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On the minimal Keller–Segel system with logistic growth
Nonlinear Analysis: Real World Applications 55 (2020) 103121
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Nonlinear Analysis: Real World Applications www.elsevier.com/locate/nonrwa
On the minimal Keller–Segel system with logistic growth✩ Aung Zaw Myint, Jianping Wang ∗, Mingxin Wang School of Mathematics, Harbin Institute of Technology, Harbin 150001, PR China
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Article history: Received 20 April 2019 Received in revised form 22 February 2020 Accepted 24 February 2020 Available online xxxx Keywords: Chemotaxis Logistic growth Global existence and boundedness
abstract This paper applies a delicate method which is inspired by Deuring (1987) and is different from those of Winkler (2010) and Yang et al. (2015) to show the known conclusion: The weak chemotactic effect can ensure the global existence and boundedness of the solutions of the minimal Keller–Segel model with logistic growth in any dimensional cases. Moreover, we obtain the explicit uniform-in-time upper bound for the global solution. It is noted that the method used in the paper may be employed to study other chemotaxis systems. © 2020 Elsevier Ltd. All rights reserved.
1. Introduction The chemotaxis system describes the movement of the bacteria which is oriented by the chemical substances in the environment. Since Keller and Segel [1,2] first proposed chemotaxis models to understand the aggregation of certain type of bacteria, lots of literatures have been devoted to clarify the dynamics of the system. The main issues of the investigations are global existence of classical/weak solutions; blow-up phenomenon; asymptotic behaviors; pattern formation; spatial spreading and front propagation dynamics when the domain is a whole space; etc. It was shown that the blow-up happens for the minimal Keller–Segel system in n-dimensions with n ≥ 2 [3]. And the collapse is caused by the chemotaxis effect. In the presence of the logistic term, the blow-up phenomenon is excluded in 2-dimensional case [3]. Hence, it is reasonable to believe that the logistic damping has the effect to suppress the blow-up. Due to the coexistence of the chemotactic effect and the damping source of logistic type, the minimal Keller–Segel model with logistic growth may exhibit many interesting dynamics which are still waiting for investigation. The aim of this paper is to provide another insight into the minimal Keller–Segel model with logistic damping. The classical ✩ This work was supported by NSFC Grant 11771110. ∗ Corresponding author. E-mail address: [email protected] (J. Wang).
https://doi.org/10.1016/j.nonrwa.2020.103121 1468-1218/© 2020 Elsevier Ltd. All rights reserved.
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A.Z. Myint, J. Wang and M. Wang / Nonlinear Analysis: Real World Applications 55 (2020) 103121
minimal Keller–Segel model with logistic growth reads ⎧ ut = d∆u − χ∇ · (u∇v) + au − bu2 , x ∈ Ω , t > 0, ⎪ ⎪ ⎪ ⎨ vt = D∆v − rv + µu, x ∈ Ω , t > 0, ∂v ∂u ⎪ = = 0, x ∈ ∂Ω , t > 0, ⎪ ⎪ ∂ν ∂ν ⎩ u(x, 0) = u0 (x), v(x, 0) = v0 (x), x ∈ Ω,
(1.1)
∂ denotes differentiation with respect where Ω is a bounded domain in Rn with smooth boundary ∂Ω , ∂ν to the outward normal vector on ∂Ω . The unknown functions u(x, t), v(x, t) represent the density of the bacterial and the concentration of the signal, respectively. We summarize the results on (1.1) as follows:
• In any dimensional cases, Winkler [4] initially proved that the strong logistic dampening effect can assert the global boundedness. Moreover, it has been found in [5] that there is θ0 > 0 such that (1.1) has a unique global bounded solution provided χ < θ0 b. • In 1,2-dimensions, it was shown in [6,7] that the solutions to (1.1) exists globally and remains bounded. In [8], Jin and Xiang investigated the chemotactic effect versus logistic damping on boundedness for (1.1) and found that the solution to (1.1) admits an upper bound which becomes singular only if b = 0. • In 3-dimension, when χ = 1 and b ≥ 23, the conclusions in [9] indicate the global existence and boundedness of the solution to (1.1). Later on, Mu and Lin [10] studied the system (1.1) with the logistic term replaced by u − buα with α ≥ 2 and claimed the existence of the globally bounded solution under the condition b1/(α−1) > 20χ which clearly improve the result obtained in [9]. Recently, Xiang [11,12] has been focusing on identifying the chemotactic aggregation versus logistic damping in (1.1). It was 9 proved in [12] that, if b > √10−2 χ, then there is an explicit uniform-in-time upper bound for the unique 9 χ. Although it is still solution of (1.1) and this upper bound has singular only in b on the line b = √10−2 unknown that whether or not the classical solution blows up if b is small, in 3-D convex domain, the weak solution has been shown to exist globally [13]. • Does the logistic damping can always prevent blow-up phenomenon? Unfortunately, the answer is not. In [14,15], Winkler claimed that the finite time blow-up happens for a parabolic–elliptic version of (1.1) despite logistic growth restriction. For other works on the dynamical properties of the chemotaxis systems with logistic source, please refer to [3,12,16–19] and references therein. Some related chemotaxis models with logistic source have attracted attentions in recent years. For the Keller–Segel model with signal-dependent motility and logistic growth, Jin et al. [20] proved the global boundedness for the solutions in two dimension and Wang and Wang [21] found that the strong logistic dampening effect can prevent blow-up in any dimensional cases. These conclusions coincide with that of (1.1). Further studies are required in this new topic. It is noted that, the above works on (1.1) are performed in the bounded domain. It would be rather interesting and meaningful to understand the dynamics of (1.1) in the whole space. Salako and Shen [22] investigated traveling wave solutions to (1.1) in the whole space and obtained some important results. Since the conclusion in this paper holds true in any dimensions, it is necessary to compare our result and method with that of [4,5]. On the one hand, it is noted that the literatures [4,5] give the conditions ensuring the global boundedness of the solution without offering the uniform-in-time upper bound for it. However, we get an explicit upper bound for the solution. On the other hand, instead of using the Maximal Sobolev Regularity or analysing the quantity ∫ q ∑ 2(q−p) cp up |∇v| dx p=0
Ω
with large q ∈ N and positive constants cp , the method we applied is inspired by [23]. To be precise, we first use Schauder-type estimates to derive a priori bounds for (1.1). And then by the Schauder’s fixed
A.Z. Myint, J. Wang and M. Wang / Nonlinear Analysis: Real World Applications 55 (2020) 103121
3
point theorem, we conclude that the solutions to (1.1) are bounded. As byproducts, sharp estimates for the solutions are obtained. The notations of the H¨ older spaces used in this paper come from [23] (see also [24, Chapter 3]). Throughout this paper the initial data u0 , v0 are supposed to satisfy ¯ ) with α ∈ (0, 1), u0 , v0 ∈ C 2+α (Ω
∂u0 ∂v0 = = 0 on ∂Ω . ∂ν ∂ν
¯, u0 , v0 ≥ 0 in Ω
(1.2)
For the convenience, we set ¯ ) : ∂ϕ = 0 on ∂Ω , ϕ ≥ 0 and ∥ϕ∥ 2+α ¯ ≤ k}. Lk = {ϕ ∈ C 2+α (Ω C (Ω) ∂ν Since it had been shown that the solution of (1.1) exist globally and remain bounded in 1,2-dimensions [6,7], the arguments in the sequel will be performed in the high dimensions, i.e., Ω ∈ Rn with n ≥ 3. The following theorem is the main result of this article. Theorem 1.1. Lemma 3.1. Let
Let n ≥ 3, D, µ, r > 0, α ∈ (0, 1), m, M ∈ R with 0 < m ≤ M , and P, R be given by d, b ∈ [m, M ], χ ∈ [0, R/2] , a ∈ [0, mM /2] ,
and u0 ∈ LM , v0 ∈ LµM/r . Then the problem (1.1) admits a unique nonnegative global solution (u, v) ∈ ¯ × [0, ∞)))2 . Furthermore, for any T > 0, we have (C 2,1 (Ω 0≤u≤M
on QT ,
∥u∥C 2+α,1+α/2 (Q
T)
≤ mM /R,
and 0 ≤ v ≤ µM /r on QT ,
∥v∥C 2+α,1+α/2 (Q
T)
≤ mM /R,
where QT = Ω × (0, T ]. We deduce from Theorem 1.1 that, for given d, a, b, D, r, µ > 0, we can define suitable m, M (and hence P, R) such that if χ ≤ R/2, then the parameters satisfy the conditions in Theorem 1.1, and hence the problem (1.1) is globally solvable. Corollary 1.1.
Assume that n ≥ 3 and d, a, b, D, r, µ > 0. Let m = min{d, b} and } { r∥v0 ∥C 2+α (Ω) ¯ 2a , ∥u0 ∥C 2+α (Ω) . M = max b, d, ¯ , m µ
Let P, R be given in Lemma 3.1. When 0 ≤ χ ≤ R/2, the problem (1.1) has a unique nonnegative and bounded global solution (u, v), and ¯ × [0, ∞)). u, v ∈ C 2,1 (Ω Furthermore, we have 0 ≤ u ≤ M, 0 ≤ v ≤ µM /r in Ω × [0, ∞), and for any T > 0, ∥u∥C 2+α,1+α/2 (Q ) , ∥v∥C 2+α,1+α/2 (Q T
T)
≤ mM /R.
¯ × (0, ∞). Remark 1.1. If we further assume that u0 , v0 ≥, ̸≡ 0, then u, v > 0 in Ω Remark 1.2. It is easy to see that M ≥ max{2a/m, ∥u0 ∥C 0 (Ω) ¯ } ≥ max{a/b, ∥u0 ∥C 0 (Ω) ¯ }, and hence our assertion is consistent with the boundedness result for the FisherKPP equation (i.e., χ = 0). Moreover, we observe from the definition of R that R is decreasing with respect to µ which is reasonable.
A.Z. Myint, J. Wang and M. Wang / Nonlinear Analysis: Real World Applications 55 (2020) 103121
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2. Preliminaries We first give a claim concerning the local-in-time existence of the classical solutions to (1.1) whose proof can be found in [4]. Lemma 2.1. Let d, a, b, D, r, µ > 0 and χ ≥ 0. Suppose that the initial data (u0 , v0 ) satisfies (1.2). There exist a Tmax ∈ (0, ∞] and a unique nonnegative solution (u, v) of (1.1) defined in [0, Tmax ) that satisfies ¯ × [0, Tmax )), u, v ∈ C 2,1 (Ω and u, v ≥ 0 in Ω × [0, Tmax ).
(2.1)
Moreover, the “existence time Tmax ” can be chosen maximal: either Tmax = ∞, or Tmax < ∞ and lim sup ∥u(·, t)∥L∞ (Ω) = ∞. t→Tmax
The following lemma plays an important role in our analysis. Lemma 2.2 ([23, Theorem 3.1]). There is a function KΩ : (0, 1) × (0, ∞)2 → (0, ∞) with the following property ¯ ), (1) Let α ∈ (0, 1), K1 ≥ K2 > 0, T ∈ [1, ∞), a, bi (i = 1, 2, . . . , n), c, f ∈ C α,α/2 (QT ), ϕ ∈ C 2+α (Ω 2,1 u ∈ C (QT ), with ∥a∥C α,α/2 (Q ) ≤ K1 , ∥bi ∥C 0 (Q ) ≤ K1 , ∥c∥C 0 (Q ) ≤ K1 , a ≥ K2 on QT , and T
T
T
⎧ n ∑ ⎪ ⎪ ⎪ u − a∆u + bi Di u − cu = f, x ∈ Ω , 0 < t ≤ T, ⎪ ⎨ t i=1 ∂u ⎪ = 0, x ∈ ∂Ω , 0 ≤ t ≤ T, ⎪ ⎪ ⎪ ⎩ ∂ν ¯. u(x, 0) = ϕ(x), x∈Ω Then ∥u∥C α,α/2 (Q
T)
≤ KΩ (α, K1 , K2 )(∥f ∥C 0 (Q
T)
+ ∥ϕ∥C 2 (Ω) ¯ + ∥u∥C 0 (Q ) ). T
2
Furthermore, there is a function LΩ : (0, 1) × (0, ∞) → (0, ∞) with the following property (2) Let α, K1 , K2 , T, a, bi , c, f, ϕ, u be given as in (1), but with the assumptions ∥bi ∥C 0 (Q ) , ∥c∥C 0 (Q ) ≤ K1 T T replaced by the stronger conditions ∥bi ∥C α,α/2 (Q ) , ∥c∥C α,α/2 (Q ) ≤ K1 . T T Then ∥u∥C 2+α,1+α/2 (Q ) ≤ LΩ (α, K1 , K2 )(∥f ∥C α,α/2 (Q ) + ∥ϕ∥C 2+α (Ω) ¯ + ∥u∥C 0 (Q ) ). T
T
T
Remark 2.1. Noticing that KΩ and LΩ are independent of T and the condition T ≥ 1 could be replaced by T ≥ T0 for any fixed T0 > 0 (See [23, Theorem 3.1, Remark]). 3. Proof of Theorem 1.1 Fixed D, r, µ > 0. Let λ = max{D, r}. We first give the following a priori estimates for u for a class of parameters, initial data and given v. Lemma 3.1.
Let T ∈ [1, ∞), α ∈ (0, 1), D, r, µ > 0, m, M ∈ R with 0 < m ≤ M , and take
P = P (α, m, M ) = 2KΩ (α, M + mM + M 2 , m),
A.Z. Myint, J. Wang and M. Wang / Nonlinear Analysis: Real World Applications 55 (2020) 103121
{ R = R(α, m, M ) = min
m m ) ( , LΩ (α, M + mM, m)(M 2 (1 + 2P ) + 2) LΩ (α, λ, D) P + 2r µ
5
} ,
where LΩ , KΩ are defined in Lemma 2.2 and λ = max{D, r}. Let d0 ∈ [m, M ], χ0 ∈ [0, M/2], a0 ∈ [0, mM/2], b0 ∈ [mM/R, M 2 /R] and φ ∈ LR . v ∥C 2+α,1+α/2 (Q ) ≤ m. Then the problem Suppose that vˆ ∈ C 2+α,1+α/2 (QT ) satisfies 0 ≤ vˆ ≤ µR r and ∥ˆ T
⎧ u ˆ − d0 ∆ˆ u + χ0 ∇ˆ v · ∇ˆ u − (a0 − χ0 ∆ˆ v − b0 u ˆ)ˆ u = 0, x ∈ Ω , 0 < t ≤ T, ⎪ ⎨ t ∂u ˆ = 0, x ∈ ∂Ω , 0 ≤ t ≤ T, ⎪ ⎩ ∂ν ¯ u ˆ(x, 0) = φ(x), x∈Ω
(3.1)
admits a unique solution u ˆ ∈ C 2+α,1+α/2 (QT ). Moreover, we have 0≤u ˆ≤R
on
QT ,
(3.2)
and ∥ˆ u∥C α,α/2 (Q
T)
≤ P R,
∥ˆ u∥C 2+α,1+α/2 (Q
T)
≤ m.
(3.3)
¯ . Hence, Proof . Step 1: Existence, uniqueness and estimate (3.2). Recall that u ˆ(x, 0) = φ(x) ≥ 0 for x ∈ Ω it can be deduced that w(x, t) ≡ 0 is a lower solution of (3.1). Let w(x, t) ≡ R on QT . Due to χ0 ∈ [0, M/2], ∥ˆ v ∥C 2+α,1+α/2 (Q
T)
≤ m, a0 ∈ [0, mM/2], b0 ∈ [mM/R, M 2 /R],
we have a0 − χ0 ∆ˆ v − b0 w ≤ 0 on QT
(3.4)
Thanks to ∥φ∥C 2+α (Ω) ¯ ≤ R, it is easy to see that w satisfies ⎧ v · ∇w − (a0 − χ0 ∆ˆ v − b0 w)w ≥ 0, x ∈ Ω , 0 < t ≤ T, ⎪ ⎨ wt − d0 ∆w + χ0 ∇ˆ ∂w ≥ 0, x ∈ ∂Ω , 0 ≤ t ≤ T, ⎪ ⎩ ∂ν ¯. w(x, 0) ≥ φ(x), x∈Ω Therefore, w is an upper solution of (3.1). In view of the upper and lower solutions method, we know that (3.1) has a unique solution u ˆ ∈ C 2+α,1+α/2 (QT ) and satisfies (3.2). Step 2: The estimate (3.3). By direct computations, it is easy to see that d0 , ∥χ0 Di vˆ∥C 0 (Q ) , ∥a0 − χ0 ∆ˆ v − b0 u ˆ∥C 0 (Q
T)
T
≤ M (1 + m + M ), i = 1, . . . , n,
(3.5)
and d0 ≥ m
(3.6)
thanks to the assumptions for d0 , χ0 , vˆ, a0 , b0 and (3.2). In view of Lemma 2.2(1), it yields ∥ˆ u∥C α,α/2 (Q
T)
≤ KΩ (α, M (1 + m + M ), m)(∥φ∥C 2 (Ω) u∥C 0 (Q ) ) ¯ + ∥ˆ T
≤ 2KΩ (α, M (1 + m + M ), m)R
(3.7)
because ∥φ∥C 2+α (Ω) ˆ ≤ R on QT . Recall the definition of P = P (α, m, M ), it follows from ¯ ≤ R and 0 ≤ u (3.7) that ∥ˆ u∥C α,α/2 (Q ) ≤ P R. T
This gives the first part of (3.3).
A.Z. Myint, J. Wang and M. Wang / Nonlinear Analysis: Real World Applications 55 (2020) 103121
6
Rewriting the system (3.1) as ⎧ u ˆ − d0 ∆ˆ u + χ0 ∇ˆ v · ∇ˆ u − (a0 − χ0 ∆ˆ v )ˆ u = −b0 u ˆ2 , x ∈ Ω , 0 < t ≤ T, ⎪ ⎨ t ∂u ˆ = 0, x ∈ ∂Ω , 0 ≤ t ≤ T, ⎪ ∂ν ⎩ ¯. u ˆ(x, 0) = φ(x), x∈Ω In light of the assumptions of d0 , χ0 , a0 , vˆ, there hold v ∥C α,α/2 (Q d0 , ∥χ0 Di vˆ∥C α,α/2 (Q ) , ∥a0 − χ0 ∆ˆ
T)
T
≤ M + mM, i = 1, . . . , n.
(3.8)
Moreover, from the first part of (3.3), we deduce ∥b0 u ˆ2 ∥C α,α/2 (Q
T)
≤ b0 (∥ˆ u∥2C 0 (Q
T)
+ 2∥ˆ u∥C 0 (Q ) ∥ˆ u∥C α,α/2 (Q ) ) ≤ M 2 R(1 + 2P ) T
(3.9)
T
where we have used the estimate [23, (4)], (3.2) and b0 ∈ [mM/R, M 2 /R]. Note that ∥φ∥C 2+α (Ω) ¯ ≤ R. Using Lemma 2.2(2) firstly and (3.2), (3.8) and (3.9) secondly, we find ∥ˆ u∥C 2+α,1+α/2 (Q
T)
≤ LΩ (α, M + mM, m)(∥b0 u ˆ2 ∥C α,α/2 (Q
T)
+ ∥φ∥C 2+α (Ω) u∥C 0 (Q ) ) ¯ + ∥ˆ T
2
≤ LΩ (α, M + mM, m)[M (1 + 2P ) + 2]R ≤ m. Here we used the definition of R. The second part of (3.3) is proved and the proof is complete. □ Let us proceed to derive the a priori estimates for v when u is given and fulfills (3.2) and (3.3). Lemma 3.2. Let T ∈ [1, ∞), α ∈ (0, 1), D, r, µ > 0, m, M ∈ R with 0 < m ≤ M , and P, R be given in the above lemma. And let ρ ∈ L(µR)/r . Suppose that u∗ ∈ C 2+α,1+α/2 (QT ) satisfies 0 ≤ u∗ ≤ R on QT , ∥u∗ ∥C α,α/2 (Q ) ≤ P R and ∥u∗ ∥C 2+α,1+α/2 (Q ) ≤ m. Then the problem T
T
⎧ ∗ v − D∆v ∗ + rv ∗ = µu∗ , x ∈ Ω , 0 < t ≤ T, ⎪ ⎨ t ∗ ∂v = 0, x ∈ ∂Ω , 0 ≤ t ≤ T, ⎪ ⎩ ∂ν ∗ ¯ v (x, 0) = ρ(x), x∈Ω
(3.10)
has a unique solution v ∗ ∈ C 2+α,1+α/2 (QT ). Moreover, we have 0 ≤ v ∗ ≤ µR/r on QT ,
(3.11)
∥v ∗ ∥C 2+α,1+α/2 (Q
(3.12)
and T)
≤ m.
¯ ), u∗ ∈ C 2+α,1+α/2 (QT ), making use of the Schauder theory, there is a unique Proof . Due to ρ ∈ C 2+α (Ω ∗ 2+α,1+α/2 solution v ∈ C (QT ). Noticing that 0 ≤ u∗ ≤ R and 0 ≤ ρ ≤ µR r . It is easy to get (3.11) by using the parabolic maximum principle. We omit the details. In view of Lemma 2.2(2), we get ∥v ∗ ∥C 2+α,1+α/2 (Q
T)
≤ LΩ (α, λ, D)(µ∥u∗ ∥C α,α/2 (Q
T)
∗ + ∥ρ∥C 2+α (Ω) ¯ + ∥v ∥C 0 (Q ) ) T
≤ LΩ (α, λ, D) (P + 2/r) µR, where we have used (3.11) and the assumptions ∥u∗ ∥C α,α/2 (Q ) ≤ P R, ∥ρ∥C 2+α (Ω) ¯ ≤ µR/r. Recalling the T definition of R, it is easy to yield (3.12). The proof is end. □
A.Z. Myint, J. Wang and M. Wang / Nonlinear Analysis: Real World Applications 55 (2020) 103121
For T ≥ 1 and D, µ, r > 0, we shall consider (1.1) in (0, T ]: ⎧ ut = d∆u − χ∇ · (u∇v) + au − bu2 , ⎪ ⎪ ⎪ ⎨ vt = D∆v − rv + µu, ∂v ∂u ⎪ = = 0, ⎪ ⎪ ∂ν ⎩ ∂ν u(x, 0) = u0 (x), v(x, 0) = v0 (x),
7
x ∈ Ω , 0 < t ≤ T, x ∈ Ω , 0 < t ≤ T, (3.13)
x ∈ ∂Ω , 0 ≤ t ≤ T, ¯. x∈Ω
Lemma 3.3. Let T ∈ [1, ∞), α ∈ (0, 1), D, r, µ > 0, m, M ∈ R with 0 < m ≤ M , and P, R be given by Lemma 3.1. Let [ ] d˜ ∈ [m, M ], χ ˜ ∈ [0, M/2] , a ˜ ∈ [0, mM /2] , ˜b ∈ mM /R, M 2 /R , and u ˜0 ∈ LR and v˜0 ∈ LµR/r . Then there exists a solution (˜ u, v˜) ∈ (C 2+α,1+α/2 (QT ))2 which solves the ˜ ˜ problem (3.13) with d, χ, a, b, u0 , v0 replaced by d, χ, ˜ a ˜, b, u ˜0 , v˜0 and satisfies 0≤u ˜ ≤ R on QT , ∥˜ u∥C α,α/2 (Q
T)
≤ P R, and ∥˜ u∥C 2+α,1+α/2 (Q
T)
≤ m,
and 0 ≤ v˜ ≤ µR/r on QT ,
∥˜ v ∥C 2+α,1+α/2 (Q
T)
≤ m.
Proof . Denote U : = {ϕ ∈ C 2+α,1+α/2 (QT ) : 0 ≤ ϕ ≤ R, ∥ϕ∥C α,α/2 (Q V : = {ϕ ∈ C
2+α,1+α/2
T)
≤ P R, ∥ϕ∥C 2+α,1+α/2 (Q
T)
(QT ) : 0 ≤ ϕ ≤ µR/r, ∥ϕ∥C 2+α,1+α/2 (Q
T)
≤ m},
≤ m}.
Define W := [C 2+α/2,1+α/4 (QT )]2 . Clearly, W is a Banach space endowed with the norm ∥(u, v)∥W = ∥u∥C 2+α/2,1+α/4 (Q
T)
+ ∥v∥C 2+α/2,1+α/4 (Q ) . T
It is easy to see that U × V is a compact, convex subset of the Banach space W . ˜ χ, ˜ For (u, v) ∈ U × V , denote by Q(v) the solution of (3.1) with d0 , χ0 , a0 , b0 , vˆ, φ replaced by d, ˜ a ˜, ˜b, v, u ˜0 ∗ ⋆ ⋆ ˜ ˜ ˜ and G(u) the solution of (3.10) with u , ρ replaced by u, v˜0 , respectively. And, let u = Q(v) and v = G(u). From Lemmas 3.1 and 3.2 we know that 0 ≤ u⋆ ≤ R, ∥u⋆ ∥C α,α/2 (Q
T)
≤ P R, ∥u⋆ ∥C 2+α,1+α/2 (Q
T)
≤ m,
(3.14)
and 0 ≤ v ⋆ ≤ µR/r, ∥v ⋆ ∥C 2+α,1+α/2 (Q
T)
≤ m.
(3.15)
˜ ˜ Set F˜ (u, v) = (u⋆ , v ⋆ ) = (Q(v), G(u)). Hence F˜ is an operator from U × V into U × V . Next we shall show that F˜ is continuous in the norm ∥·∥W of the Banach space W . For v1 , v2 ∈ V , denote ˜ i ) with i = 1, 2 and u u ˜i = Q(v ˜=u ˜1 − u ˜2 . It is easy to see that u ˜=u ˜1 − u ˜2 satisfies ⎧ ˜ u + χ∇v u ˜t − d∆˜ ˜ 1 · ∇˜ u − [˜ a − χ∆v ˜ 1 − ˜b(˜ u1 + u ˜2 )]˜ u ⎪ ⎪ ⎪ ⎨ = −χ∇˜ ˜ u2 · ∇(v1 − v2 ) − χ˜ ˜u2 ∆(v1 − v2 ), x ∈ Ω , 0 < t ≤ T, ∂u ˜ ⎪ = 0, x ∈ ∂Ω , 0 ≤ t ≤ T, ⎪ ⎪ ∂ν ⎩ ¯. u ˜(x, 0) = 0, x∈Ω ˜ χ, Moreover, u ˜i satisfy (3.14) for i = 1, 2. For simplicity, let E = max{d, ˜ a ˜, ˜b}. Thanks to 1 ≤ T < ∞, there is k > 0 which may depend on T such that ∥w∥C α/2,α/4 (Q ) ≤ k∥w∥C α,α/2 (Q ) for any w ∈ C α,α/2 (QT ). T
T
8
A.Z. Myint, J. Wang and M. Wang / Nonlinear Analysis: Real World Applications 55 (2020) 103121
˜ χ, Making use of the properties of vi , u ˜i with i = 1, 2 and the assumptions for d, ˜ a ˜, ˜b, there hold d˜ ≥ m and for j = 1, . . . , n, ˜ ∥χD a − χ∆v ˜ 1 − ˜b(˜ u1 + u ˜2 )]∥C α/2,α/4 (Q d, ˜ j v1 ∥C α/2,α/4 (Q ) , ∥[˜
T)
T
≤ E(1 + 3km).
Moreover, ∥ − χ∇˜ ˜ u2 · ∇(v1 − v2 ) − χ˜ ˜u2 ∆(v1 − v2 )∥C α/2,α/4 (Q
T)
≤ 2Ekm∥v1 − v2 ∥C 2+α/2,1+α/4 (Q ) , T
where we have used the fact ∥f g∥C α,α/2 (Q
T)
≤ ∥f ∥C α,α/2 (Q ) ∥g∥C α,α/2 (Q
T)
T
due to the definition of C α,α/2 (QT ). Hence, it follows from the parabolic Schauder theory that, there is K1 > 0 which depends on α, T, E, k, m, Ω such that ∥˜ u∥C 2+α/2,1+α/4 (Q
T)
≤ K1 ∥ − χ∇˜ ˜ u2 · ∇(v1 − v2 ) − χ˜ ˜u2 ∆(v1 − v2 )∥C α/2,α/4 (Q
T)
≤ 2kK1 mE∥v1 − v2 ∥C 2+α/2,1+α/4 (Q
T)
≤ K2 ∥v1 − v2 ∥C 2+α/2,1+α/4 (Q ) ,
(3.16)
T
where K2 = 2kK1 mE. ˜ i ) with i = 1, 2 and v˜ = v˜1 − v˜2 . It is easy to see that v˜ = v˜1 − v˜2 solves For u1 , u2 ∈ V , denote v˜i = G(v ⎧ v˜ − D∆˜ v + r˜ v = µ(u1 − u2 ), x ∈ Ω , 0 < t ≤ T, ⎪ ⎨ t ∂˜ v = 0, x ∈ ∂Ω , 0 ≤ t ≤ T, ⎪ ⎩ ∂ν ¯. v˜(x, 0) = 0, x∈Ω Again by the parabolic Schauder theory, there is K3 > 0 which depends on α, T, D, r, µ, Ω such that ∥˜ v ∥C 2+α/2,1+α/4 (Q
T)
≤ K3 ∥u1 − u2 ∥C α/2,α/4 (Q
T)
≤ K3 ∥u1 − u2 ∥C 2+α/2,1+α/4 (Q ) . T
This combined with (3.16) yields that ∥F˜ (u1 , v1 ) − F˜ (u2 , v2 )∥W = ∥(˜ u, v˜)∥W = ∥˜ u∥C 2+α/2,1+α/4 (Q
T)
+ ∥˜ v ∥C 2+α/2,1+α/4 (Q
T)
≤ max{K2 , K3 }(∥u1 − u2 ∥C 2+α/2,1+α/4 (Q
T)
+ ∥v1 − v2 ∥C 2+α/2,1+α/4 (Q ) ) T
≤ max{K2 , K3 }∥(u1 , v1 ) − (u2 , v2 )∥W . This shows that F˜ is continuous in the norm ∥ · ∥W of the Banach space W . In light of the Schauder’s fixed point theorem [25, Theorem 11.1], there exists (¯ u, v¯) ∈ U × V such that F˜ (¯ u, v¯) = (¯ u, v¯). Hence, there is a solution (¯ u, v¯) which solves the problem (3.13) with d, χ, a, b, u0 , v0 replaced ˜ χ, by d, ˜ a ˜, ˜b, u ˜0 , v˜0 . And the estimates in the lemma come from the definitions of U, V . □ Lemma 3.4. Let T ∈ [1, ∞), α ∈ (0, 1), D, µ, r > 0, m, M ∈ R with 0 < m ≤ M , and P, R be given by Lemma 3.1. Let d, b ∈ [m, M ], χ ∈ [0, R/2] , a ∈ [0, mM /2] , and u0 ∈ LM , v0 ∈ LµM/r . Then there exists a solution (u, v) ∈ (C 2,1 (QT ))2 which solves (3.13) and satisfies 0≤u≤M
on QT ,
∥u∥C 2+α,1+α/2 (Q
T)
≤ mM /R,
and 0 ≤ v ≤ µM /r on QT ,
∥v∥C 2+α,1+α/2 (Q
T)
≤ mM /R.
A.Z. Myint, J. Wang and M. Wang / Nonlinear Analysis: Real World Applications 55 (2020) 103121
9
Proof . Let
M R R M χ, a ˜ = a, ˜b = b, u ˜0 = u0 , v˜0 = v0 . d˜ = d, χ ˜= R R M M ˜ χ, Observe that these parameters d, ˜ a ˜, ˜b, u ˜0 , v˜0 satisfy the conditions in Lemma 3.3. Thus, there exists a 2 2+α,1+α/2 solution (˜ u, v˜) ∈ (C (QT )) which solves the problem (3.13) with d, χ, a, b, u0 , v0 replaced by ˜ χ, d, ˜ a ˜, ˜b, u ˜0 , v˜0 , and fulfills u∥C 2+α,1+α/2 (Q 0≤u ˜ ≤ R on QT , ∥˜
T)
≤ m,
and v ∥C 2+α,1+α/2 (Q 0 ≤ v˜ ≤ µR/r on QT , ∥˜
T)
Let u :=
M ˜ Ru
and v :=
M ˜. Rv
≤ m.
Hence, (u, v) solves (3.13) and satisfies the desired properties.
□
Proof of Theorem 1.1. Assume that Tmax < ∞. Then from Lemma 3.4, we see that there is a solution ¯ × [0, Tmax )) (u, v) ∈ C 2,1 (Ω which solves (1.1) in (0, Tmax ) and fulfills 0 ≤ u ≤ M, 0 ≤ v ≤ µM /r in Ω × [0, Tmax ). Recalling Lemma 2.1, we know that this solution (u, v) is the unique solution of (1.1). Furthermore, according to the extension rule asserted in Lemma 2.1, we derive a contradiction. Hence, Tmax = ∞ and ¯ × [0, ∞)) with (u, v) ∈ C 2,1 (Ω 0 ≤ u ≤ M, 0 ≤ v ≤ µM /r in Ω × [0, ∞), and for any T > 0, there holds ∥u∥C 2+α,1+α/2 (Q ) , ∥v∥C 2+α,1+α/2 (Q T
T)
≤ mM /R.
This finishes the proof of Theorem 1.1. Acknowledgments The authors would like to thank the editors and reviewers for their helpful comments and suggestions that significantly improve the initial version of this paper. References [1] E.F. Keller, L.A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol. 26 (1970) 399–415. [2] E.F. Keller, L.A. Segel, A model for chemotaxis, J. Theoret. Biol. 30 (1971) 225–234. [3] N. Bellomo, A. Bellouquid, Y. Tao, M. Winkler, Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci. 25 (2015) 1663–1763. [4] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations 35 (2010) 1516–1537. [5] C. Yang, X. Cao, Z. Jiang, S. Zheng, Boundedness in a quasilinear fully parabolic Keller–Segel system of higher dimension with logistic source, J. Math. Anal. Appl. 430 (2015) 585–591. [6] K. Osaki, T. Tsujikawa, A. Yagi, M. Mimura, Exponential attractor for a chemotaxis growth system of equations, Nonlinear Anal. 51 (2002) 119–144. [7] K. Osaki, A. Yagi, Finite dimensional attractor for one-dimensional Keller–Segel equations, Funkcial. Ekvac. 44 (2001) 441–469. [8] H. Jin, T. Xiang, Chemotaxis effect vs logistic damping on boundedness in the 2-D minimal Keller–Segel model, C. R. Math. Acad. Sci. Paris 356 (2018) 875–885.
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[9] Y. Tao, M. Winkler, Boundedness and decay enforced by quadratic degradation in a threedimensional chemotaxis-fluid system, Z. Angew. Math. Phys. 66 (2015) 2555–2573. [10] C. Mu, K. Lin, Global dynamics in a fully parabolic chemotaxis system with logistic source, Discrete Contin. Dyn. Syst. 36 (2016) 5025–5046. [11] T. Xiang, How strong a logistic damping can prevent blow-up for the minimal Keller–Segel chemotaxis system? J. Math. Anal. Appl. 459 (2018) 1172–1200. [12] T. Xiang, Chemotactic aggregation versus logistic damping on boundedness in the 3D minimal Keller–Segel model, SIAM J. Appl. Math. 78 (5) (2018) 2420–2438. [13] J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations 258 (2015) 1158–1191. [14] M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl. 384 (2011) 261–272. [15] M. Winkler, Finite-time blow-up in low-dimensional Keller–Segel systems with logistic-type superlinear degradation, Z. Angew. Math. Phys. 69 (2018) 69. [16] J. Lankeit, M. Mizukami, How far does small chemotactic interaction perturb the FisherKPP dynamics? J. Math. Anal. Appl. 452 (2017) 429–442. [17] Y. Tao, M. Winkler, Persistence of mass in a chemotaxis system with logistic source, J. Differential Equations 259 (2015) 6142–6161. [18] M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities? J. Nonlinear Sci. 24 (2014) 809–855. [19] T. Xiang, Boundedness and global existence in the higher-dimensional parabolic-parabolic chemotaxis system with/without growth source, J. Differential Equations 258 (2015) 4275–4323. [20] H. Jin, Y.J. Kim, Z. Wang, Boundedness, stabilization and pattern formation driven by density-suppressed motility, SIAM J. Appl. Math. 78 (2018) 1632–1657. [21] J. Wang, M. Wang, Boundedness in the higher-dimensional Keller–Segel model with signal-dependent motility and logistic growth, J. Math. Phys. 60 (2019) 1–14. [22] R.B. Salako, W. Shen, Existence of traveling wave solution of parabolic-parabolic chemotaxis systems, Nonlinear Anal. RWA 42 (2018) 93–119. [23] P. Deuring, An initial-boundary value problem for certain density dependent diffusion system, Math. Z. 194 (1987) 375–396. [24] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. [25] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin Heidelberg, New York, 2001.
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Nonlinear Analysis: Real World Applications www.elsevier.com/locate/nonrwa
On the minimal Keller–Segel system with logistic growth✩ Aung Zaw Myint, Jianping Wang ∗, Mingxin Wang School of Mathematics, Harbin Institute of Technology, Harbin 150001, PR China
article
info
Article history: Received 20 April 2019 Received in revised form 22 February 2020 Accepted 24 February 2020 Available online xxxx Keywords: Chemotaxis Logistic growth Global existence and boundedness
abstract This paper applies a delicate method which is inspired by Deuring (1987) and is different from those of Winkler (2010) and Yang et al. (2015) to show the known conclusion: The weak chemotactic effect can ensure the global existence and boundedness of the solutions of the minimal Keller–Segel model with logistic growth in any dimensional cases. Moreover, we obtain the explicit uniform-in-time upper bound for the global solution. It is noted that the method used in the paper may be employed to study other chemotaxis systems. © 2020 Elsevier Ltd. All rights reserved.
1. Introduction The chemotaxis system describes the movement of the bacteria which is oriented by the chemical substances in the environment. Since Keller and Segel [1,2] first proposed chemotaxis models to understand the aggregation of certain type of bacteria, lots of literatures have been devoted to clarify the dynamics of the system. The main issues of the investigations are global existence of classical/weak solutions; blow-up phenomenon; asymptotic behaviors; pattern formation; spatial spreading and front propagation dynamics when the domain is a whole space; etc. It was shown that the blow-up happens for the minimal Keller–Segel system in n-dimensions with n ≥ 2 [3]. And the collapse is caused by the chemotaxis effect. In the presence of the logistic term, the blow-up phenomenon is excluded in 2-dimensional case [3]. Hence, it is reasonable to believe that the logistic damping has the effect to suppress the blow-up. Due to the coexistence of the chemotactic effect and the damping source of logistic type, the minimal Keller–Segel model with logistic growth may exhibit many interesting dynamics which are still waiting for investigation. The aim of this paper is to provide another insight into the minimal Keller–Segel model with logistic damping. The classical ✩ This work was supported by NSFC Grant 11771110. ∗ Corresponding author. E-mail address: [email protected] (J. Wang).
https://doi.org/10.1016/j.nonrwa.2020.103121 1468-1218/© 2020 Elsevier Ltd. All rights reserved.
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A.Z. Myint, J. Wang and M. Wang / Nonlinear Analysis: Real World Applications 55 (2020) 103121
minimal Keller–Segel model with logistic growth reads ⎧ ut = d∆u − χ∇ · (u∇v) + au − bu2 , x ∈ Ω , t > 0, ⎪ ⎪ ⎪ ⎨ vt = D∆v − rv + µu, x ∈ Ω , t > 0, ∂v ∂u ⎪ = = 0, x ∈ ∂Ω , t > 0, ⎪ ⎪ ∂ν ∂ν ⎩ u(x, 0) = u0 (x), v(x, 0) = v0 (x), x ∈ Ω,
(1.1)
∂ denotes differentiation with respect where Ω is a bounded domain in Rn with smooth boundary ∂Ω , ∂ν to the outward normal vector on ∂Ω . The unknown functions u(x, t), v(x, t) represent the density of the bacterial and the concentration of the signal, respectively. We summarize the results on (1.1) as follows:
• In any dimensional cases, Winkler [4] initially proved that the strong logistic dampening effect can assert the global boundedness. Moreover, it has been found in [5] that there is θ0 > 0 such that (1.1) has a unique global bounded solution provided χ < θ0 b. • In 1,2-dimensions, it was shown in [6,7] that the solutions to (1.1) exists globally and remains bounded. In [8], Jin and Xiang investigated the chemotactic effect versus logistic damping on boundedness for (1.1) and found that the solution to (1.1) admits an upper bound which becomes singular only if b = 0. • In 3-dimension, when χ = 1 and b ≥ 23, the conclusions in [9] indicate the global existence and boundedness of the solution to (1.1). Later on, Mu and Lin [10] studied the system (1.1) with the logistic term replaced by u − buα with α ≥ 2 and claimed the existence of the globally bounded solution under the condition b1/(α−1) > 20χ which clearly improve the result obtained in [9]. Recently, Xiang [11,12] has been focusing on identifying the chemotactic aggregation versus logistic damping in (1.1). It was 9 proved in [12] that, if b > √10−2 χ, then there is an explicit uniform-in-time upper bound for the unique 9 χ. Although it is still solution of (1.1) and this upper bound has singular only in b on the line b = √10−2 unknown that whether or not the classical solution blows up if b is small, in 3-D convex domain, the weak solution has been shown to exist globally [13]. • Does the logistic damping can always prevent blow-up phenomenon? Unfortunately, the answer is not. In [14,15], Winkler claimed that the finite time blow-up happens for a parabolic–elliptic version of (1.1) despite logistic growth restriction. For other works on the dynamical properties of the chemotaxis systems with logistic source, please refer to [3,12,16–19] and references therein. Some related chemotaxis models with logistic source have attracted attentions in recent years. For the Keller–Segel model with signal-dependent motility and logistic growth, Jin et al. [20] proved the global boundedness for the solutions in two dimension and Wang and Wang [21] found that the strong logistic dampening effect can prevent blow-up in any dimensional cases. These conclusions coincide with that of (1.1). Further studies are required in this new topic. It is noted that, the above works on (1.1) are performed in the bounded domain. It would be rather interesting and meaningful to understand the dynamics of (1.1) in the whole space. Salako and Shen [22] investigated traveling wave solutions to (1.1) in the whole space and obtained some important results. Since the conclusion in this paper holds true in any dimensions, it is necessary to compare our result and method with that of [4,5]. On the one hand, it is noted that the literatures [4,5] give the conditions ensuring the global boundedness of the solution without offering the uniform-in-time upper bound for it. However, we get an explicit upper bound for the solution. On the other hand, instead of using the Maximal Sobolev Regularity or analysing the quantity ∫ q ∑ 2(q−p) cp up |∇v| dx p=0
Ω
with large q ∈ N and positive constants cp , the method we applied is inspired by [23]. To be precise, we first use Schauder-type estimates to derive a priori bounds for (1.1). And then by the Schauder’s fixed
A.Z. Myint, J. Wang and M. Wang / Nonlinear Analysis: Real World Applications 55 (2020) 103121
3
point theorem, we conclude that the solutions to (1.1) are bounded. As byproducts, sharp estimates for the solutions are obtained. The notations of the H¨ older spaces used in this paper come from [23] (see also [24, Chapter 3]). Throughout this paper the initial data u0 , v0 are supposed to satisfy ¯ ) with α ∈ (0, 1), u0 , v0 ∈ C 2+α (Ω
∂u0 ∂v0 = = 0 on ∂Ω . ∂ν ∂ν
¯, u0 , v0 ≥ 0 in Ω
(1.2)
For the convenience, we set ¯ ) : ∂ϕ = 0 on ∂Ω , ϕ ≥ 0 and ∥ϕ∥ 2+α ¯ ≤ k}. Lk = {ϕ ∈ C 2+α (Ω C (Ω) ∂ν Since it had been shown that the solution of (1.1) exist globally and remain bounded in 1,2-dimensions [6,7], the arguments in the sequel will be performed in the high dimensions, i.e., Ω ∈ Rn with n ≥ 3. The following theorem is the main result of this article. Theorem 1.1. Lemma 3.1. Let
Let n ≥ 3, D, µ, r > 0, α ∈ (0, 1), m, M ∈ R with 0 < m ≤ M , and P, R be given by d, b ∈ [m, M ], χ ∈ [0, R/2] , a ∈ [0, mM /2] ,
and u0 ∈ LM , v0 ∈ LµM/r . Then the problem (1.1) admits a unique nonnegative global solution (u, v) ∈ ¯ × [0, ∞)))2 . Furthermore, for any T > 0, we have (C 2,1 (Ω 0≤u≤M
on QT ,
∥u∥C 2+α,1+α/2 (Q
T)
≤ mM /R,
and 0 ≤ v ≤ µM /r on QT ,
∥v∥C 2+α,1+α/2 (Q
T)
≤ mM /R,
where QT = Ω × (0, T ]. We deduce from Theorem 1.1 that, for given d, a, b, D, r, µ > 0, we can define suitable m, M (and hence P, R) such that if χ ≤ R/2, then the parameters satisfy the conditions in Theorem 1.1, and hence the problem (1.1) is globally solvable. Corollary 1.1.
Assume that n ≥ 3 and d, a, b, D, r, µ > 0. Let m = min{d, b} and } { r∥v0 ∥C 2+α (Ω) ¯ 2a , ∥u0 ∥C 2+α (Ω) . M = max b, d, ¯ , m µ
Let P, R be given in Lemma 3.1. When 0 ≤ χ ≤ R/2, the problem (1.1) has a unique nonnegative and bounded global solution (u, v), and ¯ × [0, ∞)). u, v ∈ C 2,1 (Ω Furthermore, we have 0 ≤ u ≤ M, 0 ≤ v ≤ µM /r in Ω × [0, ∞), and for any T > 0, ∥u∥C 2+α,1+α/2 (Q ) , ∥v∥C 2+α,1+α/2 (Q T
T)
≤ mM /R.
¯ × (0, ∞). Remark 1.1. If we further assume that u0 , v0 ≥, ̸≡ 0, then u, v > 0 in Ω Remark 1.2. It is easy to see that M ≥ max{2a/m, ∥u0 ∥C 0 (Ω) ¯ } ≥ max{a/b, ∥u0 ∥C 0 (Ω) ¯ }, and hence our assertion is consistent with the boundedness result for the FisherKPP equation (i.e., χ = 0). Moreover, we observe from the definition of R that R is decreasing with respect to µ which is reasonable.
A.Z. Myint, J. Wang and M. Wang / Nonlinear Analysis: Real World Applications 55 (2020) 103121
4
2. Preliminaries We first give a claim concerning the local-in-time existence of the classical solutions to (1.1) whose proof can be found in [4]. Lemma 2.1. Let d, a, b, D, r, µ > 0 and χ ≥ 0. Suppose that the initial data (u0 , v0 ) satisfies (1.2). There exist a Tmax ∈ (0, ∞] and a unique nonnegative solution (u, v) of (1.1) defined in [0, Tmax ) that satisfies ¯ × [0, Tmax )), u, v ∈ C 2,1 (Ω and u, v ≥ 0 in Ω × [0, Tmax ).
(2.1)
Moreover, the “existence time Tmax ” can be chosen maximal: either Tmax = ∞, or Tmax < ∞ and lim sup ∥u(·, t)∥L∞ (Ω) = ∞. t→Tmax
The following lemma plays an important role in our analysis. Lemma 2.2 ([23, Theorem 3.1]). There is a function KΩ : (0, 1) × (0, ∞)2 → (0, ∞) with the following property ¯ ), (1) Let α ∈ (0, 1), K1 ≥ K2 > 0, T ∈ [1, ∞), a, bi (i = 1, 2, . . . , n), c, f ∈ C α,α/2 (QT ), ϕ ∈ C 2+α (Ω 2,1 u ∈ C (QT ), with ∥a∥C α,α/2 (Q ) ≤ K1 , ∥bi ∥C 0 (Q ) ≤ K1 , ∥c∥C 0 (Q ) ≤ K1 , a ≥ K2 on QT , and T
T
T
⎧ n ∑ ⎪ ⎪ ⎪ u − a∆u + bi Di u − cu = f, x ∈ Ω , 0 < t ≤ T, ⎪ ⎨ t i=1 ∂u ⎪ = 0, x ∈ ∂Ω , 0 ≤ t ≤ T, ⎪ ⎪ ⎪ ⎩ ∂ν ¯. u(x, 0) = ϕ(x), x∈Ω Then ∥u∥C α,α/2 (Q
T)
≤ KΩ (α, K1 , K2 )(∥f ∥C 0 (Q
T)
+ ∥ϕ∥C 2 (Ω) ¯ + ∥u∥C 0 (Q ) ). T
2
Furthermore, there is a function LΩ : (0, 1) × (0, ∞) → (0, ∞) with the following property (2) Let α, K1 , K2 , T, a, bi , c, f, ϕ, u be given as in (1), but with the assumptions ∥bi ∥C 0 (Q ) , ∥c∥C 0 (Q ) ≤ K1 T T replaced by the stronger conditions ∥bi ∥C α,α/2 (Q ) , ∥c∥C α,α/2 (Q ) ≤ K1 . T T Then ∥u∥C 2+α,1+α/2 (Q ) ≤ LΩ (α, K1 , K2 )(∥f ∥C α,α/2 (Q ) + ∥ϕ∥C 2+α (Ω) ¯ + ∥u∥C 0 (Q ) ). T
T
T
Remark 2.1. Noticing that KΩ and LΩ are independent of T and the condition T ≥ 1 could be replaced by T ≥ T0 for any fixed T0 > 0 (See [23, Theorem 3.1, Remark]). 3. Proof of Theorem 1.1 Fixed D, r, µ > 0. Let λ = max{D, r}. We first give the following a priori estimates for u for a class of parameters, initial data and given v. Lemma 3.1.
Let T ∈ [1, ∞), α ∈ (0, 1), D, r, µ > 0, m, M ∈ R with 0 < m ≤ M , and take
P = P (α, m, M ) = 2KΩ (α, M + mM + M 2 , m),
A.Z. Myint, J. Wang and M. Wang / Nonlinear Analysis: Real World Applications 55 (2020) 103121
{ R = R(α, m, M ) = min
m m ) ( , LΩ (α, M + mM, m)(M 2 (1 + 2P ) + 2) LΩ (α, λ, D) P + 2r µ
5
} ,
where LΩ , KΩ are defined in Lemma 2.2 and λ = max{D, r}. Let d0 ∈ [m, M ], χ0 ∈ [0, M/2], a0 ∈ [0, mM/2], b0 ∈ [mM/R, M 2 /R] and φ ∈ LR . v ∥C 2+α,1+α/2 (Q ) ≤ m. Then the problem Suppose that vˆ ∈ C 2+α,1+α/2 (QT ) satisfies 0 ≤ vˆ ≤ µR r and ∥ˆ T
⎧ u ˆ − d0 ∆ˆ u + χ0 ∇ˆ v · ∇ˆ u − (a0 − χ0 ∆ˆ v − b0 u ˆ)ˆ u = 0, x ∈ Ω , 0 < t ≤ T, ⎪ ⎨ t ∂u ˆ = 0, x ∈ ∂Ω , 0 ≤ t ≤ T, ⎪ ⎩ ∂ν ¯ u ˆ(x, 0) = φ(x), x∈Ω
(3.1)
admits a unique solution u ˆ ∈ C 2+α,1+α/2 (QT ). Moreover, we have 0≤u ˆ≤R
on
QT ,
(3.2)
and ∥ˆ u∥C α,α/2 (Q
T)
≤ P R,
∥ˆ u∥C 2+α,1+α/2 (Q
T)
≤ m.
(3.3)
¯ . Hence, Proof . Step 1: Existence, uniqueness and estimate (3.2). Recall that u ˆ(x, 0) = φ(x) ≥ 0 for x ∈ Ω it can be deduced that w(x, t) ≡ 0 is a lower solution of (3.1). Let w(x, t) ≡ R on QT . Due to χ0 ∈ [0, M/2], ∥ˆ v ∥C 2+α,1+α/2 (Q
T)
≤ m, a0 ∈ [0, mM/2], b0 ∈ [mM/R, M 2 /R],
we have a0 − χ0 ∆ˆ v − b0 w ≤ 0 on QT
(3.4)
Thanks to ∥φ∥C 2+α (Ω) ¯ ≤ R, it is easy to see that w satisfies ⎧ v · ∇w − (a0 − χ0 ∆ˆ v − b0 w)w ≥ 0, x ∈ Ω , 0 < t ≤ T, ⎪ ⎨ wt − d0 ∆w + χ0 ∇ˆ ∂w ≥ 0, x ∈ ∂Ω , 0 ≤ t ≤ T, ⎪ ⎩ ∂ν ¯. w(x, 0) ≥ φ(x), x∈Ω Therefore, w is an upper solution of (3.1). In view of the upper and lower solutions method, we know that (3.1) has a unique solution u ˆ ∈ C 2+α,1+α/2 (QT ) and satisfies (3.2). Step 2: The estimate (3.3). By direct computations, it is easy to see that d0 , ∥χ0 Di vˆ∥C 0 (Q ) , ∥a0 − χ0 ∆ˆ v − b0 u ˆ∥C 0 (Q
T)
T
≤ M (1 + m + M ), i = 1, . . . , n,
(3.5)
and d0 ≥ m
(3.6)
thanks to the assumptions for d0 , χ0 , vˆ, a0 , b0 and (3.2). In view of Lemma 2.2(1), it yields ∥ˆ u∥C α,α/2 (Q
T)
≤ KΩ (α, M (1 + m + M ), m)(∥φ∥C 2 (Ω) u∥C 0 (Q ) ) ¯ + ∥ˆ T
≤ 2KΩ (α, M (1 + m + M ), m)R
(3.7)
because ∥φ∥C 2+α (Ω) ˆ ≤ R on QT . Recall the definition of P = P (α, m, M ), it follows from ¯ ≤ R and 0 ≤ u (3.7) that ∥ˆ u∥C α,α/2 (Q ) ≤ P R. T
This gives the first part of (3.3).
A.Z. Myint, J. Wang and M. Wang / Nonlinear Analysis: Real World Applications 55 (2020) 103121
6
Rewriting the system (3.1) as ⎧ u ˆ − d0 ∆ˆ u + χ0 ∇ˆ v · ∇ˆ u − (a0 − χ0 ∆ˆ v )ˆ u = −b0 u ˆ2 , x ∈ Ω , 0 < t ≤ T, ⎪ ⎨ t ∂u ˆ = 0, x ∈ ∂Ω , 0 ≤ t ≤ T, ⎪ ∂ν ⎩ ¯. u ˆ(x, 0) = φ(x), x∈Ω In light of the assumptions of d0 , χ0 , a0 , vˆ, there hold v ∥C α,α/2 (Q d0 , ∥χ0 Di vˆ∥C α,α/2 (Q ) , ∥a0 − χ0 ∆ˆ
T)
T
≤ M + mM, i = 1, . . . , n.
(3.8)
Moreover, from the first part of (3.3), we deduce ∥b0 u ˆ2 ∥C α,α/2 (Q
T)
≤ b0 (∥ˆ u∥2C 0 (Q
T)
+ 2∥ˆ u∥C 0 (Q ) ∥ˆ u∥C α,α/2 (Q ) ) ≤ M 2 R(1 + 2P ) T
(3.9)
T
where we have used the estimate [23, (4)], (3.2) and b0 ∈ [mM/R, M 2 /R]. Note that ∥φ∥C 2+α (Ω) ¯ ≤ R. Using Lemma 2.2(2) firstly and (3.2), (3.8) and (3.9) secondly, we find ∥ˆ u∥C 2+α,1+α/2 (Q
T)
≤ LΩ (α, M + mM, m)(∥b0 u ˆ2 ∥C α,α/2 (Q
T)
+ ∥φ∥C 2+α (Ω) u∥C 0 (Q ) ) ¯ + ∥ˆ T
2
≤ LΩ (α, M + mM, m)[M (1 + 2P ) + 2]R ≤ m. Here we used the definition of R. The second part of (3.3) is proved and the proof is complete. □ Let us proceed to derive the a priori estimates for v when u is given and fulfills (3.2) and (3.3). Lemma 3.2. Let T ∈ [1, ∞), α ∈ (0, 1), D, r, µ > 0, m, M ∈ R with 0 < m ≤ M , and P, R be given in the above lemma. And let ρ ∈ L(µR)/r . Suppose that u∗ ∈ C 2+α,1+α/2 (QT ) satisfies 0 ≤ u∗ ≤ R on QT , ∥u∗ ∥C α,α/2 (Q ) ≤ P R and ∥u∗ ∥C 2+α,1+α/2 (Q ) ≤ m. Then the problem T
T
⎧ ∗ v − D∆v ∗ + rv ∗ = µu∗ , x ∈ Ω , 0 < t ≤ T, ⎪ ⎨ t ∗ ∂v = 0, x ∈ ∂Ω , 0 ≤ t ≤ T, ⎪ ⎩ ∂ν ∗ ¯ v (x, 0) = ρ(x), x∈Ω
(3.10)
has a unique solution v ∗ ∈ C 2+α,1+α/2 (QT ). Moreover, we have 0 ≤ v ∗ ≤ µR/r on QT ,
(3.11)
∥v ∗ ∥C 2+α,1+α/2 (Q
(3.12)
and T)
≤ m.
¯ ), u∗ ∈ C 2+α,1+α/2 (QT ), making use of the Schauder theory, there is a unique Proof . Due to ρ ∈ C 2+α (Ω ∗ 2+α,1+α/2 solution v ∈ C (QT ). Noticing that 0 ≤ u∗ ≤ R and 0 ≤ ρ ≤ µR r . It is easy to get (3.11) by using the parabolic maximum principle. We omit the details. In view of Lemma 2.2(2), we get ∥v ∗ ∥C 2+α,1+α/2 (Q
T)
≤ LΩ (α, λ, D)(µ∥u∗ ∥C α,α/2 (Q
T)
∗ + ∥ρ∥C 2+α (Ω) ¯ + ∥v ∥C 0 (Q ) ) T
≤ LΩ (α, λ, D) (P + 2/r) µR, where we have used (3.11) and the assumptions ∥u∗ ∥C α,α/2 (Q ) ≤ P R, ∥ρ∥C 2+α (Ω) ¯ ≤ µR/r. Recalling the T definition of R, it is easy to yield (3.12). The proof is end. □
A.Z. Myint, J. Wang and M. Wang / Nonlinear Analysis: Real World Applications 55 (2020) 103121
For T ≥ 1 and D, µ, r > 0, we shall consider (1.1) in (0, T ]: ⎧ ut = d∆u − χ∇ · (u∇v) + au − bu2 , ⎪ ⎪ ⎪ ⎨ vt = D∆v − rv + µu, ∂v ∂u ⎪ = = 0, ⎪ ⎪ ∂ν ⎩ ∂ν u(x, 0) = u0 (x), v(x, 0) = v0 (x),
7
x ∈ Ω , 0 < t ≤ T, x ∈ Ω , 0 < t ≤ T, (3.13)
x ∈ ∂Ω , 0 ≤ t ≤ T, ¯. x∈Ω
Lemma 3.3. Let T ∈ [1, ∞), α ∈ (0, 1), D, r, µ > 0, m, M ∈ R with 0 < m ≤ M , and P, R be given by Lemma 3.1. Let [ ] d˜ ∈ [m, M ], χ ˜ ∈ [0, M/2] , a ˜ ∈ [0, mM /2] , ˜b ∈ mM /R, M 2 /R , and u ˜0 ∈ LR and v˜0 ∈ LµR/r . Then there exists a solution (˜ u, v˜) ∈ (C 2+α,1+α/2 (QT ))2 which solves the ˜ ˜ problem (3.13) with d, χ, a, b, u0 , v0 replaced by d, χ, ˜ a ˜, b, u ˜0 , v˜0 and satisfies 0≤u ˜ ≤ R on QT , ∥˜ u∥C α,α/2 (Q
T)
≤ P R, and ∥˜ u∥C 2+α,1+α/2 (Q
T)
≤ m,
and 0 ≤ v˜ ≤ µR/r on QT ,
∥˜ v ∥C 2+α,1+α/2 (Q
T)
≤ m.
Proof . Denote U : = {ϕ ∈ C 2+α,1+α/2 (QT ) : 0 ≤ ϕ ≤ R, ∥ϕ∥C α,α/2 (Q V : = {ϕ ∈ C
2+α,1+α/2
T)
≤ P R, ∥ϕ∥C 2+α,1+α/2 (Q
T)
(QT ) : 0 ≤ ϕ ≤ µR/r, ∥ϕ∥C 2+α,1+α/2 (Q
T)
≤ m},
≤ m}.
Define W := [C 2+α/2,1+α/4 (QT )]2 . Clearly, W is a Banach space endowed with the norm ∥(u, v)∥W = ∥u∥C 2+α/2,1+α/4 (Q
T)
+ ∥v∥C 2+α/2,1+α/4 (Q ) . T
It is easy to see that U × V is a compact, convex subset of the Banach space W . ˜ χ, ˜ For (u, v) ∈ U × V , denote by Q(v) the solution of (3.1) with d0 , χ0 , a0 , b0 , vˆ, φ replaced by d, ˜ a ˜, ˜b, v, u ˜0 ∗ ⋆ ⋆ ˜ ˜ ˜ and G(u) the solution of (3.10) with u , ρ replaced by u, v˜0 , respectively. And, let u = Q(v) and v = G(u). From Lemmas 3.1 and 3.2 we know that 0 ≤ u⋆ ≤ R, ∥u⋆ ∥C α,α/2 (Q
T)
≤ P R, ∥u⋆ ∥C 2+α,1+α/2 (Q
T)
≤ m,
(3.14)
and 0 ≤ v ⋆ ≤ µR/r, ∥v ⋆ ∥C 2+α,1+α/2 (Q
T)
≤ m.
(3.15)
˜ ˜ Set F˜ (u, v) = (u⋆ , v ⋆ ) = (Q(v), G(u)). Hence F˜ is an operator from U × V into U × V . Next we shall show that F˜ is continuous in the norm ∥·∥W of the Banach space W . For v1 , v2 ∈ V , denote ˜ i ) with i = 1, 2 and u u ˜i = Q(v ˜=u ˜1 − u ˜2 . It is easy to see that u ˜=u ˜1 − u ˜2 satisfies ⎧ ˜ u + χ∇v u ˜t − d∆˜ ˜ 1 · ∇˜ u − [˜ a − χ∆v ˜ 1 − ˜b(˜ u1 + u ˜2 )]˜ u ⎪ ⎪ ⎪ ⎨ = −χ∇˜ ˜ u2 · ∇(v1 − v2 ) − χ˜ ˜u2 ∆(v1 − v2 ), x ∈ Ω , 0 < t ≤ T, ∂u ˜ ⎪ = 0, x ∈ ∂Ω , 0 ≤ t ≤ T, ⎪ ⎪ ∂ν ⎩ ¯. u ˜(x, 0) = 0, x∈Ω ˜ χ, Moreover, u ˜i satisfy (3.14) for i = 1, 2. For simplicity, let E = max{d, ˜ a ˜, ˜b}. Thanks to 1 ≤ T < ∞, there is k > 0 which may depend on T such that ∥w∥C α/2,α/4 (Q ) ≤ k∥w∥C α,α/2 (Q ) for any w ∈ C α,α/2 (QT ). T
T
8
A.Z. Myint, J. Wang and M. Wang / Nonlinear Analysis: Real World Applications 55 (2020) 103121
˜ χ, Making use of the properties of vi , u ˜i with i = 1, 2 and the assumptions for d, ˜ a ˜, ˜b, there hold d˜ ≥ m and for j = 1, . . . , n, ˜ ∥χD a − χ∆v ˜ 1 − ˜b(˜ u1 + u ˜2 )]∥C α/2,α/4 (Q d, ˜ j v1 ∥C α/2,α/4 (Q ) , ∥[˜
T)
T
≤ E(1 + 3km).
Moreover, ∥ − χ∇˜ ˜ u2 · ∇(v1 − v2 ) − χ˜ ˜u2 ∆(v1 − v2 )∥C α/2,α/4 (Q
T)
≤ 2Ekm∥v1 − v2 ∥C 2+α/2,1+α/4 (Q ) , T
where we have used the fact ∥f g∥C α,α/2 (Q
T)
≤ ∥f ∥C α,α/2 (Q ) ∥g∥C α,α/2 (Q
T)
T
due to the definition of C α,α/2 (QT ). Hence, it follows from the parabolic Schauder theory that, there is K1 > 0 which depends on α, T, E, k, m, Ω such that ∥˜ u∥C 2+α/2,1+α/4 (Q
T)
≤ K1 ∥ − χ∇˜ ˜ u2 · ∇(v1 − v2 ) − χ˜ ˜u2 ∆(v1 − v2 )∥C α/2,α/4 (Q
T)
≤ 2kK1 mE∥v1 − v2 ∥C 2+α/2,1+α/4 (Q
T)
≤ K2 ∥v1 − v2 ∥C 2+α/2,1+α/4 (Q ) ,
(3.16)
T
where K2 = 2kK1 mE. ˜ i ) with i = 1, 2 and v˜ = v˜1 − v˜2 . It is easy to see that v˜ = v˜1 − v˜2 solves For u1 , u2 ∈ V , denote v˜i = G(v ⎧ v˜ − D∆˜ v + r˜ v = µ(u1 − u2 ), x ∈ Ω , 0 < t ≤ T, ⎪ ⎨ t ∂˜ v = 0, x ∈ ∂Ω , 0 ≤ t ≤ T, ⎪ ⎩ ∂ν ¯. v˜(x, 0) = 0, x∈Ω Again by the parabolic Schauder theory, there is K3 > 0 which depends on α, T, D, r, µ, Ω such that ∥˜ v ∥C 2+α/2,1+α/4 (Q
T)
≤ K3 ∥u1 − u2 ∥C α/2,α/4 (Q
T)
≤ K3 ∥u1 − u2 ∥C 2+α/2,1+α/4 (Q ) . T
This combined with (3.16) yields that ∥F˜ (u1 , v1 ) − F˜ (u2 , v2 )∥W = ∥(˜ u, v˜)∥W = ∥˜ u∥C 2+α/2,1+α/4 (Q
T)
+ ∥˜ v ∥C 2+α/2,1+α/4 (Q
T)
≤ max{K2 , K3 }(∥u1 − u2 ∥C 2+α/2,1+α/4 (Q
T)
+ ∥v1 − v2 ∥C 2+α/2,1+α/4 (Q ) ) T
≤ max{K2 , K3 }∥(u1 , v1 ) − (u2 , v2 )∥W . This shows that F˜ is continuous in the norm ∥ · ∥W of the Banach space W . In light of the Schauder’s fixed point theorem [25, Theorem 11.1], there exists (¯ u, v¯) ∈ U × V such that F˜ (¯ u, v¯) = (¯ u, v¯). Hence, there is a solution (¯ u, v¯) which solves the problem (3.13) with d, χ, a, b, u0 , v0 replaced ˜ χ, by d, ˜ a ˜, ˜b, u ˜0 , v˜0 . And the estimates in the lemma come from the definitions of U, V . □ Lemma 3.4. Let T ∈ [1, ∞), α ∈ (0, 1), D, µ, r > 0, m, M ∈ R with 0 < m ≤ M , and P, R be given by Lemma 3.1. Let d, b ∈ [m, M ], χ ∈ [0, R/2] , a ∈ [0, mM /2] , and u0 ∈ LM , v0 ∈ LµM/r . Then there exists a solution (u, v) ∈ (C 2,1 (QT ))2 which solves (3.13) and satisfies 0≤u≤M
on QT ,
∥u∥C 2+α,1+α/2 (Q
T)
≤ mM /R,
and 0 ≤ v ≤ µM /r on QT ,
∥v∥C 2+α,1+α/2 (Q
T)
≤ mM /R.
A.Z. Myint, J. Wang and M. Wang / Nonlinear Analysis: Real World Applications 55 (2020) 103121
9
Proof . Let
M R R M χ, a ˜ = a, ˜b = b, u ˜0 = u0 , v˜0 = v0 . d˜ = d, χ ˜= R R M M ˜ χ, Observe that these parameters d, ˜ a ˜, ˜b, u ˜0 , v˜0 satisfy the conditions in Lemma 3.3. Thus, there exists a 2 2+α,1+α/2 solution (˜ u, v˜) ∈ (C (QT )) which solves the problem (3.13) with d, χ, a, b, u0 , v0 replaced by ˜ χ, d, ˜ a ˜, ˜b, u ˜0 , v˜0 , and fulfills u∥C 2+α,1+α/2 (Q 0≤u ˜ ≤ R on QT , ∥˜
T)
≤ m,
and v ∥C 2+α,1+α/2 (Q 0 ≤ v˜ ≤ µR/r on QT , ∥˜
T)
Let u :=
M ˜ Ru
and v :=
M ˜. Rv
≤ m.
Hence, (u, v) solves (3.13) and satisfies the desired properties.
□
Proof of Theorem 1.1. Assume that Tmax < ∞. Then from Lemma 3.4, we see that there is a solution ¯ × [0, Tmax )) (u, v) ∈ C 2,1 (Ω which solves (1.1) in (0, Tmax ) and fulfills 0 ≤ u ≤ M, 0 ≤ v ≤ µM /r in Ω × [0, Tmax ). Recalling Lemma 2.1, we know that this solution (u, v) is the unique solution of (1.1). Furthermore, according to the extension rule asserted in Lemma 2.1, we derive a contradiction. Hence, Tmax = ∞ and ¯ × [0, ∞)) with (u, v) ∈ C 2,1 (Ω 0 ≤ u ≤ M, 0 ≤ v ≤ µM /r in Ω × [0, ∞), and for any T > 0, there holds ∥u∥C 2+α,1+α/2 (Q ) , ∥v∥C 2+α,1+α/2 (Q T
T)
≤ mM /R.
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