# Blow-Up Solution in Logarithmic Sensitivity and Indirect Signal Generation Keller–Segel System

Authors: Jie Wu; Xinrui Xie
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this paper, we consider the following indirect signal generation and logarithmic sensitivityεnt=Δn−χ∇·n∇lncx∈Ω,t>0ct=Δc−c+wx∈Ω,t>0wt=Δw−w+nx∈Ω,t>0 under homogeneous Neumann boundary conditions in a ball domain Ω⊂RNN≥4 with smooth boundary ∂Ω. This paper considers in the singular limit ε⟶0; the result comes from the finite time blow-up of arbitrary large values of n in the corresponding nonlocal scalar parabolic equation case when N≥4 and χ>2N/N−2.


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## Body

## 1. Introduction

Keller and Segel in [1] proposed the following fully parabolic equations to describe the aggregation of certain types of bacteria.(1)τnnt=Δn−∇·nχn,c∇c,x∈Ω,t>0,ct=Δc−c+n,x∈Ω,t>0,where the unknowns n=nt,x and ct,x denote the cell density and the concentration of chemical substances, respectively. The given function χn,c represents the chemosensitivity function and physical domain Ω⊂ℝNN≥4 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:(2)τnnt=Δn−∇·nχn,c∇c,x∈Ω,t>0,0=Δc−c+n,x∈Ω,t>0.For its rigorous mathematical proof, we can see in [2]. Recently, Li et al. in [3] 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 asχn,c=χ/c. This model is proposed by the Weber-Fechner law of stimulus perception [33] and supported by experimental [34] and theoretical evidence [35]. 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 τn>0. When the dimension N=1, there is a globally bounded smooth solution for any initial data [36]. The same conclusion is N=2 and χ≤χ0 with some χ0>1.015[37] or N≥2 and χ<2/N ([38–42]). In addition, some globally generalized solutions involved in general geometry [43] with some χ<N+2/3N−4 and in radially symmetric settings [44] with some χ<N/N−2. Accordingly, the integrable global solutions of nonradial symmetry under the assumption when N=3 and χ<8 or 2≤N≠3 and χ<N/N−2[45]. 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 N≥2 and χ<N/N−2[46], the corresponding classical solution is obtained when N=2 and χ>0 or N≥3,χ<2/N−2[47]. 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 [54] that the system (1) of parabolic-elliptic τn=0 allows the radial solution to blow-up in a finite time if N≥3 and χ>2N/N−2. And, through the result of the global measure expansion of the radial solution of the classical Keller–Segel system beyond blow-up in [55], it can be inferred that there is no global L1-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. [56] established the global existence and large-time behavior in Ω⊂ℝN,N=2,3. After Wu et al. in [57] added the singular term, investigated the global boundedness and large-time behavior of the above-given problem. The global existence for N=2,3 and blow-up solutions for N≥4 were studied by Fujie and Senba in [58]. Tao and Wang [59] considered the global solvability, boundedness, blow-up, existence of nontrivial stationary solutions, and asymptotic behavior. Stinner et al. [60] gave the global existence and some basic boundedness of weak solutions for a PDE-ODE system. Li and Li [61] considered the blow-up of nonradial solutions of the parabolic-elliptic-elliptic model in two dimensions. Recently, Viglialoro [62] has investigated explicit low bounded of blow-up time for a chemotaxis system. Chiyo et al. [63] studied the blow-up phenomena of a chemotaxis system with superlinear logistic degradation in Ω⊂ℝN,N≥3.Because the more delicate analytical technique of limit in the fully parabolic framework with the logarithmic term whenτn suitably small, Winkler [64] considered how far the condition of the chemosensitivity χ plays a role in the limit process of the system (1) for τn⟶0. To motivate this idea, we study the following fully parabolic equations of the indirect signal.(3)εnεt=Δnε−χ∇·nε∇lncε,x∈Ω,t>0,cεt=Δcε−cε+wε,x∈Ω,t>0,wεt=Δw−wε+nε,x∈Ω,t>0,where the parameter χ is a positive constant, Ω⊂ℝNN≥4 is a ball, under the assumption of the no-flux Neumann boundary condition for n,c and w, i.e.,(4)∂nε∂ν=∂cε∂ν=∂wε∂ν=0,x∈∂Ω,t>0,where ν is the unit outward normal vector on ∂Ω and of the initial conditions.(5)nx,0=n0x,cx,0=c0x,wx,0=w0x,x∈Ω.satisfying(6)0≤n0x∈C0Ω¯,n0x≡0,x∈Ω¯,c0x∈W1,∞Ωsnonnegative,infx∈Ω¯c0x>0,w0∈W1,∞Ωisnonnegative,infx∈Ω¯w0x>0.Letm≔n0L1Ω. Then, m is a positive constant. The goal is to establish the identity of system (3) under the limit version in Section 3.(7)n=mcχcχL1Ω.Using the obtained identity (7) and the variation-of-constants formula, we can obtain the following equation:(8)w·,t=etΔ−1w0+m∫0tet−sΔ−1cχ·,scχ·,sL1Ωds.and(9)ct=Δc−c+etΔ−1w0+m∫0tet−sΔ−1cχ·,scχ·,sL1Ωds.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 c0 the respective limit function c should be blow-up after some finite time whenever χ>2N/N−2 in Section . Next, we give the following theorem.Theorem 1.
LetN≥4 and Ω≔BR0⊂ℝN with some R>0, and assume χ>2N/N−2 and m>0. Then, there exist c0,w0∈W1,∞ and T>0 such that c0>0,w0>0 in Ω¯, and such that for any nonnegative n0∈C0Ω¯ with ∫Ωn0=m and each ε∈0,1), it is possible to choose Tε∈0,T and functions nε,cε and wε belonging to C0Ω¯×0,Tε∩C2,1Ω¯×0,Tε) such that nε≥0 and cε>0,wε>0 in Ω¯×0,Tε, that nε,cε,wε solves (3) classically in Ω¯×0,Tε, and that(10)limsupε⟶0supt∈0,Tεnε·,tLpΩ=∞for allp>N4.and especially, giving any M>0 and ε>0 we can find ε∈0,ε0,x⋆∈Ω and t⋆∈0,Tε such that(11)nx⋆,t⋆≥M.


## 2. The Limit Procedureε⟶0 in (3)

### 2.1. Local Existence and Conditionalε-Independent Estimates

Firstly, we give the well-established the local existence of a classical solution to (3) for each fixed ε>0, along with a convenient extensibility criterion.Lemma 1.
LetN≥4 and Ω⊂ℝN be a bounded domain with smooth boundary, and let χ>0 and ε>0. Then, for any choice of n0,c0 and w0 satisfying (6), there are Tmax,ε∈0,∞ and a uniquely determined pair nε,cε,wε of functions.(12)nε∈C0Ω¯×0,Tmax,ε∩C2,1Ω¯×0,Tmax,ε,cε∈∩r>NC00,Tmax,ε;W1,rΩ∩C2,1Ω¯×0,Tmax,ε,w∈C0Ω¯×0,∞∩C2,1Ω¯×0,∞,such that nε≥0,cε>0 and wε>0 in Ω¯×0,Tmax,ε, that nε,cε,wε solves (3) classically in Ω×0,Tmax,ε, and that(13)ifTmax,ε<∞,thenlimsupt⟶Tmax,εnε·,tLpΩ=∞for allp>N4.Proof.
We can use the local existence and extensibility to complete the proof of Lemma1. We can refer to literature ([36], Lemma 10, Lemma 3.1 and 3.2) for relevant details.
The following lemma is helpful to prove the upper bounded ofwε.Lemma 2.
([60], Lemma 3.4) Let T>0, and suppose that z is a nonnegative absolutely continuous function on 0,T satisfying(14)z′t+azt≤ftfora.e.t∈0,T.
With somea>0 and a nonnegative function f∈Lloc10,T for which there exists b>0 such that(15)∫tt+1fsds≤bfor allt∈0,T−1.
Then,(16)zt≤maxz0+b,ba+2bfor allt∈0,T.
In what follows, we letTmax,εε∈0,1 and nε,cε,wεε∈0,1 be as obtained in Lemma 1. Next, we assume that p>N/4 and T>0 have the following properties.(17)Tmax,ε≥Tfor allε∈0,ε∗,supε∈0,ε∗t∈0,Tnε·,tLpΩ<∞.
We will give the pointwise lower estimates ofcε and wε, which plays an important role in the full text.Lemma 3.
If (A) holds with some T>0,p>N/4 and ε∗∈0,1, then there exist C>0 and δ0>0 such that(18)∫Ωnε·,t=∫Ωn0for allt∈0,Tmax,ε.and(19)w·,tL1Ω≤C.as well as(20)mincεx,t,wεx,t≥δ0for allx∈Ω,t∈0,T,ε∈0,ε∗.Proof.
Integrating the first equation of (3), we can obtain (18). Then, integrating the third equation of (3), we have the following equation:(21)ddt∫Ωwε+∫Ωwε=∫Ωnε.
Next, we apply Lemma2 and (18) to establish (19). Therefore, using the convexity of Ω and comparison argument ([65], Lemma 4), the following Neumann heat semigroup etΔt≥0 has properties.(22)etΔψ≥c1∫ΩψinΩfor allt>1.
In order to get the pointwise lower estimate appropriately, we employ a variation-of-constants representation ofcε and wε to see that(23)wε·,t=e−tetΔw0+∫0te−t−set−sΔnε·,sds,≥e−t·infΩw0+∫t−2+t−1+e−t−s·c1∫Ωn0ds,≥maxe−2infΩw0,c1e−2∫Ωn0,≔δ1for allt∈0,T,ε∈0,ε∗,where δ1>0 because infΩw0 and ∫Ωn0 are positive by (6). Similarly, we have the following equation:(24)cε·,t=e−tetΔc0+∫0te−t−set−sΔwε·,sds,≥e−t·infΩc0+∫t−2+t−1+e−t−s·c1∫Ωwεds,≥maxe−2infΩc0,c1e−2δ1Ω,≔δ2for allt∈0,T,ε∈0,ε∗,δ2>0.
Takingδ0≔minδ1,δ2, this entails (20).Lemma 4.
IfA holds with some T>0,p>N/4 and ε∈0,1, then there exist q>N/2 and C>0 such that(25)wε·,tLqΩ≤Cfor allt∈0,T,eachε∈0,ε∗.Proof.
Without loss of generality, we may assume thatp<N/2. Using the variation-of-constants formula and the estimate of the Neumann heat semigroup etΔt≥0 on Ω [66] give the following equation:(26)wε·,tLqΩ=etΔ−1w0+∫0tet−sΔ−1nε·,sdsLqΩ,≤c1+c1∫0t1+t−s−N/21/p−1/qe−λt−s,nε·,sLpΩds.
Since1/p−2/N<2/N, we have N/21/p−1/N/2<1. So we can take q>N/2 such that N/21/p−1/q<1. Therefore c2≔∫0∞1+σ−N/21/p−1/qe−λσdσ<∞, this ensures that(27)wε·,tLqΩ≤c1+c1c2supt∈0,Tnε·,tLpΩ,for allt∈0,T,ε∈0,ε∗.
This completes the proof of Lemma4.Lemma 5.
Suppose thatA holds with some T>0,p>N/4 and ε∈0,1. Then there exist r>N and C>0 such that(28)cε·,tW1,rΩ≤Cfor allt∈0,T,eachε∈0,ε∗.Proof.
For simplicity of expression from Lemma4, we assume that N/2<q<n. Then we can fix N<r<Nq/N−q to find c3>0 such that for all ε∈0,ε∗,(29)cε·,tW1,rΩ=etΔ−1c0+∫0tet−sΔ−1wε·,sdsW1,rΩ,≤c3+c3∫0t1+t−s−1/2−N/21/q−1/r,e−λt−swε·,sLqΩds.where 1/2+N/21/q−1/r<1/2+N/21/q−N−q/Nq=1. This entails that(30)cε·,tW1,rΩ≤c3+c2c3supt∈0,Twε·,tLqΩ,for allt∈0,T,ε∈0,ε∗.
Therefore, applying the above inequality and Lemma4, we can obtain Lemma 5.Lemma 6.
Suppose that (A) with some T>0,p>N/4 and ε∗∈0,1. Then there exists C>0 such that(31)∫0T∫Ω∇nε2≤Cfor allε∈0,ε∗.Proof.
We multiply the first equation of the system (4), integrate by parts and use Hölder’s inequality to deduce that(32)ε2ddt∫Ωnε2+∫Ω∇nε2=χ∫Ωnεcε∇nε·∇cε,≤12∫Ω∇nε2+χ22∫Ωnε2cε2∇cε2.
Using Gagliardo–Nirenberg inequality and Young’s inequality to the second term at the right end of formula (32), we can obtain the following equation:(33)εddt∫Ωnε2+∫Ω∇nε2≤c5∫Ωnε2∇cε2,≤c5nε2Lr/r−2Ω∇cε2Lr/2Ω,≤c6∇nεL2Ω2Nr+2/N+2rnεL1Ω4r−N/N+2r+c6nεL1Ω2,≤12∫Ω∇nε2+c7.
That is,(34)εddt∫Ωnε2+12∫Ω∇nε2≤c7.
Therefore, integrating the two sides of the above-given inequality, we have the following equation:(35)ε∫Ωnε2+12∫0t∫Ω∇nε2≤ε∫Ωn02+c7t,for allt∈0,T,ε∈0,ε∗.which together with (6) establishes Lemma 6.Lemma 7.
If (A) is valid with some T>0,p>N/4 and ε∗∈0,1, then there exists C>0 such that(36)∫0T∫ΩΔcε2≤Cfor allε∗∈0,ε∗.and(37)∫0T∫Ωcεt2≤Cfor allε∗∈0,ε∗.Proof.
Multiplying the third equation of (3) with w and making use of the integration by parts, we have the following equation:(38)12ddt∫Ω∇wε2+∫Ω∇wε2+∫Ωwε2≤∫Ωnεwε,≤nεL2ΩwεL2Ω,≤34∫Ωnε2+13∫Ωwε2.
Taking theL2 inner product of (4)2 with −Δcε and using the integration by parts, we deduce that(39)12ddt∫Ω∇cε2+∫ΩΔcε2=−∫Ω∇cε2−∫ΩwεΔcε,≤−∫Ω∇cε2+34∫ΩΔcε2+13∫Ωwε2.
We multiply the second equation of (3) with cε and cεt, respectively, then add them together and use Young’s inequality to get the following equation:(40)12ddt∫Ωcε2+∫Ω∇cε2+∫Ωcεt2=∫Ωcεtwε,≤34∫Ωcεt2+13∫Ωwε2.
Combining with (39)–(41), we have the following equation:(41)ddt∫Ωcε2+2∫Ω∇cε2+∫Ω∇wε2,+12∫Ωcεt2+2∫Ω∇cε2+∇wε2,+12∫ΩΔcε2≤32∫Ωnε2for allε∈0,ε∗.
We can employ the Gagliardo-Nirenberg inequality together withm≔∫Ωn0 to obtain the following equation:(42)32∫Ωnε2≤c8∇nεL2Ω2N/N+2nεL1Ω4/N+2+c4nεL1Ω2,≤12∫Ω∇nε2+c8+12c8N+2/2m2,
This ensures that(43)∫Ωcε2+2∫Ω∇cε2+∫Ω∇wε2+12∫0t∫Ωcεt2,+2∫0t∫Ω∇cε2+∇wε2+12∫0t∫ΩΔcε2,≤∫Ωc02+2∫Ω∇c02+2∫Ω∇w02+12∫0T∫Ω∇nε2,+c8+12c8N+2/2m2Tfor allε∈0,ε∗.
Thus, we can complete the proof of Lemma7 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 ofnε,cε,wε, which is to hope that the obtained limit can meet the limit version solution of the system (3). This idea comes from Winkler in [64].Lemma 8.
Assume (A) with some T>0,p>N/4 and ε∗∈0,1. Then, there exists εjj∈ℕ⊂0,ε∗ and functions.(44)n∈L20,T;H1Ω,c∈L20,T;H1Ω∩L∞Ω×0,T,w∈L20,T;H1Ω∩LqΩ×0,T,Such that εj⟶0 as j⟶∞, that n≥0,c>0 and w>0 a.e., in Ω×0,T,  that(45)1c∈L∞Ω×0,T.and that as ε=εj⟶0 we have the following equation:(46)nε⇀ninL2Ω×0,T,(47)∇nε⇀∇ninL2Ω×0,T,(48)cε⟶cinL2Ω×0,T,a.e.inΩ×0,T,(49)∇cε⟶∇cinL2Ω×0,T,(50)wε⇀winL2Ω×0,T,(51)∇wε⇀∇winL2Ω×0,T,(52)∇cεcε⟶∇ccinL2Ω×0,T.
Moreover, we have the following identities:(53)∫0T∫Ω∇n·∇φ=χ∫0T∫Ωn∇lnc·∇φ,and(54)−∫0T∫Ωcφt−∫Ωc0φ·,0+∫0T∫Ω∇c·∇φ,+∫0T∫Ωcφ=∫0T∫Ωwφ.as well as(55)−∫0T∫Ωwφt−∫Ωw0φ·,0+∫0T∫Ω∇w·∇φ,+∫0T∫Ωwφ=∫0T∫Ωnφ.
For eachφ∈C0∞Ω¯×0,T.Proof.
In light of Lemma4, Lemma 6, and (42) and together with the embedding W1,rΩ↼L∞Ω,r>N, we have the following equation:(56)nεε∈0,ε∗∈L20,T;H10,T,cεε∈0,ε∗∈L20,T;H20,T∩L∞,0,T;W1,NΩ∩L∞Ω×0,T,cεtε∈0,ε∗∈L2Ω×0,T,wεε∈0,ε∗∈L20,T;H1Ω∩L∞0,T;H1Ω.
According to Aubin–Lions lemma [67] and the standard compactness arguments, we can extract a sequence εjj∈ℝ⊂0,ε∗ along which (46)–(51) hold with some nonnegative functions n∈L20,T;H1Ω,c∈L20,T;H1Ω, and w∈L20,T;H1Ω as ε=εj⟶0. By cεε∈0,ε∗∈L∞0,T;W1,NΩ and (49), we may employ Fatou’s lemma to obtain c∈L∞0,T;W1,NΩ. Similarly, cεε∈0,ε∗∈L∞Ω×0,T and (48) entails that c∈L∞Ω×0,T. Furthermore, (48), the weak closedness of convex sets in L∞Ω×0,T, and Lemma 3 warrants that (45) and (52).
Then, testing the respective equations of system (3) with φ and using of the integration by parts, we have the following equation:(57)−ε∫0T∫Ωnεφt−ε∫Ωn0φ·,0+∫0T∫Ω∇nε,·∇φ=χ∫0T∫Ωnε∇lncε·∇φ.and(58)−∫0T∫Ωcεφt−∫Ωc0φ·,0+∫0T∫Ω∇cε·∇φ,+∫0T∫Ωcεφ=∫0T∫Ωwεφ.as well as(59)−∫0T∫Ωwεφt−∫Ωw0φ·,0+∫0T∫Ω∇wε·∇φ,+∫0T∫Ωwεφ=∫0T∫Ωnεφ.
For allφ∈C0∞Ω¯×0,T. 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 [64], thus we ignore the corresponding proof.Lemma 9.
Suppose thatA holds with some T>0,p>N/4 and ε∗∈0,1, and let n,c,w be a solution of (3) and Z⋆ be a non-Lebesgue point set of times. Then, we have the following equation:(60)n·,t=mcχ·,t∫Ωcχ·,tfor allt∈0,TZ⋆.


## 2.1. Local Existence and Conditionalε-Independent Estimates

Firstly, we give the well-established the local existence of a classical solution to (3) for each fixed ε>0, along with a convenient extensibility criterion.Lemma 1.
LetN≥4 and Ω⊂ℝN be a bounded domain with smooth boundary, and let χ>0 and ε>0. Then, for any choice of n0,c0 and w0 satisfying (6), there are Tmax,ε∈0,∞ and a uniquely determined pair nε,cε,wε of functions.(12)nε∈C0Ω¯×0,Tmax,ε∩C2,1Ω¯×0,Tmax,ε,cε∈∩r>NC00,Tmax,ε;W1,rΩ∩C2,1Ω¯×0,Tmax,ε,w∈C0Ω¯×0,∞∩C2,1Ω¯×0,∞,such that nε≥0,cε>0 and wε>0 in Ω¯×0,Tmax,ε, that nε,cε,wε solves (3) classically in Ω×0,Tmax,ε, and that(13)ifTmax,ε<∞,thenlimsupt⟶Tmax,εnε·,tLpΩ=∞for allp>N4.Proof.
We can use the local existence and extensibility to complete the proof of Lemma1. We can refer to literature ([36], Lemma 10, Lemma 3.1 and 3.2) for relevant details.
The following lemma is helpful to prove the upper bounded ofwε.Lemma 2.
([60], Lemma 3.4) Let T>0, and suppose that z is a nonnegative absolutely continuous function on 0,T satisfying(14)z′t+azt≤ftfora.e.t∈0,T.
With somea>0 and a nonnegative function f∈Lloc10,T for which there exists b>0 such that(15)∫tt+1fsds≤bfor allt∈0,T−1.
Then,(16)zt≤maxz0+b,ba+2bfor allt∈0,T.
In what follows, we letTmax,εε∈0,1 and nε,cε,wεε∈0,1 be as obtained in Lemma 1. Next, we assume that p>N/4 and T>0 have the following properties.(17)Tmax,ε≥Tfor allε∈0,ε∗,supε∈0,ε∗t∈0,Tnε·,tLpΩ<∞.
We will give the pointwise lower estimates ofcε and wε, which plays an important role in the full text.Lemma 3.
If (A) holds with some T>0,p>N/4 and ε∗∈0,1, then there exist C>0 and δ0>0 such that(18)∫Ωnε·,t=∫Ωn0for allt∈0,Tmax,ε.and(19)w·,tL1Ω≤C.as well as(20)mincεx,t,wεx,t≥δ0for allx∈Ω,t∈0,T,ε∈0,ε∗.Proof.
Integrating the first equation of (3), we can obtain (18). Then, integrating the third equation of (3), we have the following equation:(21)ddt∫Ωwε+∫Ωwε=∫Ωnε.
Next, we apply Lemma2 and (18) to establish (19). Therefore, using the convexity of Ω and comparison argument ([65], Lemma 4), the following Neumann heat semigroup etΔt≥0 has properties.(22)etΔψ≥c1∫ΩψinΩfor allt>1.
In order to get the pointwise lower estimate appropriately, we employ a variation-of-constants representation ofcε and wε to see that(23)wε·,t=e−tetΔw0+∫0te−t−set−sΔnε·,sds,≥e−t·infΩw0+∫t−2+t−1+e−t−s·c1∫Ωn0ds,≥maxe−2infΩw0,c1e−2∫Ωn0,≔δ1for allt∈0,T,ε∈0,ε∗,where δ1>0 because infΩw0 and ∫Ωn0 are positive by (6). Similarly, we have the following equation:(24)cε·,t=e−tetΔc0+∫0te−t−set−sΔwε·,sds,≥e−t·infΩc0+∫t−2+t−1+e−t−s·c1∫Ωwεds,≥maxe−2infΩc0,c1e−2δ1Ω,≔δ2for allt∈0,T,ε∈0,ε∗,δ2>0.
Takingδ0≔minδ1,δ2, this entails (20).Lemma 4.
IfA holds with some T>0,p>N/4 and ε∈0,1, then there exist q>N/2 and C>0 such that(25)wε·,tLqΩ≤Cfor allt∈0,T,eachε∈0,ε∗.Proof.
Without loss of generality, we may assume thatp<N/2. Using the variation-of-constants formula and the estimate of the Neumann heat semigroup etΔt≥0 on Ω [66] give the following equation:(26)wε·,tLqΩ=etΔ−1w0+∫0tet−sΔ−1nε·,sdsLqΩ,≤c1+c1∫0t1+t−s−N/21/p−1/qe−λt−s,nε·,sLpΩds.
Since1/p−2/N<2/N, we have N/21/p−1/N/2<1. So we can take q>N/2 such that N/21/p−1/q<1. Therefore c2≔∫0∞1+σ−N/21/p−1/qe−λσdσ<∞, this ensures that(27)wε·,tLqΩ≤c1+c1c2supt∈0,Tnε·,tLpΩ,for allt∈0,T,ε∈0,ε∗.
This completes the proof of Lemma4.Lemma 5.
Suppose thatA holds with some T>0,p>N/4 and ε∈0,1. Then there exist r>N and C>0 such that(28)cε·,tW1,rΩ≤Cfor allt∈0,T,eachε∈0,ε∗.Proof.
For simplicity of expression from Lemma4, we assume that N/2<q<n. Then we can fix N<r<Nq/N−q to find c3>0 such that for all ε∈0,ε∗,(29)cε·,tW1,rΩ=etΔ−1c0+∫0tet−sΔ−1wε·,sdsW1,rΩ,≤c3+c3∫0t1+t−s−1/2−N/21/q−1/r,e−λt−swε·,sLqΩds.where 1/2+N/21/q−1/r<1/2+N/21/q−N−q/Nq=1. This entails that(30)cε·,tW1,rΩ≤c3+c2c3supt∈0,Twε·,tLqΩ,for allt∈0,T,ε∈0,ε∗.
Therefore, applying the above inequality and Lemma4, we can obtain Lemma 5.Lemma 6.
Suppose that (A) with some T>0,p>N/4 and ε∗∈0,1. Then there exists C>0 such that(31)∫0T∫Ω∇nε2≤Cfor allε∈0,ε∗.Proof.
We multiply the first equation of the system (4), integrate by parts and use Hölder’s inequality to deduce that(32)ε2ddt∫Ωnε2+∫Ω∇nε2=χ∫Ωnεcε∇nε·∇cε,≤12∫Ω∇nε2+χ22∫Ωnε2cε2∇cε2.
Using Gagliardo–Nirenberg inequality and Young’s inequality to the second term at the right end of formula (32), we can obtain the following equation:(33)εddt∫Ωnε2+∫Ω∇nε2≤c5∫Ωnε2∇cε2,≤c5nε2Lr/r−2Ω∇cε2Lr/2Ω,≤c6∇nεL2Ω2Nr+2/N+2rnεL1Ω4r−N/N+2r+c6nεL1Ω2,≤12∫Ω∇nε2+c7.
That is,(34)εddt∫Ωnε2+12∫Ω∇nε2≤c7.
Therefore, integrating the two sides of the above-given inequality, we have the following equation:(35)ε∫Ωnε2+12∫0t∫Ω∇nε2≤ε∫Ωn02+c7t,for allt∈0,T,ε∈0,ε∗.which together with (6) establishes Lemma 6.Lemma 7.
If (A) is valid with some T>0,p>N/4 and ε∗∈0,1, then there exists C>0 such that(36)∫0T∫ΩΔcε2≤Cfor allε∗∈0,ε∗.and(37)∫0T∫Ωcεt2≤Cfor allε∗∈0,ε∗.Proof.
Multiplying the third equation of (3) with w and making use of the integration by parts, we have the following equation:(38)12ddt∫Ω∇wε2+∫Ω∇wε2+∫Ωwε2≤∫Ωnεwε,≤nεL2ΩwεL2Ω,≤34∫Ωnε2+13∫Ωwε2.
Taking theL2 inner product of (4)2 with −Δcε and using the integration by parts, we deduce that(39)12ddt∫Ω∇cε2+∫ΩΔcε2=−∫Ω∇cε2−∫ΩwεΔcε,≤−∫Ω∇cε2+34∫ΩΔcε2+13∫Ωwε2.
We multiply the second equation of (3) with cε and cεt, respectively, then add them together and use Young’s inequality to get the following equation:(40)12ddt∫Ωcε2+∫Ω∇cε2+∫Ωcεt2=∫Ωcεtwε,≤34∫Ωcεt2+13∫Ωwε2.
Combining with (39)–(41), we have the following equation:(41)ddt∫Ωcε2+2∫Ω∇cε2+∫Ω∇wε2,+12∫Ωcεt2+2∫Ω∇cε2+∇wε2,+12∫ΩΔcε2≤32∫Ωnε2for allε∈0,ε∗.
We can employ the Gagliardo-Nirenberg inequality together withm≔∫Ωn0 to obtain the following equation:(42)32∫Ωnε2≤c8∇nεL2Ω2N/N+2nεL1Ω4/N+2+c4nεL1Ω2,≤12∫Ω∇nε2+c8+12c8N+2/2m2,
This ensures that(43)∫Ωcε2+2∫Ω∇cε2+∫Ω∇wε2+12∫0t∫Ωcεt2,+2∫0t∫Ω∇cε2+∇wε2+12∫0t∫ΩΔcε2,≤∫Ωc02+2∫Ω∇c02+2∫Ω∇w02+12∫0T∫Ω∇nε2,+c8+12c8N+2/2m2Tfor allε∈0,ε∗.
Thus, we can complete the proof of Lemma7 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 ofnε,cε,wε, which is to hope that the obtained limit can meet the limit version solution of the system (3). This idea comes from Winkler in [64].Lemma 8.
Assume (A) with some T>0,p>N/4 and ε∗∈0,1. Then, there exists εjj∈ℕ⊂0,ε∗ and functions.(44)n∈L20,T;H1Ω,c∈L20,T;H1Ω∩L∞Ω×0,T,w∈L20,T;H1Ω∩LqΩ×0,T,Such that εj⟶0 as j⟶∞, that n≥0,c>0 and w>0 a.e., in Ω×0,T,  that(45)1c∈L∞Ω×0,T.and that as ε=εj⟶0 we have the following equation:(46)nε⇀ninL2Ω×0,T,(47)∇nε⇀∇ninL2Ω×0,T,(48)cε⟶cinL2Ω×0,T,a.e.inΩ×0,T,(49)∇cε⟶∇cinL2Ω×0,T,(50)wε⇀winL2Ω×0,T,(51)∇wε⇀∇winL2Ω×0,T,(52)∇cεcε⟶∇ccinL2Ω×0,T.
Moreover, we have the following identities:(53)∫0T∫Ω∇n·∇φ=χ∫0T∫Ωn∇lnc·∇φ,and(54)−∫0T∫Ωcφt−∫Ωc0φ·,0+∫0T∫Ω∇c·∇φ,+∫0T∫Ωcφ=∫0T∫Ωwφ.as well as(55)−∫0T∫Ωwφt−∫Ωw0φ·,0+∫0T∫Ω∇w·∇φ,+∫0T∫Ωwφ=∫0T∫Ωnφ.
For eachφ∈C0∞Ω¯×0,T.Proof.
In light of Lemma4, Lemma 6, and (42) and together with the embedding W1,rΩ↼L∞Ω,r>N, we have the following equation:(56)nεε∈0,ε∗∈L20,T;H10,T,cεε∈0,ε∗∈L20,T;H20,T∩L∞,0,T;W1,NΩ∩L∞Ω×0,T,cεtε∈0,ε∗∈L2Ω×0,T,wεε∈0,ε∗∈L20,T;H1Ω∩L∞0,T;H1Ω.
According to Aubin–Lions lemma [67] and the standard compactness arguments, we can extract a sequence εjj∈ℝ⊂0,ε∗ along which (46)–(51) hold with some nonnegative functions n∈L20,T;H1Ω,c∈L20,T;H1Ω, and w∈L20,T;H1Ω as ε=εj⟶0. By cεε∈0,ε∗∈L∞0,T;W1,NΩ and (49), we may employ Fatou’s lemma to obtain c∈L∞0,T;W1,NΩ. Similarly, cεε∈0,ε∗∈L∞Ω×0,T and (48) entails that c∈L∞Ω×0,T. Furthermore, (48), the weak closedness of convex sets in L∞Ω×0,T, and Lemma 3 warrants that (45) and (52).
Then, testing the respective equations of system (3) with φ and using of the integration by parts, we have the following equation:(57)−ε∫0T∫Ωnεφt−ε∫Ωn0φ·,0+∫0T∫Ω∇nε,·∇φ=χ∫0T∫Ωnε∇lncε·∇φ.and(58)−∫0T∫Ωcεφt−∫Ωc0φ·,0+∫0T∫Ω∇cε·∇φ,+∫0T∫Ωcεφ=∫0T∫Ωwεφ.as well as(59)−∫0T∫Ωwεφt−∫Ωw0φ·,0+∫0T∫Ω∇wε·∇φ,+∫0T∫Ωwεφ=∫0T∫Ωnεφ.
For allφ∈C0∞Ω¯×0,T. 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 [64], thus we ignore the corresponding proof.Lemma 9.
Suppose thatA holds with some T>0,p>N/4 and ε∗∈0,1, and let n,c,w be a solution of (3) and Z⋆ be a non-Lebesgue point set of times. Then, we have the following equation:(60)n·,t=mcχ·,t∫Ωcχ·,tfor allt∈0,TZ⋆.


## 3. Blow-Up in the Nonlocal Limit Problem

Lemma 10.
LetN≥4 and Ω=BR0⊂ℝN with some R>0, and assume χ>2N/N−2 and m>0. Then, there exists c0,w0∈W1,∞Ω,T>0 and a uniquely determined c^∈C0Ω¯×0,T∩C2,1Ω×0,T such that c0>0,w0>0 in Ω¯, and f^·,t:=etΔ−1w0+m∫0tet−sΔ−1c^χ·,s/∫Ωc^χ·,sdxds in Ω¯×0,T, that c^ solves(61)c^t=Δc^−c^+f·,t,x∈Ω,t∈0,T,∂c^∂ν=0,x∈∂Ω,t∈0,T,c^x,0=c0x,x∈Ω.in the classical sense, and that(62)limsupt⟶Tc^·,tL∞Ω=∞.Proof.
A straightforward adaptation of the reasoning from ([68], Section 44.2, also [69]) makes positive and radially symmetric c0,w0∈W1,∞ such that with some T>0 and a unique function 0<v∈C0Ω¯×0,T∩C2,1Ω¯×0,T, we have the following equation:(63)vt=Δv+g·,t,x∈Ω,t∈0,T,∂v∂ν=0,x∈∂Ω,t∈0,T,vx,0=c0x,x∈Ω,where g·,t=etf^·,t=etΔw0+m∫0tet−sΔesvχ·,s/∫Ωvχ·,sds and limsupt⟶Tv·,tL∞Ω=∞. Let vx,t=etc^x,t for x,t)∈Ω¯×0,T. As a simple calculation shows that the defined c^ satisfies the conclusion of the proposition.Proof of Theorem 1.
Letχ>2N/N−2 and m>0, we fix c0,c^ and T as given by Lemma 10, and let n0∈C0Ω¯ be a nonnegative function with ∫Ωn0=m. We now take Tmax,εε∈0,1 and nε,cε,wεε∈0,1 from Lemma 1, and let Tε:=minTmax,ε,T for ε∈0,1. 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:(64)limsupε⟶0supt∈0,Tεnε·,tLpΩ<∞for somep>p4.
First, we claim that there exists aε^∗∈0,1 such that T≤Tmax,ε for all ε∈0,ε∗. Otherwise, we can find εjj∈ℕ⊂0,1 satisfying εj⟶0 as j⟶∞ and Tmax,εj<T, then we can obtain from (13) that(65)supt∈0,Tεnεj·,tLpΩ=supt∈0,Tmax,εnεj·,tLpΩ,≥limt⟶Tmax,εjsupnεj·,tLpΩ,=∞for allj∈ℕ,which is contradictory about (64). Therefore, we deduce that Tε=T for all ε∈0,ε^∗. This ensures that (A) is valid. Thus, we can employ Lemma 8 to see that(66)c∈L20,T;H1Ω∩L∞Ω×0,T,such that c>0 a.e. in Ω×0,T, that(67)1c∈L∞Ω×0,T,and that (54) holds with some w∈LqΩ×0,T for all q>N/2. Applying the Lemma 9, we have the following equation:(68)n·,t=mcχ·,t∫Ωcχ·,ta.e.inΩ×0,T.
Then, by the third equation of the system (3), (68), and using the variation-of-constants formula gives the following equation:(69)w·,t=etΔ−1w0+m∫0tet−sΔ−1cχ·,scχ·,sL1Ωds.
In consequence,c would form a bounded generalized solution, in the standard sense specified in [70], of(70)ct=Δc−c+f·,t,x∈Ω,t∈0,T,∂c∂ν=0,x∈∂Ω,t∈0,T,cx,0=c0x,x∈Ω,with fx,t≔etΔ−1w0x+m∫0tet−sΔ−1cχx,s/cχx,sL1Ωds. Using the Neumann heat semigroup estimate, (45)2 and (66)-(67), there exists C5>0 such that(71)fx,tL∞Ω≤w0L∞Ω+mΩ1cL∞Ωχ,∫0tet−sΔ−1cχx,sL∞Ω≤C5.
Therefore, the standard results on Hölder regularity in scalar parabolic equations [71] is used to warrant that c∈Cθ,θ/2Ω¯×0,T for some θ∈0,1, whereupon classical Schauder theory [70] would ensure that f is Hölder continuous in Ω¯×0,T, thus the function c∈C2,1, and furthermore actually solve (70) in the classical sense. Therefore, the uniqueness feature in Proposition 3.1 enforces that c must be coincide with c^. This is contradictory with the boundedness claim in (66). Hence (10) must be valid. Thanks to L∞Ω⊂∩p>N/4LpΩ, we could deduce that thus also the growth property statement in (11) result.



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*Source: 8573835-2022-10-28.xml*