Elsevier

Journal of Differential Equations

Volume 297, 5 October 2021, Pages 144-174
Journal of Differential Equations

Blow-up criteria for the classical Keller-Segel model of chemotaxis in higher dimensions

Cordially dedicated to Professor Takasi Senba on his 60th birthday
https://doi.org/10.1016/j.jde.2021.06.024Get rights and content

Abstract

We study the simplest parabolic-elliptic model of chemotaxis in space dimensions N3, and show the optimal conditions on the initial data for the finite time blow-up and the global existence of solutions in terms of stationary solutions. Our argument is based on the study of the Cauchy problem for the transformed equation involving the averaged mass of the solution.

Introduction

We consider the following Cauchy problem for the parabolic-elliptic system in space dimensions N3:{ut=(uuv),xRN,t>0,0=Δv+u,xRN,t>0,u(x,0)=u0(x)0,xRN. 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 u0 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 Mp(RN), 1<p<, is defined as the space of locally integrable functions such thatuMp=supR>0,xRNRN(1/p1)|xy|<Ru(y)dy<. Let u0 be radially symmetric, i.e., u0=u0(r) with r=|x|, xRN. It was shown by Biler, Karch and Pilarczyk [4, Theorem 2.1] that, if u0(|x|) belongs to MN/2(RN)Mp(RN) with some p(N/2,N), and if|||u0|||:=supR>0R2N0RrN1u0(r)dr<2, then the solution (u,v) of (1.1) exists globally in time and satisfies|||u(t)|||=R2N0RrN1u(r,t)dr<2for allt>0. Note that (1.1) has a stationary singular solution uC(r)=2(N2)/r2 with r=|x|. By a direct calculation, we see thatR2N0RrN1uC(r)dr=2for allR>0. Hence, the condition (1.2) means that u0 is strictly below uC in the integral averaged sense, and it can be written as0RrN1u0(r)drλ0RrN1uC(r)drfor allR>0 with some λ(0,1). It was shown by [7], [10] that there exists a constant CN>0 such that, if |||u0|||>CN, then a solution cannot be global in time.

In this paper, we always assume that u0C(RN)L(RN) and u0 is radially symmetric in (1.1). Then, by Proposition A.1 in Appendix, (1.1) has a unique local in time solution. Denote by T=T[u0](0,] the maximal existence time of the solution (u,v) of (1.1). Then, for any T<T, the solution (u,v) is classical, positive, radially symmetric and bounded for 0<tT, and u(,t)L(RN) as tT if T<. When T< 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 u0 for the global existence and the finite time blow-up of solutions in terms of the stationary solutions of (1.1).

For α>0, we define U(r;α) byU(r;α)=eϕ(r;logα)forr0, where ϕ(r;α) is a solution of the initial value problem{(rN1ϕr)r+rN1eϕ=0forr>0,ϕ(0)=α,ϕr(0)=0. Note that ϕ(|x|;α) is a radially symmetric solution of the equation Δϕ+eϕ=0 in RN. By phase plane analysis, in the case N3, we see thatϕ(r;α)=2logr+log(2N4)+o(1)asr for any α>0. (See, e.g., [20, Proof of Theorem 1.1].) Then U(r;α) satisfies, for α>0,U(r;α)=(2N+4)r2(1+o(1))asr. DefineV(r;α)=ϕ(r;logα)logαforr0. Then, by a direct calculation, (U,V)=(U(|x|;α),V(|x|;α)) with xRN solves{0=(UUV)inRN,0=ΔV+UinRN, with (U(0),V(0))=(α,0).

Our first result is the following.

Theorem 1.1

For α>0, define U(r;α) by (1.3). Let u0(r) satisfyu0C[0,)L[0,),u00u00 on [0,).

(i) Assume that u0(r) is nonincreasing in r[0,), and that0RrN1u0(r)drλ0RrN1U(r;α)drfor allR>0 with some λ(0,1]. Then the solution (u,v) of (1.1) exists globally in time and u satisfies u(,t)L[0,)0 as t if λ(0,1), and0RrN1u(r,t)dr0RrN1U(r;α)drfor all t>0 and R>0 if λ=1.

(ii) Assume that0RrN1u0(r)drλ0RrN1U(r;α)for allR>0 with some λ>1. Then the solution (u,v) 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 α>0, define U(r;α) by (1.3). Let u0(r) satisfy (1.8).

(i) Assume that u0(r) is nonincreasing in r[0,), and that 0u0(r)λU(r;α) for all r0 with some λ(0,1]. Then the solution (u,v) of (1.1) exists globally in time and u(,t)L[0,)0 as t if λ(0,1), and (1.10) holds if λ=1.

(ii) Assume that u0(r)λU(r;α) for all r0 with some λ>1. Then the solution (u,v) 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 u0 is in the Lebesgue spaces L1(RN) or LN/2(RN). From (1.6), we do not require that u0(|x|) belongs to neither L1(RN) nor LN/2(RN) 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 N10, and let >2. Then there exists u0 satisfying (1.8) such thatsupR>0R2N0RrN1u0(r)dr=, and that the solution (u,v) of (1.1) blows up in finite time

(ii) Let 3N9. Then there exists u0 satisfying (1.8) such thatsupR>0R2N0RrN1u0(r)dr>2, and that the solution (u,v) 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 N10, while the situation is different in the case 3N9.

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 (u,v) be a radially symmetric solution of (1.1) for 0t<T. Definew(r,t)=1rN0rsN1u(s,t)dsforr>0,0t<T. Then w satisfieswt=wrr+N+1rwr+rwrw+Nw2forr>0,0t<T. It can be seen that w=w(r,t) is a radially symmetric solution of the equationwt=Δw+(xw)w+Nw2forxRN+2,0<t<T, where Δ and ∇, respectively, are Laplacian and the spatial gradient in N+2 space variables. In the proof of Theorem 1.1, we will study the Cauchy problem for the transformed scalar equation{wt=Δw+(xw)w+Nw2,xRN+2,t>0,w(x,0)=w00,xRN+2, where w0C(RN+2)L(RN+2). We first show a basic maximum principle for solutions of the problemwtΔw+b(x,t)w+c(x,t)w,xRN+2,0tT, and derive the comparison principle for solutions to (1.14). Employing the comparison principle, together with properties of radially symmetric solutions to the stationary equationΔw+(xw)w+Nw2=0inRN+2, we will obtain Theorem 1.1. Combining the criteria obtained by Corollary 1.2 and the asymptotic properties of R2N0RsN1U(s;α)ds for R>0, we will obtain Theorem 1.4 with u0(r)=λU(r;α) for some λ>0.

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 QT0=RN+2×(0,T0] for T0>0. We first show the following comparison principle.

Proposition 2.1

Assume that u,vC2,1(QT0)C(RN+2×[0,T0]) satisfytuΔu(xu)uNu2tvΔv(xv)vNv2 in QT0, where Δ and, respectively, are Laplacian and the spatial gradient in RN+2. Assume that supQT0|u|< and supQT0|v|<, and that either xu or xv is bounded above in QT0. If u(x,0)v(x,0) for xRN+2 then uv in QT0.

To prove Proposition 2.1 we need the following lemma.

Lemma 2.2

Assume that wC2,1(QT0)C(RN+2×[0,T0]) satisfies supQ

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 w(x,t) is either nonincreasing or nondecreasing in t0 for each fixed xRN+2, and put W(x)=limtw(x,t) for each xRN+2. Then WLloc1(RN+2)Lloc2(RN+2) and W solvesΔW+(xW)W+NW2=0 for xRN+2 in the sense of distribution. Furthermore, the following (i) and (ii) hold.

(i) If WL(RN+2) then WC2(RN+2) is a classical solution of (3.1) in RN+2.

(ii) If WL(RN+2) and if W=W(r), r=|x|,

Properties of stationary solutions

In this section, we consider the following ordinary differential equationwrr+N+1rwr+rwrw+Nw2=0forr>0, where N>0. 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 wC2(0,)C1[0,) is a solution of (4.1) satisfying w(0)=α>0 if and only if w(r) is given byw(r)=1rN0rsN1eϕ(s;log(Nα))dsforr0, where ϕ(r;α) is the solution of (1.4).

(ii) Let wC2(0,)C1[0,) be a solution of (4.1) with w(0)>0. Then wr(0)=0 and wC2[0,).

Proof of Theorem 1.1

Let (u,v) be a radially symmetric solution of (1.1), and let T be the maximal existence time of (u,v). Definew(r,t)=1rN0rsN1u(s,t)dsfor0<r<,0t<T. By a direct calculation, we see that w satisfieswt=wrr+N+1rwr+rwrw+Nw2for0<r<,0<t<T, and wr(0,t)=0 for 0t<T. (See [19, Proposition 3.2].)

We will show the following lemma.

Lemma 5.1

(i) Let T<T. Then rwr(r,t) and w(r,t) are bounded for 0r< and 0tT.

(ii) Assume, in addition, that u(r,t) is nonincreasing in r0 for each t[0,T). Then w(r,t) is

Proof of Theorem 1.4

Let U(r;α) be defined by (1.3). First we consider the case N10.

Lemma 6.1

Let N10. Then, for any α>0,supR>0R2N0RrN1U(s;α)ds=2.

Proof

Let α>0. From (1.5), we obtain (1.6). It was shown by [20, Proof of Theorem 1.1] that, if N10, thenϕ(r;α)<2logr+log(2N4)for allr>0. Then U(r;α) satisfiesU(r;α)<(2N+4)r2for allr>0. From (1.6) and L'Hospital's rule, we havelimRR2N0RrN1U(r;α)dr=limRR2U(R;α)N2=2. From (6.2) it follows thatR2N0RrN1U(r;α)dr<2for allR>0. 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.

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