Abstract
In this paper, we consider the following indirect signal generation and singular sensitivity in a bounded domain with smooth boundary . Under the nonflux boundary conditions for , , and , we first eliminate the singularity of by using the Neumann heat semigroup and then establish the global boundedness and rates of convergence for solution.
1. 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 and denote the cell density and chemical concentration, respectively. The given function is the chemotactic sensitivity. The physical domain 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
Another important chemotaxis model is formed with singular sensitivity function, such as . 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 [5–9].
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 [10–15] and the references therein. There are also some models involving nonlinear diffusion and rotation terms, which can be referred to [16–19].
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 is a positive constant and is a known function. On the other hand, the case ofis a bounded domain, under the assumption of the no-flux Neumann boundary condition for , and , i.e., where is the unit outward normal vector on and of the initial conditions satisfy
There are some sensitivity functions satisfying the fourth conditions of (6). For example, or, 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 be a bounded domain with smooth boundary. Suppose that satisfy (6). Then, for any , systems (3)–(4) possess a global classical solution which enjoys the regularity properties: Moreover, this solution is uniformly bounded in the sense that with some positive constant .
Theorem 2. Let be a bounded domain with smooth boundary. Suppose that (6) holds. Then, there exists such that if satisfies for some , the solution of (3) has the following decay estimates: where and is Lebesgue measure.
2. 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 , let be a bounded domain with smooth boundary. Assume that satisfy (6). Then, there exist and a classical solution of (3)–(4) in such that
Proof. Let . With adaptations of the methods akin to those used in [24] and ([25], Thm. 2.3 i) to deal with the singular sensitivity, and to be specified below, in Banach’s space we consider the closed set and introduce a mapping on by defining for and . Using the reasoning (see [26], Lemma 1) based on Banach’s fixed point theorem applied in a closed bounded set in for suitably small , the following regularity arguments, proving this local existence and uniqueness result.☐
In order to get time-independent pointwise lower bounds of and , we need to use the L1-conservation of . The purpose of this method is to eliminate the singularity of the function at zero.
Lemma 2. For any , there exist , , and such that Moreover, we have
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 is a positive constant and . In the same way, we see that
where is a positive constant. Taking , we get (16).
We integrate the third equation of (3) to obtain
Applying Lemma 3.4 in [23], we obtain (17). In a similar way, we can get (18).☐
Lemma 3. Let For any , there exists constant such that Moreover, if , then,
Proof. We represent according to Using the properties of fractional powers with a dense domain , in [28], we see from that where and , , are constants. If , we can take the time large enough such that☐
Lemma 4. For any , there exists constant such that Moreover, if , then
Proof. By applying the representation formula, we have We apply to both sides of equation (29) to obtain Then by If , taking the time t large enough and by virtue of Lemma 3, we can complete the proof.☐☐
Lemma 5. For any , there exists constant such that with some fixed .
Proof. Multiplying by the first equation of (3) and integration by parts, using Hölder’s inequality and Young inequality, we have that
That is,
To handle the right-hand side of (34), we use Hölder’s inequality and Gagliardo-Nirenberg inequality to get
where is constant and .
Similarly, using the Gagliardo-Nirenberg inequality, there is such that
From (35) and (36), we obtain such that
We now substitute (37)–(38) into (34) to obtain that
Applying Gronwall’s inequality, we see that
with some fixed . Due to being uniformly bounded, we can obtain (32) immediately.☐
Lemma 6. For any , there exists constant such that
Proof. Using the variation-of-constant formula for again, we obtain Therefore, the estimate of provides us with and , for any satisfying wherein the last integral is finite since . Similarly, we can deduce that with some , where we can select some such that Thus, by virtue of (43) and (44), we finish the proof of Lemma 6.☐☐
Proof of Theorem 1. In light of the prior estimates obtained in Lemma 2–Lemma 6 and the local existence results obtained in Lemma 1, we can complete the proof of Theorem 1.☐
3. Asymptotic Behavior
To simplify notation, we shall abbreviate the deviations from the nonzero homogeneous steady state by the following transformation: for all and . Through simple calculation, we see that satisfies the following initial boundary value problem:
In order to prove Theorem 2, we need several lemmas.
Lemma 7. For any , there exists constant such that
Proof. By using the variation-of-constant representation, for all , we obtain For , there is a constant such that Noticing that , we have For , taking , using the estimate of Neumann heat semigroup and Hölder’s inequality, we obtain where are constants. We now substitute (51)–(52) into (49) to complete the proof.☐☐
Next, we want to extend to infinity. Applying the Lemma 7, we can select to obtain for some .
For any , one has
By combining Lemma 3 and (45), we see that
Applying the Lemma 4, we can get
We now choose small enough such that
It is easy to see that
Let where is a given positive constant. Then, is well-defined since (49), (51), and (58). In order to extend to infinity, we give the following lemmas.
Lemma 8. For any, there exists a constantsatisfying
Proof. We first use (46) to represent according to and the fact that and (55) to estimate Furthermore, using Hölder’s inequality and the definitions of¨ and entails that Thus, substituting (62) and (63) into (61), we obtain the Lemma 8.☐☐
Lemma 9. For any , there exists constant such that
Proof. By means of the variation-of-constant representation for , combined with (56) and Lemma 8, we show that with some .☐☐
Lemma 10. Let denote the first nonzero eigenvalue of in under Neumann boundary conditions. Then, there exists constant such that
Proof. Notice that the fact of has the following estimate:
Furthermore, we can use (45) to obtain
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
for all and is a constant.
Thus, substituting (70) and (71) into (69), we have
where is a positive constant. Then, we select as sufficiently small to fulfilling
In conjunction with (57) and (73), this yields
By the continuity of , we can extend . So, we complete the proof.☐☐
Lemma 11. Let . Then, there is constant satisfying for all .
Proof. Let . From the second equation of (3), we can get the following system: Let be the solution of the following initial value problem: Using the comparison principle in [29], we see that is a supersolution of the system (76), and thus, Similarly, we have . Hence, we furthermore obtain that On the other side, direct computation shows that there are some constants and such that Thus, we can deduce that In a similar way, we can get the convergence of . Thus, we complete the proof.☐☐
Proof of Theorem 1. Using the estimates of Lemma 10 and Lemma 11, we obtain the decay estimates of , , and . Hence, the proof is completed.
Data Availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Conflicts of Interest
The authors declare that they have no competing interests.
Acknowledgments
The authors are very grateful to the referees for their detailed comments and valuable suggestions, which greatly improved the manuscript. The research of J. Wu was supported by Scientific Research Funds of Chengdu University under grant No. 2081921030. The research of H. Pan was supported by the National Natural Science Foundation of China with Grant 71971031.