Blow-Up Solution in Logarithmic Sensitivity and Indirect Signal Generation Keller–Segel System
In this paper, we consider the following indirect signal generation and logarithmic sensitivity under homogeneous Neumann boundary conditions in a ball domain with smooth boundary . This paper considers in the singular limit ; the result comes from the finite time blow-up of arbitrary large values of in the corresponding nonlocal scalar parabolic equation case when and .
Keller and Segel in  proposed the following fully parabolic equations to describe the aggregation of certain types of bacteria.where the unknowns and denote the cell density and the concentration of chemical substances, respectively. The given function represents the chemosensitivity function and physical domain is a bounded domain with a smooth boundary. This model describes a biological process in which cells move towards their preferred environment and the signal is produced by the cells themselves. When the diffusion of chemical signals is much faster than that of cells, the system can be simplified as follows:
For its rigorous mathematical proof, we can see in . Recently, Li et al. in  have considered the stability analysis of the Keller–Segel model under fluid action in the two-dimensional case and has given the corresponding numerical experiments. For more references about the chemotaxis-fluid system, the corresponding global solvability of classical solutions has been investigated by [4–12] in two or three-dimensional situations. We also mention complicated variants, e.g., involving rotational flux [13–20] and logistic source terms [21–26] as well as nonlinear diffusion [4, 9, 16, 27–32].
Another important chemotaxis model is formed with a singular sensitivity function, such as . This model is proposed by the Weber-Fechner law of stimulus perception  and supported by experimental  and theoretical evidence . This fully parabolic logarithmic Keller–Segel system evidently lacks some good structures, which weakens the corresponding analysis skills. It is worth noting that this knowledge seems very fragmented, but it is essentially reduced to the relevant initial boundary value problems, and the assumptions allowing global solvability are based on . When the dimension , there is a globally bounded smooth solution for any initial data . The same conclusion is and with some  or and ([38–42]). In addition, some globally generalized solutions involved in general geometry  with some and in radially symmetric settings  with some . Accordingly, the integrable global solutions of nonradial symmetry under the assumption when and or and . In the similar parabolic-elliptic case, removing the technical assumption under the three-dimensional condition can also prove the global existence and integrability of the solution in the nonradial case when and , the corresponding classical solution is obtained when and or . For the quasilinear chemotaxis-Navier–Stokes system of this problem, there are lots of good results in [48–53].
On the other hand, based on the simplification of the scalar parabolic equation, it can be shown in  that the system (1) of parabolic-elliptic allows the radial solution to blow-up in a finite time if and . And, through the result of the global measure expansion of the radial solution of the classical Keller–Segel system beyond blow-up in , it can be inferred that there is no global -solution in this parameter region. The research on blow-up model has a strong physical background, such as gash healing, expansion, and collapse of geometric flow and energy released by stars in the universe.
An indirect signal generation without sensitivity function is also a very important Keller–Segel types model. Lin et al.  established the global existence and large-time behavior in . After Wu et al. in  added the singular term, investigated the global boundedness and large-time behavior of the above-given problem. The global existence for and blow-up solutions for were studied by Fujie and Senba in . Tao and Wang  considered the global solvability, boundedness, blow-up, existence of nontrivial stationary solutions, and asymptotic behavior. Stinner et al.  gave the global existence and some basic boundedness of weak solutions for a PDE-ODE system. Li and Li  considered the blow-up of nonradial solutions of the parabolic-elliptic-elliptic model in two dimensions. Recently, Viglialoro  has investigated explicit low bounded of blow-up time for a chemotaxis system. Chiyo et al.  studied the blow-up phenomena of a chemotaxis system with superlinear logistic degradation in .
Because the more delicate analytical technique of limit in the fully parabolic framework with the logarithmic term when suitably small, Winkler  considered how far the condition of the chemosensitivity plays a role in the limit process of the system (1) for . To motivate this idea, we study the following fully parabolic equations of the indirect signal.where the parameter is a positive constant, is a ball, 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.satisfying
Using the obtained identity (7) and the variation-of-constants formula, we can obtain the following equation:and
Applying the scalar parabolic problems of the type obtained in (9), the analysis method of a well-known nonlocal parabolic problems for suitable chosen radial initial data the respective limit function should be blow-up after some finite time whenever in Section . Next, we give the following theorem.
Theorem 1. Let and with some , and assume and . Then, there exist and such that in , and such that for any nonnegative with and each ), it is possible to choose and functions and belonging to ) such that and in , that solves (3) classically in , and thatand especially, giving any and we can find and such that
2. The Limit Procedure in (3)
2.1. Local Existence and Conditional -Independent Estimates
Firstly, we give the well-established the local existence of a classical solution to (3) for each fixed , along with a convenient extensibility criterion.
Lemma 1. Let and be a bounded domain with smooth boundary, and let and . Then, for any choice of and satisfying (6), there are and a uniquely determined pair of functions.such that and in , that solves (3) classically in , and that
Proof. We can use the local existence and extensibility to complete the proof of Lemma 1. We can refer to literature (, Lemma 10, Lemma 3.1 and 3.2) for relevant details.
The following lemma is helpful to prove the upper bounded of .
Lemma 2. (, Lemma 3.4) Let , and suppose that is a nonnegative absolutely continuous function on satisfyingWith some and a nonnegative function for which there exists such thatThen,In what follows, we let and be as obtained in Lemma 1. Next, we assume that and have the following properties.We will give the pointwise lower estimates of and , which plays an important role in the full text.
Lemma 3. If (A) holds with some and , then there exist and such thatandas well as
Proof. Integrating the first equation of (3), we can obtain (18). Then, integrating the third equation of (3), we have the following equation:Next, we apply Lemma 2 and (18) to establish (19). Therefore, using the convexity of and comparison argument (, Lemma 4), the following Neumann heat semigroup has properties.In order to get the pointwise lower estimate appropriately, we employ a variation-of-constants representation of and to see thatwhere because and are positive by (6). Similarly, we have the following equation:Taking , this entails (20).
Lemma 4. If holds with some and , then there exist and such that
Proof. Without loss of generality, we may assume that . Using the variation-of-constants formula and the estimate of the Neumann heat semigroup on  give the following equation:Since , we have . So we can take such that . Therefore , this ensures thatThis completes the proof of Lemma 4.
Lemma 5. Suppose that holds with some and . Then there exist and such that
Proof. For simplicity of expression from Lemma 4, we assume that . Then we can fix to find such that for all ,where . This entails thatTherefore, applying the above inequality and Lemma 4, we can obtain Lemma 5.
Lemma 6. Suppose that (A) with some and . Then there exists such that
Proof. We multiply the first equation of the system (4), integrate by parts and use Hölder’s inequality to deduce thatUsing Gagliardo–Nirenberg inequality and Young’s inequality to the second term at the right end of formula (32), we can obtain the following equation:That is,Therefore, integrating the two sides of the above-given inequality, we have the following equation:which together with (6) establishes Lemma 6.
Lemma 7. If (A) is valid with some and , then there exists such thatand
Proof. Multiplying the third equation of (3) with and making use of the integration by parts, we have the following equation:Taking the inner product of (4)2 with and using the integration by parts, we deduce thatWe multiply the second equation of (3) with and , respectively, then add them together and use Young’s inequality to get the following equation:Combining with (39)–(41), we have the following equation:We can employ the Gagliardo-Nirenberg inequality together with to obtain the following equation:This ensures thatThus, we can complete the proof of Lemma 7 from Lemma 6.
2.2. Passing to the Singular Limit
With the above-given important prior estimation, we can carry out the following limit process. The purpose of this part is to take the limit of , which is to hope that the obtained limit can meet the limit version solution of the system (3). This idea comes from Winkler in .
Lemma 8. Assume (A) with some and . Then, there exists and functions.Such that as , that and a.e., in , thatand that as we have the following equation:Moreover, we have the following identities:andas well asFor each .
Proof. In light of Lemma 4, Lemma 6, and (42) and together with the embedding , we have the following equation:According to Aubin–Lions lemma  and the standard compactness arguments, we can extract a sequence along which (46)–(51) hold with some nonnegative functions , and as . By and (49), we may employ Fatou’s lemma to obtain . Similarly, and (48) entails that . Furthermore, (48), the weak closedness of convex sets in , and Lemma 3 warrants that (45) and (52).
Then, testing the respective equations of system (3) with and using of the integration by parts, we have the following equation:andas well asFor all . We apply (57), (46)–(47), and (52) to obtain (53). Similarly, using (48)–(50) and (58), we derive (54). And, thanks to (46) and (50)-(51), we deduce that (55).
2.3. Identical Equation
Next, we give an important identity equation under the regular time. These techniques and methods are similar to the literature , thus we ignore the corresponding proof.
Lemma 9. Suppose that holds with some and , and let be a solution of (3) and be a non-Lebesgue point set of times. Then, we have the following equation:
3. Blow-Up in the Nonlocal Limit Problem
Lemma 10. Let and with some , and assume and . Then, there exists and a uniquely determined such that in , and in , that solvesin the classical sense, and that
Proof. A straightforward adaptation of the reasoning from (, Section 44.2, also ) makes positive and radially symmetric such that with some and a unique function , we have the following equation:where and . Let for . As a simple calculation shows that the defined satisfies the conclusion of the proposition.
Proof of Theorem 1. Let and , we fix and as given by Lemma 10, and let be a nonnegative function with . We now take and from Lemma 1, and let for . Next, we claim that (11) holds. We use the proof by contradiction to prove it.
If (11) is false, then we have the following equation:First, we claim that there exists a such that for all . Otherwise, we can find satisfying as and , then we can obtain from (13) thatwhich is contradictory about (64). Therefore, we deduce that for all . This ensures that (A) is valid. Thus, we can employ Lemma 8 to see thatsuch that a.e. in , thatand that (54) holds with some for all . Applying the Lemma 9, we have the following equation:Then, by the third equation of the system (3), (68), and using the variation-of-constants formula gives the following equation:In consequence, would form a bounded generalized solution, in the standard sense specified in , ofwith . Using the Neumann heat semigroup estimate, (45)2 and (66)-(67), there exists such that