# Global Existence of a Chemotactic Movement with Singular Sensitivity by Two Stimuli

Authors: Heping Ma
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

In this study, we deal with the chemotaxis system with singular sensitivity by two stimuli under homogeneous Neumann boundary conditions in a bounded domain with smooth boundary. Under appropriate regularity assumptions on the initial data, we show that the system possesses global classical solution. Our results generalize and improve previously known ones.


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## Body

## 1. Introduction

Chemotaxis is a well-known biological phenomenon describing the collective motion of cells or the evolution of density of bacteria driven by chemicals, including embryo development, skin wound healing, cancer invasion, and metastasis. The pioneering works of the chemotaxis model was introduced by Keller and Segel in [1], describing the aggregation of cellular slime mold toward a higher concentration of a chemical signal, which reads(1)ut=Δu−∇⋅u∇v,x∈Ω,t>0,vt=Δv−v+u,x∈Ω,where Ω⊂ℝNN≥1 is a bounded domain with smooth boundary. The mathematical analysis of (1) and the variants thereof mainly concentrate on the boundedness and blow-up of the solutions (refer to [2–6] and the references therein), that is, it is well-known that for all suitably regular initial data u0,v0, an associated Neumann initial boundary value problem, posed in a smooth N-dimensional domain Ω, Osaki and Yagi [4] proved that system (1) always possesses a global bounded classical solution in one-dimensional bounded domain, Nagai et al. [3] showed that system (1) also admits a global bounded classical solution in two-dimensional bounded domain if u0L1Ω is small, as N≥3, Winkler [5] proved that for each q>N/2, p>N, one can find ε0>0, such that if the initial data u0,v0 fulfill u0LqΩ<ε and v0LpΩ<ε for some ε<ε0, then the solution is global in time and bounded; this is complemented by corresponding findings on the occurrence of finite-time blow-up of some solutions emanating from smooth but appropriately large initial data [2, 6]. In the past few decades, system (1) has attracted extensive attentions.Keller and Segel [7] introduced a phenomenological model of the wave-like solution behavior without any type of cell kinetics, a prototypical version of which is given by(2)ut=Δu−χ∇⋅uv∇v,x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0.where u represents the density of bacteria and v denotes the concentration of the nutrient. The second equation models consumption of the signal. In the first equation, the chemotactic sensitivity is determined according to the Weber–Fechner law, which says that the chemotactic sensitivity is proportional to the reciprocal of signal density. Winkler [8] proved that if initial data satisfy appropriate regularity assumptions, system (2) possesses at least one global generalized solution in two-dimensional bounded domains. Moreover, he took into account asymptotic behavior of solutions to system (2) and proved that v⋅,t⇀∗0 in L∞Ω and v⋅,t⟶0 in LpΩ as t⟶∞ provided ∫Ωu0≤m, −∫Ωlnv0/v0L∞Ω≤M, where m,M are the positive constants. When uv is replaced by guv, g∈C1ℝ, and 0≤gu≤uβ, β∈0,1, χ∈0,1, and any sufficiently regular initial data, Lankeit and Viglialoro [9] showed that system (2) has a global classical solution. Moreover, if additionally m=u0L1Ω is sufficiently small, then also their boundedness is achieved. When system (2) has a logistic source fu, Lankeit and Lankeit [10] showed that system (2) possesses a global generalized solution for any χ≥0, r≥0, and μ>0 if fu=κu−μu2. As fu=ru−μuk, and 1/v is replaced by ϕv, ϕv∈C10,∞, satisfying ϕv⟶0 as v⟶∞. Zhao and Zheng [11] proved that system (2) possesses a unique positive global classical solution provided k>1 with N=1 or k>1+N/2 with N≥2. When 1/v is replaced by 1/vα, α≥1 and Δv turned into εΔv, Zhao et al. [12] obtained the global existence of classical solutions with N=1. Moreover, for any global classical solution u,v to the case of N, it is shown that v converges to 0 in the L∞-norm as t⟶∞ with decay rate established whenever ε∈ε0,1 with ε0=max0,1−χ/αv0L∞Ωα−1. When u is replaced by fu and 0<fu≤Ku+1α, Liu [13] showed that for any sufficiently smooth initial data, system (2) admits a global classical solution when either N=1 and α<2 or N≥2 and α<1−N/4. When Δu is replaced by Δumm≥1, Lankeit [14] proved that if m>1+N/4, system (2) admits a global classical solution or global locally bounded weak solution.This study deals with a chemotaxis system with singular sensitivity by two stimuli, which is given by(3)ut=∇⋅u+1m−1∇u−∇⋅ϕ1uv∇v+∇⋅ϕ2uw∇w,x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0,wt=Δw−uw,x∈Ω,t>0,∂u∂ν=∂v∂ν=∂w∂ν=0,x∈∂Ω,t>0,ux,0=u0x,vx,0=v0x,wx,0=w0x,x∈Ω,where Ω⊂ℝNN≥2 is a bounded domain with smooth boundary ∂Ω, ∂/∂ν denotes the derivative with respect to the outer normal of ∂Ω, and ux,t, vx,t, and wx,t represent the density of the cell population, the concentration of the chemoattractant substances, and the concentration of the chemorepellent substances, respectively. We assume that m≥1, ϕ1u,ϕ2u∈C1+σ0,∞, where σ∈0,1 and satisfies(4)0<ϕ1s≤Γ1sα1,ϕ10=0,0<ϕ2s≤Γ2sα2,ϕ20=0,for alls>0,with Γ1,Γ2>0, and α1,α2<2α1≠α2 are the constants. Furthermore, we assume that the initial data u0,v0,w0 satisfy(5)u0∈C0Ω¯,u0≥0inΩandu0≡0as well as,v0,w0∈W1,∞Ω,v0x,w0x>0inΩ¯.It is different from model (3); the following is the attraction-repulsion Keller–Segel model where the signal is produced and not consumed by the cells:(6)ut=∇⋅Duu−χ∇⋅u∇v+ξ∇⋅u∇w,x∈Ω,t>0,τ1vt=Δv+αu−βv,x∈Ω,t>0,τ2wt=Δw+γu−δw,x∈Ω,t>0,which was proposed in [15] to describe how the combination of chemical might interact to produce aggregates of cells. In two space dimensions, when Du=1, τ1=1, and τ2=0, Jin et al. [16] proved that if χα−ξγ≤0, then system (6) possesses a unique global uniformly in-time bounded classical solution, and if χα−ξγ>0, ∫Ωu0x<4π/χα−ξγ, the same result is also obtained; to the contrary, if ∫Ωu0x>4π/χα−ξγ, the solutions blow up in finite or infinite time. Ulteriorly, Xu et al. [17] showed that if ξγβ≥χαδ, the global classical solution of system (6) converged to the unique constant state u¯0,α/βu¯0,γ/δu¯0 as t⟶∞, where u¯0=1/Ω∫Ωu0. When Du=1, τ1=τ2=0, Li et al. [18] proved that if ∫Ωu0xx−x02 is sufficiently small, either of the following cases holds: (i) χα>ξγ, δ≥β, and ∫Ωu0x>8π/χα−ξγ; (ii) χαδ>ξγβ, δ<β, and ∫Ωu0x>8π/χαδ−ξγβ, the corresponding solution of (6) blows up in finite time. Hu et al. [19] showed that if χα>ξγ and ∫Ωu0x>8π/χα−ξγ hold with ∫Ωu0xx−x02 small enough, the solution of (6) blows up in finite time. In high dimensions, when Du=1, τ1=τ2=1, Jin [20] proved that if ξγ>χα, system (6) possesses global classical solution in dimensions and weak solution in three dimensions with large initial data. When τ1=1, τ2=0, Ds≥0, s≥0, and Ds=D0ss−θ for all s>0 with some D0>0 and θ∈ℝ, Lin et al. [21] showed that if θ<2/N−1, for any nonnegative initial data, system (6) admits a unique classical solution which was global and bounded; if θ>2/N−1, the radially symmetric solutions may blow up in finite time. When τ1,τ2>0, Du>0 for u≥0 and Du≥dum−1 with d>0 and m≥1 for all u>1, Li et al. [22] proved that the corresponding initial boundary value problem possesses a unique global bounded classical solution for m>2−2/N. In particular, in the case τ1=τ2 and χα=ξγ, the solution is globally bounded if m>2−2/N−N+2/N2−N+2. When the system has a logistic source, the relevant results can be found in [23–25].To the best of our knowledge, Dong et al. [26] first put forward the following chemotactic model with general rotational sensitivity caused by two stimuli:(7)ut=Δu−∇⋅uS1x,u,v,w∇v+∇⋅uS2x,u,v,w∇w,x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0,wt=Δw−uw,x∈Ω,t>0,where Ω⊂ℝNN≥1 is a bounded domain with smooth boundary ∂Ω. Under mild assumptions on S1x,u,v,w, S2x,u,v,w, system (7) admits at least one global generalized solution.Throughout above analysis, compared with system (6), the theory of system (3) is so fragmentary. To the best of our knowledge, the global classical solution of model (3) in N≥2 has never been touched. No matter biological relevance or mathematical meaning, we find it is worth addressing the basic solvability theory of the model (3). Inspired by the arguments in previous studies [8, 13, 14, 26, 27], we mainly investigate the global classical solution in a chemotactic movement with singular sensitivity by two stimuli. Theorem 1 partially generalizes and improves previously known ones.In this study, we use symbolsCi and cii=1,2,… as some generic positive constants which may vary in the context. For simplicity, ux,t is written as u, the integral ∫Ωuxdx is written as ∫Ωux, and ∫0t∫Ωuxdxdt is written as ∫0t∫Ωux.The rest of this study is organized as follows. InSection 2, we summarize some useful lemmata in order to prove the main result. In Section 3, we give some fundamental estimates for the solution to system (3) and proof of Theorem 1.


## 2. Preliminaries and Main Result

In this section, we give the main theorem and the local existence of the classical solution to (3) and also summarize some useful lemmata in order to prove the main result. Noting the singular chemotaxis term, we let(8)n1≔−lnvx,tv0L∞Ω,n2≔−lnwx,tw0L∞Ω,x,t∈Ω¯×0,∞.Then, we can rewrite (3) as(9)ut=∇⋅u+1m−1∇u+∇⋅ϕ1u∇n1−∇⋅ϕ2u∇n2,x∈Ω,t>0,n1t=Δn1−∇n12+u,x∈Ω,t>0,n2t=Δn2−∇n22+u,x∈Ω,t>0,∂u∂ν=∂n1∂ν=∂n2∂ν=0,x∈∂Ω,t>0,ux,0=u0x,n1x,0=n10x≔−lnvx,tv0L∞Ω,n2x,0=n20x≔−lnwx,tw0L∞Ω,x∈Ω.At first, we give the main result of global existence of the classical solution to (3).Theorem 1.
LetΩ⊂ℝNN≥2 be a bounded domain with smooth boundary. Assume that min1−2/N,2−6/N+2<m≤2−2/N, ϕ1u, ϕ2u satisfy (4) and(10)α1<1−1−mN4,α2<1−1−mN4.Then, for any choice of the initial datau0,v0,w0 fulfilling (5), there exists a triple u,v,w∈C0Ω¯×0,∞∩C2,1Ω¯×0,∞3 which solves (3) classically. Moreover, we have u≥0 and v,w>0 in Ω¯×0,∞.Remark 1.
Theorem1 shows that system (3) admits a global classical solution nothing to do with Γ1 and Γ2.Remark 2.
Theorem1 partially generalizes and improves the results in ([14], Theorem 1.1) and ([13], Theorem 1.1).Remark 3.
If we replaced the termsuv and uw in system (3) by uf1v,w and uf2v,w, respectively, Theorem 1 still holds provided that both f1 and f2 are nonnegative differentiable functions satisfying f10,0=0 and f20,0=0.In the sequel, we will consider system (9) to obtain the local boundedness of v,w and then come back to system (3) to prove the main theorem. Under the framework of fixed point theorem, we will prove the local existence of classical solution to system (3) in the following lemma. The proof is quite standard, and a more detailed display of a similar reasoning in a related circumstance can be found in [14].Lemma 1.
LetΩ⊂ℝNN≥2 be a bounded domain with smooth boundary. Assume that m≥1, ϕ1u, ϕ2u satisfy (4) and α1<2,α2<2. Then, for any initial data u0,v0,w0 fulfilling (5), there exist Tmax∈0,∞ and a triple u,v,w∈C0Ω¯×0,Tmax∩C2,1Ω¯×0,Tmax3 solving (3) classically in Ω×0,Tmax, where Tmax denotes the maximal existence time. Moreover, the solution u,v,w satisfies(11)∫Ωux,t=∫Ωu0x,for allt>0,0<v<v0L∞Ω,0<w<w0L∞ΩinΩ×0,∞,if Tmax<+∞, then(12)limsupt↗Tmaxu⋅,tL∞Ω=∞.Lemma 2.
The solution of (9) satisfies(13)∫0t∫Ω∇n12≤∫Ωn10+Ct,for allt∈0,Tmax,(14)∫0t∫Ω∇n22≤∫Ωn20+Ct,for allt∈0,Tmax,where C=∫Ωu0x.Proof.
Integrating the second equation of (9) with respect to x,t∈Ω×0,Tmax, we have(15)∫Ωn1−∫Ωn10=−∫0t∫Ω∇n12+∫0t∫Ωu≤−∫0t∫Ω∇n12+Ct.
Due to the positivity ofn1, we obtain (13) immediately. Likewise, we get (14). The proof is complete.Lemma 3.
(Gagliardo–Nirenberg interpolation inequality [28]). Let 0<θ≤p≤2N/N−2. There exists a positive constant CGN, such that for all u∈W1,2Ω∩LθΩ,(16)uLpΩ≤CGN∇uL2ΩauLθΩ1−a+uLθΩ,is valid with a=N/θ−N/p/1−N/2+N/θ∈0,1.Lemma 4.
(See [29]). Let T>0 and y∈C00,T∩C10,T be such that(17)y′t+ayt≤gt,for allt∈0,T,where g∈Lloc1ℝ has the property that(18)1τ∫tt+τgsds≤b,for allt∈0,T,with some τ>0 and b>0. Then,(19)yt≤y0+bτ1−e−aτ,for allt∈0,T.


## 3. Proof of Theorem1

In this section, we establish some priori estimates for solutions to system (9); we first establish a bound for n1x,t and n2x,t in the one-dimensional case, which differs from that in the multidimensional settings.Lemma 5.
LetN≥2 and suppose that min1−2/N,2−6/N+2<m≤2−2/N, (4) holds with σ∈0,1, Γ1,Γ2>0, and α1<1−1−mN/4, α2<1−1−mN/4. Then, there exists C3,C4>0, such that the solution of (9) satisfies(20)∫Ωu1−2α1≤C3,∫Ωu1−2α2≤C4,for allt∈0,Tmax,(21)∫0t∫Ωum−2α1−1∇u2≤C31+t,∫0t∫Ωum−2α2−1∇u2≤C41+t,for allt∈0,Tmax.Proof.
Multiplying the first equation in (9) by 21−α1u1−2α1, integrating by parts, using (4) and Young’s inequality, we have(22)ddt∫Ωu2−2α1=21−α1∫Ωu1−2α1∇⋅u+1m−1∇u+∇⋅ϕ1u∇n1−∇⋅ϕ2u∇n2≤−L1∫Ωum−2α1−1∇u2−2L1∫Ωu−2α1ϕ1u∇u⋅∇n1+2L1∫Ωu−2α1ϕ2u∇u⋅∇n2≤−2L1∫Ωum−2α1−1∇u2+12L1∫Ωum−2α1−1∇u2+2L1∫Ωu1−2α1−mϕ12u∇n12+12L1∫Ωum−2α1−1∇u2+2L1∫Ωu1−2α1−mϕ22u∇n22≤−L1∫Ωu1−2α1−m∇u2+2L1Γ1∫Ωu1−m∇n12+2L1Γ2∫Ωu1+2α2−2α1−m∇n22,for all t∈0,Tmax, where L1=1−α11−2α1, and we have used the facts that L1>0 due to α1<1−1−mN/4; since α1≠α2, without loss of generality, we let α1>α2; by Lemma 1, we know that there exists a constant δ1>0, such that u>δ1, and with some rearrangements, we have(23)ddt∫Ωu2−2α1+L1∫Ωu−2α1∇u2≤2δ11+2α2−2α1−mL1Γ2∫Ω∇n22+2δ11−mL1Γ1∫Ω∇n12.
Multiplying the first equation in (9) by 21−α2u1−2α2, similar to (23), we have(24)ddt∫Ωu2−2α2+L2∫Ωum−2α2−1∇u2≤2L2Γ1∫Ωu1+2α1−2α2−m∇n12+2δ11−mL2Γ2∫Ω∇n22,where by Lemma 2, we know that there exists a constant c1>0, such that uL1Ω≤c1.(25)u1+2α1−2α2−m≤c11+2α1−2α2−m,ifm≤1+2α1−2α2,δ11+2α1−2α2−m,ifm>1+2α1−2α2.
Letc2=maxc11+2α1−2α2−m,δ11+2α1−2α2−m, we have(26)ddt∫Ωu2−2α2+L2∫Ωum−2α2−1∇u2≤2c2L2Γ1∫Ω∇n12+2δ11−mL2Γ2∫Ω∇n22.
Combining with (23) and (26), we obtain that(27)ddt∫Ωu2−2α1+ddt∫Ωu2−2α2+L1∫Ωum−2α2−1∇u2+L2∫Ωum−2α2−1∇u2≤2Γ1δ11−mL1+c2L2∫Ω∇n12+2Γ2δ11−mL2+δ11+2α2−2α1−mL1∫Ω∇n22.
By the Gagliardo–Nirenberg inequality with (16), there exist c3,c4>0, such that(28)∫Ωu2−2α1=um−2α2+1/2L22−2α1/m−2α2+1Ω22−2α1/m−2α2+1≤c3∇um−2α2+1/2L2Ω22−2α1/m−2α2+1θ1um−2α2+1/2L2/m−2α2+1Ω22−2α1/m−2α2+11−θ1+c3um−2α2+1/2L2/m−2α2+1Ω22−2α1/m−2α2+1≤c4∫Ωum−2α1−1∇u2+c4,with(29)θ1=m−2α2−1N/2−m−2α2−1N/41−α11−N/2+m−2α2−1N/2∈0,1,due to α1<1−1−mN/4. Similar to (28), there exists c5>0, such that(30)∫Ωu2−2α2≤c5∫Ωum−2α2−1∇u2+c5.
Choose appropriatec4,c5, such that c4≤L1 and c5≤L2. Let yt=∫Ωu2−2α1+∫Ωu2−2α2, gt=∫Ω∇n12+∫Ω∇n22,(31)c6=minL1−c4,L2−c5,c7=max2Γ1δ11−mL1+c2L2,2Γ2δ11−mL2+δ11+2α2−2α1−mL1.
By (26), we have(32)y′t+yt+c6∫Ωu−2α1∇u2+c6∫Ωu−2α2∇u2≤c7gt.
As from Lemma3 and Lemma 4, we readily obtain (28). (21) follows by integrating (32) in time. The proof is complete.Lemma 6.
LetN≥2 and suppose that min1−2/N,2−6/N+2<m≤2−2/N, (4) holds with σ∈0,1, Γ1,Γ2>0 and α1<1−1−mN/4, α2<1−1−mN/4. Then, there exist p>1 and q>1, such that(33)∫0tuLpΩq≤Ct+1,for allt∈0,Tmax.Proof.
By the Gagliardo–Nirenberg inequality, there existp>1 and q>1, such that(34)uLpΩq=um−2α1+1/2L2p/m−2α1+1Ω2q/m−2α1+1≤c8∇um−2α1+1/2L2Ω2q/m−2α1+1θ2um−2α1+1/2L41−α1/m−2α1+1Ω2q/m−2α1+11−θ2+c8um−2α1+1/2L41−α1/m−2α1+1Ω2q/m−2α1+1,(35)uLpΩq=um−2α2+1/2L2p/m−2α2+1Ω2q/m−2α2+1≤c9∇um−2α2+1/2L2Ω2q/m−2α2+1θ3um−2α2+1/2L41−α2/m−2α2+1Ω2q/m−2α2+11−θ3+c9um−2α2+1/2L41−α2/m−2α2+1Ω2q/m−2α2+1,for all t∈0,Tmax with some c5,c6>0 and(36)θ2=m−2α1+1N/41−α1−m−2α1+1N/2p1−N/2+m−2α1+1N/41−α1∈0,1,as well as(37)θ3=m−2α2+1N/41−α2−m−2α2+1N/2p1−N/2+m−2α2+1N/41−α2∈0,1.
In accordance with Lemma5, there exist some c10,c11>0, such that(38)∫Ωu2−2α1≤c10,∫Ωu2−2α2≤c10,for allt∈0,Tmax,∫0t∫Ωum−2α1−1∇u2≤c111+t,∫0t∫Ωum−2α2−1∇u2≤c111+t,for allt∈0,Tmax.
Substituting above four inequalities into (34) and (35), we have(39)∫0tuLpΩq≤c8∫0t∇um−2α1+1/2L2Ω2q/m−2α1+1θ2um−2α1+1/2L41−α1/m−2α1+1Ω2q/m−2α1+11−θ2+c8∫0tum−2α1+1/2L41−α1/m−2α1+1Ω2q/m−2α1+11−θ2≤c8c10q1−θ2/2m−2α1+1∫0t∇um−2α1+1/2L2Ω2q/m−2α1+1θ2+c8c10q/2m−2α1+1⋅t≤c12t+c12,∫0tuLpΩq≤c9∫0t∇um−2α1+1/2L2Ω2q/m−2α2+1θ3um−2α2+1/2L41−α2/m−2α2+1Ω2q/m−2α2+11−θ3+c9∫0tum−2α2+1/2L41−α2/m−2α2+1Ω2q/m−2α2+11−θ3≤c9c11q1−θ3/2m−2α2+1∫0t∇um−2α2+1/2L2Ω2q/m−2α2+1θ2+c9c11q/2m−2α2+1⋅t≤c13t+c13,for all t∈0,Tmax with some c12,c13>0, where we used the fact that qθ2/m−2α1+1<2 and qθ3/m−2α2+1<2, and the proof similar to Lemma 2.6 in [13]. The proof is complete.
According to the proof of [14], we have from Lemma 6 that(40)n1x,t≤c141+t,n2x,t≤c151+t,for allx,t∈Ω×0,Tmax,with some c14,c15>0. We claim that for any N≥2 and for all T∈0,Tmax, there exist some Υ1T,Υ2T>0, such that n1x,t≤Υ1T and n2x,t≤Υ2T in 0,T×Ω. This together with the definition of n1,n2 entails v≥v0L∞Ωe−Υ1T and w≥w0L∞Ωe−Υ2T in 0,T×Ω. With this in hand, we next consider system (3).Lemma 7.
LetN≥2, T∈0,Tmax and suppose that min1−2/N,2−6/N+2<m≤2−2/N, (4) holds with σ∈0,1, Γ1,Γ2>0, and(41)α1<1−1−mN4,α2<1−1−mN4.Then, for allp>1, there exists C≔CT>0, such that(42)ddt∫Ωup+2pp−1p+m−12∫Ω∇up+m−1/22≤C∫Ωup+2α1−m−1∇v2+C∫Ωup+2α2−m−1∇w2.Proof.
Multiplying the first equation in (3) by pup−1, integrating by parts and Young’s inequality, we have(43)ddt∫Ωup=p∫Ωup−1∇⋅u+1m−1∇u+∇⋅ϕ1uv∇v−∇⋅ϕ2uw∇w≤−pp−1∫Ωup+m−3∇u2+pp−1∫Ωup−2ϕ1uv∇u⋅∇v−pp−1∫Ωup−2ϕ2uw∇u⋅∇w≤−pp−1∫Ωup+m−3∇u2+pp−14∫Ωup−2∇u2+pp−1∫Ωup−m−1ϕ12uv2∇v2+pp−14∫Ωup+m−3∇u2+pp−1∫Ωup−m−1ϕ22uw2∇w2≤−pp−12∫Ωup+m−3∇u2+pp−1Γ12e2Υ1Tv0L∞Ω2∫Ωup+2α1−2∇v2+pp−1Γ22e2Υ2Tw0L∞Ω2∫Ωup+2α2−2∇w2,for all t∈0,T, here we used the fact that v≥v0L∞Ωe−Υ1T and w≥w0L∞Ωe−Υ2T in 0,T×Ω. We obtain (42) if we let(44)C≔maxpp−1Γ12e2Υ1Tv0L∞Ω2,pp−1Γ22e2Υ2Tw0L∞Ω2.
The proof is complete.
Applying the same arguments as in the proof of Lemma 2.9 and Lemma 2.10 in [13], we have the following Lemma.Lemma 8.
Assume thatΩ⊂ℝNN≥1 be a bounded domain with smooth boundary. Then, for all q≥2, there exists C>0, such that(45)1qddt∫Ω∇v2q+∫Ω∇v2q−2D2v2≤C∫Ωu2∇v2q−2+C,for allt∈0,Tmax,1qddt∫Ω∇w2q+∫Ω∇w2q−2D2w2≤C∫Ωu2∇w2q−2+C,for allt∈0,Tmax.Lemma 9.
LetN≥2. Assume min1−2/N,2−6/N+2<m≤2−2/N, (3) holds with σ∈0,1, Γ1,Γ2>0, and(46)α1<1−1−mN4,α2<1−1−mN4.Letp and q be the same as in Lemmas 7 and 8. Then, for all T∈0,Tmax, one can find some C>0 fulfilling(47)ux,tLpΩ≤C,∇vx,tL2qΩ≤C,as well as(48)∇wx,tL2qΩ≤C.Proof.
We first consider the dimensionalN≥2. Combining with Lemmas 7 and 8, there exists some c16≔c16T>0, such that(49)ddt∫Ωup+1q∫Ω∇v2q+1q∫Ω∇w2q+2pp−1p+m−12∫Ω∇up+m−1/22+∫Ω∇v2q−2D2v2+∫Ω∇w2q−2D2w2≤c16∫Ωup+2α1−m−1∇v2+c16∫Ωup+2α2−m−1∇w2+c16∫Ωu2∇v2q−2+c16∫Ωu2∇w2q−2,for all t∈0,T. By the Hölder inequality, we have(50)∫Ωup+2α1−m−1∇v2≤∫Ωup+2α1−m−1τ11/τ1∫Ω∇v2τ1′1/τ1′,(51)∫Ωup+2α2−m−1∇w2≤∫Ωup+2α2−m−1τ21/τ2∫Ω∇w2τ2′1/τ2′,for all t∈0,T, where τ1,τ2≔q+1/q and τ1′,τ2′≔q+1. Moreover, by the Gagliardo–Nirenberg inequality, there exist some constants c17,c18>0, such that(52)∫Ωup+2α1−m−1τ11/τ1=up+m−1/2L2p+2α1−m−1τ1/p+m−1Ω2p+2α1−m−1/p+m−1≤c17∇up+m−1/2L2Ω2p+2α1−m−1/pθ4up+m−1/2L22−2α1/p+m−1Ω2p+2α1−m−1/p+m−11−θ4+c17up+m−1/2L22−2α1/p+m−1Ω2p+2α1−m−1/p+m−1,(53)∫Ωup+2α2−m−1τ11/τ2=up+m−1/2L2p+2α2−m−1τ2/p+m−1Ω2p+2α2−m−1/p+m−1≤c18∇up+m−1/2L2Ω2p+2α2−m−1/pθ5up+m−1/2L22−2α2/p+m−1Ω2p+2α2−m−1/p+m−11−θ5+c18up+m−1/2L22−2α2/p+m−1Ω2p+2α2−m−1/p+m−1,with(54)θ4=p+m−1N/22−2α1−p+m−1N/2p+2α1−m−1τ11−N/2+p+m−1N/22−2α1∈0,1,θ5=p+m−1N/22−2α2−p+m−1N/2p+2α2−m−1τ21−N/2+p+m−1N/22−2α2∈0,1,the reason analogous to (2.42) and (2.43) in ([13], Lemma 2.11). Therefore, combining with (20) and (52), we get(55)∫Ωup+2α1−m−1τ11/τ1≤c19∫Ω∇up+m−1/22Np+2α1−m−1/41−α1−N/2τ1/1−N/2+p+m−1N/41−α1+c19,(56)∫Ωup+2α2−m−1τ21/τ2≤c20∫Ω∇up+m−1/22Np+2α2−m−1/41−α2−N/2τ2/1−N/2+p+m−1N/41−α2+c20,for all t∈0,T, where c19≔c17C3p+2α1−m−1/21−α1 and c20≔c18C4p+2α2−m−1/21−α2. Along with (55), (56) follows from ([13], Lemma 2.9), (50), and (51), and we deduce(57)∫Ωup+2α1−m−1∇v2≤c21∫Ω∇up+m−1/22Np+2α1−m−1/41−α1−N/2τ1/1−N/2+p+m−1N/41−α1∫Ω∇v2q−2D2v21/q+1+c21∫Ω∇v2q−2D2v21/q+1,(58)∫Ωup+2α2−m−1∇w2≤c22∫Ω∇up+m−1/22Np+2α2−m−1/41−α2−N/2τ2/1−N/2+p+m−1N/41−α2∫Ω∇w2q−2D2w21/q+1+c22∫Ω∇w2q−2D2w21/q+1,for all t∈0,T, with some constants c21,c22>0 determined by T. Similar to (2.45) in ([13], Lemma 2.11), we know that(59)Np+2α1−m−1/41−α1−qN/2q+11−N/2+p+m−1N/41−α1+1q+1<1,Np+2α2−m−1/41−α2−qN/2q+11−N/2+p+m−1N/41−α2+1q+1<1.
Therefore, using the Young inequality, we derive(60)c21∫Ω∇up+m−1/22Np+2α1−m−1/41−α1−N/2τ1/1−N/2+p+m−1N/41−α1∫Ω∇v2q−2D2v21/q+1≤p−14c16p∫Ω∇up+m−1/22+18c16∫Ω∇v2q−2D2v2+c23,(61)c22∫Ω∇up+m−1/22Np+2α2−m−1/41−α2−N/2τ2/1−N/2+p+m−1N/41−α2∫Ω∇w2q−2D2w21q+1≤p−14c16p∫Ω∇up+m−1/22+18c16∫Ω∇w2q−2D2w2+c24,for all t∈0,T, with some constants c23,c24>0 determined by T. Once more, employing Young’s inequality, there exist some constants c25,c26>0 determined by T, such that(62)c21∫Ω∇v2q−2D2v21/q+1≤18c16∫Ω∇v2q−2D2v2+c25,as well as(63)c22∫Ω∇w2q−2D2w21/q+1≤18c16∫Ω∇w2q−2D2w2+c26,for all t∈0,T. Combining with (57)–(63), we obtain(64)∫Ωup+2α1−2∇v2≤p−14c16p∫Ω∇up+m−1/22+14c16∫Ω∇v2q−2D2v2+c23+c25,(65)∫Ωup+2α2−2∇w2≤p−14c16p∫Ω∇up+m−1/22+14c16∫Ω∇w2q−2D2w2+c24+c26,for all t∈0,T. Analogous to (64) and (65), we have(66)∫Ωu2∇v2q−2≤p−14c16p∫Ω∇up+m−1/22+14c16∫Ω∇v2q−2D2v2+c27,(67)∫Ωu2∇w2q−2≤p−14c16p∫Ω∇up+m−1/22+14c16∫Ω∇w2q−2D2w2+c28,for all t∈0,T, with some constants c27,c28>0 determined by T. By the Gagliardo–Nirenberg inequality, there exists a constant c29>0, such that(68)∫Ωup≤p−12c16p∫Ω∇up+m−1/22+c29.
Together with (49) and (64)–(68), we have(69)ddt∫Ωup+1q∫Ω∇v2q+1q∫Ω∇w2q≤c30,for all t∈0,T with some constants c30>0. The proof is complete.
Finally, we prove the main theorem.Proof.
of Theorem1. From Lemma 9 and the Moser–Alikakos iterative technique in Lemma A.1 of [30], we have for all T∈0,Tmax,(70)u.,tL∞Ω≤C,for all t∈0,T with some constant C>0 determined by T. Thus, the global existence is a consequence of (70) and the extensibility criterion provided by Lemma 1.



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*Source: 9323338-2021-08-05.xml*