# Boundedness and Asymptotic Behavior to a Chemotaxis System with Indirect Signal Generation and Singular Sensitivity

Authors: Jie Wu; Li Zhao; Heping Pan
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

In this paper, we consider the following indirect signal generation and singular sensitivitynt=Δn+χ∇⋅n/φc∇c,x∈Ω,t>0,ct=Δc−c+w,x∈Ω,t>0,wt=Δw−w+n,x∈Ω,t>0, in a bounded domain Ω⊂RNN=2,3 with smooth boundary ∂Ω. Under the nonflux boundary conditions for n, c, and w, we first eliminate the singularity of φc by using the Neumann heat semigroup and then establish the global boundedness and rates of convergence for solution.


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## Body

## 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
(1)nt=Δn−∇⋅nχn,c∇c,x∈Ω,t>0,ct=Δc−c+n,x∈Ω,t>0.Here, the unknownsn=nt,x and ct,x denote the cell density and chemical concentration, respectively. The given function χn,c is the chemotactic sensitivity. The physical domain Ω⊂ℝNN=2,3 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
(2)nt=Δn−∇⋅nχn,c∇c,x∈Ω,t>0,0=Δc−c+n,x∈Ω,t>0.Another important chemotaxis model is formed with singular sensitivity function, such asχn,c=χ/c. 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 systemConsidering the singular sensitivity function, we study the following singular chemotaxis model of indirect signal generation(3)nt=Δn+χ∇⋅nφc∇c,x∈Ω,t>0,ct=Δc−c+w,x∈Ω,t>0,wt=Δw−w+n,x∈Ω,t>0,where the parameter χ is a positive constant and φ is a known function. On the other hand, the case ofΩ⊂ℝNN=2,3is a bounded domain, under the assumption of the no-flux Neumann boundary condition for n,c and w, i.e.,
(4)∂n∂ν=∂c∂ν=∂w∂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∈Ωsatisfy
(6)0≤n0x∈C0Ω¯andn0x≢0,x∈Ω¯,c0x∈W1,∞Ωisnonnegativeandinfx∈Ωc0x>0,w0∈W1,∞Ωisnonnegative,φx∈C10,+∞,φ′x>0,x∈0,+∞andlimx⟶0+φx=0.There are some sensitivity functionsφ satisfying the fourth conditions of (6). For example, φx=xα,α>0orφx=log1+x,φx=arctanx,φx=xαlog1+x,φx=∫0xταlog1+τdτ, 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.
LetΩ⊂ℝN be a bounded domain with smooth boundary. Suppose that n0,c0,w0,φ satisfy (6). Then, for any q>1, systems (3)–(4) possess a global classical solution n,c,w which enjoys the regularity properties:
(7)nϵC0Ω¯×0,∞∩c2,1Ω¯×0,∞,cϵC0Ω¯×0,∞∩c2,1Ω¯×0,∞∩L∞0,∞;W1,qΩ,wϵC0Ω¯×0,∞∩c2,1Ω¯×0,∞∩L∞0,∞;W1,qΩ.
Moreover, this solution is uniformly bounded in the sense that(8)n·,tL∞Ω+c·,tW1,qΩ+w·,tW1,qΩ≤C,forallt∈0,∞,with some positive constant C.Theorem 2.
LetΩ⊂ℝN be a bounded domain with smooth boundary. Suppose that (6) holds. Then, there exists ϵ0>0 such that if m satisfies
(9)m<ϵfor some 0<ϵ<ϵ0, the solution of (3) has the following decay estimates:
(10)n·,t−mΩL∞Ω⟶0,c·,t−mΩL∞Ω⟶0,w·,t−mΩL∞Ω⟶0,where m≔n0·L1Ω and Ω 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.
ForN∈2,3, let Ω⊂ℝN be a bounded domain with smooth boundary. Assume that n0,c0,w0,φ satisfy (6). Then, there exist Tmax∈0,∞ and a classical solution n,c,w of (3)–(4) in Ω×0,Tmax such that
(11)n∈C0Ω¯×0,Tmax∩C2,1Ω¯×0,Tmax,c∈C0Ω¯×0,Tmax∩C2,1Ω¯×0,Tmax∩L∞0,∞;W1,qΩ,w∈C0Ω¯×0,Tmax∩C2,1Ω¯×0,Tmax∩L∞0,∞;W1,qΩ,Tmax=∞orlimt⟶Tmaxn.,tL∞Ω+c·,tW1,qΩ+w·,tW1,qΩ=∞.Proof.
Letc∗=1/einfx∈Ωc0x>0. With adaptations of the methods akin to those used in [24] and ([25], Thm. 2.3 i) to deal with the singular sensitivity, R>0 and T∈0,1 to be specified below, in Banach’s space
(12)X≔L∞O,T;C0Ω×W1,qΩ×W1,qΩ,forallq>0,we consider the closed set
(13)S≔n,c,w∈X∣nL∞Ω+cW1,qΩ+ww1,qΩ≤R,fora.e.t∈0,T°and introduce a mapping Φ=Φ1,Φ2,Φ3 on S by defining
(14)Φ1n,c,w≔et∆n0−χ∫t0tet−sΔ∇∙nφc∇cds,Φ2n,c,w≔et∆−1c0+∫t0tet−sΔ−1w·,sds,Φ3n,c,w≔Φ2n,c,w≔et∆−1w0+∫t0tet−sΔ−1n·,sds,for n,c,w∈S and t∈0,T. Using the reasoning (see [26], Lemma 1) based on Banach’s fixed point theorem applied in a closed bounded set in L∞0,T;C0Ω¯×W1,qΩ×W1,qΩ for suitably small T>0, the following regularity arguments, proving this local existence and uniqueness result.☐In order to get time-independent pointwise lower bounds ofw and c, we need to use the L1-conservation of n. The purpose of this method is to eliminate the singularity of the function 1/φC at zero.Lemma 2.
For anyt∈0,Tmax, there exist C>0, η>0, and m>0 such that
(15)n·,tL1Ω=n0·L1Ω,(16)minw.,t,c.,t≥η.
Moreover, we have(17)w·,tL1Ω≤m+w0xL1Ω·e−t,(18)c·,tL1Ω≤m+c0xL1Ω·e−t.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
(19)w⋅,t=etΔ−1u0+∫0tet−sΔ−1n⋅,sds≥∫0t14πt−sn/2e−t−s+diamΩ2/4t−s⋅n⋅,sL1Ωds=m∫0t14πt−sn/2e−t−s+diamΩ2/4t−sds=m∫0t014πτn/2e−τ+diamΩ2/4τ≔η1>0,where η1 is a positive constant and diamΩ≔maxx,y∈Ωx−y. In the same way, we see that
(20)c⋅,t=etΔ−1c0+∫0tet−sΔ−1w⋅,sds≥∫0t14πt−sn/2e−t−s+diamΩ2/4t−s⋅w⋅,sL1Ωds≥η1Ω∫0t14πt−sn/2e−t−s+diamΩ2/4t−sds=η1Ω∫0t014πτn/2e−τ+diamΩ2/4τdτ≔η2>0,where η2 is a positive constant. Taking η=minη1,η2>0, we get (16).
We integrate the third equation of (3) to obtain
(21)ddt∫Ωwx,tdx+∫Ωwx,tdx=∫Ωnx,tdx=m.
Applying Lemma 3.4 in [23], we obtain (17). In a similar way, we can get (18).☐Lemma 3.
Let(22)p¯=+∞,N=2,3,N=3.
For anyp∈0,p¯, there exists constant C such that
(23)w·,tLpΩ≤C,forallt∈0,Tmax.
Moreover, ifTmax=∞, then,
(24)w·,tLpΩ≤Cm,ast⟶∞.Proof.
We representw according to
(25)w⋅,t=etΔ−1u0+∫0tet−sΔ−1n⋅,sds,forall0<t<Tmax.
Using the properties of fractional powers−∆+1θ with a dense domain D−∆+1θ, θ∈0,1 in [28], we see from N/21−1/p<1 that
(26)w⋅,tLpΩ≤C1−Δ+1θw⋅,tLpΩ≤C1−Δ+1θetΔ−1w0LpΩ+C1∫0t−Δ+1θet−sΔ−1n⋅,sLpΩds≤C2t−θe−λ1tw0LpΩ+C2∫0+∞t−s−θ−N/21−1/pe−λ1t−snL1Ωds≤C3t−θe−λ1t+n0L1Ω,where λ1∈0,1 and C1, C2, C3>0 are constants. If Tmax=∞, we can take the time t large enough such thatw·,tLpΩ≤Cm.☐Lemma 4.
For anyq∈0,+∞, there exists constant C such that
(27)cW1,qΩ≤C,forallt∈0,Tmax.
Moreover, ifTmax=∞, then
(28)cW1,qΩ≤Cm,ast⟶∞.Proof.
By applying the representation formula, we have(29)c·,t=et∆−1c0+∫0tet−s∆−1w·,sds,t>0.
We apply−∆+1θ to both sides of equation (29) to obtain
(30)c⋅,tLqΩ≤C1−Δ+1θc⋅,tLqΩ≤C1−Δ+1θetΔ−1c0LqΩ≤C1∫0t−Δ+1θet−sΔ−1w⋅,sLqΩds≤C2t−θe−λ1tc0LqΩ≤C2∫0+∞t−s−θ−N/21/p−1/qe−λ1t−swLpΩds≤C3t−θe−λ1t+wLpΩ.
Then by(31)∇c⋅,tLqΩ≤C1∇etΔ+1c0LqΩ+C1∫0t∇et−sΔ+1w⋅,sLqΩds≤C11+t−1/2e−λ1tc0LqΩ+C2∫0+∞1+t−s−1/2−N/21/p−1/qe−λ1t−swLpΩds≤C31+t−1/2e−λ1t+wLqΩ.
IfTmax=∞, taking the time t large enough and by virtue of Lemma 3, we can complete the proof.☐☐Lemma 5.
For anyr>1, there exists constant C such that
(32)nLrΩ≤Cm2q−N/Nqr−1+4q−2N1+mNqr−1+2/2Nqr−1+4q−2N+m2Nqr−1+4q/2Nqr−1+4q−2N+n⋅,t0LrΩe−1/rt−t0,forallt≥t0,with some fixed t0>0.Proof.
Multiplyingnr−1 by the first equation of (3) and integration by parts, using Hölder’s inequality and Young inequality, we have that
(33)ddt∫Ωnrdx+4r−1r∫Ω∇nr/22dx=χrr−1∫Ωnr−1φc∇n⋅∇cdx≤2χr−11φη∇nr/2L2Ωnr/2∇cL2Ω≤2r−1r∫Ω∇nr/22dx+χ2rr−12φ2η∫Ωnr/2∇c2dx.
That is,(34)ddt∫Ωnrdx+2r−1r∫Ω∇nr/22dx≤χ2rr−12φ2η∫Ωnr/2∇c2dx.
To handle the right-hand side of (34), we use Hölder’s inequality and Gagliardo-Nirenberg inequality to get
(35)nr/2∇cL2Ω≤nr/2L2q/q−2Ω∇cLqΩ≤CGN∇nr/2L2Ω2N+Nqr−1/2q+Nqr−1nr/2L2Ω2r−1/2q+Nqr−1+CGNnr/2L2/rΩ∇cLqΩ≤CGN∇nr/2L2Ω2N+Nqr−1/2q+Nqr−1nL1Ωrq−N/2q+Nqr−1∇cLqΩ+CGNnL1Ωr/2∇cLqΩ=CGNmrq−N/2q+Nqr−1∇cLqΩ∇nr/2L2Ω2N+Nqr−1/2q+Nqr−1+CGNmr/2∇cLqΩ,where CGN>0 is constant and q>n.
Similarly, using the Gagliardo-Nirenberg inequality, there isCGN>0 such that
(36)nL2Ωr=nr/2L2Ω2≤CGNnr/2L1Ω2r/Nr−1+2∇nr/2L2Ω2Nr−1/Nr−1+2+CGNnL1Ωr=CGNnL1Ω2r/Nr−1+2∇nr/2L2Ω2Nr−1/Nr−1+2+CGNnL1Ωr=CGNm2r/Nr−1+2∇nr/2L2Ω2Nr−1/Nr−1+2+mr.
From (35) and (36), we obtain C4>0 such that
(37)nr/2∇cL2Ω2≤2φ2ηχ2rr−1r−12r∇nr/2L2Ω2+C4m2rq−N/Nqr−1+4q−2N1+mrNqr−1+2/2Nqr−1+4q−2N,(38)nLrΩr≤r−12r∇nr/2L2Ω2≤C4mr.
We now substitute (37)–(38) into (34) to obtain that
(39)ddtnLrΩr+nLrΩr+r−1r∇nr/2LrΩ2≤C4m2rq−N/Nqr−1+4q−2N1+mrNqr−1+2/2Nqr−1+4q−2N+mr2Nqr−1+4q/2Nqr−1+4q−2N.
Applying Gronwall’s inequality, we see that(40)nLrΩr≤C4m2rq−N/Nqr−1+4q−2N1+mrNqr−1+2rNqr−1+2/2Nqr−1+4q−2N+mr2Nqr−1+4q/2Nqr−1+4q−2N+n⋅,t0LrΩre−t−t0,forallt≥t0,with some fixed t0>0. Due to nLrΩ being uniformly bounded, we can obtain (32) immediately.☐Lemma 6.
For anyp∈0,∞, there exists constant C such that
(41)w·,tW1,pΩ≤C,forallt∈0,Tmax.Proof.
Using the variation-of-constant formula forw again, we obtain
(42)w·,t=et∆−1u0+∫0tet−s∆−1n·,sds,forall0<t<Tmax.
Therefore, the estimate ofnLrΩ provides us with C5>0 and C6>0, for any t∈0,Tmax satisfying
(43)w⋅,tLpΩ≤e−tetΔw0LpΩ+∫0te−t−set−sΔn⋅,sLr1Ωds≤C5w0L∞Ω+C5∫0te−t−s1+t−s−1/2−N/s1/r2−1/r1n⋅,sLrΩds≤C6+C6∫0te−t−s1+t−s−1/2−N/s1/r−1/r1ds,wherein the last integral is finite since 1/2+N/21/r−1/r1<1/2. Similarly, we can deduce that
(44)∇w⋅,tLpΩ≤C5∇etΔ−1w0LpΩ+C5∫0t∇et−sΔ−1n⋅,sLpΩds≤C51+t−1/2e−λ1tw0LpΩ+C6∫0+∞1+t−s−1/2−N/21/r−1/pe−λ1t−snLpΩds≤C7,forallt∈0,Tmax,with some C7>0, where we can select some p>r>1 such that N/21/r−1/p<1/2 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 Lemma2–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:(45)Ux,t=nx,t−mΩ,Vx,t=cx,t−mΩ,Wx,t=wx,t−mΩ,for all x∈Ω and t>0. Through simple calculation, we see that U,V,W satisfies the following initial boundary value problem:
(46)Ut=ΔU+∇⋅nφc∇V,x∈Ω,t>0,Vt=ΔV−V+W,x∈Ω,t>0,Wt=ΔW−W+U,x∈Ω,t>0,∂U∂ν=∂V∂ν=∂W∂ν=0,x∈∂Ω,t>0,Ux,0=n0x−mΩ,Vx,0=c0x−mΩ,Wx,0=u0x−mΩ,x∈Ω.In order to prove Theorem2, we need several lemmas.Lemma 7.
For anyr>1,q>N, there exists constant C such that
(47)limt⟶∞U·,tL∞Ω≤Cm1+2q−N/Nqr−1+4q−2N.Proof.
By using the variation-of-constant representation,(48)U⋅,t=et−t2ΔU⋅,t2−∫t2tet−sΔ∇⋅n⋅,sφc⋅,s∇V⋅,sds,for all t>t2, we obtain
(49)U⋅,tL∞Ω=et−t2ΔU⋅,t2L∞Ω+∫t2tet−sΔ∇⋅n⋅,sφc⋅,s∇V⋅,sL∞Ωds≔I1+I2,forallt>t2.
ForI1, there is a constant c1>0 such that
(50)U⋅,t2LrΩ=nx,t2−mΩLrΩ≤nx,t2LrΩ+mΩLrΩ≤c1.
Noticing that∫ΩU⋅,tdx=0, we have
(51)I1=et−t2ΔU⋅,t2L∞Ω≤c11+t−t2−N/2re−λ1t−t2=U⋅,t2LrΩ⟶0,ast⟶∞.
ForI2, taking r>r1>N,q>N, using the estimate of Neumann heat semigroup and Hölder’s inequality, we obtain
(52)I2=∫t2tet−sΔ∇⋅n⋅,sφc∇V⋅,sL∞Ωds≤c2∫t2t1+t−s−1/2−N/2r1e−λ1t−sn⋅,sφc∇V⋅,sLr1Ωds≤c2η∫t2t1+t−s−1/2−N/2r1e−λ1t−sn⋅,sLrΩ∇V⋅,srr1/Lr−r1Ωds≤c2η∫t2t1+t−s−1/2−N/2r1e−λ1t−sn⋅,sLrΩc⋅,sW1rr1/r−r1Ωds≤c3m1+2q−N/Nqr−1+4q−2N,where c2,c3>0 are constants. We now substitute (51)–(52) into (49) to complete the proof.☐☐Next, we want to extendT~0 to infinity. Applying the Lemma 7, we can select t3=t3n,c,u>0 to obtain
(53)U⋅,tL∞Ω≤2c3m1+2q−N/Nqr−1+4q−2N,for some r>1,q>N.For anyp∈1,p¯, one has
(54)W⋅,tLPΩ≤et−t2Δ−1W⋅,tLPΩ+∫t2tet−t2Δ−1U⋅,tLPΩds≤c3t−t2−θeλ1tW⋅,t2LPΩ+c3∫t2tt−s−θ−n/21−1/pe−λ1t−sU⋅,sL1Ωds=c3t−t2−θe−λ1tW⋅,t2LPΩ⟶0,ast⟶∞.By combining Lemma3 and (45), we see that
(55)W·,tLpΩ≤2c3m,forallt>t3.Applying the Lemma4, we can get
(56)∇V·,tLpΩ=∇c·,tLpΩ≤c3m,forallt>t3.We now choosem small enough such that
(57)c3m2q−N/Nqr−1+4q−2N≤12.It is easy to see that(58)U·,t3L∞Ω≤12ϵ,forallt≥t3.Let(59)T~0≔T≥t3∣U·,tL∞Ω≤ϵe−λ1t−t3,forallt∈t3,T0,where T0 is a given positive constant. Then, T~0 is well-defined since (49), (51), and (58). In order to extend T~0 to infinity, we give the following lemmas.Lemma 8.
For anyp∈1,p¯, there exists a constantc4>0satisfying(60)W·,tLpΩ≤2c4ϵe−λ1t−t3,forallt∈t3,T.Proof.
We first use (46) to represent W according to
(61)W·,t=et−t3∆−1W·,t1+∫t3tet−s∆−1U·,sdsand the fact that λ1<1 and (55) to estimate
(62)et−t3Δ−1W⋅,t3LpΩ≤e−t−t3et−t3ΔW⋅,t3LpΩ≤c4ϵe−λ1t−t3,forallt>t3.
Furthermore, using Hölder’s inequality and the definitions of¨T and c5 entails that
(63)∫t3tet−sΔ−1U⋅,sdsLpΩ≤c3∫t3te−t−set−sΔU⋅,sLpΩds≤c3∫t3t1+t−s−N/21−1/pe−λ1+1t−sU⋅,sL1Ωds≤c3Ω∫t3t1+t−s−N/21−1/pe−λ1+1t−sU⋅,sL∞Ωds≤c3ϵΩ∫t3t1+t−s−N/21−1/pe−λ1+1t−se−λ1s−t3ds≤c3ϵΩ∫0t−t31+t−σ−t3−N/21−1/pe−λ1+1t−σ−t3e−λ1σdσ≤c3ϵΩe−λ1+1t−t3∫0t−t31+t−σ−t3−N/21−1/peσdσ≤c4ϵe−λ1t−t3forallt∈t3,T.
Thus, substituting (62) and (63) into (61), we obtain the Lemma 8.☐☐Lemma 9.
For anyq∈1,+∞, there exists constant c5 such that
(64)∇V·,tLqΩ≤c5ϵe−λ11t−t3,forallt∈t3,T.Proof.
By means of the variation-of-constant representation forV, combined with (56) and Lemma 8, we show that
(65)∇V⋅,tLqΩ≤∇et−t3Δ−1V⋅,t3LqΩ+∫t3t∇et−sΔ−1W⋅,sLqΩds=e−t−t3∇et−t3ΔV⋅,t3LqΩ+∫t3te−t−s∇et−sΔW⋅,sLqΩds≤c1e−λ1+1t−t3∇V⋅,t3LqΩ+c2∫t3t1+t−s−1/2−N/21/p−1/qe−λ1+1t−sW⋅,sLpΩds≤c1c3∫e−λ1+1t−t3+c2∫t3t1+t−s−1/2−N/21/p−1/qe−λ1+1t−s2c4∫e−λ1s−t3ds≤c1c3∫e−λ1+1t−t3+2c2c4∫e−λ1s−t3c2∫0t−t31+σ−1/2−N/21/p−1/qe−σds≤c5∫e−λ1t−t3,forallt∈t3,T,with some c5>0.☐☐Lemma 10.
Letλ1>0 denote the first nonzero eigenvalue of −∆ in Ω under Neumann boundary conditions. Then, there exists constant c6 such that
(66)U·,tL∞Ω≤c6e−λ1t−t3,forallt>t3.Proof.
Notice that the fact ofU has the following estimate:
(67)U·,tL∞Ω≤c6e−λ1t−t3,forallt∈t3,T.
Furthermore, we can use (45) to obtain
(68)n∙,tL∞Ω=U∙,t+mΩL∞Ω≤U∙,tL∞Ω+mΩ≤ϵe−λ1t−t3+mΩ.
We next write(69)U·,tL∞Ω≤et−t3∆U·,t3L∞Ω∫t3tet−S∆∇n·,sφc∇V·,sL∞Ωdsand employ the estimate (53) to obtain
(70)et−t3ΔU⋅,t3L∞Ω≤c5e−λ1t−t3U⋅,t3L∞Ω≤2c3c5m1+2q−N/Nqr−1+4q−2Ne−λ1t−t3≤2c3c5∫1+2q−N/Nqr−1+4q−2Ne−λ1t−t3.
We next recall (18) and (45) and employ the estimates (64) and (68) to see that
(71)∫t3tet−sΔ∇⋅n⋅,sφc∇V⋅,sL∞Ωds≤c5∫t3t1+t−s−1/2−N/2re−λ1t−sn⋅,sφc∇V⋅,sLrΩds≤c5φη∫t3t1+t−s−1/2−N/2re−λ1t−sn⋅,sL∞Ω∇V⋅,sLrΩds≤c5φη∫t3t1+t−s−1/2−N/2re−λ1t−s∫e−λ1s−t3+1Ωc5∫e−λ1s−t3ds≤c52∫2φηe−λ1t−t3∫t3t1+t−s−1/2−N/2re−λ1s−t3+1Ωds≤c52∫2φηe−2λ1t−t3∫0t−t31+σ−1/2−N/2reλ1σ+1Ωdσ≤c52c7∫2φηe−λ1t−t3,for all r>N and c7> is a constant.
Thus, substituting (70) and (71) into (69), we have
(72)U∙,tL∞Ω≤12c8ϵ1+2q−N/Nqr−1+4q−2Ne−λ1t−t3,forallt∈t3,T,where c8 is a positive constant. Then, we select ϵ0>0 as sufficiently small to fulfilling
(73)c8ϵ1+2q−N/Nqr−1+4q−2N≤12.
In conjunction with (57) and (73), this yields
(74)U∙,tL∞Ω≤12ϵe−λ1t−t3,forallt∈t3,T.
By the continuity ofU, we can extend T~0=∞. So, we complete the proof.☐☐Lemma 11.
Letλ1∈0,1. Then, there is constant c9>0 satisfying
(75)c⋅,t−mΩL∞Ω≤c9e−λ1/2t,w⋅,t−mΩL∞Ω≤c9e−λ1/2t.for all t>0.Proof.
Letx,t≔cx,t−m/Ω. From the second equation of (3), we can get the following system:
(76)ψt−Δψ+ψ=u−mΩ,x∈Ω,t>0,∂ψ∂ν=0,x∈∂Ω,t>0,ψx,0=c0x−mΩ≔ψ0x,x∈Ω.
Letψ∗ be the solution of the following initial value problem:
(77)ψt∗+ψ∗=c10e−λ1t,t>0,ψ∗0=ψ∗L∞Ω.
Using the comparison principle in [29], we see that ψ∗t is a supersolution of the system (76), and thus,
(78)ψx,t≤ψ∗t,forallx∈Ω,t>0.
Similarly, we haveψx,t≥−ψ∗tforallx∈Ω,t>0. Hence, we furthermore obtain that
(79)ψx,t≤ψ∗t,forallx∈Ω,t>0.
On the other side, direct computation shows that there are some constantsc11 and c12 such that
(80)0≤ψ∗t≤c111+ψ∗L∞Ωe−λ1tc12e−λ1/2t,forallt>0.
Thus, we can deduce that(81)c⋅,t−mΩL∞Ω=ψ⋅,tL∞Ω≤ψ∗tc12e−λ1/2t,forallt>0.
In a similar way, we can get the convergence ofw. Thus, we complete the proof.☐☐Proof of Theorem 1.
Using the estimates of Lemma10 and Lemma 11, we obtain the decay estimates of n, c, and w. Hence, the proof is completed.



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*Source: 5130409-2021-07-31.xml*