Blow-up criteria for the classical Keller-Segel model of chemotaxis in higher dimensions
Introduction
We consider the following Cauchy problem for the parabolic-elliptic system in space dimensions : This system arises as a simplified model of chemotaxis, where u and v stand for, respectively, the density of the bacterial population and of the secreted chemoattractant. This system is also known as a model of gravitational interaction of particles (see [1], [2], [8], [9]), and attracted considerable attention from the mathematical point of view (see [3] for references).
It was shown by [12], [13], [15] that the system (1.1) is locally well posed and has global in time solutions with small initial data in the scaling critical spaces. It was also shown that, under certain condition on the initial data, (1.1) possesses local in time solutions which cannot be continued to the global in time, see [13], [15], [19]. Corrias-Perthame-Zaag [13] showed that, if is well concentrated, then the solution blows up in a finite time. Ogawa and Wakui [15] obtained blow-up criteria by employing the logarithmic entropy functional and a generalized version of the Shannon inequality. Recently, the final blow-up profile of blow-up solutions was clarified by Souplet and Winkler [19]. A short proof of finite time blow-up of solutions was also obtained by [19, Proposition 3.4].
Recently, criteria for the global existence, as well as the finite time blow-up of radial solutions were derived in terms of suitable Morrey spaces norms by [4], [5], [6], [7], [10]. Recall that the homogeneous Morrey spaces , , is defined as the space of locally integrable functions such that Let be radially symmetric, i.e., with , . It was shown by Biler, Karch and Pilarczyk [4, Theorem 2.1] that, if belongs to with some , and if then the solution of (1.1) exists globally in time and satisfies Note that (1.1) has a stationary singular solution with . By a direct calculation, we see that Hence, the condition (1.2) means that is strictly below in the integral averaged sense, and it can be written as with some . It was shown by [7], [10] that there exists a constant such that, if , then a solution cannot be global in time.
In this paper, we always assume that and is radially symmetric in (1.1). Then, by Proposition A.1 in Appendix, (1.1) has a unique local in time solution. Denote by the maximal existence time of the solution of (1.1). Then, for any , the solution is classical, positive, radially symmetric and bounded for , and as if . When we say that the solution blows up in finite time. It is a natural question whether the solution exists globally in time or blows up in a finite time. We will show the optimal conditions on for the global existence and the finite time blow-up of solutions in terms of the stationary solutions of (1.1).
For , we define by where is a solution of the initial value problem Note that is a radially symmetric solution of the equation in . By phase plane analysis, in the case , we see that for any . (See, e.g., [20, Proof of Theorem 1.1].) Then satisfies, for , Define Then, by a direct calculation, with solves with .
Our first result is the following.
Theorem 1.1 For , define by (1.3). Let satisfy (i) Assume that is nonincreasing in , and that with some . Then the solution of (1.1) exists globally in time and u satisfies as if , and if . (ii) Assume that with some . Then the solution of (1.1) blows up in finite time.
As a typical case of Theorem 1.1, the following criteria follow immediately.
Corollary 1.2 For , define by (1.3). Let satisfy (1.8). (i) Assume that is nonincreasing in , and that for all with some . Then the solution of (1.1) exists globally in time and as if , and (1.10) holds if . (ii) Assume that for all with some . Then the solution of (1.1) blows up in finite time.
Remark 1.3 In [13], [15], the existence and nonexistence results of global in time solutions were obtained in the case where is in the Lebesgue spaces or . From (1.6), we do not require that belongs to neither nor in Corollary 1.2.
Next, we consider the optimality of the condition (1.2) for the global existence of solutions to (1.1).
Theorem 1.4 (i) Let , and let . Then there exists satisfying (1.8) such that and that the solution of (1.1) blows up in finite time (ii) Let . Then there exists satisfying (1.8) such that and that the solution of (1.1) exists globally in time.
Remark 1.5 Theorem 1.4 implies that the condition (1.2) is optimal for the global existence of solutions in the case , while the situation is different in the case .
The idea of the proof of the theorems is to reduce the system (1.1) into a scalar equation involving the averaged mass of u (see, e.g., [11], [14], [19]). Let be a radially symmetric solution of (1.1) for . Define Then w satisfies It can be seen that is a radially symmetric solution of the equation where Δ and ∇, respectively, are Laplacian and the spatial gradient in space variables. In the proof of Theorem 1.1, we will study the Cauchy problem for the transformed scalar equation where . We first show a basic maximum principle for solutions of the problem and derive the comparison principle for solutions to (1.14). Employing the comparison principle, together with properties of radially symmetric solutions to the stationary equation we will obtain Theorem 1.1. Combining the criteria obtained by Corollary 1.2 and the asymptotic properties of for , we will obtain Theorem 1.4 with for some .
The paper is organized as follows. In Section 2, we derive a comparison principle for (1.14), and show the convergence properties solutions in Section 3. In Section 4, we study the properties of radially symmetric solutions to (1.15), and prove Theorem 1.1, Theorem 1.4 in Sections 5 and 6, respectively. We show the local existence of solutions to (1.1) in the appendix.
Section snippets
Comparison principle and monotonicity of solutions in time
We set for . We first show the following comparison principle.
Proposition 2.1 Assume that satisfy in , where Δ and ∇, respectively, are Laplacian and the spatial gradient in . Assume that and , and that either or is bounded above in . If for then in .
To prove Proposition 2.1 we need the following lemma.
Lemma 2.2 Assume that satisfies
Convergence properties of solutions
In this section, we will show the following proposition.
Proposition 3.1 Let w be a global in time solution of (1.14). Assume that is either nonincreasing or nondecreasing in for each fixed , and put for each . Then and W solves for in the sense of distribution. Furthermore, the following (i) and (ii) hold. (i) If then is a classical solution of (3.1) in . (ii) If and if , ,
Properties of stationary solutions
In this section, we consider the following ordinary differential equation where . First we show that regular solutions of (4.1) can be represented by making use of the solution ϕ of (1.4).
Proposition 4.1 (i) A function is a solution of (4.1) satisfying if and only if is given by where is the solution of (1.4). (ii) Let be a solution of (4.1) with . Then and .
Proof of Theorem 1.1
Let be a radially symmetric solution of (1.1), and let be the maximal existence time of . Define By a direct calculation, we see that w satisfies and for . (See [19, Proposition 3.2].)
We will show the following lemma.
Lemma 5.1 (i) Let . Then and are bounded for and . (ii) Assume, in addition, that is nonincreasing in for each . Then is
Proof of Theorem 1.4
Let be defined by (1.3). First we consider the case .
Lemma 6.1 Let . Then, for any ,
Proof Let . From (1.5), we obtain (1.6). It was shown by [20, Proof of Theorem 1.1] that, if , then Then satisfies From (1.6) and L'Hospital's rule, we have From (6.2) it follows that Thus (6.1) holds. □
Next we consider the case
Acknowledgement
The author wish to thank Takasi Senba for useful suggestions and discussions. The author was supported by JSPS KAKENHI Grant Number 17K05333 and 20K03685. This work was also supported by Research Institute for Mathematical Sciences, a Joint Usage/Research Center located in Kyoto University.
References (20)
- et al.
Global radial solutions in classical Keller-Segel model of chemotaxis
J. Differ. Equ.
(2019) - et al.
Large global-in-time solutions to a nonlocal model of chemotaxis
Adv. Math.
(2018) Stability of steady states of the Cauchy problem for the exponential reaction-diffusion equation
J. Math. Anal. Appl.
(2006)The Cauchy problem and self-similar solutions for a nonlinear parabolic equation
Stud. Math.
(1995)Existence and nonexistence of solutions for a model of gravitational interaction of particles III
Colloq. Math.
(1995)Singularities of Solutions to Chemotaxis Systems
(2019)- et al.
Optimal criteria for blowup of radial and N-symmetric solutions of chemotaxis systems
Nonlinearity
(2015) - et al.
Morrey spaces norms and criteria for blowup in chemotaxis models
Netw. Heterog. Media
(2016) - et al.
Existence and nonexistence of solutions for a model of gravitational interactions of particles I
Colloq. Math.
(1994) - et al.
A nonlocal singular parabolic problem modelling gravitational interaction of particles
Adv. Differ. Equ.
(1998)
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