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Research article Special Issues

Well defined extinction time of solutions for a class of weak-viscoelastic parabolic equation with positive initial energy

  • Received: 08 December 2020 Accepted: 01 February 2021 Published: 08 February 2021
  • MSC : 35B44, 35K45

  • In the present paper, we consider an important problem from the point of view of application in sciences and mechanic, namely, a class of p(x)-Laplacian type parabolic equation with weak-viscoelasticity. Here, we are concerned with global in time non-existence under suitable conditions on the exponents q(x) and p(x) with positive initial energy. We show that the weak-memory term is unable to stabilize problem (1.2) under conditions (1.5) and (1.7). Our main interest in this paper arose in the first place in consequence of a query to blow-up phenomenon.

    Citation: Ahmed Himadan. Well defined extinction time of solutions for a class of weak-viscoelastic parabolic equation with positive initial energy[J]. AIMS Mathematics, 2021, 6(5): 4331-4344. doi: 10.3934/math.2021257

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  • In the present paper, we consider an important problem from the point of view of application in sciences and mechanic, namely, a class of p(x)-Laplacian type parabolic equation with weak-viscoelasticity. Here, we are concerned with global in time non-existence under suitable conditions on the exponents q(x) and p(x) with positive initial energy. We show that the weak-memory term is unable to stabilize problem (1.2) under conditions (1.5) and (1.7). Our main interest in this paper arose in the first place in consequence of a query to blow-up phenomenon.



    Emergent dynamics in interacting multi-agent systems are commonly observed in nature. Natural phenomena, including animal migration [1], bacterial movement [2], and synchronization of coupled cells [3] and fireflies [4], exhibit collective behaviors. For instance, in the field of ecology, collective behaviors can facilitate population reproduction, predator evasion, and the reduction of competition among individuals. Ultimately, these behaviors can enhance the population's safety coefficient. Therefore, studying collective behaviors is of significant importance and meaning.

    To investigate aggregation phenomena, biophysicist T. Vicsek et al. conducted numerical experiments to elucidate the mechanisms underlying collective motion [5]. A. Jadbabaie subsequently verified these experiments through analytical methods [6]. Following the pioneering work of T. Vicsek et al. numerous mathematical models have been developed to study emergent behavior. Professors Cucker and Smale proposed the Cucker-Smale model, which characterizes aggregation phenomena [7]. The Cucker-Smale model describes a flocking dynamic system with position and velocity following Newtonian dynamics. For the ith particle in the Cucker-Smale model, let xiRd and viRd denote its position and velocity. Here, Rd represents d-dimensional Euclidean space. The Cucker-Smale model is governed by

    {dxidt=vi,t>0,i[N]:={1,2,,N},dvidt=κN1jiϕ(||xixj||)(vjvi),(xi(0),vi(0))=(x0i,v0i)Rd×Rd, (1.1)

    where N represents the number of particles, κ denotes the non-negative coupling strength, and |||| denotes the standard l2-norm. ϕ signifies the communication weight. The Cucker-Smale model offers unique advantages in mathematical analysis due to its high degree of symmetry. Additionally, the solution's large-time behavior is determined solely by the initial conditions and the interaction function ϕ. Since its proposal, the Cucker-Smale model has been the subject of extensive research, with scholars exploring diverse communication weights ϕ(r) tailored to specific application contexts. For instance, the authors of [8] adopted the communication weight ϕ(r) as ϕ(r)=1rβ, known as the singular kernel, to conduct a detailed analysis of its clustering behavior.

    However, the Cucker-Smale model cannot describe the aggregative behaviors influenced by external factors, including light and temperature. For instance, Bhaya et al. [9] and Jakob et al. [10] observed that cyanobacteria actively migrate toward light sources under certain conditions. Ha et al. [11] investigated the effect of temperature on aggregative behavior, resulting in the development of a thermodynamic Cucker-Smale (in short, TCS) model. Since then, the TCS model has also been a subject of extensive research. Two sufficient frameworks for the emergence of mono-cluster flocking on a digraph for the continuous and discrete models were presented in [12]. The emergent behaviors of a TCS ensemble confined in a harmonic potential field was studied in [13]. The coupling of a kinetic TCS equation and viscous fluid system was proposed and considered in [14,15]. Based on system (1.1), we set Ti to denote the temperature of the ith particle, and then the TCS model is governed by

    {dxidt=vi,t>0,i[N]:={1,2,,N},dvidt=κ1N1jiϕ(||xixj||)(vjTjviTi),ddt(Ti+12||vi||2)=κ2N1jiζ(||xixj||)(1Ti1Tj),(xi(0),vi(0),Ti(0))=(x0i,v0i,T0i)Rd×Rd×(0,+), (1.2)

    where N denotes the number of particles, while κ1 and κ2 represent strictly positive coupling strengths. Moreover, ϕ,ζ, which are mappings from (0,+)(0,+), serve as the communication weights. These functions are non-negative, locally Lipschitz continuous, and monotonically decreasing.

    Based on the conceptual framework for the unit-speed Cucker-Smale model presented in [16], Ahn modified the velocity coupling term to guarantee that each velocity possesses a constant positive modulus [17], as follows:

    {dxidt=vi,t>0,i[N]:={1,2,,N},dvidt=κ1N1jiϕ(||xixj||)(vjTjvi,vjviTj||vi||2),ddt(Ti+12||vi||2)=κ2N1jiζ(||xixj||)(1Ti1Tj),(xi(0),vi(0),Ti(0))=(x0i,v0i,T0i)Rd×Sd1×(0,+), (1.3)

    where N, κ1, κ2, and communication weights ϕ,ζ are defined as above. The term Sd1 denotes the unit (d1)-sphere. However, the author of [17] only addressed the scenarios where the communication weights ϕ and ζ are non-negative, bounded, locally Lipschitz continuous, and monotonically decreasing.

    Furthermore, the author of [18] employed suitable subdivided configurations {Zα}nα=1 and demonstrated that the velocity and temperature of all agents within each cluster group converge to identical values. In addition, based on the results of [17], the authors of [19] proved that asymptotic flocking occurs when the communication weights ϕ,ζ are singular kernels.

    This article considers the multi-cluster flocking dynamics of the thermodynamic Cucker-Smale model with a unit-speed constraint (TCSUS) under a singular kernel. The system (1.3) is reorganized into a multi-cluster framework, which is governed by,

    {dxαidt=vαi,t>0,i[Nα]:={1,2,,Nα},α[n]:={1,2,n},n3,dvαidt=κ1N1Nαj=1jiϕ(||xαixαj||)(vαjTαjvαi,vαjvαiTαj||vαi||2)+κ1N1βαNβj=1ϕ(||xαixβj||)(vβjTβjvαi,vβjvαiTβj||vαi||2),ddt(Tαi+12||vαi||2)=κ2N1Nαj=1jiζ(||xαixαj||)(1Tαi1Tαj)+κ2N1βαNβj=1ζ(||xαixβj||)(1Tαi1Tβj),(xαi(0),vαi(0),Tαi(0))=(x0αi,v0αi,T0αi)Rd×Sd1×(0,+), (1.4)

    where t represents time, n represents the number of clusters, N represents the number of particles, and Nα represents the number of particles in the αth cluster. Additionally, xαi, vαi, and Tαi denote the position, velocity, and temperature of the ith particle in the αth cluster, respectively. Furthermore, κ1 and κ2 represent strictly positive coupling strengths, and Sd1 is the unit (d1)-sphere. Specifically, we assume that when the communication weights ϕ,ζ are singular kernels, they will take the following explicit assumption: ϕ(r)=1rλ,ζ(r)=1rμ(λ,μ>0).

    In fact, system (1.4) corresponds exactly to system (1.3). The formulation of system (1.4) specifically highlights the influence of different clusters on particle dynamics.

    Previous studies in [17,18] thoroughly explored mono-cluster, bi-cluster, and multi-cluster flocking behaviors in system (1.4) under a standard kernel. More recently, studies cited in [19] have concentrated on mechanical flocking and thermal homogenization within the TCSUS model under a singular kernel. However, a comprehensive study about the multi-cluster flocking of TCSUS under a singular kernel remains largely unexplored. In this paper, we mainly focus on studying the multi-cluster flocking under a strong singular kernel and provide relevant conclusions and estimates.

    For simplicity, we apply the following notation:

    Notation 1.1. For the vector uRd, we denote by ||u|| and ui the Euclidean l2norm of u and its ith component, respectively. The standard inner product of two vectors u,vRd is denoted by u,v. The distance between two sets A and B is denoted by d(A,B):=inf{d(x,y):xA,yB}. For simplicity, we define [N]:={1,2,,N}. Given fixed real sequences {ai}ni=1 and {bi}ni=1, we define the family of sets {Iα}nα=1 such that Iα:=[aα,bα]. The solution of system (1.3) is denoted by A:=(a1,a2,,aN), where A{X,V,T}, and a{x,v,T}. After we reorganize the system (1.3) into a multi-cluster configuration (1.4), if we denote the solution of the αth cluster as Aα:=(aα1,aα2,,aαNα), where A{X,V,T} and a{x,v,T}, it becomes evident that A=(a1,a2,,aN)=(A1,A2,,An). In addition, we define a0:=a(0). The generic constant C may differ from line to line. We define the centers for position, velocity, and temperature of the αth cluster as xcenα:=1Nαi[Nα]xαi, vcenα:=1Nαi[Nα]vαi, and Tcenα:=1Nαi[Nα]Tαi, respectively. Furthermore, we define the minimum temperature of the αth cluster as Tαm(t):=mini[Nα]Tαi(t) and the maximum temperature of the αth cluster as TαM(t):=maxi[Nα]Tαi(t). We define the minimum temperature of the whole system as Tm(t):=minα[n]Tαm(t) and the maximum temperature of the whole system as TM(t):=maxα[n]TαM(t). The minimum inner product throughout the system is denoted as A(v):=minα,β[n]i[Nα],j[Nβ]vαi,vβj.

    First, we define the L diameters for position, velocity, and temperature of each cluster group Zα:={(xαi,vαi,Tαi)}Nαi=1 as follows:

    ● (position-velocity-temperature diameters for the αth cluster group)

    DXα:=maxi,j[Nα]||xαixαj||, DVα:=maxi,j[Nα]||vαivαj||, DTα:=maxi,j[Nα]|TαiTαj|,

    ● (position-velocity-temperature diameters for the whole system)

    DX:=nα=1DXα, DV:=nα=1DVα, DT:=nα=1DTα.

    Then, we define the following three configuration vectors for each cluster group:

    Xα:=(xα1,xα2,,xαNα), Vα:=(vα1,vα2,,vαNα), Tα:=(Tα1,Tα2,,TαNα), where 1αn.

    Next, we introduce the definition of the multi-cluster flocking behavior of system (1.4):

    Definition 1.1. Let Z={(xi,vi,Ti)}Ni=1 be a solution to system (1.4). Then, the configuration Z exhibits multi-cluster flocking if there exist n cluster groups Zα={(xαi,vαi,Tαi)}Nαi=1 such that the following assertions hold for 2nN and 1αn:

    |Zα|=Nα1,nα=1|Zα|=nα=1Nα=N,nα=1Zα=Z,

    α[n],sup0t<max1k,lNα||xαkxαl||<, limtmax1k,lNα||vαkvαl||=0, limtmax1k,lNα|TαkTαl|=0,

    inf0t<mink,l||xαkxβl||=,1kNα, 1lNβ,1αβn.

    To describe adequate frameworks for multi-cluster flocking estimation, we display the admissible data and conditions (H) as follows:

    (H):={(X(0),V(0),T(0))R2dN×(0,+)N|(H0),(H1),(H2), and (H3) hold.}

    (i)(H0) (Notation): For brevity, we have the following notation:

    Tm:=Tm(0),TM:=TM(0),δ0:=inf0tmin1ijN||xi(t)xj(t)||,r0:=minα<β,i,j(xkβj(0)xkαi(0)),R0:=maxα<β,i,j(xkβj(0)xkαi(0)) for some fixed1kd,Λ0(DX):=κ1min(N1,,Nα)A(v0)ϕ(DX)(N1)TM,ˉΛ0(DX):=κ2(min(N1,,Nα)2)ζ(DX)N(TM)2,Λ:=DV(0)Λ0+4(n1)Nκ1(N1)TmΛ20ϕ(r02)+4(n1)Nκ1(N1)min1αn1d(Iα,Iα+1)TmΛ0r02ϕ(s)ds,Λα:=κ1(Nα1)ϕ(δ0)(N1)TmΛ+κ1(NNα)(N1)Tm(min1αn1d(Iα,Iα+1))r02ϕ(s)ds.

    (ii)(H1) (Well prepared conditions): There exists a strictly positive number DX>0 such that

    DXDX(0)+max(Λ,DV(0)TMκ1A(v0)ϕ(DX)),andλ,μ>1(strong kernel).

    (iii)(H2) (Separated initial data): For fixed 1kd in (H0), there exist real sequences {ai}ni=1 and {bi}ni=1 such that the initial data and system parameters are appropriately split as follows:

    r0>0,a1<b1<a2<b2<an<bn,Iα=[aα,bα][1,1],IαIβ=(βα),[vkαi(0)Λα,vkαi(0)+Λα]Iα=[aα,bα][1,1],α,β=1,,n,i=1,,Nα.

    (iv)(H3) (Small fluctuations): The local velocity perturbation for all cluster groups is sufficiently small as follows:

    DV(0)κ1A(v0)TMDXDX(0)ϕ(s)ds. (1.5)

    Finally, we present the main theorems of this article.

    Theorem 1.1. Assume that Zα={(xαi,vαi,Tαi)}Nαi=1 is a solution to system (1.4). Suppose that (H) holds. It follows that

    minαβ,i,jxαi(t)xβj(t)(min1αn1d(Iα,Iα+1))t+r02,t(0,+). (1.6)

    Theorem 1.2. Assume that Zα={(xαi,vαi,Tαi)}Nαi=1 is a solution to system (1.4). Suppose that (H) holds. Then, it follows that for t(0,+):

    1. (Velocity alignment for each cluster group)

    DV(t)DV(0)exp(Λ0t)+2κ1(n1)NTmΛ0(N1)exp(Λ02t)ϕ(r02)+2κ1(n1)NTmΛ0(N1)ϕ((min1αn1d(Iα,Iα+1))t+r02). (1.7)

    2. (Temperature equilibrium for each cluster group)

    DT(t)DT(0)exp(ˉΛ0t)+2κ2(n1)NN1(1Tm1TM)exp(ˉΛ02t)ζ(r02)+2κ2(n1)NN1(1Tm1TM)ζ((min1αn1d(Iα,Iα+1))t+r02). (1.8)

    Remark 1.1. It is evident that Theorems 1.1 and 1.2 demonstrate that system (1.4) exhibits the phenomenon of multi-cluster flocking.

    Theorem 1.3. Assume that Zα:={(xαi,vαi,Tαi)}Nαi=1 is a solution to system (1.4). Then, under the sufficient frameworks (H), there exist some strictly positive convergence constants V1,V2 and T1,T2 that satisfy the subsequent criteria for t(0,+).

    1. (Velocity convergence value for each cluster group) If we define vα:=limtvcenα, then the existence of vα is guaranteed, and the two values V1 and V2 satisfy the following inequality for all α[n] and iα[Nα].

    V1tλ1nα=1vαiα(t)vαV2tλ1,t. (1.9)

    2. (Temperature convergence value for each cluster group) If we define Tα:=limtTcenα, then the existence of Tα is guaranteed, and the two values T1 and T2 satisfy the following inequality for all α[n] and iα[Nα].

    T1tμ1nα=1Tαiα(t)TαT2tμ1,t. (1.10)

    Remark 1.2. In [18], Ahn proved that when the communication weights ϕ and ζ are standard kernels, which are non-negative, bounded, locally Lipschitz continuous, monotonically decreasing, and integrable, the multi-cluster flocking of the system (1.4) is exhibited under some sufficient framework. However, when the communication weights ϕ and ζ are singular kernels, they will blow up as r approaches 0, meaning they will not be bounded and integrable over the interval (0,+). In [19], Ahn et al. proved the mono-cluster flocking of system (1.4) under the singular kernel. However, the multi-cluster flocking of system (1.4) under a singular kernel remains unexplored. In this article, by employing a new sufficient framework, we address the non-regularity of the singular kernel at r=0 and obtain the multi-cluster flocking of the system (1.4) under the singular kernels based on the work of [19].

    This article is organized as follows. In Section 2, several basic results of the TCSUS model are briefly reviewed initially. Subsequently, some previous results related to the TCSUS model under a singular kernel in [19] are reviewed to prepare for the proof of multi-cluster flocking. In Section 3, several fundamental frameworks for achieving multi-cluster flocking in system (1.4) with a strongly singular kernel are provided, and appropriate dissipative differential inequalities for position, velocity, and temperature are derived. Then, by using self-consistent parameters for these inequalities, we derive sufficient conditions to ensure multi-cluster flocking of system (1.4) based on initial data and system parameters.

    This section reviews several basic results for the TCSUS to guarantee its multi-cluster flocking. These estimates will be crucial throughout this paper.

    Proposition 2.1. For τ(0,+), let (X,V,T) be a solution to system (1.4) in the time-interval (0,τ). Then, the following assertions hold:

    (1) (Conservation laws): The modulus of velocities and the total sum of temperatures are conserved.

    ddtnα=1Nααi=1Tαi(t)=0,||vαi(t)||=1,t(0,τ).

    (2) (Monotonicity of temperature): External temperatures Tm(t) and TM(t) are monotonically increasing and decreasing, respectively, and one has positivity and uniform boundedness.

    0<TmTαi(t)TM,α[n],i[Nα],t[0,τ).

    (3) (Monotonicity of A(v)): If the initial data satisfies that A(v0):=minv0αi,v0βj>0, then A(v) is monotonically increasing: If 0st<τ, then 0<A(v0)A(v)(s)A(v)(t)1.

    Proof. (1) To demonstrate speed conservation, we take the inner product of the second equation in system (1.4) with 2vαi to obtain

    2vαi,dvαidt=2κ1N1Nαj=1jiϕ(||xαixαj||)(vαi,vαjTαjvαi,vαjvαi,vαiTαj||vαi||2)+2κ1N1βαNβj=1ϕ(||xαixβj||)(vβj,vαiTβjvαi,vβjvαi,vαiTβj||vαi||2)=0.

    This implies that d||vαi||2dt=0, i.e., ||vαi(t)||=||vαi(0)||=1. Then, we employ ζ(||xixj||)=ζ(||xjxi||) and exchange αiαj and αiβj respectively to get

    ddtnα=1Nααi=1Tαi(t)=κ2N1nα=1Nααi=1Nαj=1jiζ(||xαixαj||)(1Tαi1Tαj)+κ2N1nα=1Nααi=1βαNβj=1ζ(||xαixβj||)(1Tαi1Tβj),=κ2N1nα=1Nααj=1Nαi=1ijζ(||xαjxαi||)(1Tαj1Tαi)+κ2N1nβ=1Nββj=1αβNαi=1ζ(||xαjxβi||)(1Tβj1Tαi)=0. (2.1)

    (2) We induce that αtit depends on time t[0,τ) satisfying Tm(t)=Tαtit(t), and then we have

    dTαtitdt=κ2N1Nαtj=1jiζ(||xαtitxαtj||)>0(1Tαtit1Tαtj)0+κ2N1βαtNβj=1ζ(||xαtitxβj||)>0(1Tαtit1Tβj)0.

    Therefore, dTαtitdt0, i.e., Tm(t) is increasing. By the same token, we get that TM(t) is decreasing. This implies that 0<TmTαi(t)TM,α[n],i[Nα],t[0,τ).

    (3) We split the proof into two steps:

    ● First, we show that the functional A(v) is strictly positive in the time interval (0,τ),A(v)>0.

    ● Second, we verify that in the time interval (0,τ), ddtA(v)0.

    Step A: For fixed t(0,τ), we choose two indices αtit, βtjt (α,β[n],i[Nα],j[Nβ]) such that vαtit,vβtjt=A(v)(t). Then, we define a temporal set S1 and its supremum τ1 as

    S1:={t(0,τ)|A(v)(t)>0},τ1:=supS1.

    Since A(v0)>0, and A(v) is continuous, the set S1 is an open set, and 0<τ1τ. Next, we claim τ1=τ.

    Suppose the contrary holds, i.e., τ1<τ, which implies A(v)(τ10)=0. We differentiate A(v) with respect to time t(0,τ1) to find

    ddtA(v)=˙vαtit,vβtjt+vαtit,˙vβtjt=κ1N1Nαtj=1jitϕ(||xαtitxαj||)(vαj,vβtjtvαtit,vαjvαtit,vβtjtTαtj)+κ1N1βαtNβj=1ϕ(||xαtitxβj||)(vβtjt,vβjvαtit,vβjvαtit,vβtjtTβj)+κ1N1Nβtj=1jjtϕ(||xβtjtxβtj||)(vαtitvβtjvβtjt,vβtjvαtitvβtjtTβtj)+κ1N1ββtNβj=1ϕ(||xβtjtxβj||)(vαtitvβjvβtjt,vβjvαtitvβtjtTβj). (2.2)

    For any vαi,vβj which are the components of V, the unit-speed constraint yields

    |vαi,vβj|||vαi||||vβj||1.

    Thus, the positivity and minimality of A(v) lead to

    vγk,vβtjtA(v):=vαtit,vβtjtvαtit,vβtjtvαtit,vγk

    and

    vγk,vαtitA(v):=vαtit,vβtjtvαtit,vβtjtvβtjt,vγk,γ[n].

    Since each temperature is bounded below by a positive constant, each summand of Eq (2.2) is non-negative, and one has ddtA(v)0,t(0,τ1). Therefore, A(v)(τ10)A(v)(0)>0, which is contradictory to A(v)(τ10)=0. Finally, we have τ1=τ and A(v)>0,t(0,τ).

    Step B: It follows from Eq (2.2) that ddtA(v)0,t(0,τ).

    Remark 2.1. Based on Proposition 2.1, it can be immediately inferred that the system (1.4) can be simplified into the following system:

    {dxαidt=vαi,t>0,i[Nα]:={1,2,,Nα},α[n]:={1,2,n},n3,dvαidt=κ1N1Nαj=1jiϕ(||xαixαj||)(vαjvαi,vαjvαiTαj)+κ1N1βαNβj=1ϕ(||xαixβj||)(vβjvαi,vβjvαiTβj)dTαidt=κ2N1Nαj=1jiζ(||xαixαj||)(1Tαi1Tαj)+κ2N1βαNβj=1ζ(||xαixβj||)(1Tαi1Tβj),(xαi(0),vαi(0),Tαi(0))=(x0αi,v0αi,T0αi)Rd×Sd1×(0,+).

    In this subsection, we will give some previous results about the mono-cluster flocking of system (1.3) under a singular kernel. These results are necessary for later sections.

    Definition 2.1. We suppose that t0(0,+) is the first collision time of the system (1.3) ensemble, and the lth particle is one of the such colliding particles at time t0. Then, we denote by [l] the collection of all particles colliding with the lth particle at time t0,

    [l]:={i[N]|limtt0||xi(t)xl(t)||=0}.

    For t[0,t0) and i[l], we define the constant δ such that δ is a strictly positive real number satisfying ||xl(t)xi(t)||δ>0. Then, we define the following L diameters as follows:

    DX,[l]:=maxi,j[l]||xixj||,DV,[l]:=maxi,j[l]||vivj||,A[l](v):=mini,j[l]vi,vj.

    Proposition 2.2. [19] Let (X,V,T) be a solution to system (1.3). If A(v0)>0, then sub-ensemble diameters satisfy the following system of dissipative differential inequalities: For a.e. t(0,t0),

    {|ddtDX,[l]|DV,[l],t>0,ddtDV,[l]κ1|[l]|A[l](v0)(N1)TMϕ(DX,[l])DV,[l]+4κ1(N|[l]|)ϕ(δ)(N1)Tm,t>0,dDTdtκ2(N2)ζ(DX)(N1)(TM)2DT. (2.3)

    Proposition 2.3 (Collision avoidance). [19] We suppose that communication weight and initial data satisfy λ>1,A(v0)>0,min1ijN||xi(0)xj(0)||>0, and let (X,V,T) be a solution to the system (1.3). Then, collision avoidance occurs, i.e., xi(t)xj(t) (ij,i,j[N],t0).

    Proposition 2.4. [19] We suppose that communication weight and initial data satisfy the following conditions:

    (1) The parameter λ and initial configuration satisfy λ>1,A(v0)>0,min1ijN||xi(0)xj(0)||>0.

    (2) If there exists a positive constant DX such that DX(0)+DV(0)TMκ1A(v0)ϕ(DX)<DX and (X,V,T) is a solution to system (1.3), then thermodynamic flocking emerges asymptotically:

    (ⅰ) sup0t<+DX(t)DX, DV(t)DV(0)exp(κ1A(v0)ϕ(DX)TM).

    (ⅱ) DT(t)DT(0)exp(κ2ζ(DX)(N2)|TM|2(N1)).

    (ⅲ) (A uniform positive lower bound for relative distances) We suppose that DV(0)κ1A(v0)TMDXDX(0)ϕ(s)ds holds, and (X,V,T) is a global solution of the system (1.3). Then, there exists a strictly positive lower bound for the relative spatial distances, i.e., δ0:=inf0tmin1ijN||xi(t)xj(t)||>0.

    In this section, we derive suitable dissipative differential inequalities initially with respect to position, velocity, and temperature. By employing a bootstrapping technique with these inequalities, we apply appropriate sufficient conditions based on the initial conditions and system parameters to ensure multi-cluster flocking within system (1.4).

    In the following, we derive several dissipative differential inequalities with respect to position-velocity-temperature to establish adequate frameworks based on system parameters and initial conditions.

    Lemma 3.1 (Dissipative structure). Assume that Zα={(xαi,vαi,Tαi)}Nαi=1 is a solution to the system (1.4). We define ϕM(t):=maxαβ,i,jϕ(xβjxαi) and ζM(t):=maxαβ,i,jζ(xβjxαi).

    Then, we have the following three differential inequalities for a.e. t(0,+):

    1.|dDXdt|DV,

    2.dDTdtκ2(min(N1,,Nα)2)ζ(DX)(N1)(TM)2DT+2κ2(n1)NN1ζM(1Tm1TM),

    3.dDVdtκ1min(N1,,Nα)A(v0)ϕ(DX)(N1)TMDV+2κ1(n1)NϕM(N1)Tm.

    Proof. The first assertion follows directly from the Cauchy-Schwarz inequality. To prove the second assertion, we select two indices, Mt and mt depending on t, such that

    DTα(t)=TαMt(t)Tαmt(t),1mt,MtNα.

    Then, for a.e. t(0,+), one can show that by using the definitions of Mt and mt,

    dDTαdt=κ2N1Nαj=1jiζ(xαMtxαj)(1TαMt1Tαj)κ2N1Nαj=1jiζ(xαmtxαj)(1Tαmt1Tαj)+κ2N1βαNβj=1ζ(xαMtxβj)(1TαMt1Tβj)κ2N1βαNβj=1ζ(xαmtxβj)(1Tαmt1Tβj)=:I1+I2+I3+I4.

    (ⅰ) (Estimate of I1+I2) In the same method as the proof of Proposition 2.2, the following inequality holds for a.e. t(0,+):

    I1+I2κ2(Nα2)ζ(DXα)(N1)(TM)2DTα.

    (ⅱ) (Estimate of I3+I4) From Proposition 2.1 and the definitions of ζM and ζm, we derive the following inequality for a.e. t(0,+):

    I3+I4κ2N1|βαNβj=1ζ(xαMtxβj)(1TαMt1Tβj)|+κ2N1|βαNβj=1ζ(xαmtxβj)(1Tαmt1Tβj)|2κ2(NNα)ζMN1(1Tm1TM).

    Thus, combining I1+I2 and I3+I4 yields that, for a.e. t(0,+),

    dDTαdtκ2(Nα2)ζ(DXα)(N1)(TM)2DTα+2κ2(NNα)ζMN1(1Tm1TM).

    Therefore, summing for α from 1 to n to the above inequality, we obtain that for a.e. t(0,+),

    dDTdtκ2(min(N1,,Nα)2)ζ(DX)(N1)(TM)2DT+2κ2(n1)NN1ζM(1Tm1TM).

    To verify the third assertion, we select two indices Mt and mt depending on t which satisfy

    DVα(t)=vαMt(t)vαmt(t),1mt,MtNα.

    Hence, we attain that for a.e. t(0,+),

    12dD2Vαdt=vαMtvαmt,˙vαMt˙vαmt=vαMtvαmt,κ1N1Nαj=1ϕαMtj((vαjvαMt,vαjvαMt)Tαj)κ1N1Nαj=1ϕαmtj((vαjvαmt,vαjvαmt)Tαj)+vαMtvαmt,κ1N1βαNαj=1ϕ(||xαMtxβj||)((vβjvαMt,vβjvαMt)Tβj)κ1N1βαNαj=1ϕ(||xαmtxβj||)((vβjvαmt,vβjvαmt)Tβj)=:J1+J2.

    (iii) (Estimate of J1) Replacing [l] with [Nα] and in the same method as the proof of Proposition 2.2, for a.e.t(0,+), we have

    J1κ1NαA(v0)(N1)TMϕ(DXα)D2Vα.

    (iv) (Estimate of J2) We employ the identities

    ||vβjvαMt,vβjvαMt||1,||vβjvαmt,vβjvαmt||1

    with the Cauchy-Schwarz inequality and Proposition 2.3 to estimate that for a.e. t(0,+),

    J2κ1DVαN1||βαNβj=1ϕ(||xαMtxβj||)(vβjvαMt,vβjvαMtTβj)||+κ1DVαN1||βαNβj=1ϕ(||xαmtxβj||)(vβjvαmt,vβjvαmtTβj)||2κ1(NNα)ϕMDVα(N1)Tm.

    Then, we combine J1 and J2 to derive that for a.e. t(0,+),

    dDVαdtκ1NαA(v0)(N1)TMϕ(DXα)DVα+2κ1(NNα)ϕM(N1)Tm.

    Summing the above inequality from α=1 to n, we obtain that

    dDVdtκ1min(N1,,Nα)A(v0)ϕ(DX)(N1)TMDV+2κ1(n1)NϕM(N1)Tm,

    since the monotonicity of ϕ implies that min(ϕ(DX1),,ϕ(DXα))ϕ(DX). Finally, we demonstrate the third assertion.

    The proofs of Theorems 1.1–1.3 are presented in this subsection. First of all, we give a brief comment regarding (H).

    The assumption (H1) is the sufficient condition which guarantees group formation within each cluster. The assumption (H2) implies that the initial positions for each cluster group should be sufficiently separated from each other to achieve a multi-cluster flocking result. In fact, if vkαi(0) is covered by Iα:=[aα,bα], then we can take sufficiently small κ1 such that [vkαi(0)Λα,vkαi(0)+Λα]Iα because Λα is linearly proportional to κ1.

    The assumption (H3) ensures that a uniformly strictly positive lower bound exists for relative distances. Here, we can find the admissible data meeting the assumption (H3) requirements when κ1 is sufficiently small. Moreover, under sufficiently large r0, suitable temperature initial data and small coupling strength regime, we can check that the sufficient framework (H) is admissible data.

    Lemma 3.2. Assume that Zα={(xαi,vαi,Tαi)}Nαi=1 is a solution to the system (1.4) and suppose that (H) holds. We define the following set:

    S2:={s>0|minαβ,i,j||xαi(t)xβj(t)||(min1αn1d(Iα,Iα+1))t+r02,t[0,s)}.

    Then, S2 is nonempty, and it follows that DX(t)DX,t[0,T], where T:=supS2.

    Proof. We observe that S2 is nonempty due to the assumption (H2) and the continuity of ||xαi(t)xβj(t)||. Then, we just need to prove DX(t)DX,t[0,T]. First of all, we consider the following set:

    S3:={s>0t[0,s],DX(t)DX,sT}. (3.1)

    We set supS3=:T. Then, we have DX(T)=DX and suppose that T<T for the proof by contradiction. Then, for t[0,T], one has

    κ1min(N1,,Nα)ϕ(DX)(N1)TMκ1min(N1,,Nα)ϕ(DX)(N1)TM. (3.2)

    Thus, for a.e. t(0,T), the second assertion of Lemma 3.1 and the above estimates lead to the following inequalities:

    dDVdtκ1min(N1,,Nα)A(v0)ϕ(DX)(N1)TMDV+2κ1(n1)NϕM(N1)Tmκ1min(N1,,Nα)A(v0)ϕ(DX)(N1)TMDV+2κ1(n1)NϕM(N1)Tm=Λ0DV+2κ1(n1)NϕM(N1)Tm. (3.3)

    For t[0,T], we integrate inequality (3.3) from time 0 to t through multiplying both sides of the inequality by the integral factor exp(Λ0t).

    DV(t)DV(0)exp(Λ0t)+t02κ1(n1)NϕM(N1)Tmexp(Λ0(st))ds=DV(0)exp(Λ0t)+(t20+tt2)2κ1(n1)NϕM(N1)Tmexp(Λ0(st))dsDV(0)exp(Λ0t)+2κ1(n1)N(N1)Λ0Tmϕ(r02)[exp(Λ02t)exp(Λ0t)]+2κ1(n1)N(N1)Λ0Tmϕ((min1αn1d(Iα,Iα+1))t+r02)[1exp(Λ02t)]DV(0)exp(Λ0t)+2κ1(n1)N(N1)Λ0Tmϕ(r02)exp(Λ02t)+2κ1(n1)N(N1)Λ0Tmϕ((min1αn1d(Iα,Iα+1))t+r02), (3.4)

    where we used the definition of S3 and the fact that ϕMϕ((min1αn1d(Iα,Iα+1))t+r02).

    In the latter case, we estimate from inequality (3.4) that for t[0,T],

    DX(t)DX(0)+t0DV(s)dsDX(0)+t0[DV(0)exp(Λ0s)+2κ1N(n1)(N1)TmΛ0exp(Λ02s)ϕ(r02)+2κ1N(n1)(N1)TmΛ0ϕ((min1αn1d(Iα,Iα+1))s+r0)2)]ds<DX(0)+ΛDX. (3.5)

    Accordingly, DX(T)<DX, which is contradictory to DX(T)=DX. Finally, supS3=T=T. We have reached the desired lemma.

    Proof of Theorem 1.1. Following Lemma 3.2, we just need to prove that T=, which is equivalent to

    minαβ,i,jxαi(t)xβj(t)(min1αn1d(Iα,Iα+1))t+r02,t(0,+). (3.6)

    For the proof by contradiction, we suppose that T<. From the definition of S2, we select four indices that satisfy

    1α<βn,i{1,,Nα} and j{1,,Nβ} (3.7)

    such that

    xαi(T)xβj(T)=(min1αn1d(Iα,Iα+1))T+r02. (3.8)

    Then, we show that for the k{1,,d} chosen in(H0),

    xαi(T)xβj(T)xkβj(T)xkαi(T)=xkβj(0)xkαi(0)+T0(vkβj(t)vkαi(t))dtr0+T0(vkβj(t)vkαi(t))dt.

    From the third assertion of Proposition 2.3, we set a positive number δ0 such that δ0:=inf0tmin1ijN||xi(t)xj(t)||>0.

    Next, we integrate the second equation of system (1.4) and employ the following relation:

    1vαi,vαj=||vαivαj||22

    and

    vαjvαi,vαjvαi2=1vαi,vαj2=(1vαi,vαj)(1+vαi,vαj)D2Vα

    to attain that for t[0,T],

    |vkαi(t)vkαi(0)|vαi(t)vαi(0)t0˙vαidsκ1(Nα1)ϕ(δ0)(N1)Tmt0DVα(s)ds+κ1(NNα)(N1)Tmt0ϕM(s)dsκ1(Nα1)ϕ(δ0)(N1)TmDVα(s)ds+κ1(NNα)(N1)Tm0ϕ(min1αn1d(Iα,Iα+1)s+r02)dsκ1(Nα1)ϕ(δ0)(N1)TmDV(s)ds+κ1(NNα)(N1)Tm0ϕ(min1αn1d(Iα,Iα+1)s+r02)dsκ1(Nα1)ϕ(δ0)(N1)TmΛ+κ1(NNα)(N1)Tm0ϕ(min1αn1d(Iα,Iα+1)s+r02)ds=κ1(Nα1)ϕ(δ0)(N1)TmΛ+κ1(NNα)(N1)Tm(min1αn1d(Iα,Iα+1))r02ϕ(s)ds=:Λα,

    where we used ϕϕ(δ0), vβjvαi,vβjvαi1, and Λ was estimated in inequality (3.5). Therefore, it follows by (H2) that for α=1,,n,

    vkαi(0)+Λαvkαi(0)+|vkαi(t)vkαi(0)|vkαi(t)=vkαi(0)+vkαi(t)vkαi(0)vkαi(0)|vkαi(t)vkαi(0)|vkαi(0)Λαvkαi(t)Iα.

    By using the assumption (H2), we derive that

    xαi(T)xβj(T)r0+T0(vkβj(t)vkαi(t))dt>r02+min1αn1d(Iα,Iα+1)T,

    which gives a contradiction to T<. Consequently, we conclude that T=. Subsequently, we claim that T=, which is crucial to derive the multi-cluster flocking estimate of the system (1.4).

    Proof of Theorem 1.2. We apply the second assertion of Lemma 3.1, the definition of the set S2, and Theorem 1.1 to have that for a.e. t(0,+),

    dDVdtΛ0DV+2κ1(n1)NTm(N1)ϕMΛ0DV+2κ1(n1)NTm(N1)ϕ(r02+min1αn1d(Iα,Iα+1)t). (3.9)

    Similar to inequality (3.4), we recall that for t(0,+),

    DV(t)DV(0)exp(Λ0t)+2κ1(n1)NTmΛ0(N1)exp(Λ02t)ϕ(r02)+2κ1(n1)NTmΛ0(N1)ϕ((min1αn1d(Iα,Iα+1))t+r02). (3.10)

    Hence, we reach the desired first assertion.

    To prove the second assertion, we employ the third assertion of Lemma 3.1 and Theorem 1.1 to get that for t(0,+),

    dDTdtκ2(min(N1,,Nα)2)ζ(DX)(N1)(TM)2DT+2κ2(n1)NN1ζM(1Tm1TM)ˉΛ0DT+2κ2(n1)NN1(1Tm1TM)ζ((min1αn1d(Iα,Iα+1))t+r02). (3.11)

    We use Gronwall's lemma to yield that for t(0,+),

    DT(t)DT(0)exp(ˉΛ0t)+t02κ2(n1)NζMN1(1Tm1TM)exp(ˉΛ0(st))ds=DT(0)exp(ˉΛ0t)+(t20+tt2)2κ2(n1)NζMN1(1Tm1TM)exp(ˉΛ0(st))dsDT(0)exp(ˉΛ0t)+2κ2(n1)NN1(1Tm1TM)ζ(r02)[exp(ˉΛ02t)exp(ˉΛ0t)]+2κ2(n1)NN1(1Tm1TM)ζ((min1αn1d(Iα,Iα+1))t+r02)[1exp(ˉΛ02t)]DT(0)exp(ˉΛ0t)+2κ2(n1)NN1(1Tm1TM)exp(ˉΛ02t)ζ(r02)+2κ2(n1)NN1(1Tm1TM)ζ((min1αn1d(Iα,Iα+1))t+r02). (3.12)

    We conclude the desired second assertion.

    As a direct consequence, we present the following result that the velocity and temperature of each agent in each cluster group converge to some same non-negative value, respectively. We prove the following lemma at first.

    Lemma 3.3. Assume that Zα={xαi,vαi,Tαi}Nαi=1 is a solution to the system (1.4). Each local average (xcenα,νcenα,Tcenα) then satisfies the following relations:

    {dxcenαdt=vcenα,t>0,α{1,,n},n3,Nα˙vcenα=κ1N11ijNαϕ(xαixαj)vαivαjvαi22Tαj+κ1N1βαNαi=1Nβj=1ϕ(xαixβj)(vβjvαi+vαivαjvαi22)1Tβj,Nα˙Tcenα=κ2N1βαNαi=1Nβj=1ζ(xαixβj)(1Tαi1Tβj). (3.13)

    Proof. The first assertion is trivial. For the second assertion, we take Nαi=1 to ˙vαi and use the standard trick of interchanging i and j to obtain that

    vαivαj22=12vαivαj,vαivαj=12(vαi,vαi+vαj,vαj2vαi,vαj)=1vαi,vαj. (3.14)

    Therefore, we have

    Nα˙vcenα=κ1N11ijNαϕ(||xαixαj||)(vαjvαi+vαivαi,vαjvαiTαj)+κ1N1βαNαi=1Nβj=1ϕ(||xαixβj||)(vβjvαi+vαivαi,vβjvαiTβj)=κ1N11ijNαϕ(xαixαj)(vαjvαi+vαivαjvαi22)1Tαj+κ1N1βαNαi=1Nβj=1ϕ(xαixβj)(vβjvαi+vαivαjvαi22)1Tβj=κ1N11ijNαϕ(xαixαj)vαivαjvαi22Tαj+κ1N1βαNαi=1Nβj=1ϕ(xαixβj)(vβjvαi+vαivαjvαi22)1Tβj. (3.15)

    For the third assertion, we take Nαi=1 to ˙Tαi and again use the standard trick as above. Finally, we prove the lemma.

    Proof of Theorem 1.3. According to Lemma 3.3,

    Nα˙vcenα=κ1N11ijNαϕ(xαixαj)vαivαjvαi22Tαj+κ1N1βαNαi=1Nβj=1ϕ(xαixβj)(vβjvαi+vαivβjvαi22)1Tβj, (3.16)

    and thus we have

    vcenα(t)=vcenα(0)+κ1(N1)Nα1ijNαt0ϕ(xαixαj)vαivαjvαi22Tαjds+κ1(N1)NαβαNαi=1Nβj=1t0ϕ(xαixβj)(vβjvαi+vαivβjvαi22)1Tβjds. (3.17)

    For t1,t2(0,+), we have

    vcenα(t2)vcenα(t1)κ1(N1)Nα1ijNαt2t1ϕ(xαixαj)vαivαjvαi22Tαjds+κ1(N1)NαβαNαi=1Nβj=1t2t1ϕ(xαixβj)(vβjvαi+vαivβjvαi22)1Tβjdsκ1(N1)Nα1ijNαt2t1ϕ(δ0)D2V2Tmds+κ1(N1)NαβαNαi=1Nβj=1t2t1ϕ(xαixβj)2+1TmdsCκ1ϕ(δ0)2Tm(N1)Nα1ijNαt2t1ϕ2((min1αn1d(Iα,Iα+1))s+r02)ds+(2+1)κ1Tm(N1)NαβαNαi=1Nβj=1t2t1ϕ((min1αn1d(Iα,Iα+1))s+r02)ds. (3.18)

    By employing the Cauchy convergence criterion and the existence of 0ϕ((min1αn1d(Iα,Iα+1))s+r02)ds and 0ϕ2((min1αn1d(Iα,Iα+1))s+r02)ds, it is straightforward to observe that vcenα(t2)vcenα(t1) can be arbitrarily small when both t1 and t2 are sufficiently large. Therefore, the existence of limtvcenα(t) is guaranteed.

    By employing vα:=limtvcenα(t) and

    vcenα=vcenα(0)+κ1(N1)Nα1ijNαt0ϕ(xαixαj)vαivαjvαi22Tαjds+κ1(N1)NαβαNαi=1Nβj=1t0ϕ(xαixβj)(vβjvαi+vαivβjvαi22)1Tβjds,

    we have that

    vcenα(t)vακ1(N1)Nα1ijNαtϕ(xαixαj)vαivαjvαi22Tαjds+κ1(N1)NαβαNαi=1Nβj=1tϕ(xαixβj)(vβjvαi+vαivβjvαi22)1Tβjds. (3.19)

    Then, the multi-flocking estimate studied in Theorem 1.1 and Theorem 1.2 and the monotonicity and non-negativity of ϕ imply that

    vcenα(t)vαO(tϕ((min1αn1d(Iα,Iα+1))s+r02)ds)O(1)1tλ10,t. (3.20)

    Drawing from from Theorem 1.1 and Theorem 1.2, we observe that

    vαi(t)vcenα(t)=O(exp(Λ02t)+ϕ((min1αn1d(Iα,Iα+1))s+r02))O(1)1tλ,t. (3.21)

    We combine the above estimates to derive that for all α[n] and i[Nα],

    vαi(t)vαvcenα(t)vα+vαi(t)vcenα(t)=O(1)1tλ1+O(1)1tλO(1)1tλ1,t. (3.22)

    Conversely, it is evident that for α,β[n], and i[Nα],j[Nβ],

    xαixβjxαi(0)xβj(0)+t0vαivβjdtR0+t0DV(s)dsR0+(DV(0)+C0)t, (3.23)

    where C0:=4κ1(n1)NTmΛ0(N1)ϕ(r02). Therefore, the multi-flocking estimate studied in Theorem 1.2 and the monotonicity and non-negativity of ϕ imply that for α[Nα]

    vcenα(t)vαO(tϕ(R0+(DV(0)+C0)t)dsexp(Λ0t))O(1)1tλ10,t. (3.24)

    Then, we combine the above estimates to derive that for all α[n] and i[Nα],

    vαi(t)vαvcenα(t)vαvαi(t)vcenα(t)=O(1)1tλ1O(1)1tλO(1)1tλ1,t. (3.25)

    Finally, there exist 2n strictly positive values Vα1,Vα2 such that

    Vα1tλ1vαi(t)vαVα2tλ1,α[n],i[Nα]. (3.26)

    Therefore, there exist two strictly positive values V1,V2 such that for all α[n] and iα[Nα],

    V1tλ1nα=1vαiα(t)vαV2tλ1,t. (3.27)

    Similar to the previous proof, the existence of Tα can be demonstrated, and there exist two positive values T1 and T2 such that for all α[n] and iα[Nα],

    T1tμ1nα=1Tαiα(t)TαT2tμ1,t. (3.28)

    We conclude the desired results.

    This study provides proof for the fundamental properties and multi-cluster flocking behaviors of the TCSUS system (1.4) under a singular kernel.

    Specifically, Propositions 2.1–2.4 establish the foundational characteristics of the TCSUS model and present essential findings that facilitate the investigation of multi-cluster flocking within the TCSUS framework. Lemma 3.1 establishes the dissipative structure of the TCSUS system as derived from its configuration.

    Subsequently, the bootstrapping technique is utilized to derive the multi-cluster flocking outcome within a finite time interval. Furthermore, in Theorem 1.1, by enforcing particular initial velocity conditions and applying bootstrapping methods, we ascertain that the divergence rate of distinct clusters is bounded below by a linear function of time.

    Theorem 1.2 provides estimates of the position-velocity-temperature L-diameters for all cluster groups by using Gronwall inequalities. Consequently, it is also demonstrated that the velocities and temperatures of all clusters converge to common values, respectively.

    Lemma 3.3 establishes the differential equalities for the central velocity and temperature of a cluster, derived by summing the velocities and temperatures of its constituent particles. Finally, Theorem 1.3 provides the convergence values for velocity and temperature within each cluster group by asserting Lemma 3.3 and Theorem 1.2.

    Shenglun Yan: Methodology, analysis, calculation, and writing original draft; Wanqian Zhang: Discussion, review and editing; Weiyuan Zou: Supervision, validation, and revision.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The work of Shenglun Yan was supported by the Innovation and Entrepreneurship Projects for College Students in Beijing University of Chemical Technology (X202410010342), and the work of Weiyuan Zou was supported by the National Natural Science Foundation of China (NSFC)12001033.

    The authors declare no conflict of interest.



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