1
∫ (v − Ω
ξ2 α 2 ) + β 8(1 − ϵ1 )
∫ (w − Ω
γ 2 ) ≤ E(t) ≤ E(T1 )e−ϵ4 (t−T1 ) , t > T1 . δ
By using the Gagliardo–Nirenberg inequality (4.9) again together with (4.1), we have ϵ4
∥u(·, t) − 1∥L∞ (Ω) ≤ c0 e− N +2 (t−T1 ) , ϵ4 α ∥v(·, t) − ∥L∞ (Ω) ≤ c0 e− N +2 (t−T1 ) , β ϵ4 γ ∥w(·, t) − ∥L∞ (Ω) ≤ c0 e− N +2 (t−T1 ) , δ
(4.10) (4.11) (4.12)
with t > T1 and c0 > 0. For the case of τ1 = 1 and τ2 = 0, define E1 (t) = A(t) + B(t). Multiplying by w − γδ on (1.1)3 and integrating by parts, we get ∫ ∫ ∫ ∫ ∫ γ δ γ γ2 γ 2 (w − )2 + (u − 1)2 , |∇w| = γ (u − 1)(w − ) − δ (w − )2 ≤ − δ δ 2 δ 2δ Ω Ω Ω Ω Ω and thus
∫ (w − Ω
Taking η1 =
χ2 8ϵ1
γ 2 γ2 ) ≤ 2 δ δ
∫
(u − 1)2 .
Ω
in (4.4), together with (4.3), we have
∫ ∫ ∫ d ξ2γ γ ξ2δ γ E1 (t) ≤ (w − )(u − 1) − (w − )2 − µ (u − 1)2 dt 4(1 − ϵ1 ) Ω δ 4(1 − ϵ1 ) Ω δ Ω ∫ ∫ χ2 α α χ2 β α 2 + (v − )(u − 1) − (v − ) 4ϵ1 Ω β 4ϵ1 Ω β ∫ ∫ χ2 βϵ2 α 2 ( 1 ( χ2 α2 ξ 2 γ 2 )) ≤− (v − ) − µ − + (u − 1)2 4ϵ1 Ω β 16 ϵ1 β(1 − ϵ2 ) (1 − ϵ1 )δ Ω ∫ ∫ ( χ2 α2 α 2 ( 1 ξ 2 γ 2 )) χ2 βϵ2 ≤− (v − ) − µ − + (u − 1)2 . 4ϵ1 Ω β 16(1 − ϵ2 ) ϵ1 β (1 − ϵ1 )δ Ω Similar to the procedure of the case τ1 = τ2 = 1, we know with µ > µ1 that ∫ ∫ 1 χ2 α 2 (u − 1) + (v − )2 ≤ E1 (t) ≤ E1 (T1 )e−ϵ4 (t−T1 ) , t > T1 . 4 Ω 8ϵ1 Ω β It follows from (4.13) that ∫ (w − Ω
γ 2 4γ 2 ) ≤ 2 E1 (T1 )e−ϵ4 (t−T1 ) , t > T1 . δ δ
Then (4.10)–(4.12) can be derived by (4.9) and (4.1). For case of τ1 = 0 and τ2 = 1, the simple result from (1.1)2 that ∫ ∫ ∫ ∫ ∫ α α β α α2 2 |∇v| = α (u − 1)(v − ) − β (v − )2 ≤ − (v − )2 + (u − 1)2 β β 2 β 2β Ω Ω Ω Ω Ω
(4.13)
X. He, M. Tian and S. Zheng / Nonlinear Analysis: Real World Applications 54 (2020) 103095
implies
α2 α (v − )2 ≤ 2 β β Ω
∫
∫
(u − 1)2 .
13
(4.14)
Ω
This concludes (4.10)–(4.12) as well. If τ1 = τ2 = 0, (4.13) and (4.14) are obviously true. So we have ∫ ∫ ∫ d χ2 α α χ2 β α 2 γ ξ2γ A(t) ≤ (v − )(u − 1) − (v − ) + (w − )(u − 1) dt 4ϵ1 Ω β 4ϵ1 Ω β 4(1 − ϵ1 ) Ω δ ∫ ∫ 2 ξ δ γ − (w − )2 − µ (u − 1)2 4(1 − ϵ1 ) Ω δ Ω ∫ ( 2 2 ( 1 χ α ξ 2 γ 2 )) ≤− µ− + (u − 1)2 . 16 ϵ1 β (1 − ϵ1 )δ Ω By a similar procedure as that for the case of τ1 = τ2 = 1, we know with µ > µ1 that ∫ 1 (u − 1)2 ≤ A(t) ≤ A(T1 )e−ϵ4 (t−T1 ) , t > T1 . 4 Ω Together with (4.13) and (4.14), we obtain ∫ ∫ γ 2 4γ 2 α 4α2 −ϵ4 (t−T1 ) (w − ) ≤ 2 A(T1 )e , (v − )2 ≤ 2 A(T1 )e−ϵ4 (t−T1 ) , t > T1 , δ δ β β Ω Ω and hence the desired (4.10)–(4.12) by (4.9) and (4.1).
□
Acknowledgment The authors would like to express their sincere appreciation to the anonymous reviewer for his/her significant suggestions to complete the present version of the paper. References [1] M.A. Herrero, J.J.L. Vel´ azquez, A blow-up mechanism for a chemotaxis model, Ann. Sc. Norm. Super. 24 (1997) 633–683. [2] D. Horstmann, From 1970 until present: The Keller–Segel model in chemotaxis and its consequences I, Jber. DMV 105 (2003) 103–165. [3] D. Horstmann, M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations 215 (2005) 52–107. [4] W. J¨ ager, S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc. 329 (1992) 819–824. [5] M. Winkler, Aggregation. vs, Aggregation vs global diffusive behavior in the higher-dimensional Keller–Segel model, J. Differential Equations 248 (2010) 2889–2905. [6] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller–Segel system, J. Math. Pures Appl. 100 (2013) 748–767. [7] J.I. Tello, M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations 32 (2007) 849–877. [8] K. Osaki, A. Yagi, Global existence of a chemotaxis-growth system in R2 , Adv. Math. Sci. Appl. 12 (2002) 587–606. [9] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations 35 (2010) 1516–1537. [10] X.R. Cao, Boundedness in a quasilinear parabolic-parabolic Keller–Segel system with logistic source, J. Math. Anal. Appl. 412 (2014) 181–188. [11] M. Winkler, K.C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filing effect, Nonlinear Anal. 72 (2010) 1044–1064. [12] X.R. Cao, S.N. Zheng, Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source, Math. Methods Appl. Sci. 37 (2014) 2326–2330. [13] Y. Tao, M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller–Segel system with subcritical sensitivity, J. Differential Equations 252 (2012) 692–715. [14] C.B. Yang, X.R. Cao, Z.X. Jiang, S.N. Zheng, Boundedness in a quasilinear fully parabolic Keller–Segel system of higher dimension with logistic source, J. Math. Anal. Appl. 430 (2015) 585–591.
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