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Large time behavior of solutions to a fully parabolic attraction–repulsion chemotaxis system with logistic source
Nonlinear Analysis: Real World Applications 39 (2018) 261–277
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Nonlinear Analysis: Real World Applications www.elsevier.com/locate/nonrwa
Large time behavior of solutions to a fully parabolic attraction–repulsion chemotaxis system with logistic source Jing Lia , Yuanyuan Keb , Yifu Wangc, * a
College of Science, Minzu University of China, Beijing, 100081, PR China School of Information, Renmin University of China, Beijing, 100872, PR China c School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, PR China b
article
info
Article history: Received 14 May 2017 Received in revised form 5 July 2017 Accepted 8 July 2017 Available online 7 August 2017 Keywords: Chemotaxis Attraction–repulsion Boundedness asymptotic behavior
abstract This paper deals with an attraction–repulsion chemotaxis system with logistic source
{
ut = ∆u − χ∇ · (u∇v) + ξ∇ · (u∇w) + f (u), vt = ∆v + αu − βv, wt = ∆w + γu − δw,
x ∈ Ω , t > 0, x ∈ Ω , t > 0, x ∈ Ω, t > 0
under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ RN (N ≥ 1), where parameters χ, ξ, α, β, γ and δ are positive and f (s) = κs − µs1+k with κ ∈ R, µ > 0 and k ≥ 1. It is shown that the corresponding system possesses a unique global bounded classical solution in the cases k > 1 or k = 1 with µ > CN µ∗ for some µ∗ , CN > 0. Moreover, the large time behavior of solutions to the problem is also investigated. Specially speaking, when κ < 0 (resp. κ = 0), the corresponding solution of the system decays to (0, ( 0, 0) exponentially (resp. algebraically), and when κ > 0 the solution converges ) to
( κ )1/k µ
,
α β
( κ )1/k µ
,
γ δ
( κ )1/k µ
exponentially if µ is larger.
© 2017 Elsevier Ltd. All rights reserved.
1. Introduction This paper is concerned with the following attraction–repulsion chemotaxis system ⎧ ut = ∆u − χ∇ · (u∇v) + ξ∇ · (u∇w) + f (u), x ∈ Ω , t > 0, ⎪ ⎪ ⎪ ⎪ τ v = ∆v + αu − βv, x ∈ Ω , t > 0, t ⎪ ⎪ ⎨τ w = ∆w + γu − δw, x ∈ Ω , t > 0, t ∂v ∂w ∂u ⎪ ⎪ ⎪ = = = 0, x ∈ ∂Ω , t > 0, ⎪ ⎪ ∂ν ∂ν ∂ν ⎪ ⎩ u(x, 0) = u0 (x), τ v(x, 0) = v0 (x), τ w(x, 0) = w0 (x), x ∈ Ω
*
Corresponding author. E-mail address: [email protected] (Y. Wang).
http://dx.doi.org/10.1016/j.nonrwa.2017.07.002 1468-1218/© 2017 Elsevier Ltd. All rights reserved.
(1.1)
262
J. Li et al. / Nonlinear Analysis: Real World Applications 39 (2018) 261–277
in a bounded domain Ω ⊂ RN (N ≥ 1) with smooth boundary ∂Ω . The model (1.1) was proposed in [1] to describe the aggregation of microglia in Alzheimer’s disease, where unknown functions u, v and w denote the concentrations of Microglia, chemoattractant and chemorepellent which are produced by Microglia, respectively. The positive parameters χ and ξ are the chemotactic coefficients, α, β, γ and δ are chemical production and degradation rates. Here τ = 0, 1 is parameter, and the logistic source f describes the cell proliferation and death. It is noticed that (1.1) was also introduced in [2] to describe the quorum sensing effect in the chemotactic process. The first two equations of (1.1) with ξ = 0 comprise the Keller–Segel chemotaxis system { ut = ∆u − χ∇ · (u∇v) + f (u), x ∈ Ω , t > 0, (1.2) τ vt = ∆v + αu − βv, x ∈ Ω , t > 0. The most characteristic ingredient of this system is that it reflect the attractive chemotaxis through a nonlinear cross-diffusive term. Since the blow-up is an extreme facet of bacterial aggregation, considerable efforts have been devoted to identifying circumstances under which explosions are precluded and thus generates pattern formation which is applicable in realistic situation (see [3–12]). It is verified that the logistic source f (u) = µu(1 − u) with µ > 0 exerts a certain dampening influence which may suppress blow-up in many relevant situations: all of the solutions to (1.2) remain uniformly bounded globally in two dimensional bounded domain [13,14], alternatively in higher dimensions (N ≥ 3), the uniform-in-time boundedness of classical solutions to (1.2) with τ = 0, 1 is valid when f (u) ≤ a − µu2 for some a ≥ 0 and suitably large µ > 0 [14,15]. In particular, a more generalized logistic term f (u) = κu − µu2 with κ ∈ R and µ > 0 was considered in [3], where the existence of global weak solution to a three-dimensional chemotaxis system was proved for arbitrarily small values of µ > 0, which becomes the classical solution after some time if κ is not too large. As source term f is controlled by −c0 (u + uα ) and a − µuα with some a ≥ 0, µ, c0 > 0 and α < 2, the uniform-in-time boundedness of some weak solutions to (1.2) with τ = 1 is also shown in the recent paper [16]. It should be mentioned that the blow-up is possible in a slightly modified version of system (1.2) with logistic source [17]. On the other hand, in the absence of chemoattractant, the first and third equations of (1.1) with χ = 0 form a repulsive Keller–Segel model { ut = ∆u + ξ∇ · (u∇w) + f (u), x ∈ Ω , t > 0, (1.3) τ wt = ∆w + γu − δw, x ∈ Ω , t > 0. It should be mentioned that based on a Lyapunov functional identity, Cie´slak, Lauren¸cot and C. MoralesRodrigo [18] showed the global existence of classical solutions to (1.3) in two dimensions and weak solutions in three and four dimensions. Compared to the classical chemotaxis model, the mathematical analysis on the boundedness of the attraction–repulsion chemotaxis model (1.1) is much harder due to the complicated interactions between three species u, v and w and the difficulty of constructing a Lyapunov functional. For the model (1.1) with τ = 0 and f (u) ≡ 0, Tao and Wang [19] obtained the global boundedness of the solution in high-dimensions if ξγ > χα; while if ξγ < χα, the solution might blow up in the two-dimensional case if the cell mass is larger than a threshold number. Recently, Jin and Wang [20] investigated the boundedness, blowup and critical mass phenomenon in the parabolic–parabolic–elliptic case of model (1.1) (i.e., τ = 1 in the second equation of (1.1), while τ = 0 in the third one). As for the model (1.1) with logistic source, based on the Lp energy estimates and Moser iteration, Zhang and Li [21] proved the existence of global classical solutions (N −2)+ of (1.1) with τ = 0, f (u) = µu(1 − u) whenever µ > (χα − ξγ), and also discussed the asymptotic N behavior of solutions when µ > 2χα. It is also noticed that for (1.1) with τ = 0, f (u) ≤ a − µuk+1 , Li and Xiang [22] proved that the √ classical solution will exist globally for all N ≥ 2 in the case k > 1. Moreover, N 2 +4N −N
when ξγ = χα and k > , the classical solution is uniformly bounded. Additionally, in [23–25] the 2 global boundedness were considered for the attraction–repulsion chemotaxis system with nonlinear diffusion.
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To the best of our knowledge, although there is only difference of the second and third equations between the full attraction–repulsion chemotaxis model (1.1) with τ = 1 and the model (1.1) with τ = 0, the mathematical analysis of the full model seems to be more difficult. For model (1.1) with τ = 1 and f (u) ≡ 0, Tao and Wang [19] obtained the convergence of solutions to the trivial stationary solution in two dimensions when ξγ > χα and β = δ, while in the case β ̸= δ, they used an entropy inequality to establish the ∫ boundedness of solutions with the small initial mass Ω u0 (x)dx. It should be remarked that the smallness assumption on the initial mass is removed in [26,27], respectively; however, when N ≥ 3 and ξγ > χα, the global existence of classical solutions for (1.1) is still open. As for the critical case ξγ = χα, Lin et al. [23,28] investigated that the solution of (1.1) is globally bounded, and proved that the global solution of small ∫ γ 1 initial data converges to (u0 , α β u0 , δ u0 ) where u0 = |Ω| Ω u0 (x)dx for N = 2, 3. It seems that up to now, the result on (1.1) with f (u) ̸≡ 0 is far from satisfactory [5,29,30]. In one-dimension, Jin and Wang [31] derived a uniform-in-time bound of solutions to (1.1) and proved that the model possesses a global attractor. The existence of global bounded solutions to (1.1) in two-dimensional case was obtained in [22,32] by the different method, respectively. As for N = 3 and f (u) = u(1 − µuk ) with k ≥ 1 and µ > 0, it is shown in [29] that (1.1) admits a unique global bounded solution under the conditions {( )k ( )k } 1 1 41 41 β≥ , δ≥ and µ ≥ max χα + 9ξγ , ξγ + 9χα . (1.4) 2 2 2 2 In addition, the convergence of the solution was discussed when k ∈ N and u0 ̸≡ 0; however, the convergence rate was left as an open problem there. In the higher-dimensional setting N ≥ 3, Zheng et al. [5] proved that (1.1) admits a unique globally bounded classical solution provided that β = δ and µ > θ0 (χα + ξγ) for some constant θ0 > 0. It is also noted that when χα = ξγ, the boundedness and large time behavior of solutions to (1.1) were considered in the recent paper [30]. The purpose of this paper is to mathematically understand the effect of the attractive signal, the repulsive one and the logistic source on the global boundedness and asymptotic behavior of solutions to the full model (1.1) in the higher-dimensional setting. Unless otherwise stated, we will always consider model (1.1) with τ = 1 in the sequel, and assume that the nonnegative initial data u0 , v0 , w0 ∈ W 1,∞ (Ω ) with u0 ̸≡ 0, and the smooth function f satisfies f (s) = κs − µsk+1
with κ ∈ R, µ > 0 and k ≥ 1 for s ≥ 0.
(1.5)
Here the term κs being the difference between birth rate and death rate of the population, is used to describe population growth, and the term −µsk+1 models additional overcrowding preventing effects. We now state the main results of this paper. Our first theorem indicates that the boundedness of solutions can be ensured by the sufficiently strong dampening effect. Theorem 1.1. Let (1.5) hold with k > 1 and N ≥ 3. Then for any µ > 0, (1.1) with τ = 1 admits a unique global bounded classical solution. Theorem 1.2. Let k = 1 in (1.5) and N ≥ 3. Then there exists CN > 0 such that for µ > CN µ∗ with { max{|χα − ξγ|, αχ(β − δ)}, if β ≥ δ, ∗ µ = max{|χα − ξγ|, ξγ(δ − β)}, if β < δ, (1.1) with τ = 1 admits a unique global bounded classical solution. It is observed that the chemorepellent w has a dispersal effect in chemotactic movement in contrast to the aggregation effect of the chemoattractant v. Indeed, it follows from Theorem 1.2 that if β = δ, then
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(1.1) possesses a unique global bounded solution when µ > CN |χα − ξγ|. Hence Theorem 1.2 improves the results of [5,29] in some sense. The following two theorems are concerning the asymptotic of solutions of (1.1) with τ = 1 in the cases of κ ≤ 0 and κ > 0, respectively. Theorem 1.3. Let (1.5) hold with κ ≤ 0 and k ≥ 1 and (u, v, w) be the solution of (1.1). Then if { 0, if k > 1, µ> CN µ∗ , if k = 1, where CN and µ∗ defined in Theorem 1.2, there exists C > 0 such that for all t > 0, ( ∥u(·, t)∥L∞ (Ω) + ∥v(·, t)∥L∞ (Ω) + ∥w(·, t)∥L∞ (Ω) ≤ C
µkt |Ω |
k
+ (∫ Ω
)−
1
1 (N +1)k
)k u0 (x)
for κ = 0 and κt
∥u(·, t)∥L∞ (Ω) ≤ Ce N +1 , ∥v(·, t)∥L∞ (Ω) ≤ Ce
max{κ,−β}t N +1
and ∥w(·, t)∥L∞ (Ω) ≤ Ce
max{κ,−δ}t N +1
for κ < 0. As κ > 0, inspired by the arguments in [30,33], we prove that the solution of (1.1) converges ((for) the case ( )1/k ( )1/k ) 1/k α κ κ ,β µ , γδ µκ exponentially if µ is larger, which solves the problem proposed by the to µ authors of [29]. Theorem 1.4. Let (1.5) hold with κ > 0 and k ≥ 1 and (u, v, w) be the solution of (1.1). Then if { µ0 , if k > 1, µ> max{CN µ∗ , µ0 }, if k = 1, where CN and µ∗ are defined in Theorem 1.2, and ( µ0 =
(χα − ξγ)2 2(αχ)2 (β − δ)2 + 2δ δβ 2
)k/2
κ1−k/2 ,
there exist some C > 0 and D > 0 such that for all t > 0, ( )1/k ( )1/k ( )1/k κ α κ γ κ + v(·, t) − + w(·, t) − u(·, t) − ∞ ∞ µ β µ δ µ L
(Ω)
L
(Ω)
Dt
≤ Ce− N +2 .
L∞ (Ω)
This paper is organized as follows. In Section 2, we recall some preliminary results. In Section 3, with the aid of the maximal Sobolev regularity, we consider the boundedness of solutions to (1.1) in the case k > 1 and k = 1, respectively. In Section 4, we show the uniform convergence of solution to (1.1) with convergence rate in the case κ ≤ 0 and κ > 0, respectively. Throughout this paper, we use C and Ci (i ∈ N) to denote generic constants which may vary in the ∫ ∫ context. Moreover, for simplicity, the integral Ω f (x)dx is written as Ω f (x).
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2. Local existence and basic properties Firstly, we state a result concerning the local existence of classical solutions, which can be proved by well-established methods involving standard parabolic regularity theory and an appropriate fixed point framework (see [19]). Lemma 2.1. Let Ω ⊂ RN ( N ≥ 3) be a bounded domain with smooth boundary, and fix q ∈ (N, ∞). Suppose that the nonnegative functions u0 , v0 and w0 ∈ W 1,∞ (Ω ). There exist Tmax ∈ (0, ∞] and a unique triple of non-negative functions (u, v, w) satisfying the inclusions u ∈ C 0 ([0, Tmax ); C 0 (Ω )) ∩ C 2,1 (Ω × (0, Tmax )), 1,q (v, w) ∈ [C 0 ([0, Tmax ); C 0 (Ω )) ∩ L∞ (Ω )) ∩ C 2,1 (Ω × (0, Tmax ))]2 , loc ([0, Tmax ); W
which solves (1.1) classically in Ω × (0, Tmax ). Moreover, if Tmax < ∞, then ( ) lim ∥u(·, t)∥L∞ (Ω) + ∥v(·, t)∥W 1,∞ (Ω) + ∥w(·, t)∥W 1,∞ (Ω) = ∞. t→Tmax
As in Lemma 2.2 of [6], by introducing ω(x, t) = eεt z(x, t) and applying the maximal Sobolev regularity (Theorem 3.1 of [34]) to ω, we obtain the following lemma, which will play an important role in the proof of our main result. Lemma 2.2 ([7, Lemma 2.1] [6, Lemma 2.2]). Let r ⎧ ⎪ ⎪zt = ∆z − εz + f, ⎨ ∂z = 0, ⎪ ∂ν ⎪ ⎩ z(x, 0) = z0 (x),
∈ (N, ∞) and consider the following evolution equation (x, t) ∈ Ω × (0, t), (x, t) ∈ ∂Ω × (0, t),
(2.1)
x ∈ Ω.
r r 0 Then for each z0 ∈ W 2,r (Ω ) with ∂z ∂ν = 0 on ∂Ω and any f ∈ L ((0, t), L (Ω )), (2.1) admits a unique mild solution z ∈ W 1,r ((0, t); Lr (Ω )) ∩ Lr ((0, t); W 2,r (Ω )), which is given by ∫ τ −τ τ ∆ z(τ ) = e e z0 + e−(τ −s) e(τ −s)∆ f (s)ds, τ ∈ [0, t], 0 τ∆
where e is the semigroup generated by the Neumann Laplacian. Moreover, there exists Cr > 0, such that 0) if τ0 ∈ [0, t), z(·, τ0 ) ∈ W 2,r (Ω ) (r > N ) with ∂z(·,τ = 0, then ∂n ∫ t∫ ∫ t∫ r r r eεrτ (|z| + |∆z| ) ≤ Cr eεrτ |f | + Cr eεrt (∥z(·, τ0 )∥rLr (Ω) + ∥∆z(·, τ0 )∥rLr (Ω) ). (2.2) τ0
Ω
τ0
Ω
The following extended version of the Gagliardo–Nirenberg inequality will be frequently used in the later analysis (see [35]). Lemma 2.3. Assume that r ∈ (0, p), 1 ≤ p, q ≤ ∞ and satisfying (N − q)p < N q. Then, for any ψ ∈ W 1,q (Ω ) ∩ Lr (Ω ), there exists a constant CGN > 0 only depending on N, q, r and Ω such that ( ) ∗ ∗ r ∥ψ∥Lp (Ω) ≤ CGN ∥∇ψ∥λLq (Ω) ∥ψ∥1−λ (2.3) Lr (Ω) + ∥ψ∥L (Ω) holds with λ∗ ∈ (0, 1) satisfying λ∗ =
N r
1−
− N q
N p
+
N r
.
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The following Lemma shows the boundedness criterion of solution for the system (1.1), and we refer the readers to Lemma 2.4 of [8] for details. Lemma 2.4. Let (u, v, w) be a solution of (1.1) defined on [0, Tmax ). If there exists p0 > N/2 such that ∥u(·, t)∥Lp0 (Ω) ≤ C
for all t ∈ [0, Tmax ),
there exists some constant C > 0 independent of t such that ∥u(·, t)∥L∞ (Ω) + ∥v(·, t)∥W 1,∞ (Ω) + ∥w(·, t)∥W 1,∞ (Ω) ≤ C
for all t ∈ [0, Tmax ).
(2.4)
3. Boundedness To address our approaches, we see that under the transformation h := χv − ξw, when β > δ, the system (1.1) becomes ⎧ ut = ∆u − ∇ · (u∇h) + f (u), x ∈ Ω , t > 0, ⎪ ⎪ ⎪ ⎪ h = ∆h + (χα − ξγ)u − δh − χ(β − δ)v, x ∈ Ω , t > 0, t ⎪ ⎪ ⎨v = ∆v + αu − βv ≤ ∆v + αu − δv, x ∈ Ω , t > 0, t (3.1) ∂u ∂h ∂v ⎪ ⎪ ⎪ = = = 0, x ∈ ∂Ω , t > 0, ⎪ ⎪ ∂ν ∂ν ⎪ ⎩ ∂ν u(x, 0) = u0 (x), h(x, 0) = h0 (x) = χv0 (x) − ξw0 (x), v(x, 0) = v0 (x), x ∈ Ω . When β = δ, (1.1) is reduced to the subsystem ⎧ ut = ∆u − ∇ · (u∇h) + f (u), x ∈ Ω , t > 0, ⎪ ⎪ ⎪ ⎪ h = ∆h + (χα − ξγ)u − δh, x ∈ Ω , t > 0, ⎨ t ∂u ∂h ⎪ = = 0, x ∈ ∂Ω , t > 0, ⎪ ⎪ ∂ν ∂ν ⎪ ⎩ u(x, 0) = u0 (x), h(x, 0) = h0 (x) = χv0 (x) − ξw0 (x), x ∈ Ω . When β < δ, the system (1.1) becomes ⎧ ⎪ ⎪ut = ∆u − ∇ · (u∇h) + f (u), ⎪ ⎪ ⎪ ⎪ht = ∆h + (χα − ξγ)u − βh − ξ(β − δ)w, ⎨ wt = ∆w + γu − δw ≤ ∆w + γu − βw, ∂h ∂w ∂u ⎪ ⎪ ⎪ = = = 0, ⎪ ⎪ ∂ν ∂ν ∂ν ⎪ ⎩ u(x, 0) = u0 (x), h(x, 0) = h0 (x) = χv0 (x) − ξw0 (x),
x ∈ Ω , t > 0, x ∈ Ω , t > 0, x ∈ Ω , t > 0,
(3.2)
(3.3)
x ∈ ∂Ω , t > 0, w(x, 0) = w0 (x),
x ∈ Ω.
Now we are going to prove the results on boundedness of solutions to (1.1). Since the regularity obtained in Lemma 2.2 requires that the initial data to be in the space W 2,r (Ω ) and satisfy homogeneous Neumann boundary conditions, we will perform a small time shift and thus use the positive time as the “initial time”. Lemma 3.1. Let r ∈ (1, ∞). There exists Cr > 0 and C > 0, such that if τ0 ∈ [0, Tmax ), h(·, τ0 ) ∈ W 2,r (Ω ) 0) (r > N ) with ∂h(·,τ = 0, then for β ≥ δ, for any t ∈ (τ0 , Tmax ), we have ∂n ∫ t∫ ∫ t∫ r r r eδrτ |∆h| ≤ Cr (|χα − ξγ| + |χα(β − δ)| ) eδrτ ur τ0 Ω τ0 Ω ( + Cr eδrt ∥h(·, τ0 )∥rLr (Ω) + ∥∆h(·, τ0 )∥rLr (Ω) ) r + |χ(β − δ)| (∥v(·, τ0 )∥rLr (Ω) + ∥∆v(·, τ0 )∥rLr (Ω) ) . (3.4)
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While for β < δ, ∫
t
∫ t∫ r r r eβrτ |∆h| ≤ Cr (|χα − ξγ| + |ξγ(β − δ)| ) eβrτ ur Ω τ0 Ω ( + Cr eβrt ∥h(·, τ0 )∥rLr (Ω) + ∥∆h(·, τ0 )∥rLr (Ω)
∫
τ0
) r + |ξ(β − δ)| (∥w(·, τ0 )∥rLr (Ω) + ∥∆w(·, τ0 )∥rLr (Ω) ) .
(3.5)
Proof . For β = δ, from (3.2) and Lemma 2.2, we obtain ∫ t∫ ∫ t∫ r r eδrτ |∆h| ≤ Cr |χα − ξγ| eδrτ ur τ0
τ0
Ω
+ Cr e
δrt
Ω (∥h(·, τ0 )∥rLr (Ω)
+ ∥∆h(·, τ0 )∥rLr (Ω) ).
For β > δ, from (3.1) and Lemma 2.2, we obtain ∫ ∫ t∫ ∫ t∫ r r r eδrτ ur + Cr |χ(β − δ)| eδrτ |∆h| ≤ Cr |χα − ξγ| Ω
τ0
Ω δrt Cr e (∥h(·, τ0 )∥rLr (Ω)
Now we try to estimate the term
∫t ∫ τ0
Ω
∫
eδrτ v r
Ω
τ0
τ0
+
t
+ ∥∆h(·, τ0 )∥rLr (Ω) ).
eδrτ v r . Denote s(x, t) the solution of
⎧ ⎪ ⎪st = ∆s + αu − δs, (x, t) ∈ Ω × (τ0 , Tmax ), ⎨ ∂s = 0, (x, t) ∈ ∂Ω × (τ0 , Tmax ), ⎪ ∂ν ⎪ ⎩ s(x, τ0 ) = v(x, τ0 ), x ∈ Ω .
(3.6)
Using again Lemma 2.2, we obtain for any t ∈ (τ0 , Tmax ), ∫ t∫ ∫ t∫ δrτ r r e s ≤ Cr α eδrτ ur + Cr eδrt (∥v(·, τ0 )∥rLr (Ω) + ∥∆v(·, τ0 )∥rLr (Ω) ). τ0
Ω
τ0
Ω
On the other hand, noticing the nonnegativity of function v, ∂v = ∆v + αu − βv ≤ ∆v + αu − δv, ∂t
(x, t) ∈ Ω × (τ0 , Tmax ),
we obtain v ≤ s in Ω × [0, Tmax ] by comparison principle, and thus ∫ t∫ ∫ t∫ ∫ t∫ eδrτ v r ≤ eδrτ sr ≤ Cr αr eδrτ ur + Cr eδrt (∥v(·, τ0 )∥rLr (Ω) + ∥∆v(·, τ0 )∥rLr (Ω) ), τ0
Ω
τ0
Ω
τ0
Ω
from which (3.4) follows. For β < δ, by the similar procedure, it follows from (3.3) that ∫ t∫ ∫ t∫ ∫ r r r eβrτ |∆h| ≤ Cr |χα − ξγ| eβrτ ur + Cr |ξ(β − δ)| τ0
τ0
Ω βrt Cr e (∥h(·, τ0 )∥rLr (Ω)
Ω
+
t
τ0
∫
eβrτ wr
Ω
+ ∥∆h(·, τ0 )∥rLr (Ω) )
and ∫
t
∫ e
τ0
Ω
Then (3.5) follows.
βrτ
r
w ≤ Cr γ
r
∫
t
τ0
∫ Ω
eβrτ ur + Cr eβrt (∥w(·, τ0 )∥rLr (Ω) + ∥∆w(·, τ0 )∥rLr (Ω) ).
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Lemma 3.2. Let (1.5) hold with k > 1. For any p > 1, there is C = C(p) > 0 such that for t ∈ (0, Tmax ) ∫ up (x, t)dx ≤ C. Ω
Proof . Taking up−1 as a test function for the first equation of (1.1) and let h = χv − ξw, we obtain ∫ ∫ ∫ ∫ 1 d up = − ∇up−1 ∇u + ∇up−1 · u∇h + up (κ − µuk ) p dt Ω Ω Ω Ω ∫ ∫ p−1 ≤− up ∆h + up (κ − µuk ) p Ω [ Ω ∫ ] ∫ ∫ p 1 p−1 p+1 p+1 u + |∆h| + ≤ up (κ − µuk ). (3.7) p p+1 Ω p+1 Ω Ω ∫ p u on both sides of (3.7), we obtain For the case of β ≥ δ, adding (p+1)δ p Ω ( )∫ ∫ ∫ ∫ 1 d (p + 1)δ (p + 1)δ p−1 up + up ≤ κ + up + up+1 p dt Ω p p p + 1 Ω Ω Ω ∫ ∫ p−1 p+1 p+k |∆h| −µ u + p(p + 1) Ω Ω and then t
[ ∫ u ≤e u (τ0 ) + e (pκ + (p + 1)δ) up Ω Ω τ0 Ω ] ∫ ∫ ∫ (p − 1)p p−1 p+1 p+1 p+k + u − µp u + |∆h| . p+1 Ω p+1 Ω Ω
∫
p
−δ(p+1)(t−τ0 )
∫
∫
p
−δ(p+1)(t−τ )
(3.8)
Then from (3.4), we obtain ∫ ∫ p−1( p −δ(p+1)(t−τ0 ) u ≤e up (τ0 ) + Cp+1 ∥h(·, τ0 )∥pLp (Ω) + ∥∆h(·, τ0 )∥pLp (Ω) p + 1 Ω Ω [ ∫ ) ∫ t p + |χ(β − δ)| (∥v(·, τ0 )∥pLp (Ω) + ∥∆v(·, τ0 )∥pLp (Ω) ) + e−δ(p+1)(t−τ ) (pκ + (p + 1)δ) up Ω τ0 ] ∫ ( ( )) p − 1 ∫ p+1 p+1 p+k p+1 u − pµ u . (3.9) + p + Cp+1 |χα − ξγ| + (αχ(β − δ)) p+1 Ω Ω Similarly, for β < δ, we obtain ∫ ∫ p−1( up ≤ e−β(p+1)(t−τ0 ) up (τ0 ) + Cp+1 ∥h(·, τ0 )∥pLp (Ω) + ∥∆h(·, τ0 )∥pLp (Ω) p+1 Ω Ω [ ∫ ) ∫ t p + |ξ(β − δ)| (∥w(·, τ0 )∥pLp (Ω) + ∥∆w(·, τ0 )∥pLp (Ω) ) + e−β(p+1)(t−τ ) (pκ + (p + 1)β) up τ0 Ω ] ∫ ( ( )) p − 1 ∫ p+1 + p + Cp+1 |χα − ξγ| + (ξγ(δ − β))p+1 up+1 − pµ up+k . (3.10) p+1 Ω Ω Noticing by Young’s inequality, for any ε > 0, there exists C(ε) > 0 such that ∫ ∫ up ≤ ε up+k + C(ε)|Ω | Ω
(3.11)
Ω
and ∫ Ω
up+1 ≤ ε
∫ Ω
up+k + C(ε)|Ω |.
(3.12)
J. Li et al. / Nonlinear Analysis: Real World Applications 39 (2018) 261–277
By choosing ε sufficiently small in (3.9) and (3.10), there exists C > 0 such that proved.
∫ Ω
269
up ≤ C. The lemma is
Lemma 3.3. Let (1.5) hold with k = 1 and N ≥ 3. There exists CN > 0 and { max{|χα − ξγ|, αχ(β − δ)}, if β ≥ δ, µ∗ = max{|χα − ξγ| + ξγ(δ − β)}, if β < δ such that if µ > CN µ∗ , ∫
uN (x, t)dx ≤ C
Ω
for t ∈ (0, Tmax ) and some constant C > 0. Proof . By introducing ε to Young’s inequality in (3.7), then instead of (3.9) and (3.10), we obtain ∫ ∫ p−1( ∥h(·, τ0 )∥pLp (Ω) + ∥∆h(·, τ0 )∥pLp (Ω) up ≤ e−δ(p+1)(t−τ0 ) up (τ0 ) + Cp+1 p+1 Ω Ω [ ∫ ) ∫ t p p p −δ(p+1)(t−τ ) + |χ(β − δ)| (∥v(·, τ0 )∥Lp (Ω) + ∥∆v(·, τ0 )∥Lp (Ω) ) + e (pκ + (p + 1)δ) up τ0 Ω ) ] ( ∫ ∫ p+1 |χα − ξγ| + (αχ(β − δ))p+1 p − 1 up+1 − pµ up+1 (3.13) + pε + Cp+1 εp p+1 Ω Ω for β ≥ δ and ∫ ∫ p−1( ∥h(·, τ0 )∥pLp (Ω) + ∥∆h(·, τ0 )∥pLp (Ω) up ≤ e−β(p+1)(t−τ0 ) up (τ0 ) + Cp+1 p+1 Ω Ω [ ∫ ) ∫ t p + |ξ(β − δ)| (∥w(·, τ0 )∥pLp (Ω) + ∥∆w(·, τ0 )∥pLp (Ω) ) + e−β(p+1)(t−τ ) (pκ + (p + 1)β) up τ0 Ω ) ] ( ∫ ∫ p+1 |χα − ξγ| + (ξγ(δ − β))p+1 p − 1 p+1 p+1 u − pµ u (3.14) + pε + Cp+1 εp p+1 Ω Ω for β < δ. Denote ⎧[ ( )] p − 1 ⎪ ⎨ pεp+1 + Cp+1 |χα − ξγ|p+1 + (χα(β − δ))p+1 − pµεp , if β ≥ δ, p + 1 ( )] p − 1 g(ε) = [ ⎪ ⎩ pεp+1 + Cp+1 |χα − ξγ|p+1 + (ξγ(δ − β))p+1 − pµεp , if β < δ, p+1 then g attains its minimum at ε∗ = ∗
µ =
pµ p+1 .
{
Hence let
max{|χα − ξγ|, αχ(β − δ)}, if β ≥ δ, max{|χα − ξγ|, ξγ(δ − β)}, if β < δ,
one can see that for p = N , there exists a constant CN > 0 such that g(ε∗ ) < 0 provided that µ > CN µ∗ , ∫ and thereby the coefficient of Ω uN +1 in the right side of (3.14) is negative, so there is C > 0 such that ∫ uN ≤ C Ω
for all t ∈ (0, Tmax ), and Lemma 3.3 is thus proved. Now we can collect our previous findings to arrive at the main results.
J. Li et al. / Nonlinear Analysis: Real World Applications 39 (2018) 261–277
270
Proof of Theorem 1.1 and Theorem 1.2. In view of Lemmas 2.1 and 2.4, Theorems 1.1 and 1.2 are direct consequences of Lemmas 3.2 and 3.3, respectively.
4. Asymptotic behavior This section is devoted to show the uniform convergence of solution to (1.1) with convergence rate in the case κ ≤ 0 and κ > 0, respectively. 4.1. Proof of Theorem 1.3 (The case κ ≤ 0) Let ϕ(t) :=
∫ Ω
ϕ(x, t)dx. Integrating the equations of (1.1) over Ω , we obtain ⎧ ∫ du µuk+1 ⎪ ⎪ , = κu − µ uk+1 ≤ κu − ⎪ ⎪ k ⎪ dt |Ω | Ω ⎪ ⎨ dv = αu − βv, ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎩ dw = γu − δw. dt
(4.1)
Firstly we consider the case κ = 0. From (4.1), we obtain ( u(t) ≤
µkt
1 + k k u (0) |Ω |
v(t) = e−β(t−τ0 ) v(τ0 ) + α
∫
≤ e−β(t−τ0 ) v(τ0 ) + α
∫
)−1/k ,
(4.2)
t
e−β(t−s) u(s)ds
τ0
(
t
e−β(t−s)
τ0
µks
1 + k k u (0) |Ω |
)−1/k ds
(4.3)
ds.
(4.4)
and w(t) = e
−δ(t−τ0 )
∫
t
e−δ(t−s) u(s)ds
w(τ0 ) + γ τ0
≤e
−δ(t−τ0 )
∫
(
t
w(τ0 ) + γ
−δ(t−s)
e τ0
Choosing τ0 =
2 βk
µks
1 + k k u (0) |Ω |
)−1
we obtain )−1/k 1 e−β(t−s) + k ds k u (0) |Ω | τ0 )−1/k ∫ ( 1 t µks 1 = + k de−β(t−s) β τ0 |Ω |k u (0) t
1 + k k u (0) |Ω |
in (4.3), then for s ∈ [τ0 , t], noticing (
∫
µks
(
µks
k
≤
k
|Ω | β|Ω | = , µkτ0 2µ
)−1/k
J. Li et al. / Nonlinear Analysis: Real World Applications 39 (2018) 261–277
1 = β
(
µks
1 + k k u (0) |Ω |
)−1/k s=t e−β(t−s) ⏐s=τ 0
⏐
∫
µ
+
(
t −β(t−s)
e
k
β|Ω | τ0 ( )−1/k ( )−1/k 1 µkt 1 1 1 µkτ0 ≤ + k + k − e−β(t−τ0 ) β |Ω |k β |Ω |k u (0) u (0) )−1/k ( ∫ 1 1 t −β(t−s) µks ds, e + k + k 2 τ0 u (0) |Ω |
µks
1 + k k u (0) |Ω |
271
)− k+1 k
ds
from which we obtain ∫
(
t
e−β(t−s)
τ0
µks
1 + k k u (0) |Ω |
)−1/k
2 ds ≤ β
(
µkt
1 + k k u (0) |Ω |
)−1/k .
(4.5)
Substituting (4.5) into (4.3) yields ( )−1/k 2α µkt 1 v(t) ≤ e−β(t−τ0 ) v(τ0 ) + + k β |Ω |k u (0) )−1/k ( 1 µkt + k . ≤C k u (0) |Ω | Similarly, we obtain ( )−1/k 2γ µkt 1 w(t) ≤ e w(τ0 ) + + k δ |Ω |k u (0) ( )−1/k 1 µkt + k ≤C . k u (0) |Ω | −δ(t−τ0 )
Furthermore, based on the regularities of solution (u, v, w), one can readily get a constant C1 > 0 (e.g. see [30, Lemma 3.14] or [3, Lemma 4.4]) such that ∥u∥W 1,∞ (Ω) ≤ C1 ,
∥v∥W 1,∞ (Ω) ≤ C1 ,
∥w∥W 1,∞ (Ω) ≤ C1
for all t > τ0 .
Thus from Lemma 2.3 and (4.2) it follows that ) ( N 1 N +1 N +1 ∥u(·, t)∥L∞ (Ω) ≤ CGN ∥u(·, t)∥W 1,∞ (Ω) ∥u(·, t)∥L1 (Ω) + ∥u(·, t)∥L1 (Ω) 1
≤ Cu N +1 ( )− 1 (N +1)k µkt 1 + ≤C k uk (0) |Ω | for all t > 0. Similarly, we obtain ∥v(·, t)∥L∞ (Ω)
(
µkt
(
µkt
1 ≤C + k k u (0) |Ω |
)−
1 (N +1)k
)−
1 (N +1)k
and ∥w(·, t)∥L∞ (Ω)
1 ≤C + k k u (0) |Ω |
.
272
J. Li et al. / Nonlinear Analysis: Real World Applications 39 (2018) 261–277
Now we turn to consider the case κ < 0. From the first equation of (4.1), we obtain d u ≤ κu dt and thus u(t) ≤ eκ(t−τ0 ) u(τ0 ). On the other hand, from the second and the third equation of (4.1), we obtain ∫ t v(t) = e−β(t−τ0 ) v(τ0 ) + α e−β(t−s) u(s)ds τ0 t
≤ e−β(t−τ0 ) v(τ0 ) + α
∫
e−β(t−s) eκ(s−τ0 ) u(τ0 )ds
τ0
= e−β(t−τ0 ) v(τ0 ) + αu(τ0 )
eκ(t−τ0 ) − e−β(t−τ0 ) β+κ
(4.6)
and w(t) = e−δ(t−τ0 ) w(τ0 ) + γ
∫
t
e−δ(t−s) u(s)ds
τ0 t
≤e
−δ(t−τ0 )
∫ w(τ0 ) + γ
e−δ(t−s) eκ(s−τ0 ) u(τ0 )ds
τ0
= e−δ(t−τ0 ) w(τ0 ) + γu(τ0 )
eκ(t−τ0 ) − e−δ(t−τ0 ) . δ+κ
(4.7)
Hence there exists C > 0 such that u(t) ≤ Ceκt , v(t) ≤ Cemax{κ,−β}t and w(t) ≤ Cemax{κ,−δ}t . Following the same procedure as that in the case κ = 0, we can obtain κt
∥u(·, t)∥L∞ (Ω) ≤ Ce N +1 , ∥v(·, t)∥L∞ (Ω) ≤ Ce
max{κ,−β}t N +1
and ∥w(·, t)∥L∞ (Ω) ≤ Ce
max{κ,−δ}t N +1
.
The proof of Theorem 1.3 is complete. 4.2. Proof of Theorem 1.4 (The case κ > 0) It is noted that in contrast to the solution behavior of the diffusive Fisher–KPP problem, the interaction effects between chemotactic cross-diffusion and cell kinetics may result in a quite colorful dynamics of chemotaxis–growth system [36–38]. In particular, the solution of (1.2) approaches to the unique nontrivial spatially homogeneous equilibrium when the strength of the overcrowding preventing effect µ is sufficiently
J. Li et al. / Nonlinear Analysis: Real World Applications 39 (2018) 261–277
273
large compared with that of the chemotactic sensitivity χ [39]. In this subsection, motivated by the ideas from [30,33,40], we consider the convergence of the solution in the case κ > 0. For simplicity, denote h = χv − ξw and h0 = χv0 − ξw0 , U :=
( µ )1/k κ
( H := h −
u,
χα ξγ − β δ
) ( )1/k κ , µ
( )1/k α κ V := v − . β µ
Then system (1.1) becomes ⎧ Ut = ∆U − ∇ · (U ∇H) + κU (1 − U k ), x ∈ Ω , t > 0, ⎪ ⎪ ( )1/k ⎪ ⎪ ⎪ κ ⎪ ⎪ (U − 1) − δH − χ(β − δ)V, x ∈ Ω , t > 0, Ht = ∆H + (χα − ξγ) ⎪ ⎪ µ ⎪ ⎪ ⎪ ( )1/k ⎪ ⎪ ⎪ κ ⎪ ⎪ Vt = ∆V + α (U − 1) − βV, x ∈ Ω , t > 0, ⎪ ⎪ µ ⎨ ∂H ∂V ∂U ⎪ = = = 0, x ∈ ∂Ω , t > 0, ⎪ ⎪ ∂ν ∂ν ∂ν ⎪ ⎪ ) ( )1/k ( ⎪ ( µ )1/k ⎪ ⎪ κ χα ξγ ⎪ ⎪ U (x, 0) = − , u (x), H(x, 0) = h (x) − 0 0 ⎪ ⎪ κ β δ µ ⎪ ⎪ ⎪ ( )1/k ⎪ ⎪ α κ ⎪ ⎪ ⎩V (x, 0) = v0 (x) − , x ∈ Ω. β µ
(4.8)
Lemma 4.1. For κ > 0, let (1.5) hold and (u, v, w) be the global solution of (1.1). Suppose ( µ>
2(αχ)2 (β − α)2 (χα − ξγ)2 + 2δ δβ 2
)k/2
κ1−k/2 .
Then for all t > 0 the function ∫
1 F (t) := (U − 1 − ln U ) + 2 Ω
2χ2 (β − δ)2 H + δβ Ω
∫
∫
2
V2
Ω
satisfies F ′ (t) ≤ −D(t), where {∫ D(t) := D
1 (U − 1) + 2 Ω 2
2χ2 (β − δ)2 H + δβ Ω
∫
2
∫ W
2
}
Ω
and D is defined in (4.12). ¯ × (0, ∞). Proof . According to the strong maximum principle and the assumption U0 ̸≡ 0, U is positive in Ω 1 Testing the first equation of (4.8) by 1 − U , we obtain d dt
∫ ∫ 2 |∇U | ∇U ∇H (U − 1 − ln U ) = − + − κ (U − 1)(U k − 1) U2 U Ω Ω Ω Ω ∫ ∫ ∫ ∫ 2 2 |∇U | 1 |∇U | 1 2 ≤− + + |∇H| − κ (U − 1)2 U2 2 Ω U2 2 Ω Ω Ω ∫ ∫ 1 2 2 ≤ |∇H| − κ (U − 1) , 2 Ω Ω
∫
∫
(4.9)
J. Li et al. / Nonlinear Analysis: Real World Applications 39 (2018) 261–277
274
where we have used the Young inequality. Testing the second and the third equation of (4.8) by H and V , respectively, and by the Young inequality, we obtain 1 d 2 dt
∫ Ω
( )1/k ∫ ∫ ∫ κ 2 (U − 1)H − δ H 2 − χ(β − δ) |∇H| + (χα − ξγ) VH µ Ω Ω Ω Ω ( )2/k ∫ ∫ ∫ ∫ δ χ2 (β − δ)2 (χα − ξγ)2 κ 2 2 2 (U − 1) − H + V 2 (4.10) ≤− |∇H| + 2δ µ 4 Ω δ Ω Ω Ω
H2 = −
∫
and 1 d 2 dt
∫
( )1/k ∫ ∫ κ (U − 1)V − β V2 µ Ω Ω Ω ( )2/k ∫ ∫ ∫ 2 κ α β 2 2 V 2. |∇V | + (U − 1) − ≤− 2β µ 2 Ω Ω Ω
V2 =−
Ω
∫
2
|∇V | + α
(4.11)
The combination of (4.9), (4.10) and (4.11) gives [ ( ) ( )2/k ] ∫ (χα − ξγ)2 2(χα)2 (β − δ)2 κ ′ F (t) ≤ − κ − + (U − 1)2 2δ β2δ µ Ω ∫ 2 2 ∫ δ χ (β − δ) − H2 − V 2. 4 Ω δ Ω Choosing {
(
D := min κ −
(χα − ξγ)2 2(χα)2 (β − δ)2 + 2δ β2δ
} ) ( )2/k κ δ β , , µ 2 2
(4.12)
and ( µ>
(χα − ξγ)2 2(αχ)2 (β − δ)2 + 2δ β2δ
)k/2
κ1−k/2 ,
we have D > 0 and thus ′
{∫
F (t) ≤ −D
1 (U − 1) + 2 Ω 2
2χ2 (β − δ)2 H + δβ Ω
∫
2
∫ V
2
} = −D(t).
Ω
Now we are at the position to show the proof of Theorem 1.4. Denote h(s) = s − 1 − ln s. Noticing that h′ (s) = 1 − 1s and h′′ (s) = s12 > 0 for all s > 0, we obtain that h(s) ≥ h(1) = 0 and F (t) is nonnegative. From Lemma 4.1, we have F ′ (t) ≤ −D(t) and then ∫
t
D(τ )dτ ≤ F (τ0 ) − F (t) ≤ F (τ0 ) τ0
for all t > τ0 , from which we have } ∫ t {∫ ∫ ∫ 2χ2 (β − δ)2 1 2 2 2 (U − 1) + H + V < ∞. 2 Ω δβ Ω τ0 Ω Using the similar argument as in Lemma 3.10 of [33], we can obtain the uniform convergence of solutions, namely ∥U − 1∥L∞ (Ω) → 0,
∥H∥L∞ (Ω) → 0,
∥V ∥L∞ (Ω) → 0
J. Li et al. / Nonlinear Analysis: Real World Applications 39 (2018) 261–277
275
as t → ∞. Then there exists t0 > 0 such that for all t > t0 , ∥U − 1∥L∞ (Ω) ≤ 12 , which together with the fact that 1 (s − 1)2 ≤ h(s) ≤ (s − 1)2 3 for all s >
1 2
implies that ∫ ∫ ∫ 1 1 2χ2 (β − δ)2 1 (U − 1)2 + H2 + V 2 ≤ F (t) ≤ D(t) 3 Ω 2 Ω δβ D Ω
(4.13)
for all t > t0 . Hence F ′ (t) ≤ −D(t) ≤ −DF (t), from which we obtain F (t) ≤ F (t0 )e−D(t−t0 ) .
(4.14)
Substituting (4.14) into (4.13), we obtain ∫ ∫ ∫ 1 2χ2 (β − δ)2 1 2 2 (U − 1) + H + V 2 ≤ F (t0 )e−D(t−t0 ) , 3 Ω 2 Ω δβ Ω which implies that there exists C > 0 such that for all t > t0 , ∥U (·, t) − 1∥L2 (Ω) ≤ Ce−Dt/2 , ∥H(·, t)∥L2 (Ω) ≤ Ce−Dt/2 and ∥V (·, t)∥L2 (Ω) ≤ Ce−Dt/2 . Furthermore, noticing that there exists a constant C1 > 0 such that ∥U − 1∥W 1,∞ (Ω) ≤ C1 ,
∥H∥W 1,∞ (Ω) ≤ C1 ,
∥V ∥W 1,∞ (Ω) ≤ C1
for all t > τ0 .
Thus the Gagliardo–Nirenberg inequality (Lemma 2.3) yields ( ) N 2 N +2 N +2 ∥U (·, t) − 1∥L∞ (Ω) ≤ C ∥U (·, t) − 1∥W ∥U (·, t) − 1∥ + ∥U (·, t) − 1∥ 2 1,∞ (Ω) L (Ω) L2 (Ω) 2
Dt
≤ Ce− N +2 ≤ C∥U (·, t) − 1∥LN2+2 (Ω) for all t > 0. Similarly, we can obtain Dt
∥H(·, t)∥L∞ (Ω) ≤ Ce− N +2 and Dt
∥V (·, t)∥L∞ (Ω) ≤ Ce− N +2 . Therefore the proof of Theorem 1.4 is complete. Acknowledgments The authors are grateful to the referee whose suggestions improve the exposition of the paper. This work is partially supported by NSFC (Nos. 11571363, 61620106002) and the Beijing Natural Science Foundation (No. 9152013) and the Research Grant Funds of Minzu University of China.
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Nonlinear Analysis: Real World Applications www.elsevier.com/locate/nonrwa
Large time behavior of solutions to a fully parabolic attraction–repulsion chemotaxis system with logistic source Jing Lia , Yuanyuan Keb , Yifu Wangc, * a
College of Science, Minzu University of China, Beijing, 100081, PR China School of Information, Renmin University of China, Beijing, 100872, PR China c School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, PR China b
article
info
Article history: Received 14 May 2017 Received in revised form 5 July 2017 Accepted 8 July 2017 Available online 7 August 2017 Keywords: Chemotaxis Attraction–repulsion Boundedness asymptotic behavior
abstract This paper deals with an attraction–repulsion chemotaxis system with logistic source
{
ut = ∆u − χ∇ · (u∇v) + ξ∇ · (u∇w) + f (u), vt = ∆v + αu − βv, wt = ∆w + γu − δw,
x ∈ Ω , t > 0, x ∈ Ω , t > 0, x ∈ Ω, t > 0
under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ RN (N ≥ 1), where parameters χ, ξ, α, β, γ and δ are positive and f (s) = κs − µs1+k with κ ∈ R, µ > 0 and k ≥ 1. It is shown that the corresponding system possesses a unique global bounded classical solution in the cases k > 1 or k = 1 with µ > CN µ∗ for some µ∗ , CN > 0. Moreover, the large time behavior of solutions to the problem is also investigated. Specially speaking, when κ < 0 (resp. κ = 0), the corresponding solution of the system decays to (0, ( 0, 0) exponentially (resp. algebraically), and when κ > 0 the solution converges ) to
( κ )1/k µ
,
α β
( κ )1/k µ
,
γ δ
( κ )1/k µ
exponentially if µ is larger.
© 2017 Elsevier Ltd. All rights reserved.
1. Introduction This paper is concerned with the following attraction–repulsion chemotaxis system ⎧ ut = ∆u − χ∇ · (u∇v) + ξ∇ · (u∇w) + f (u), x ∈ Ω , t > 0, ⎪ ⎪ ⎪ ⎪ τ v = ∆v + αu − βv, x ∈ Ω , t > 0, t ⎪ ⎪ ⎨τ w = ∆w + γu − δw, x ∈ Ω , t > 0, t ∂v ∂w ∂u ⎪ ⎪ ⎪ = = = 0, x ∈ ∂Ω , t > 0, ⎪ ⎪ ∂ν ∂ν ∂ν ⎪ ⎩ u(x, 0) = u0 (x), τ v(x, 0) = v0 (x), τ w(x, 0) = w0 (x), x ∈ Ω
*
Corresponding author. E-mail address: [email protected] (Y. Wang).
http://dx.doi.org/10.1016/j.nonrwa.2017.07.002 1468-1218/© 2017 Elsevier Ltd. All rights reserved.
(1.1)
262
J. Li et al. / Nonlinear Analysis: Real World Applications 39 (2018) 261–277
in a bounded domain Ω ⊂ RN (N ≥ 1) with smooth boundary ∂Ω . The model (1.1) was proposed in [1] to describe the aggregation of microglia in Alzheimer’s disease, where unknown functions u, v and w denote the concentrations of Microglia, chemoattractant and chemorepellent which are produced by Microglia, respectively. The positive parameters χ and ξ are the chemotactic coefficients, α, β, γ and δ are chemical production and degradation rates. Here τ = 0, 1 is parameter, and the logistic source f describes the cell proliferation and death. It is noticed that (1.1) was also introduced in [2] to describe the quorum sensing effect in the chemotactic process. The first two equations of (1.1) with ξ = 0 comprise the Keller–Segel chemotaxis system { ut = ∆u − χ∇ · (u∇v) + f (u), x ∈ Ω , t > 0, (1.2) τ vt = ∆v + αu − βv, x ∈ Ω , t > 0. The most characteristic ingredient of this system is that it reflect the attractive chemotaxis through a nonlinear cross-diffusive term. Since the blow-up is an extreme facet of bacterial aggregation, considerable efforts have been devoted to identifying circumstances under which explosions are precluded and thus generates pattern formation which is applicable in realistic situation (see [3–12]). It is verified that the logistic source f (u) = µu(1 − u) with µ > 0 exerts a certain dampening influence which may suppress blow-up in many relevant situations: all of the solutions to (1.2) remain uniformly bounded globally in two dimensional bounded domain [13,14], alternatively in higher dimensions (N ≥ 3), the uniform-in-time boundedness of classical solutions to (1.2) with τ = 0, 1 is valid when f (u) ≤ a − µu2 for some a ≥ 0 and suitably large µ > 0 [14,15]. In particular, a more generalized logistic term f (u) = κu − µu2 with κ ∈ R and µ > 0 was considered in [3], where the existence of global weak solution to a three-dimensional chemotaxis system was proved for arbitrarily small values of µ > 0, which becomes the classical solution after some time if κ is not too large. As source term f is controlled by −c0 (u + uα ) and a − µuα with some a ≥ 0, µ, c0 > 0 and α < 2, the uniform-in-time boundedness of some weak solutions to (1.2) with τ = 1 is also shown in the recent paper [16]. It should be mentioned that the blow-up is possible in a slightly modified version of system (1.2) with logistic source [17]. On the other hand, in the absence of chemoattractant, the first and third equations of (1.1) with χ = 0 form a repulsive Keller–Segel model { ut = ∆u + ξ∇ · (u∇w) + f (u), x ∈ Ω , t > 0, (1.3) τ wt = ∆w + γu − δw, x ∈ Ω , t > 0. It should be mentioned that based on a Lyapunov functional identity, Cie´slak, Lauren¸cot and C. MoralesRodrigo [18] showed the global existence of classical solutions to (1.3) in two dimensions and weak solutions in three and four dimensions. Compared to the classical chemotaxis model, the mathematical analysis on the boundedness of the attraction–repulsion chemotaxis model (1.1) is much harder due to the complicated interactions between three species u, v and w and the difficulty of constructing a Lyapunov functional. For the model (1.1) with τ = 0 and f (u) ≡ 0, Tao and Wang [19] obtained the global boundedness of the solution in high-dimensions if ξγ > χα; while if ξγ < χα, the solution might blow up in the two-dimensional case if the cell mass is larger than a threshold number. Recently, Jin and Wang [20] investigated the boundedness, blowup and critical mass phenomenon in the parabolic–parabolic–elliptic case of model (1.1) (i.e., τ = 1 in the second equation of (1.1), while τ = 0 in the third one). As for the model (1.1) with logistic source, based on the Lp energy estimates and Moser iteration, Zhang and Li [21] proved the existence of global classical solutions (N −2)+ of (1.1) with τ = 0, f (u) = µu(1 − u) whenever µ > (χα − ξγ), and also discussed the asymptotic N behavior of solutions when µ > 2χα. It is also noticed that for (1.1) with τ = 0, f (u) ≤ a − µuk+1 , Li and Xiang [22] proved that the √ classical solution will exist globally for all N ≥ 2 in the case k > 1. Moreover, N 2 +4N −N
when ξγ = χα and k > , the classical solution is uniformly bounded. Additionally, in [23–25] the 2 global boundedness were considered for the attraction–repulsion chemotaxis system with nonlinear diffusion.
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To the best of our knowledge, although there is only difference of the second and third equations between the full attraction–repulsion chemotaxis model (1.1) with τ = 1 and the model (1.1) with τ = 0, the mathematical analysis of the full model seems to be more difficult. For model (1.1) with τ = 1 and f (u) ≡ 0, Tao and Wang [19] obtained the convergence of solutions to the trivial stationary solution in two dimensions when ξγ > χα and β = δ, while in the case β ̸= δ, they used an entropy inequality to establish the ∫ boundedness of solutions with the small initial mass Ω u0 (x)dx. It should be remarked that the smallness assumption on the initial mass is removed in [26,27], respectively; however, when N ≥ 3 and ξγ > χα, the global existence of classical solutions for (1.1) is still open. As for the critical case ξγ = χα, Lin et al. [23,28] investigated that the solution of (1.1) is globally bounded, and proved that the global solution of small ∫ γ 1 initial data converges to (u0 , α β u0 , δ u0 ) where u0 = |Ω| Ω u0 (x)dx for N = 2, 3. It seems that up to now, the result on (1.1) with f (u) ̸≡ 0 is far from satisfactory [5,29,30]. In one-dimension, Jin and Wang [31] derived a uniform-in-time bound of solutions to (1.1) and proved that the model possesses a global attractor. The existence of global bounded solutions to (1.1) in two-dimensional case was obtained in [22,32] by the different method, respectively. As for N = 3 and f (u) = u(1 − µuk ) with k ≥ 1 and µ > 0, it is shown in [29] that (1.1) admits a unique global bounded solution under the conditions {( )k ( )k } 1 1 41 41 β≥ , δ≥ and µ ≥ max χα + 9ξγ , ξγ + 9χα . (1.4) 2 2 2 2 In addition, the convergence of the solution was discussed when k ∈ N and u0 ̸≡ 0; however, the convergence rate was left as an open problem there. In the higher-dimensional setting N ≥ 3, Zheng et al. [5] proved that (1.1) admits a unique globally bounded classical solution provided that β = δ and µ > θ0 (χα + ξγ) for some constant θ0 > 0. It is also noted that when χα = ξγ, the boundedness and large time behavior of solutions to (1.1) were considered in the recent paper [30]. The purpose of this paper is to mathematically understand the effect of the attractive signal, the repulsive one and the logistic source on the global boundedness and asymptotic behavior of solutions to the full model (1.1) in the higher-dimensional setting. Unless otherwise stated, we will always consider model (1.1) with τ = 1 in the sequel, and assume that the nonnegative initial data u0 , v0 , w0 ∈ W 1,∞ (Ω ) with u0 ̸≡ 0, and the smooth function f satisfies f (s) = κs − µsk+1
with κ ∈ R, µ > 0 and k ≥ 1 for s ≥ 0.
(1.5)
Here the term κs being the difference between birth rate and death rate of the population, is used to describe population growth, and the term −µsk+1 models additional overcrowding preventing effects. We now state the main results of this paper. Our first theorem indicates that the boundedness of solutions can be ensured by the sufficiently strong dampening effect. Theorem 1.1. Let (1.5) hold with k > 1 and N ≥ 3. Then for any µ > 0, (1.1) with τ = 1 admits a unique global bounded classical solution. Theorem 1.2. Let k = 1 in (1.5) and N ≥ 3. Then there exists CN > 0 such that for µ > CN µ∗ with { max{|χα − ξγ|, αχ(β − δ)}, if β ≥ δ, ∗ µ = max{|χα − ξγ|, ξγ(δ − β)}, if β < δ, (1.1) with τ = 1 admits a unique global bounded classical solution. It is observed that the chemorepellent w has a dispersal effect in chemotactic movement in contrast to the aggregation effect of the chemoattractant v. Indeed, it follows from Theorem 1.2 that if β = δ, then
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(1.1) possesses a unique global bounded solution when µ > CN |χα − ξγ|. Hence Theorem 1.2 improves the results of [5,29] in some sense. The following two theorems are concerning the asymptotic of solutions of (1.1) with τ = 1 in the cases of κ ≤ 0 and κ > 0, respectively. Theorem 1.3. Let (1.5) hold with κ ≤ 0 and k ≥ 1 and (u, v, w) be the solution of (1.1). Then if { 0, if k > 1, µ> CN µ∗ , if k = 1, where CN and µ∗ defined in Theorem 1.2, there exists C > 0 such that for all t > 0, ( ∥u(·, t)∥L∞ (Ω) + ∥v(·, t)∥L∞ (Ω) + ∥w(·, t)∥L∞ (Ω) ≤ C
µkt |Ω |
k
+ (∫ Ω
)−
1
1 (N +1)k
)k u0 (x)
for κ = 0 and κt
∥u(·, t)∥L∞ (Ω) ≤ Ce N +1 , ∥v(·, t)∥L∞ (Ω) ≤ Ce
max{κ,−β}t N +1
and ∥w(·, t)∥L∞ (Ω) ≤ Ce
max{κ,−δ}t N +1
for κ < 0. As κ > 0, inspired by the arguments in [30,33], we prove that the solution of (1.1) converges ((for) the case ( )1/k ( )1/k ) 1/k α κ κ ,β µ , γδ µκ exponentially if µ is larger, which solves the problem proposed by the to µ authors of [29]. Theorem 1.4. Let (1.5) hold with κ > 0 and k ≥ 1 and (u, v, w) be the solution of (1.1). Then if { µ0 , if k > 1, µ> max{CN µ∗ , µ0 }, if k = 1, where CN and µ∗ are defined in Theorem 1.2, and ( µ0 =
(χα − ξγ)2 2(αχ)2 (β − δ)2 + 2δ δβ 2
)k/2
κ1−k/2 ,
there exist some C > 0 and D > 0 such that for all t > 0, ( )1/k ( )1/k ( )1/k κ α κ γ κ + v(·, t) − + w(·, t) − u(·, t) − ∞ ∞ µ β µ δ µ L
(Ω)
L
(Ω)
Dt
≤ Ce− N +2 .
L∞ (Ω)
This paper is organized as follows. In Section 2, we recall some preliminary results. In Section 3, with the aid of the maximal Sobolev regularity, we consider the boundedness of solutions to (1.1) in the case k > 1 and k = 1, respectively. In Section 4, we show the uniform convergence of solution to (1.1) with convergence rate in the case κ ≤ 0 and κ > 0, respectively. Throughout this paper, we use C and Ci (i ∈ N) to denote generic constants which may vary in the ∫ ∫ context. Moreover, for simplicity, the integral Ω f (x)dx is written as Ω f (x).
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2. Local existence and basic properties Firstly, we state a result concerning the local existence of classical solutions, which can be proved by well-established methods involving standard parabolic regularity theory and an appropriate fixed point framework (see [19]). Lemma 2.1. Let Ω ⊂ RN ( N ≥ 3) be a bounded domain with smooth boundary, and fix q ∈ (N, ∞). Suppose that the nonnegative functions u0 , v0 and w0 ∈ W 1,∞ (Ω ). There exist Tmax ∈ (0, ∞] and a unique triple of non-negative functions (u, v, w) satisfying the inclusions u ∈ C 0 ([0, Tmax ); C 0 (Ω )) ∩ C 2,1 (Ω × (0, Tmax )), 1,q (v, w) ∈ [C 0 ([0, Tmax ); C 0 (Ω )) ∩ L∞ (Ω )) ∩ C 2,1 (Ω × (0, Tmax ))]2 , loc ([0, Tmax ); W
which solves (1.1) classically in Ω × (0, Tmax ). Moreover, if Tmax < ∞, then ( ) lim ∥u(·, t)∥L∞ (Ω) + ∥v(·, t)∥W 1,∞ (Ω) + ∥w(·, t)∥W 1,∞ (Ω) = ∞. t→Tmax
As in Lemma 2.2 of [6], by introducing ω(x, t) = eεt z(x, t) and applying the maximal Sobolev regularity (Theorem 3.1 of [34]) to ω, we obtain the following lemma, which will play an important role in the proof of our main result. Lemma 2.2 ([7, Lemma 2.1] [6, Lemma 2.2]). Let r ⎧ ⎪ ⎪zt = ∆z − εz + f, ⎨ ∂z = 0, ⎪ ∂ν ⎪ ⎩ z(x, 0) = z0 (x),
∈ (N, ∞) and consider the following evolution equation (x, t) ∈ Ω × (0, t), (x, t) ∈ ∂Ω × (0, t),
(2.1)
x ∈ Ω.
r r 0 Then for each z0 ∈ W 2,r (Ω ) with ∂z ∂ν = 0 on ∂Ω and any f ∈ L ((0, t), L (Ω )), (2.1) admits a unique mild solution z ∈ W 1,r ((0, t); Lr (Ω )) ∩ Lr ((0, t); W 2,r (Ω )), which is given by ∫ τ −τ τ ∆ z(τ ) = e e z0 + e−(τ −s) e(τ −s)∆ f (s)ds, τ ∈ [0, t], 0 τ∆
where e is the semigroup generated by the Neumann Laplacian. Moreover, there exists Cr > 0, such that 0) if τ0 ∈ [0, t), z(·, τ0 ) ∈ W 2,r (Ω ) (r > N ) with ∂z(·,τ = 0, then ∂n ∫ t∫ ∫ t∫ r r r eεrτ (|z| + |∆z| ) ≤ Cr eεrτ |f | + Cr eεrt (∥z(·, τ0 )∥rLr (Ω) + ∥∆z(·, τ0 )∥rLr (Ω) ). (2.2) τ0
Ω
τ0
Ω
The following extended version of the Gagliardo–Nirenberg inequality will be frequently used in the later analysis (see [35]). Lemma 2.3. Assume that r ∈ (0, p), 1 ≤ p, q ≤ ∞ and satisfying (N − q)p < N q. Then, for any ψ ∈ W 1,q (Ω ) ∩ Lr (Ω ), there exists a constant CGN > 0 only depending on N, q, r and Ω such that ( ) ∗ ∗ r ∥ψ∥Lp (Ω) ≤ CGN ∥∇ψ∥λLq (Ω) ∥ψ∥1−λ (2.3) Lr (Ω) + ∥ψ∥L (Ω) holds with λ∗ ∈ (0, 1) satisfying λ∗ =
N r
1−
− N q
N p
+
N r
.
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The following Lemma shows the boundedness criterion of solution for the system (1.1), and we refer the readers to Lemma 2.4 of [8] for details. Lemma 2.4. Let (u, v, w) be a solution of (1.1) defined on [0, Tmax ). If there exists p0 > N/2 such that ∥u(·, t)∥Lp0 (Ω) ≤ C
for all t ∈ [0, Tmax ),
there exists some constant C > 0 independent of t such that ∥u(·, t)∥L∞ (Ω) + ∥v(·, t)∥W 1,∞ (Ω) + ∥w(·, t)∥W 1,∞ (Ω) ≤ C
for all t ∈ [0, Tmax ).
(2.4)
3. Boundedness To address our approaches, we see that under the transformation h := χv − ξw, when β > δ, the system (1.1) becomes ⎧ ut = ∆u − ∇ · (u∇h) + f (u), x ∈ Ω , t > 0, ⎪ ⎪ ⎪ ⎪ h = ∆h + (χα − ξγ)u − δh − χ(β − δ)v, x ∈ Ω , t > 0, t ⎪ ⎪ ⎨v = ∆v + αu − βv ≤ ∆v + αu − δv, x ∈ Ω , t > 0, t (3.1) ∂u ∂h ∂v ⎪ ⎪ ⎪ = = = 0, x ∈ ∂Ω , t > 0, ⎪ ⎪ ∂ν ∂ν ⎪ ⎩ ∂ν u(x, 0) = u0 (x), h(x, 0) = h0 (x) = χv0 (x) − ξw0 (x), v(x, 0) = v0 (x), x ∈ Ω . When β = δ, (1.1) is reduced to the subsystem ⎧ ut = ∆u − ∇ · (u∇h) + f (u), x ∈ Ω , t > 0, ⎪ ⎪ ⎪ ⎪ h = ∆h + (χα − ξγ)u − δh, x ∈ Ω , t > 0, ⎨ t ∂u ∂h ⎪ = = 0, x ∈ ∂Ω , t > 0, ⎪ ⎪ ∂ν ∂ν ⎪ ⎩ u(x, 0) = u0 (x), h(x, 0) = h0 (x) = χv0 (x) − ξw0 (x), x ∈ Ω . When β < δ, the system (1.1) becomes ⎧ ⎪ ⎪ut = ∆u − ∇ · (u∇h) + f (u), ⎪ ⎪ ⎪ ⎪ht = ∆h + (χα − ξγ)u − βh − ξ(β − δ)w, ⎨ wt = ∆w + γu − δw ≤ ∆w + γu − βw, ∂h ∂w ∂u ⎪ ⎪ ⎪ = = = 0, ⎪ ⎪ ∂ν ∂ν ∂ν ⎪ ⎩ u(x, 0) = u0 (x), h(x, 0) = h0 (x) = χv0 (x) − ξw0 (x),
x ∈ Ω , t > 0, x ∈ Ω , t > 0, x ∈ Ω , t > 0,
(3.2)
(3.3)
x ∈ ∂Ω , t > 0, w(x, 0) = w0 (x),
x ∈ Ω.
Now we are going to prove the results on boundedness of solutions to (1.1). Since the regularity obtained in Lemma 2.2 requires that the initial data to be in the space W 2,r (Ω ) and satisfy homogeneous Neumann boundary conditions, we will perform a small time shift and thus use the positive time as the “initial time”. Lemma 3.1. Let r ∈ (1, ∞). There exists Cr > 0 and C > 0, such that if τ0 ∈ [0, Tmax ), h(·, τ0 ) ∈ W 2,r (Ω ) 0) (r > N ) with ∂h(·,τ = 0, then for β ≥ δ, for any t ∈ (τ0 , Tmax ), we have ∂n ∫ t∫ ∫ t∫ r r r eδrτ |∆h| ≤ Cr (|χα − ξγ| + |χα(β − δ)| ) eδrτ ur τ0 Ω τ0 Ω ( + Cr eδrt ∥h(·, τ0 )∥rLr (Ω) + ∥∆h(·, τ0 )∥rLr (Ω) ) r + |χ(β − δ)| (∥v(·, τ0 )∥rLr (Ω) + ∥∆v(·, τ0 )∥rLr (Ω) ) . (3.4)
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While for β < δ, ∫
t
∫ t∫ r r r eβrτ |∆h| ≤ Cr (|χα − ξγ| + |ξγ(β − δ)| ) eβrτ ur Ω τ0 Ω ( + Cr eβrt ∥h(·, τ0 )∥rLr (Ω) + ∥∆h(·, τ0 )∥rLr (Ω)
∫
τ0
) r + |ξ(β − δ)| (∥w(·, τ0 )∥rLr (Ω) + ∥∆w(·, τ0 )∥rLr (Ω) ) .
(3.5)
Proof . For β = δ, from (3.2) and Lemma 2.2, we obtain ∫ t∫ ∫ t∫ r r eδrτ |∆h| ≤ Cr |χα − ξγ| eδrτ ur τ0
τ0
Ω
+ Cr e
δrt
Ω (∥h(·, τ0 )∥rLr (Ω)
+ ∥∆h(·, τ0 )∥rLr (Ω) ).
For β > δ, from (3.1) and Lemma 2.2, we obtain ∫ ∫ t∫ ∫ t∫ r r r eδrτ ur + Cr |χ(β − δ)| eδrτ |∆h| ≤ Cr |χα − ξγ| Ω
τ0
Ω δrt Cr e (∥h(·, τ0 )∥rLr (Ω)
Now we try to estimate the term
∫t ∫ τ0
Ω
∫
eδrτ v r
Ω
τ0
τ0
+
t
+ ∥∆h(·, τ0 )∥rLr (Ω) ).
eδrτ v r . Denote s(x, t) the solution of
⎧ ⎪ ⎪st = ∆s + αu − δs, (x, t) ∈ Ω × (τ0 , Tmax ), ⎨ ∂s = 0, (x, t) ∈ ∂Ω × (τ0 , Tmax ), ⎪ ∂ν ⎪ ⎩ s(x, τ0 ) = v(x, τ0 ), x ∈ Ω .
(3.6)
Using again Lemma 2.2, we obtain for any t ∈ (τ0 , Tmax ), ∫ t∫ ∫ t∫ δrτ r r e s ≤ Cr α eδrτ ur + Cr eδrt (∥v(·, τ0 )∥rLr (Ω) + ∥∆v(·, τ0 )∥rLr (Ω) ). τ0
Ω
τ0
Ω
On the other hand, noticing the nonnegativity of function v, ∂v = ∆v + αu − βv ≤ ∆v + αu − δv, ∂t
(x, t) ∈ Ω × (τ0 , Tmax ),
we obtain v ≤ s in Ω × [0, Tmax ] by comparison principle, and thus ∫ t∫ ∫ t∫ ∫ t∫ eδrτ v r ≤ eδrτ sr ≤ Cr αr eδrτ ur + Cr eδrt (∥v(·, τ0 )∥rLr (Ω) + ∥∆v(·, τ0 )∥rLr (Ω) ), τ0
Ω
τ0
Ω
τ0
Ω
from which (3.4) follows. For β < δ, by the similar procedure, it follows from (3.3) that ∫ t∫ ∫ t∫ ∫ r r r eβrτ |∆h| ≤ Cr |χα − ξγ| eβrτ ur + Cr |ξ(β − δ)| τ0
τ0
Ω βrt Cr e (∥h(·, τ0 )∥rLr (Ω)
Ω
+
t
τ0
∫
eβrτ wr
Ω
+ ∥∆h(·, τ0 )∥rLr (Ω) )
and ∫
t
∫ e
τ0
Ω
Then (3.5) follows.
βrτ
r
w ≤ Cr γ
r
∫
t
τ0
∫ Ω
eβrτ ur + Cr eβrt (∥w(·, τ0 )∥rLr (Ω) + ∥∆w(·, τ0 )∥rLr (Ω) ).
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Lemma 3.2. Let (1.5) hold with k > 1. For any p > 1, there is C = C(p) > 0 such that for t ∈ (0, Tmax ) ∫ up (x, t)dx ≤ C. Ω
Proof . Taking up−1 as a test function for the first equation of (1.1) and let h = χv − ξw, we obtain ∫ ∫ ∫ ∫ 1 d up = − ∇up−1 ∇u + ∇up−1 · u∇h + up (κ − µuk ) p dt Ω Ω Ω Ω ∫ ∫ p−1 ≤− up ∆h + up (κ − µuk ) p Ω [ Ω ∫ ] ∫ ∫ p 1 p−1 p+1 p+1 u + |∆h| + ≤ up (κ − µuk ). (3.7) p p+1 Ω p+1 Ω Ω ∫ p u on both sides of (3.7), we obtain For the case of β ≥ δ, adding (p+1)δ p Ω ( )∫ ∫ ∫ ∫ 1 d (p + 1)δ (p + 1)δ p−1 up + up ≤ κ + up + up+1 p dt Ω p p p + 1 Ω Ω Ω ∫ ∫ p−1 p+1 p+k |∆h| −µ u + p(p + 1) Ω Ω and then t
[ ∫ u ≤e u (τ0 ) + e (pκ + (p + 1)δ) up Ω Ω τ0 Ω ] ∫ ∫ ∫ (p − 1)p p−1 p+1 p+1 p+k + u − µp u + |∆h| . p+1 Ω p+1 Ω Ω
∫
p
−δ(p+1)(t−τ0 )
∫
∫
p
−δ(p+1)(t−τ )
(3.8)
Then from (3.4), we obtain ∫ ∫ p−1( p −δ(p+1)(t−τ0 ) u ≤e up (τ0 ) + Cp+1 ∥h(·, τ0 )∥pLp (Ω) + ∥∆h(·, τ0 )∥pLp (Ω) p + 1 Ω Ω [ ∫ ) ∫ t p + |χ(β − δ)| (∥v(·, τ0 )∥pLp (Ω) + ∥∆v(·, τ0 )∥pLp (Ω) ) + e−δ(p+1)(t−τ ) (pκ + (p + 1)δ) up Ω τ0 ] ∫ ( ( )) p − 1 ∫ p+1 p+1 p+k p+1 u − pµ u . (3.9) + p + Cp+1 |χα − ξγ| + (αχ(β − δ)) p+1 Ω Ω Similarly, for β < δ, we obtain ∫ ∫ p−1( up ≤ e−β(p+1)(t−τ0 ) up (τ0 ) + Cp+1 ∥h(·, τ0 )∥pLp (Ω) + ∥∆h(·, τ0 )∥pLp (Ω) p+1 Ω Ω [ ∫ ) ∫ t p + |ξ(β − δ)| (∥w(·, τ0 )∥pLp (Ω) + ∥∆w(·, τ0 )∥pLp (Ω) ) + e−β(p+1)(t−τ ) (pκ + (p + 1)β) up τ0 Ω ] ∫ ( ( )) p − 1 ∫ p+1 + p + Cp+1 |χα − ξγ| + (ξγ(δ − β))p+1 up+1 − pµ up+k . (3.10) p+1 Ω Ω Noticing by Young’s inequality, for any ε > 0, there exists C(ε) > 0 such that ∫ ∫ up ≤ ε up+k + C(ε)|Ω | Ω
(3.11)
Ω
and ∫ Ω
up+1 ≤ ε
∫ Ω
up+k + C(ε)|Ω |.
(3.12)
J. Li et al. / Nonlinear Analysis: Real World Applications 39 (2018) 261–277
By choosing ε sufficiently small in (3.9) and (3.10), there exists C > 0 such that proved.
∫ Ω
269
up ≤ C. The lemma is
Lemma 3.3. Let (1.5) hold with k = 1 and N ≥ 3. There exists CN > 0 and { max{|χα − ξγ|, αχ(β − δ)}, if β ≥ δ, µ∗ = max{|χα − ξγ| + ξγ(δ − β)}, if β < δ such that if µ > CN µ∗ , ∫
uN (x, t)dx ≤ C
Ω
for t ∈ (0, Tmax ) and some constant C > 0. Proof . By introducing ε to Young’s inequality in (3.7), then instead of (3.9) and (3.10), we obtain ∫ ∫ p−1( ∥h(·, τ0 )∥pLp (Ω) + ∥∆h(·, τ0 )∥pLp (Ω) up ≤ e−δ(p+1)(t−τ0 ) up (τ0 ) + Cp+1 p+1 Ω Ω [ ∫ ) ∫ t p p p −δ(p+1)(t−τ ) + |χ(β − δ)| (∥v(·, τ0 )∥Lp (Ω) + ∥∆v(·, τ0 )∥Lp (Ω) ) + e (pκ + (p + 1)δ) up τ0 Ω ) ] ( ∫ ∫ p+1 |χα − ξγ| + (αχ(β − δ))p+1 p − 1 up+1 − pµ up+1 (3.13) + pε + Cp+1 εp p+1 Ω Ω for β ≥ δ and ∫ ∫ p−1( ∥h(·, τ0 )∥pLp (Ω) + ∥∆h(·, τ0 )∥pLp (Ω) up ≤ e−β(p+1)(t−τ0 ) up (τ0 ) + Cp+1 p+1 Ω Ω [ ∫ ) ∫ t p + |ξ(β − δ)| (∥w(·, τ0 )∥pLp (Ω) + ∥∆w(·, τ0 )∥pLp (Ω) ) + e−β(p+1)(t−τ ) (pκ + (p + 1)β) up τ0 Ω ) ] ( ∫ ∫ p+1 |χα − ξγ| + (ξγ(δ − β))p+1 p − 1 p+1 p+1 u − pµ u (3.14) + pε + Cp+1 εp p+1 Ω Ω for β < δ. Denote ⎧[ ( )] p − 1 ⎪ ⎨ pεp+1 + Cp+1 |χα − ξγ|p+1 + (χα(β − δ))p+1 − pµεp , if β ≥ δ, p + 1 ( )] p − 1 g(ε) = [ ⎪ ⎩ pεp+1 + Cp+1 |χα − ξγ|p+1 + (ξγ(δ − β))p+1 − pµεp , if β < δ, p+1 then g attains its minimum at ε∗ = ∗
µ =
pµ p+1 .
{
Hence let
max{|χα − ξγ|, αχ(β − δ)}, if β ≥ δ, max{|χα − ξγ|, ξγ(δ − β)}, if β < δ,
one can see that for p = N , there exists a constant CN > 0 such that g(ε∗ ) < 0 provided that µ > CN µ∗ , ∫ and thereby the coefficient of Ω uN +1 in the right side of (3.14) is negative, so there is C > 0 such that ∫ uN ≤ C Ω
for all t ∈ (0, Tmax ), and Lemma 3.3 is thus proved. Now we can collect our previous findings to arrive at the main results.
J. Li et al. / Nonlinear Analysis: Real World Applications 39 (2018) 261–277
270
Proof of Theorem 1.1 and Theorem 1.2. In view of Lemmas 2.1 and 2.4, Theorems 1.1 and 1.2 are direct consequences of Lemmas 3.2 and 3.3, respectively.
4. Asymptotic behavior This section is devoted to show the uniform convergence of solution to (1.1) with convergence rate in the case κ ≤ 0 and κ > 0, respectively. 4.1. Proof of Theorem 1.3 (The case κ ≤ 0) Let ϕ(t) :=
∫ Ω
ϕ(x, t)dx. Integrating the equations of (1.1) over Ω , we obtain ⎧ ∫ du µuk+1 ⎪ ⎪ , = κu − µ uk+1 ≤ κu − ⎪ ⎪ k ⎪ dt |Ω | Ω ⎪ ⎨ dv = αu − βv, ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎩ dw = γu − δw. dt
(4.1)
Firstly we consider the case κ = 0. From (4.1), we obtain ( u(t) ≤
µkt
1 + k k u (0) |Ω |
v(t) = e−β(t−τ0 ) v(τ0 ) + α
∫
≤ e−β(t−τ0 ) v(τ0 ) + α
∫
)−1/k ,
(4.2)
t
e−β(t−s) u(s)ds
τ0
(
t
e−β(t−s)
τ0
µks
1 + k k u (0) |Ω |
)−1/k ds
(4.3)
ds.
(4.4)
and w(t) = e
−δ(t−τ0 )
∫
t
e−δ(t−s) u(s)ds
w(τ0 ) + γ τ0
≤e
−δ(t−τ0 )
∫
(
t
w(τ0 ) + γ
−δ(t−s)
e τ0
Choosing τ0 =
2 βk
µks
1 + k k u (0) |Ω |
)−1
we obtain )−1/k 1 e−β(t−s) + k ds k u (0) |Ω | τ0 )−1/k ∫ ( 1 t µks 1 = + k de−β(t−s) β τ0 |Ω |k u (0) t
1 + k k u (0) |Ω |
in (4.3), then for s ∈ [τ0 , t], noticing (
∫
µks
(
µks
k
≤
k
|Ω | β|Ω | = , µkτ0 2µ
)−1/k
J. Li et al. / Nonlinear Analysis: Real World Applications 39 (2018) 261–277
1 = β
(
µks
1 + k k u (0) |Ω |
)−1/k s=t e−β(t−s) ⏐s=τ 0
⏐
∫
µ
+
(
t −β(t−s)
e
k
β|Ω | τ0 ( )−1/k ( )−1/k 1 µkt 1 1 1 µkτ0 ≤ + k + k − e−β(t−τ0 ) β |Ω |k β |Ω |k u (0) u (0) )−1/k ( ∫ 1 1 t −β(t−s) µks ds, e + k + k 2 τ0 u (0) |Ω |
µks
1 + k k u (0) |Ω |
271
)− k+1 k
ds
from which we obtain ∫
(
t
e−β(t−s)
τ0
µks
1 + k k u (0) |Ω |
)−1/k
2 ds ≤ β
(
µkt
1 + k k u (0) |Ω |
)−1/k .
(4.5)
Substituting (4.5) into (4.3) yields ( )−1/k 2α µkt 1 v(t) ≤ e−β(t−τ0 ) v(τ0 ) + + k β |Ω |k u (0) )−1/k ( 1 µkt + k . ≤C k u (0) |Ω | Similarly, we obtain ( )−1/k 2γ µkt 1 w(t) ≤ e w(τ0 ) + + k δ |Ω |k u (0) ( )−1/k 1 µkt + k ≤C . k u (0) |Ω | −δ(t−τ0 )
Furthermore, based on the regularities of solution (u, v, w), one can readily get a constant C1 > 0 (e.g. see [30, Lemma 3.14] or [3, Lemma 4.4]) such that ∥u∥W 1,∞ (Ω) ≤ C1 ,
∥v∥W 1,∞ (Ω) ≤ C1 ,
∥w∥W 1,∞ (Ω) ≤ C1
for all t > τ0 .
Thus from Lemma 2.3 and (4.2) it follows that ) ( N 1 N +1 N +1 ∥u(·, t)∥L∞ (Ω) ≤ CGN ∥u(·, t)∥W 1,∞ (Ω) ∥u(·, t)∥L1 (Ω) + ∥u(·, t)∥L1 (Ω) 1
≤ Cu N +1 ( )− 1 (N +1)k µkt 1 + ≤C k uk (0) |Ω | for all t > 0. Similarly, we obtain ∥v(·, t)∥L∞ (Ω)
(
µkt
(
µkt
1 ≤C + k k u (0) |Ω |
)−
1 (N +1)k
)−
1 (N +1)k
and ∥w(·, t)∥L∞ (Ω)
1 ≤C + k k u (0) |Ω |
.
272
J. Li et al. / Nonlinear Analysis: Real World Applications 39 (2018) 261–277
Now we turn to consider the case κ < 0. From the first equation of (4.1), we obtain d u ≤ κu dt and thus u(t) ≤ eκ(t−τ0 ) u(τ0 ). On the other hand, from the second and the third equation of (4.1), we obtain ∫ t v(t) = e−β(t−τ0 ) v(τ0 ) + α e−β(t−s) u(s)ds τ0 t
≤ e−β(t−τ0 ) v(τ0 ) + α
∫
e−β(t−s) eκ(s−τ0 ) u(τ0 )ds
τ0
= e−β(t−τ0 ) v(τ0 ) + αu(τ0 )
eκ(t−τ0 ) − e−β(t−τ0 ) β+κ
(4.6)
and w(t) = e−δ(t−τ0 ) w(τ0 ) + γ
∫
t
e−δ(t−s) u(s)ds
τ0 t
≤e
−δ(t−τ0 )
∫ w(τ0 ) + γ
e−δ(t−s) eκ(s−τ0 ) u(τ0 )ds
τ0
= e−δ(t−τ0 ) w(τ0 ) + γu(τ0 )
eκ(t−τ0 ) − e−δ(t−τ0 ) . δ+κ
(4.7)
Hence there exists C > 0 such that u(t) ≤ Ceκt , v(t) ≤ Cemax{κ,−β}t and w(t) ≤ Cemax{κ,−δ}t . Following the same procedure as that in the case κ = 0, we can obtain κt
∥u(·, t)∥L∞ (Ω) ≤ Ce N +1 , ∥v(·, t)∥L∞ (Ω) ≤ Ce
max{κ,−β}t N +1
and ∥w(·, t)∥L∞ (Ω) ≤ Ce
max{κ,−δ}t N +1
.
The proof of Theorem 1.3 is complete. 4.2. Proof of Theorem 1.4 (The case κ > 0) It is noted that in contrast to the solution behavior of the diffusive Fisher–KPP problem, the interaction effects between chemotactic cross-diffusion and cell kinetics may result in a quite colorful dynamics of chemotaxis–growth system [36–38]. In particular, the solution of (1.2) approaches to the unique nontrivial spatially homogeneous equilibrium when the strength of the overcrowding preventing effect µ is sufficiently
J. Li et al. / Nonlinear Analysis: Real World Applications 39 (2018) 261–277
273
large compared with that of the chemotactic sensitivity χ [39]. In this subsection, motivated by the ideas from [30,33,40], we consider the convergence of the solution in the case κ > 0. For simplicity, denote h = χv − ξw and h0 = χv0 − ξw0 , U :=
( µ )1/k κ
( H := h −
u,
χα ξγ − β δ
) ( )1/k κ , µ
( )1/k α κ V := v − . β µ
Then system (1.1) becomes ⎧ Ut = ∆U − ∇ · (U ∇H) + κU (1 − U k ), x ∈ Ω , t > 0, ⎪ ⎪ ( )1/k ⎪ ⎪ ⎪ κ ⎪ ⎪ (U − 1) − δH − χ(β − δ)V, x ∈ Ω , t > 0, Ht = ∆H + (χα − ξγ) ⎪ ⎪ µ ⎪ ⎪ ⎪ ( )1/k ⎪ ⎪ ⎪ κ ⎪ ⎪ Vt = ∆V + α (U − 1) − βV, x ∈ Ω , t > 0, ⎪ ⎪ µ ⎨ ∂H ∂V ∂U ⎪ = = = 0, x ∈ ∂Ω , t > 0, ⎪ ⎪ ∂ν ∂ν ∂ν ⎪ ⎪ ) ( )1/k ( ⎪ ( µ )1/k ⎪ ⎪ κ χα ξγ ⎪ ⎪ U (x, 0) = − , u (x), H(x, 0) = h (x) − 0 0 ⎪ ⎪ κ β δ µ ⎪ ⎪ ⎪ ( )1/k ⎪ ⎪ α κ ⎪ ⎪ ⎩V (x, 0) = v0 (x) − , x ∈ Ω. β µ
(4.8)
Lemma 4.1. For κ > 0, let (1.5) hold and (u, v, w) be the global solution of (1.1). Suppose ( µ>
2(αχ)2 (β − α)2 (χα − ξγ)2 + 2δ δβ 2
)k/2
κ1−k/2 .
Then for all t > 0 the function ∫
1 F (t) := (U − 1 − ln U ) + 2 Ω
2χ2 (β − δ)2 H + δβ Ω
∫
∫
2
V2
Ω
satisfies F ′ (t) ≤ −D(t), where {∫ D(t) := D
1 (U − 1) + 2 Ω 2
2χ2 (β − δ)2 H + δβ Ω
∫
2
∫ W
2
}
Ω
and D is defined in (4.12). ¯ × (0, ∞). Proof . According to the strong maximum principle and the assumption U0 ̸≡ 0, U is positive in Ω 1 Testing the first equation of (4.8) by 1 − U , we obtain d dt
∫ ∫ 2 |∇U | ∇U ∇H (U − 1 − ln U ) = − + − κ (U − 1)(U k − 1) U2 U Ω Ω Ω Ω ∫ ∫ ∫ ∫ 2 2 |∇U | 1 |∇U | 1 2 ≤− + + |∇H| − κ (U − 1)2 U2 2 Ω U2 2 Ω Ω Ω ∫ ∫ 1 2 2 ≤ |∇H| − κ (U − 1) , 2 Ω Ω
∫
∫
(4.9)
J. Li et al. / Nonlinear Analysis: Real World Applications 39 (2018) 261–277
274
where we have used the Young inequality. Testing the second and the third equation of (4.8) by H and V , respectively, and by the Young inequality, we obtain 1 d 2 dt
∫ Ω
( )1/k ∫ ∫ ∫ κ 2 (U − 1)H − δ H 2 − χ(β − δ) |∇H| + (χα − ξγ) VH µ Ω Ω Ω Ω ( )2/k ∫ ∫ ∫ ∫ δ χ2 (β − δ)2 (χα − ξγ)2 κ 2 2 2 (U − 1) − H + V 2 (4.10) ≤− |∇H| + 2δ µ 4 Ω δ Ω Ω Ω
H2 = −
∫
and 1 d 2 dt
∫
( )1/k ∫ ∫ κ (U − 1)V − β V2 µ Ω Ω Ω ( )2/k ∫ ∫ ∫ 2 κ α β 2 2 V 2. |∇V | + (U − 1) − ≤− 2β µ 2 Ω Ω Ω
V2 =−
Ω
∫
2
|∇V | + α
(4.11)
The combination of (4.9), (4.10) and (4.11) gives [ ( ) ( )2/k ] ∫ (χα − ξγ)2 2(χα)2 (β − δ)2 κ ′ F (t) ≤ − κ − + (U − 1)2 2δ β2δ µ Ω ∫ 2 2 ∫ δ χ (β − δ) − H2 − V 2. 4 Ω δ Ω Choosing {
(
D := min κ −
(χα − ξγ)2 2(χα)2 (β − δ)2 + 2δ β2δ
} ) ( )2/k κ δ β , , µ 2 2
(4.12)
and ( µ>
(χα − ξγ)2 2(αχ)2 (β − δ)2 + 2δ β2δ
)k/2
κ1−k/2 ,
we have D > 0 and thus ′
{∫
F (t) ≤ −D
1 (U − 1) + 2 Ω 2
2χ2 (β − δ)2 H + δβ Ω
∫
2
∫ V
2
} = −D(t).
Ω
Now we are at the position to show the proof of Theorem 1.4. Denote h(s) = s − 1 − ln s. Noticing that h′ (s) = 1 − 1s and h′′ (s) = s12 > 0 for all s > 0, we obtain that h(s) ≥ h(1) = 0 and F (t) is nonnegative. From Lemma 4.1, we have F ′ (t) ≤ −D(t) and then ∫
t
D(τ )dτ ≤ F (τ0 ) − F (t) ≤ F (τ0 ) τ0
for all t > τ0 , from which we have } ∫ t {∫ ∫ ∫ 2χ2 (β − δ)2 1 2 2 2 (U − 1) + H + V < ∞. 2 Ω δβ Ω τ0 Ω Using the similar argument as in Lemma 3.10 of [33], we can obtain the uniform convergence of solutions, namely ∥U − 1∥L∞ (Ω) → 0,
∥H∥L∞ (Ω) → 0,
∥V ∥L∞ (Ω) → 0
J. Li et al. / Nonlinear Analysis: Real World Applications 39 (2018) 261–277
275
as t → ∞. Then there exists t0 > 0 such that for all t > t0 , ∥U − 1∥L∞ (Ω) ≤ 12 , which together with the fact that 1 (s − 1)2 ≤ h(s) ≤ (s − 1)2 3 for all s >
1 2
implies that ∫ ∫ ∫ 1 1 2χ2 (β − δ)2 1 (U − 1)2 + H2 + V 2 ≤ F (t) ≤ D(t) 3 Ω 2 Ω δβ D Ω
(4.13)
for all t > t0 . Hence F ′ (t) ≤ −D(t) ≤ −DF (t), from which we obtain F (t) ≤ F (t0 )e−D(t−t0 ) .
(4.14)
Substituting (4.14) into (4.13), we obtain ∫ ∫ ∫ 1 2χ2 (β − δ)2 1 2 2 (U − 1) + H + V 2 ≤ F (t0 )e−D(t−t0 ) , 3 Ω 2 Ω δβ Ω which implies that there exists C > 0 such that for all t > t0 , ∥U (·, t) − 1∥L2 (Ω) ≤ Ce−Dt/2 , ∥H(·, t)∥L2 (Ω) ≤ Ce−Dt/2 and ∥V (·, t)∥L2 (Ω) ≤ Ce−Dt/2 . Furthermore, noticing that there exists a constant C1 > 0 such that ∥U − 1∥W 1,∞ (Ω) ≤ C1 ,
∥H∥W 1,∞ (Ω) ≤ C1 ,
∥V ∥W 1,∞ (Ω) ≤ C1
for all t > τ0 .
Thus the Gagliardo–Nirenberg inequality (Lemma 2.3) yields ( ) N 2 N +2 N +2 ∥U (·, t) − 1∥L∞ (Ω) ≤ C ∥U (·, t) − 1∥W ∥U (·, t) − 1∥ + ∥U (·, t) − 1∥ 2 1,∞ (Ω) L (Ω) L2 (Ω) 2
Dt
≤ Ce− N +2 ≤ C∥U (·, t) − 1∥LN2+2 (Ω) for all t > 0. Similarly, we can obtain Dt
∥H(·, t)∥L∞ (Ω) ≤ Ce− N +2 and Dt
∥V (·, t)∥L∞ (Ω) ≤ Ce− N +2 . Therefore the proof of Theorem 1.4 is complete. Acknowledgments The authors are grateful to the referee whose suggestions improve the exposition of the paper. This work is partially supported by NSFC (Nos. 11571363, 61620106002) and the Beijing Natural Science Foundation (No. 9152013) and the Research Grant Funds of Minzu University of China.
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