Boundedness and Asymptotic Behavior to a Chemotaxis System with Indirect Signal Generation and Singular Sensitivity

Here, the unknowns n = nðt, xÞ and cðt, xÞ denote the cell density and chemical concentration, respectively. The given function χðn, cÞ is the chemotactic sensitivity. The physical domainΩ ⊂RNðN = 2, 3Þ is a bounded domain with smooth boundary. This model describes a biological process in which cells move towards their preferred environment and a signal being produced by the cells themselves. When the diffusion of chemical signals is much faster than that of cells, the system can be simplified as


Introduction
One of the first mathematical models of chemotaxis was introduced by Keller and Segel [1] to describe the aggregation of certain types of bacteria. In mathematics, it is described as a fully parabolic system Here, the unknowns n = nðt, xÞ and cðt, xÞ denote the cell density and chemical concentration, respectively. The given function χðn, cÞ is the chemotactic sensitivity. The physical domain Ω ⊂ ℝ N ðN = 2, 3Þ is a bounded domain with smooth boundary. This model describes a biological process in which cells move towards their preferred environment and a signal being produced by the cells themselves. When the diffusion of chemical signals is much faster than that of cells, the system can be simplified as n t = Δn−∇ ⋅ nχ n, c ð Þ∇c ð Þ , x ∈ Ω, t > 0, Another important chemotaxis model is formed with singular sensitivity function, such as χðn, cÞ = χ/c. This model is proposed by the Weber-Fechner law of stimulus perception [2] and supported by experimental [3] and theoretical evidence [4]. The articles about singular sensitive function can be referred to reference [5][6][7][8][9].
Considering the proliferation and death of cells, many scholars have done corresponding research on the above model to add the logistic source. We refer the reader to the survey [10][11][12][13][14][15] and the references therein. There are also some models involving nonlinear diffusion and rotation terms, which can be referred to [16][17][18][19].
It is also important to consider the indirect signal model because the attractive signal and repulsive signal exist simultaneously in some Keller-Segel models. Lin-Mu-Wang established the global existence and large-time behavior in [20].
The blow-up solution was studied by Fujie and Senba in [21]. Tao and Wang [22] considered the global solvability, boundedness, blow-up, existence of nontrivial stationary solutions, and asymptotic behavior. Stinner et al. [23] have given the global existence and some basic boundedness of weak solutions for a PDE-ODE system Considering the singular sensitivity function, we study the following singular chemotaxis model of indirect signal generation where the parameter χ is a positive constant and φ is a known function. On the other hand, the case ofΩ ⊂ ℝ N ðN = 2, 3Þis a bounded domain, under the assumption of the no-flux Neumann boundary condition for n,c and w, i.e., where ν is the unit outward normal vector on ∂Ω and of the initial conditions There are some sensitivity functions φ satisfying the fourth conditions of (6). For example, φðxÞ = x α , α > 0 or-φðxÞ = log ð1 + xÞ, φðxÞ = arctan x, φðxÞ = x α log ð1 + xÞ, φð xÞ = Ð x 0 τ α log ð1 + τÞdτ, and so on are all satisfied with conditions of (6).
Under these assumptions, we give the well-posedness and asymptotic behavior results as follows.

Theorem 1.
Let Ω ⊂ ℝ N be a bounded domain with smooth boundary. Suppose that n 0 , c 0 , w 0 , φ satisfy (6). Then, for any q > 1, systems (3)-(4) possess a global classical solution ðn, c, wÞ which enjoys the regularity properties: Moreover, this solution is uniformly bounded in the sense that with some positive constant C.
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Preliminaries and Bounded Estimates
We first establish the local existence result; then the global existence of the solutions is obtained by using a priori estimate.

Lemma 1.
For N ∈ f2, 3g, let Ω ⊂ ℝ N be a bounded domain with smooth boundary. Assume that n 0 , c 0 , w 0 , φ satisfy (6). Then, there exist T max ∈ ð0,∞ and a classical solution ðn, c, wÞ of (3)- (4) in Ω × ð0, T max Þ such that Proof. Let c * = ð1/eÞ inf x∈Ω c 0 ðxÞ > 0. With adaptations of the methods akin to those used in [24] and ( [25], Thm. 2.3 i) to deal with the singular sensitivity, R > 0 and T ∈ ð0, 1Þ to be specified below, in Banach's space we consider the closed set and introduce a mapping Φ = ðΦ 1 , Φ 2 , Φ 3 Þ on S by defining for ðn, c, wÞ ∈ S and t ∈ ð0, TÞ. Using the reasoning (see [26], Lemma 1) based on Banach's fixed point theorem applied in a closed bounded set in L ∞ ðð0, TÞ ; C 0 ð ΩÞ × W 1,q ðΩÞ × W 1,q ðΩÞÞ for suitably small T > 0, the following regularity arguments, proving this local existence and uniqueness result. ☐ In order to get time-independent pointwise lower bounds of w and c, we need to use the L 1 -conservation of n. The purpose of this method is to eliminate the singularity of the function 1/φðCÞ at zero.

Lemma 2. For any
min w :,t ð Þ,c:,tÞ f g≥ η: ð16Þ Proof. Integrate the first equation of (3) to obtain (15). Using the representation formula of Neumann heat semigroup and point lower bound estimation in [27], we have where η 1 is a positive constant and diamΩ ≔ max In the same way, we see that

Lemma 3.
Let For any p ∈ ð0, pÞ, there exists constant C such that Proof. We represent w according to Using the properties of fractional powers ð−Δ + 1Þ θ with a dense domain Dðð−Δ+1ÞθÞ, θ ∈ ð0, 1Þ in [28], we see from where λ 1 ∈ ð0, 1Þ and C 1 , C 2 , C 3 > 0 are constants. If T max = ∞, we can take the time t large enough such thatkwð·, tÞk L p ðΩÞ ≤ Cm: ☐ Lemma 4. For any q ∈ ð0,+∞Þ, there exists constant C such that Proof. By applying the representation formula, we have We apply ð−Δ + 1Þ θ to both sides of equation (29) to obtain

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Then by If T max = ∞, taking the time t large enough and by virtue of Lemma 3, we can complete the proof.
☐ ☐ Lemma 5. For any r > 1, there exists constant C such that with some fixed t 0 > 0.
Proof. Multiplying n r−1 by the first equation of (3) and integration by parts, using Hölder's inequality and Young inequality, we have that d dt That is, To handle the right-hand side of (34), we use Hölder's inequality and Gagliardo-Nirenberg inequality to get n r/2 ∇c where C GN > 0 is constant and q > n. Similarly, using the Gagliardo-Nirenberg inequality, there is C GN > 0 such that From (35) and (36), we obtain C 4 > 0 such that We now substitute (37)-(38) into (34) to obtain that

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Applying Gronwall's inequality, we see that with some fixed t 0 > 0. Due to knk L r ðΩÞ being uniformly bounded, we can obtain (32) immediately. ☐ Lemma 6. For any p ∈ ð0,∞Þ, there exists constant C such that Proof. Using the variation-of-constant formula for w again, we obtain Therefore, the estimate of knk L r ðΩÞ provides us with C 5 > 0 and C 6 > 0, for any t ∈ ð0, T max Þ satisfying wherein the last integral is finite since 1/2 + N/2ðð1/rÞ − ð1/ r 1 ÞÞ < ð1/2Þ. Similarly, we can deduce that with some C 7 > 0, where we can select some p > r > 1 such that N/2ðð1/rÞ − ð1/pÞÞ < ð1/2Þ Thus, by virtue of (43)

Asymptotic Behavior
To simplify notation, we shall abbreviate the deviations from the nonzero homogeneous steady state by the following transformation: for all x ∈ Ω and t > 0. Through simple calculation, we see that ðU, V, WÞ satisfies the following initial boundary value problem: Advances in Mathematical Physics In order to prove Theorem 2, we need several lemmas.

Lemma 7.
For any r > 1, q > N, there exists constant C such that Proof. By using the variation-of-constant representation, for all t > t 2 , we obtain For I 1 , there is a constant c 1 > 0 such that Noticing that Ð Ω Uð⋅ ,tÞdx = 0, we have For I 2 , taking r > r 1 > N, q > N, using the estimate of Neumann heat semigroup and Hölder's inequality, we obtain where c 2 , c 3 > 0 are constants. We now substitute (51)-(52) into (49) to complete the proof. ☐ ☐ Next, we want to extendT 0 to infinity. Applying the Lemma 7, we can select t 3 = t 3 ðn, c, uÞ > 0 to obtain for some r > 1, q > N. For any p ∈ ð1, pÞ, one has By combining Lemma 3 and (45), we see that Applying the Lemma 4, we can get We now choose m small enough such that It is easy to see that Let

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where T 0 is a given positive constant. Then,T 0 is welldefined since (49), (51), and (58). In order to extendT 0 to infinity, we give the following lemmas. Lemma 8. For any p ∈ ð1, pÞ, there exists a constant c 4 > 0 satisfying Proof. We first use (46) to represent W according to and the fact that λ 1 < 1 and (55) to estimate Furthermore, using Hölder's inequality and the definitions of¨T and c 5 entails that Thus, substituting (62) and (63) into (61), we obtain the Lemma 8.
☐ ☐ Lemma 9. For any q ∈ ð1, +∞Þ, there exists constant c 5 such that Proof. By means of the variation-of-constant representation for V, combined with (56) and Lemma 8, we show that Proof. Notice that the fact of U has the following estimate: Furthermore, we can use (45) to obtain

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We next write and employ the estimate (53) to obtain We next recall (18) and (45) and employ the estimates (64) and (68) to see that