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Research article

Analysis of a COVID-19 model with media coverage and limited resources


  • Received: 12 December 2023 Revised: 20 February 2024 Accepted: 29 February 2024 Published: 06 March 2024
  • The novel coronavirus disease (COVID-19) pandemic has profoundly impacted the global economy and human health. The paper mainly proposed an improved susceptible-exposed-infected-recovered (SEIR) epidemic model with media coverage and limited medical resources to investigate the spread of COVID-19. We proved the positivity and boundedness of the solution. The existence and local asymptotically stability of equilibria were studied and a sufficient criterion was established for backward bifurcation. Further, we applied the proposed model to study the trend of COVID-19 in Shanghai, China, from March to April 2022. The results showed sensitivity analysis, bifurcation, and the effects of critical parameters in the COVID-19 model.

    Citation: Tao Chen, Zhiming Li, Ge Zhang. Analysis of a COVID-19 model with media coverage and limited resources[J]. Mathematical Biosciences and Engineering, 2024, 21(4): 5283-5307. doi: 10.3934/mbe.2024233

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  • The novel coronavirus disease (COVID-19) pandemic has profoundly impacted the global economy and human health. The paper mainly proposed an improved susceptible-exposed-infected-recovered (SEIR) epidemic model with media coverage and limited medical resources to investigate the spread of COVID-19. We proved the positivity and boundedness of the solution. The existence and local asymptotically stability of equilibria were studied and a sufficient criterion was established for backward bifurcation. Further, we applied the proposed model to study the trend of COVID-19 in Shanghai, China, from March to April 2022. The results showed sensitivity analysis, bifurcation, and the effects of critical parameters in the COVID-19 model.



    Chemotaxis is a physiological phenomenon of organisms seeking benefits and avoiding harm, which has been widely concerned in the fields of both mathematics and biology. In order to depict such phenomena, in 1970, Keller and Segel [1] established the first mathematical model (also called the Keller-Segel model). The general form of this model is described as follows

    {ut=Δuχ(uv)+f(u), xΩ, t>0,τvt=Δvv+g(u), xΩ, t>0,u(x,0)=u0(x),v(x,0)=v0(x), xΩ, (1.1)

    where ΩRn(n1) is a bounded domain with smooth boundary, the value of τ can be chosen by 0 or 1 and the chemotaxis sensitivity coefficient χ>0. Here, u is the density of cell or bacteria and v stands for the concentration of chemical signal secreted by cell or bacteria. The functions f(u) and g(u) are used to characterize the growth and death of cells or bacteria and production of chemical signals, respectively.

    Over the past serval decades, considerable efforts have been done on the dynamical behavior (including the global existence and boundedness, the convergence as well as the existence of blow-up solutions) of the solutions to system (1.1) (see [2,3,4,5,6,7]). Let us briefly recall some contributions among them in this direction. For example, assume that f(u)=0 and g(u)=u. For τ=1, it has been shown that the classical solutions to system (1.1) always remain globally bounded when n=1 [8]. Additionally, there will be a critical mass phenomenon to system (1.1) when n=2, namely, if the initial data u0 fulfill Ωu0dx<4πχ, the classical solutions are globally bounded [9]; and if Ωu0dx>4πχ, the solutions will blow up in finite time [10,11]. However, when n3, Winkler [12,13] showed that though the initial data satisfy some smallness conditions, the solutions will blow up either in finite or infinite time. Assume that the system (1.1) involves a non-trivial logistic source and g(u)=u. For τ=0 and f(u)aμu2 with a0 and μ>0, Tello and Winkler [14] obtained that there exists a unique global classical solution for system (1.1) provided that n2,μ>0 or n3 and suitably large μ>0. Furthermore, for τ>0 and n1, suppose that Ω is a bounded convex domain. Winkler [15] proved that the system (1.1) has global classical solutions under the restriction that μ>0 is sufficiently large. When τ=1 and f(u)=uμu2, Winkler [16] showed that nontrivial spatially homogeneous equilibrium (1μ,1μ) is globally asymptotically stable provided that the ratio μχ is sufficiently large and Ω is a convex domain. Later, based on maximal Sobolev regularity, Cao [17] also obtained the similar convergence results by removing the restrictions τ=1 and the convexity of Ω required in [16]. In addition, for the more related works in this direction, we mention that some variants of system (1.1), such as the attraction-repulsion systems (see [18,19,20,21]), the chemotaxis-haptotaxis models (see [22,23,24]), the Keller-Segel-Navier-Stokes systems (see [25,26,27,28,29,30]) and the pursuit-evasion models (see [31,32,33]), have been deeply investigated.

    Recently, the Keller-Segel model with nonlinear production mechanism of the signal (i.e. g(u) is a nonlinear function with respect to u) has attracted widespread attention from scholars. For instance, when the second equation in (1.1) satisfies vt=Δvv+g(u) with 0g(u)Kuα for K,α>0, Liu and Tao [34] obtained the global existence of classical solutions under the condition that 0<α<2n. When f(u)u(abus) and the second equation becomes 0=Δvv+uk with k,s>0, Wang and Xiang [35] showed that if either s>k or s=k with kn2knχ<b, the system (1.2) has global classical solutions. When the second equation in (1.1) turns into 0=Δv1|Ω|Ωg(u)+g(u) for g(u)=uκ with κ>0, Winkler [36] showed that the system has a critical exponent 2n such that if κ>2n, the solution blows up in finite time; conversely, if κ<2n, the solution is globally bounded with respect to t. More results on Keller-Segel model with logistic source can be found in [6,37,38,39,40].

    In addition, previous contributions also imply that diffusion functions may lead to colorful dynamic behaviors. The corresponding model can be given by

    {ut=(D(u)u)(S(u)v), xΩ, t>0,vt=Δvv+u, xΩ, t>0, (1.2)

    where D(u) and S(u) are positive functions that are used to characterize the strength of diffusion and chemoattractants, respectively. When D(u) and S(u) are nonlinear functions of u, Tao-Winkler [41] and Winkler [42] proved that the existence of global classical solutions or blow-up solutions depend on the value of S(u)D(u). Namely, if S(u)D(u)cuα with α>2n,n2 and c>0 for all u>1, then for any M>0 there exist solutions that blow up in either finite or infinite time with mass Ωu0=M in [42]. Later, Tao and Winkler [41] showed that such a result is optimal, i.e., if S(u)D(u)cuα with α<2n,n1 and c>0 for all u>1, then the system (1.2) possesses global classical solutions, which are bounded in Ω×(0,). Furthermore, Zheng [43] studied a logistic-type parabolic-elliptic system with ut=((u+1)m1u)χ(u(u+1)q1v)+aubur and 0=Δvv+u for m1,r>1,a0,b,q,χ>0. It is shown that when q+1<max{r,m+2n}, or b>b0=n(rm)2(rm)n+2(r2)χ if q+1=r, then for any sufficiently smooth initial data there exists a classical solution that is global in time and bounded. For more relevant results, please refer to [38,44,45,46].

    In the Keller-Segel model mentioned above, the chemical signals are secreted by cell population, directly. Nevertheless, in reality, the production of chemical signals may go through very complex processes. For example, signal substance is not secreted directly by cell population but is produced by some other signal substance. Such a process may be described as the following system involving an indirect signal mechanism

    {ut=Δu(uv)+f(u),   xΩ, t>0,τvt=Δvv+w,τwt=Δww+u,   xΩ, t>0, (1.3)

    where u represents the density of cell, v and w denote the concentration of chemical signal and indirect chemical signal, respectively. For τ=1, assume that f(u)=μ(uuγ) with μ,γ>0, Zhang-Niu-Liu [47] showed that the system has global classical solutions under the condition that γ>n4+12 with n2. Such a boundedness result was also extended to a quasilinear system in [48,49]. Ren [50] studied system (1.3) and obtained the global existence and asymptotic behavior of generalized solutions. For τ=0, Li and Li [51] investigated the global existence and long time behavior of classical solutions for a quasilinear version of system (1.3). In [52], we extended Li and Li's results to a quasilinear system with a nonlinear indirect signal mechanism. More relevant results involving indirect signal mechanisms can be found in [53,54,55,56].

    In the existing literatures, the indirect signal secretion mechanism is usually a linear function of u. However, there are very few papers that study the chemotaxis system, where chemical signal production is not only indirect but also nonlinear. Considering the complexity of biological processes, such signal production mechanisms may be more in line with the actual situation. Thus, in this paper, we study the following chemotaxis system

    {ut=Δuχ(φ(u)v)ξ(ψ(u)w)+f(u),  xΩ, t>0,0=Δvv+vγ11, 0=Δv1v1+uγ2,  xΩ, t>0,0=Δww+wγ31, 0=Δw1w1+uγ4,  xΩ, t>0,uν=vν=wν=v1ν=w1ν=0,  xΩ, t>0, (1.4)

    where ΩRn(n1) is a smoothly bounded domain and ν denotes the outward unit normal vector on Ω, the parameters χ,ξ,γ2,γ4>0, and γ1,γ31. The initial data u(x,0)=u0(x) satisfy some smooth conditions. Here, the nonlinear functions are assumed to satisfy

    φ,ψC2([0,)), φ(ϱ)ϱ(ϱ+1)θ1 and ψ(ϱ)ϱ(ϱ+1)l1 for all ϱ0, (1.5)

    with θ,lR. The logistic source fC([0,)) is supposed to satisfy

    f(0)0 and f(ϱ)aϱbϱs for all ϱ0, (1.6)

    with a,b>0 and s>1. The purpose of this paper is to detect the influence of power exponents (instead of the coefficients and space dimension n) of the system (1.4) on the existence and boundedness of global classical solutions.

    We state our main result as follows.

    Theorem 1.1. Let ΩRn(n1) be a bounded domain with smooth boundary and the parameters fulfill ξ,χ,γ2,γ4>0 and γ1,γ31. Assume that the nonlinear functions φ,ψ and f satisfy the conditions (1.5) and (1.6) with a,b>0,s>1 and θ,lR. If smax{γ1γ2+θ,γ3γ4+l}, then for any nonnegative initial data u0W1,(Ω), the system (1.4) has a nonnegative global classical solution

    (u,v,v1,w,w1)(C0(¯Ω×[0,))C2,1(¯Ω×(0,)))×(C2,0(¯Ω×(0,)))4.

    Furthermore, this solution is bounded in Ω×(0,), in other words, there exists a constant C>0 such that

    u(,t)L(Ω)+(v(,t),v1(,t),w(,t),w1(,t))W1,(Ω)<C

    for all t>0.

    The system (1.4) is a bi-attraction chemotaxis model, which can somewhat be seen as a variant of the classical attraction-repulsion system proposed by Luca [57]. In [58], Hong-Tian-Zheng studied an attraction-repulsion model with nonlinear productions and obtained the buondedness conditions which not only depend on the power exponents of the system, but also rely on the coefficients of the system as well as space dimension n. Based on [58], Zhou-Li-Zhao [59] further improved such boundedness results to some critical conditions. Compared to [58] and [59], the boundedness condition developed in Theorem 1.1 relies only on the power exponents of the system, which removes restrictions on the coefficients of the system and space dimension n. The main difficulties in the proof of Theorem 1.1 are how to reasonably deal with the integrals with power exponents in obtaining the estimate of Ω(u+1)p in Lemma 3.1. Based on a prior estimates of solutions (Lemma 2.2) and some scaling techniques of inequalities, we can overcome these difficulties and then establish the conditions of global boundedness.

    The rest of this paper is arranged as follows. In Sec.2, we give a result on local existence of classical solutions and get some estimates of solutions. In Sec.3, we first prove the boundedness of Ω(u+1)p and then complete the proof of Theorem 1.1 based on the Moser iteration [41, Lemma A.1].

    To begin with, we state a lemma involving the local existence of classical solutions and get some estimates on the solutions of system (1.4).

    Lemma 2.1. Let ΩRn(n1) be a bounded domain with smooth boundary and the parameters fulfill ξ,χ,γ2,γ4>0 and γ1,γ31. Assume that the nonlinear functions φ,ψ and f satisfy the conditions (1.5) and (1.6) with a,b>0,s>1 and θ,lR. For any nonnegative initial data u0W1,(Ω), there exists Tmax(0,] and nonnegative functions

    (u,v,v1,w,w1)(C0(¯Ω×[0,Tmax))C2,1(¯Ω×(0,Tmax)))×(C2,0(¯Ω×(0,Tmax)))4,

    which solve system (1.4) in classical sense. Furthermore,

    if  Tmax<, then limtTmaxsupu(,t)L(Ω)=. (2.1)

    Proof. The proof relies on the Schauder fixed point theorem and partial differential regularity theory, which is similar to [60, Lemma 2.1]. For convenience, we give a proof here. For any T(0,1) and the nonnegative initial data u0W1,, we set

    X:=C0(¯Ω×[0,T]) and S:={uX|u(,t)L(Ω)R for all t[0,T]},

    where R:=u0L(Ω)+1. We can pick smooth functions φR,ψR on [0,) such that φRφ and ψRψ when 0ϱR and φRR and ψRR when ϱR. It is easy to see that S is a bounded closed convex subset of X. For any ˆuS, let v,v1,w and w1 solve

    {Δv+v=vγ11,xΩ, t(0,T),vν=0,xΩ, t(0,T), and {Δv1+v1=ˆuγ2,xΩ, t(0,T),v1ν=0,xΩ, t(0,T), (2.2)

    as well as

    {Δw+w=wγ31,xΩ, t(0,T),wν=0,xΩ, t(0,T), and {Δw1+w1=ˆuγ4,xΩ, t(0,T),w1ν=0,xΩ, t(0,T), (2.3)

    respectively, in turn, let u be a solution of

    {ut=Δuχ(φR(u)v)ξ(ψR(u)w)+f(u),  xΩ, t(0,T),uν=vν=wν=v1ν=w1ν=0,  xΩ, t(0,T),u(x,0)=u0(x),  xΩ. (2.4)

    Thus, we introduce a map Φ:ˆu(S)u defined by Φ(ˆu)=u. We shall show that for any T>0 sufficiently small, Φ has a fixed point in S. Using the elliptic regularity [61, Theorem 8.34] and Morrey's theorem [62], for a certain fixed ˆuS, we conclude that the solutions to (2.2) satisfy v1(,t)C1+δ(Ω) and v(,t)C3+δ(Ω) for all δ(0,1), as well as the solutions to (2.3) satisfy w1(,t)C1+δ(Ω) and w(,t)C3+δ(Ω) for all δ(0,1). From the Sobolev embedding theorem and Lpestimate, there exist mi>0,i=1,...,4 such that

    v1L((0,T);Cδ(Ω))m1v1L((0,T);W2,p(Ω))m2ˆuγ2L((0,T)×Ω)

    and

    w1L((0,T);Cδ(Ω))m3w1L((0,T);W2,p(Ω))m4ˆuγ4L((0,T)×Ω)

    for p>max{1,nγ1γ2,nγ3γ4}. Furthermore, we can also find mi>0,i=5,...,10 such that

    vL((0,T);Cδ(Ω))m5vL((0,T);W2,p(Ω))m6vγ11L((0,T)×Ω)m7ˆuγ1γ2L((0,T)×Ω)

    and

    wL((0,T);Cδ(Ω))m8wL((0,T);W2,p(Ω))m9wγ31L((0,T)×Ω)m10ˆuγ3γ4L((0,T)×Ω)

    for p>max{1,nγ1γ2,nγ3γ4}. Since v,wL((0,T)×Ω) and u0Cδ(¯Ω) for all δ(0,1) due to the Sobolev embedding W1,(Ω)Cδ(Ω), we can infer from [63, Theorem V1.1] that uCδ,δ2(¯Ω×[0,T]) and

    uCδ,δ2(¯Ω×[0,T])m11 for all δ(0,1), (2.5)

    with some m11>0 depending only on vL((0,T);Cδ(Ω)),wL((0,T);Cδ(Ω)) and R. Thus, we have

    u(,t)L(Ω)u0L(Ω)+u(,t)u0L(Ω)u0L(Ω)+m11tδ2. (2.6)

    Hence if T<(1m11)2δ, we can obtain

    u(,t)L(Ω)u0L(Ω)+1 (2.7)

    for all t[0,T], which implies that uS. Thus, we derive that Φ maps S into itself. We deduce that Φ is continuous. Moreover, we get from (2.5) that Φ is a compact map. Hence, by the Schauder fixed point theorem there exists a fixed point uS such that Φ(u)=u.

    Applying the regularity theory of elliptic equations, we derive that v1(,t)C2+δ(Ω), v(,t)C4+δ(Ω) and w1(,t)C2+δ(Ω), w(,t)C4+δ(Ω) for all δ(0,1). Recalling (2.5), we get v1(x,t)C2+δ,δ2(Ω×[ι,T]), v(x,t)C4+δ,δ2(Ω×[ι,T]) and w1(x,t)C2+δ,δ2(Ω×[ι,T]), w(x,t)C4+δ,δ2(Ω×[ι,T]) for all δ(0,1) and ι(0,T). We use the regularity theory of parabolic equation [63, Theorem V6.1] to get

    u(x,t)C2+δ,1+δ2(¯Ω×[ι,T])

    for all ι(0,T). The solution may be prolonged in the interval [0,Tmax) with either Tmax= or Tmax<, where in the later case

    u(.t)L(Ω)  as  tTmax.

    Additionally, since f(0)0, we thus get from the parabolic comparison principle that u is nonnegative. By employing the elliptic comparison principle to the second, the third, the fourth and the fifth equations in (1.4), we conclude that v,v1,w,w1 are also nonnegative. Thus, we complete the proof of Lemma 2.1.

    Lemma 2.2. Let ΩRn(n1) be a bounded domain with smooth boundary and the parameters fulfill ξ,χ,γ2,γ4>0 and γ1,γ31. Assume that the nonlinear functions φ,ψ and f satisfy the conditions (1.5) and (1.6) with a,b>0,s>1 and θ,lR. For any η1,η2,η3,η4>0 and τ>1, we can find c1,c2,c3,c4>0 which depend only on γ1,γ2,γ3,γ4,η1,η2,η3,η4,τ, such that

    Ωwτ1η2Ω(u+1)γ4τ+c1 and Ωwτη1η2Ω(u+1)γ3γ4τ+c2, (2.8)

    as well as

    Ωvτ1η4Ω(u+1)γ2τ+c3 and Ωvτη3η4Ω(u+1)γ1γ2τ+c4 (2.9)

    for all t(0,Tmax).

    Proof. Integrating the first equation of system (1.4) over Ω and using Hölder's inequality, it is easy to get that

    ddtΩudxΩaubusaΩub|Ω|s1(Ωu)s  for all t(0,Tmax). (2.10)

    Employing the standard ODE comparison theory, we conclude

    Ωumax{Ωu0,(ab)1s1|Ω|}  for all t(0,Tmax). (2.11)

    Moreover, integrating the fifth equation of system (1.4) over Ω, one may get

    w1L1(Ω)=uγ4L1(Ω)(u+1)γ4L1(Ω)  for all t(0,Tmax). (2.12)

    For any τ>1, multiplying the fifth equation of system(1.4) with wτ11, we can get by integration by parts that

    4(τ1)τ2Ω|wτ21|2+Ωwτ1=Ωuγ4wτ11τ1τΩwτ1+1τΩuγ4τ, (2.13)

    where Young's inequality has been used. Thus, we deduce

    w1Lτ(Ω)uγ4Lτ(Ω)(u+1)γ4Lτ(Ω)  for all t(0,Tmax), (2.14)

    and

    4(τ1)τΩ|wτ21|2Ωuγ4τΩ(u+1)γ4τ  for all t(0,Tmax). (2.15)

    Using Ehrling's lemma, we know that for any η2>0 and τ>1, there exists c5=c5(η2,τ)>0 such that

    ϕ2L2(Ω)η2ϕ2W1,2(Ω)+c5ϕ2L2τ(Ω) for all ϕW1,2(Ω). (2.16)

    Let ϕ=wτ21. Combining (2.12) with (2.14), (2.15), there exists c6=c6(η2,γ4,τ)>0 such that

    Ωwτ1η2Ω(u+1)γ4τ+c6(u+1)γ4τL1(Ω). (2.17)

    If γ4(0,1], by (2.11) and Hölder's inequality, we can derive

    (u+1)γ4τL1(Ω)c7, (2.18)

    with c7=c7(η2,τ,γ4)>0. If γ4(1,), invoking interpolation inequality and Young's inequality, we can get from (2.11) that

    (u+1)γ4τL1(Ω)(u+1)γ4τρLτ(Ω)(u+1)γ4τ(1ρ)L1γ4(Ω)η2Ω(u+1)γ4τ+c8, (2.19)

    where ρ=γ41γ41τ(0,1) and c8=c8(η2,τ,γ4)>0. Collecting (2.17)–(2.19), we can directly infer that the first inequality of (2.8) holds. Integrating the fourth equation of system (1.4) over Ω, we have wL1(Ω)=wγ31L1(Ω)  for all t(0,Tmax). Due to γ31, from the first inequality of (2.8), it is easy to see that

    wL1(Ω)=Ωwγ31η2Ω(u+1)γ3γ4+˜c1 (2.20)

    for all  t(0,Tmax), where ˜c1=˜c1(η2,γ3,γ4)>0. By the same procedures as in (2.13)-(2.19), we thus can obtain for any η1>0 and τ>1 that

    Ωwτη1Ωwγ3τ1+c9  for all t(0,Tmax), (2.21)

    where c9=c9(η1,τ,γ3)>0. Recalling γ31 and using the first inequality of (2.8) again, we get that

    Ωwγ3τ1η2Ω(u+1)γ3γ4τ+c10  for all t(0,Tmax), (2.22)

    with c10>0. Hence, the second inequality of (2.8) can be obtained from (2.21) and (2.22). In addition, we can employ the same processes as above to prove (2.9). Here, we omit the detailed proof. Thus, the proof of Lemma 2.2 is complete.

    In order to prove the global existence and uniform boundedness of classical solutions to system (1.4), we established the following Lpestimate for component u.

    Lemma 3.1. Let ΩRn(n1) be a bounded domain with smooth boundary and the parameters fulfill ξ,χ,γ2,γ4>0 and γ1,γ31. Assume that the nonlinear functions φ,ψ and f satisfy the conditions (1.5) and (1.6) with a,b>0,s>1 and θ,lR. If smax{γ1γ2+θ,γ3γ4+l}, for any p>max{1,1θ,1l,γ1γ2s+1,γ3γ4s+1}, there exists C>0 such that

    Ω(u+1)pC (3.1)

    for all t(0,Tmax).

    Proof. For any p>1, we multiply the first equation of system (1.4) with (u+1)p1 and use integration by parts over Ω to obtain

    1pddtΩ(u+1)p4(p1)p2Ω|(u+1)p2|2+χ(p1)Ω(u+1)p2φ(u)uv+ξ(p1)Ω(u+1)p2ψ(u)uw+aΩu(u+1)p1bΩus(u+1)p1 (3.2)

    for all t(0,Tmax). Let Ψ1(y)=y0(ζ+1)p2ψ(ζ)dζ and Ψ2(y)=y0(ζ+1)p2φ(ζ)dζ. It is easy to get

    Ψ1(u)=(u+1)p2ψ(u)u (3.3)

    and

    Ψ2(u)=(u+1)p2φ(u)u (3.4)

    for all t(0,Tmax). Furthermore, by a simple calculation, one can get

    |Ψ1(u)|1p+l1(u+1)p+l1 (3.5)

    and

    |Ψ2(u)|1p+θ1(u+1)p+θ1 (3.6)

    for all t(0,Tmax). Thus, the second term on the right-hand side of (3.2) can be estimated as

    χ(p1)Ω(u+1)p2φ(u)uv=χ(p1)ΩΨ2(u)vχ(p1)ΩΨ2(u)|Δv|χ(p1)p+θ1Ω(u+1)p+θ1|Δv| (3.7)

    for all t(0,Tmax). Similarly, we can deduce

    ξ(p1)Ω(u+1)p2ψ(u)uwξ(p1)p+l1Ω(u+1)p+l1|Δw| (3.8)

    for all t(0,Tmax). From the basic inequality (u+1)s2s(us+1) with s>0 and u0, we can get

    bΩus(u+1)p1b2sΩ(u+1)p+s1+bΩ(u+1)p1 (3.9)

    for all t(0,Tmax). Set m=max{a,b}. From (3.7)–(3.9), the (3.2) can be rewritten as

    1pddtΩ(u+1)pχ(p1)p+θ1Ω(u+1)p+θ1|vvγ11|+ξ(p1)p+l1Ω(u+1)p+l1|wwγ31|+mΩ(u+1)pb2sΩ(u+1)p+s1χ(p1)p+θ1Ω(u+1)p+θ1v+χ(p1)p+θ1Ω(u+1)p+θ1vγ11+ξ(p1)p+l1Ω(u+1)p+l1w+ξ(p1)p+l1Ω(u+1)p+l1wγ31+mΩ(u+1)pb2sΩ(u+1)p+s1 (3.10)

    for all t(0,Tmax), where we have used the equations 0=Δvv+vγ11 and 0=Δw1w1+uγ4 in system (1.4). In the following, we shall establish the Lp estimate of component u.

    Case (ⅰ) s>max{γ1γ2+θ,γ3γ4+l}.

    It follows from Young's inequality that

    Ω(u+1)p+θ1vγ11b(p+θ1)2s+4χ(p1)Ω(u+1)p+s1+c11Ωv(p+s1)γ1sθ1 (3.11)

    for all t(0,Tmax), with c11=(2s+4χ(p1)b(p+θ1))p+θ1sθ>0. Due to sθ>γ1γ2, we infer from Young's inequality and Lemma 2.2 by choosing τ=p+s1γ2 that

    Ωv(p+s1)γ1sθ1b(p+θ1)2s+4χη4(p1)c11Ωvp+s1γ21+c12b(p+θ1)2s+4χ(p1)c11Ω(u+1)p+s1+c13 (3.12)

    for all t(0,Tmax), with c12=(2s+4χη4(p1)c11b(p+θ1))γ1γ2sθγ1γ2|Ω| and c13=c12+c3. According to Young's inequality, we can find c14=(2s+4χ(p1)b(p+θ1))p+θ1sθ>0 such that

    Ω(u+1)p+θ1vb(p+θ1)2s+4χ(p1)Ω(u+1)p+s1+c14Ωvp+s1sθ (3.13)

    for all t(0,Tmax). For sθ>γ1γ2, we use Lemma 2.2 with τ=p+s1sθ and Young's inequality to get

    Ωvp+s1sθη3η4Ω(u+1)γ1γ2(p+s1)sθ+c4b(p+θ1)2s+4χ(p1)c14Ω(u+1)p+s1+c15 (3.14)

    for all t(0,Tmax), with c15=(η3η4)sθsθγ1γ2(2s+4χ(p1)c14b(p+θ1))γ1γ2sθγ1γ2+c4. Analogously, we have

    Ω(u+1)p+l1wγ31b(p+l1)2s+4ξ(p1)Ω(u+1)p+s1+c16Ωw(p+s1)γ3sl1 (3.15)

    for all t(0,Tmax), where c16=(2s+4ξ(p1)b(p+l1))p+l1sl. Since sl>γ3γ4, it follows from Young's inequality and Lemma 2.2 with τ=p+s1γ4 that

    Ωw(p+s1)γ3sl1b(p+l1)2s+4ξ(p1)c16η2Ωwp+s1γ41+c17b(p+l1)2s+4ξ(p1)c16Ω(u+1)p+s1+c18 (3.16)

    for all t(0,Tmax), where c17=(2s+4ξ(p1)c16η2b(p+l1))γ3γ4slγ3γ4|Ω| and c18=c17+c1. Similarly, there exists c19=(2s+4ξ(p1)b(p+l1))p+l1sl>0 such that

    Ω(u+1)p+l1wb(p+l1)2s+4ξ(p1)Ω(u+1)p+s1+c19Ωwp+s1sl (3.17)

    for all t(0,Tmax). Using Lemma 2.2 once more, one may obtain

    Ωwp+s1slη1η2Ω(u+1)γ3γ4(p+s1)sl+c2b(p+l1)2s+4ξ(p1)c19Ω(u+1)p+s1+c20 (3.18)

    for all t(0,Tmax), where c20=(η1η2)slslγ3γ4(2s+4ξ(p1)c19b(p+l1))γ3γ4slγ3γ4+c2. Due to s>1, we thus have

    Ω(u+1)pc21Ω(u+1)p+s1+c22 (3.19)

    for all t(0,Tmax), where c21=b2s+2(m+1) and c22=(2s+2(m+1)b)ps1|Ω|. From (3.11)-(3.19), the inequality (3.10) can be estimated as

    1pddtΩ(u+1)p+Ω(u+1)pχ(p1)p+θ1[b(p+θ1)2s+2χ(p1)Ω(u+1)p+s1+c11c13++c14c15]+ξ(p1)p+l1[b(p+l1)2s+2ξ(p1)Ω(u+1)p+s1+c16c18+c19c20]b2sΩ(u+1)p+s1+b2s+2Ω(u+1)p+s1+c22(m+1)b2s+2Ω(u+1)p+s1+c23 (3.20)

    for all t(0,Tmax), where c23=(c11c13+c14c15)χ(p1)p+θ1+(c16c18+c19c20)ξ(p1)p+l1+c22(m+1). Hence, we can derive (3.1) easily by using the ODE comparison principle.

    Case (ⅱ) s=max{γ1γ2+θ,γ3γ4+l}.

    (a) s=γ1γ2+θ=γ3γ4+l. Recalling (3.11), (3.13), (3.15) and (3.17), there hold

    Ω(u+1)p+θ1vγ11b(p+θ1)2s+4χ(p1)Ω(u+1)p+s1+c11Ωv(p+s1)γ1sθ1 (3.21)

    and

    Ω(u+1)p+θ1vb(p+θ1)2s+4χ(p1)Ω(u+1)p+s1+c14Ωvp+s1sθ (3.22)

    and

    Ω(u+1)p+l1wγ31b(p+l1)2s+4ξ(p1)Ω(u+1)p+s1+c16Ωw(p+s1)γ3sl1 (3.23)

    as well as

    Ω(u+1)p+l1wb(p+l1)2s+4ξ(p1)Ω(u+1)p+s1+c19Ωwp+s1sl (3.24)

    for all t\in (0, T_{\max}).

    Since s-\theta = \gamma_{1}\gamma_{2} and s-l = \gamma_{3}\gamma_{4}. Thus, we can directly get from Lemma 2.2 that

    \begin{align} \int_{\Omega}w_{1}^\frac{(p+s-1)\gamma_{3}}{s-l} = \int_{\Omega}w_{1}^\frac{p+s-1}{\gamma_{4}} \leq\eta_{2}\int_{\Omega}(u+1)^{p+s-1}+c_{1} \end{align} (3.25)

    and

    \begin{align} \int_{\Omega}w^\frac{p+s-1}{s-l} = \int_{\Omega}w^\frac{p+s-1}{\gamma_{3}\gamma_{4}} \leq\eta_{1}\eta_{2}\int_{\Omega}(u+1)^{p+s-1}+c_{2}, \end{align} (3.26)

    and

    \begin{align} \int_{\Omega}v_{1}^\frac{(p+s-1)\gamma_{1}}{s-\theta} = \int_{\Omega}v_{1}^\frac{p+s-1}{\gamma_{2}} \leq\eta_{4}\int_{\Omega}(u+1)^{p+s-1}+c_{3} \end{align} (3.27)

    as well as

    \begin{align} \int_{\Omega}v^\frac{p+s-1}{s-\theta} = \int_{\Omega}v^\frac{p+s-1}{\gamma_{1}\gamma_{2}} \leq\eta_{3}\eta_{4}\int_{\Omega}(u+1)^{p+s-1}+c_{4} \end{align} (3.28)

    for all t\in (0, T_{\max}). For the arbitrariness of \eta_{1}, \eta_{2}, \eta_{3}, \eta_{4} > 0, we choose \eta_{2} = \frac{b(p+l-1)}{2^{s+4}c_{16}\xi(p-1)}, \eta_{1}\eta_{2} = \frac{b(p+l-1)}{2^{s+4}c_{19}\xi(p-1)}, \eta_{4} = \frac{b(p+\theta-1)}{2^{s+4}c_{11}\chi(p-1)} and \eta_{3}\eta_{4} = \frac{b(p+\theta-1)}{2^{s+4}c_{14}\chi(p-1)} in(3.25)-(3.28), respectively. Combining (3.19) with (3.21)-(3.28), the inequality (3.10) can be rewritten as

    \begin{align} \frac{1}{p}\frac{d}{dt}\int_{\Omega}(u+1)^{p}+\int_{\Omega}(u+1)^{p} \leq& \frac{\chi(p-1)}{p+\theta-1} \bigg[\frac{b(p+\theta-1)}{2^{s+2}\chi(p-1)}\int_{\Omega}(u+1)^{p+s-1} +c_{3}c_{11}+c_{4}c_{14}\bigg] \\ &+\frac{\xi(p-1)}{p+l-1} \bigg[\frac{b(p+l-1)}{2^{s+2}\xi(p-1)}\int_{\Omega}(u+1)^{p+s-1} +c_{1}c_{16}+c_{2}c_{19}\bigg] \\ &-\frac{b}{2^{s}}\int_{\Omega}(u+1)^{p+s-1}+\frac{b}{2^{s+2}} \int_{\Omega}(u+1)^{p+s-1}+c_{22}(m+1) \\ \leq& -\frac{b}{2^{s+2}}\int_{\Omega}(u+1)^{p+s-1}+c_{24} \end{align} (3.29)

    for all t\in (0, T_{\max}), where c_{24} = \big(c_{3}c_{11}+c_{4}c_{14}\big)\cdot\frac{\chi(p-1)}{p+\theta-1}+ \big(c_{1}c_{16}+c_{2}c_{19}\big)\cdot\frac{\xi(p-1)}{p+l-1}+c_{22}(m+1). From the ODE comparison principle, we can get the desired conclusion (3.1).

    (b) s = \gamma_{1}\gamma_{2}+\theta > \gamma_{3}\gamma_{4}+l. Recalling (3.11) and (3.13) again, there hold

    \begin{align} \int_{\Omega}(u+1)^{p+\theta-1}v_{1}^{\gamma_{1}} \leq \frac{b(p+\theta-1)}{2^{s+4}\chi(p-1)}\int_{\Omega}(u+1)^{p+s-1} +c_{11}\int_{\Omega}v_{1}^{\frac{(p+s-1)\gamma_{1}}{s-\theta}} \end{align} (3.30)

    and

    \begin{align} \int_{\Omega}(u+1)^{p+\theta-1}v \leq\frac{b(p+\theta-1)}{2^{s+4}\chi(p-1)}\int_{\Omega}(u+1)^{p+s-1} +c_{14}\int_{\Omega}v^{\frac{p+s-1}{s-\theta}}. \end{align} (3.31)

    For s = \gamma_{1}\gamma_{2}+\theta, we can get from Lemma 2.2 that

    \begin{align} \int_{\Omega}v_{1}^\frac{(p+s-1)\gamma_{1}}{s-\theta} = \int_{\Omega}v_{1}^\frac{p+s-1}{\gamma_{2}} \leq\eta_{4}\int_{\Omega}(u+1)^{p+s-1}+c_{3} \end{align} (3.32)

    and

    \begin{align} \int_{\Omega}v^\frac{p+s-1}{s-\theta} = \int_{\Omega}v^\frac{p+s-1}{\gamma_{1}\gamma_{2}} \leq\eta_{3}\eta_{4}\int_{\Omega}(u+1)^{p+s-1}+c_{4} \end{align} (3.33)

    for all t\in (0, T_{\max}). Here, we also choose \eta_{4} = \frac{b(p+\theta-1)}{2^{s+4}c_{11}\chi(p-1)} in (3.32) and \eta_{3}\eta_{4} = \frac{b(p+\theta-1)}{2^{s+4}c_{14}\chi(p-1)} in (3.33). For s > \gamma_{3}\gamma_{4}+l, we can conclude from (3.15)-(3.18) that

    \begin{align} \int_{\Omega}(u+1)^{p+l-1}w_{1}^{\gamma_{3}}&\leq \frac{b(p+l-1)}{2^{s+4}\xi(p-1)}\int_{\Omega}(u+1)^{p+s-1} +c_{16}\int_{\Omega}w_{1}^\frac{(p+s-1)\gamma_{3}}{s-l} \end{align} (3.34)

    for all t\in (0, T_{\max}), with c_{16} = \left(\frac{2^{s+4}\xi(p-1)}{b(p+l-1)}\right)^{\frac{p+l-1}{s-l}}. Moreover, using Lemma 2.2, it is easy to get

    \begin{align} \int_{\Omega}w_{1}^\frac{(p+s-1)\gamma_{3}}{s-l} \leq \frac{b(p+l-1)}{2^{s+4}\xi(p-1)c_{16}\eta_{2}}\int_{\Omega} w_{1}^{\frac{p+s-1}{\gamma_{4}}}+c_{17} \leq \frac{b(p+l-1)}{2^{s+4}\xi(p-1)c_{16}}\int_{\Omega}(u+1)^{p+s-1}+c_{18} \end{align} (3.35)

    for all t\in (0, T_{\max}), where c_{17} = \left(\frac{2^{s+4}\xi(p-1)c_{16}\eta_{2}}{b(p+l-1)}\right) ^{\frac{\gamma_{3}\gamma_{4}}{s-l-\gamma_{3}\gamma_{4}}}|\Omega| and c_{18} = c_{17}+c_{1}. By a simple calculation, we know

    \begin{align} \int_{\Omega}(u+1)^{p+l-1}w \leq \frac{b(p+l-1)}{2^{s+4}\xi(p-1)}\int_{\Omega}(u+1)^{p+s-1} +c_{19}\int_{\Omega}w^{\frac{p+s-1}{s-l}} \end{align} (3.36)

    and

    \begin{align} \int_{\Omega}w^{\frac{p+s-1}{s-l}}&\leq \eta_{1}\eta_{2}\int_{\Omega}(u+1)^{\frac{\gamma_{3}\gamma_{4} (p+s-1)}{s-l}}+c_{2} \leq \frac{b(p+l-1)}{2^{s+4}\xi(p-1)c_{19}}\int_{\Omega}(u+1)^{p+s-1}+c_{20} \end{align} (3.37)

    for all t\in (0, T_{\max}), where c_{19} = \left(\frac{2^{s+4}\xi(p-1)}{b(p+l-1)}\right) ^\frac{p+l-1}{s-l} and c_{20} = (\eta_{1}\eta_{2})^{\frac{s-l}{s-l-\gamma_{3}\gamma_{4}}}\cdot\left(\frac{2^{s+4}\xi(p-1) c_{19}}{b(p+l-1)}\right) ^{\frac{\gamma_{3}\gamma_{4}}{s-l-\gamma_{3}\gamma_{4}}}+c_{2}. Recalling (3.19), for s > 1, we have

    \begin{align} \int_{\Omega}(u+1)^{p}\leq c_{21}\int_{\Omega}(u+1)^{p+s-1}+c_{22} \end{align} (3.38)

    for all t\in (0, T_{\max}), where c_{21} = \frac{b}{2^{s+2}(m+1)} and c_{22} = \left(\frac{2^{s+2}(m+1)}{b}\right)^{\frac{p}{s-1}}|\Omega|. Collecting (3.30)-(3.38), it can be deduced from (3.10) that

    \begin{align} \frac{1}{p}\frac{d}{dt}\int_{\Omega}(u+1)^{p}+\int_{\Omega}(u+1)^{p} \leq& \frac{\chi(p-1)}{p+\theta-1} \bigg[\frac{b(p+\theta-1)}{2^{s+2}\chi(p-1)}\int_{\Omega}(u+1)^{p+s-1} +c_{3}c_{11}+c_{4}c_{14}\bigg] \\ &+\frac{\xi(p-1)}{p+l-1} \bigg[\frac{b(p+l-1)}{2^{s+2}\xi(p-1)}\int_{\Omega}(u+1)^{p+s-1} +c_{16}c_{18}+c_{19}c_{20}\bigg] \\ &-\frac{b}{2^{s}}\int_{\Omega}(u+1)^{p+s-1}+\frac{b}{2^{s+2}} \int_{\Omega}(u+1)^{p+s-1}+c_{22}(m+1) \\ \leq& -\frac{b}{2^{s+2}}\int_{\Omega}(u+1)^{p+s-1}+c_{25} \end{align} (3.39)

    for all t\in (0, T_{\max}), where c_{25} = \big(c_{3}c_{11}+c_{4}c_{14}\big)\cdot\frac{\chi(p-1)}{p+\theta-1}+ \big(c_{16}c_{18}+c_{19}c_{20}\big)\cdot\frac{\xi(p-1)}{p+l-1}+c_{22}(m+1). In view of the ODE comparison principle, we conclude (3.1), directly.

    (c) s = \gamma_{3}\gamma_{4}+l > \gamma_{1}\gamma_{2}+\theta. The proof of this case is similar to the case (b). Using (3.15) and (3.17) again, we get

    \begin{align} \int_{\Omega}(u+1)^{p+l-1}w_{1}^{\gamma_{3}}&\leq \frac{b(p+l-1)}{2^{s+4}\xi(p-1)}\int_{\Omega}(u+1)^{p+s-1} +c_{16}\int_{\Omega}w_{1}^\frac{(p+s-1)\gamma_{3}}{s-l} \end{align} (3.40)

    and

    \begin{align} \int_{\Omega}(u+1)^{p+l-1}w \leq \frac{b(p+l-1)}{2^{s+4}\xi(p-1)}\int_{\Omega}(u+1)^{p+s-1} +c_{19}\int_{\Omega}w^{\frac{p+s-1}{s-l}} \end{align} (3.41)

    for all t\in (0, T_{\max}). Since s = \gamma_{3}\gamma_{4}+l, it is easy to deduce from Lemma 2.2 that

    \begin{align} \int_{\Omega}w_{1}^\frac{(p+s-1)\gamma_{3}}{s-l} = \int_{\Omega}w_{1}^\frac{p+s-1}{\gamma_{4}} \leq\eta_{2}\int_{\Omega}(u+1)^{p+s-1}+c_{1} \end{align} (3.42)

    and

    \begin{align} \int_{\Omega}w^\frac{p+s-1}{s-l} = \int_{\Omega}v^\frac{p+s-1}{\gamma_{3}\gamma_{4}} \leq\eta_{1}\eta_{2}\int_{\Omega}(u+1)^{p+s-1}+c_{2} \end{align} (3.43)

    for all t\in(0, T_{\max}). Due to the arbitrariness of \eta_{1} and \eta_{2}, here we let \eta_{2} = \frac{b(p+l-1)}{2^{s+4}c_{16}\xi(p-1)} in (3.42) and \eta_{1}\eta_{2} = \frac{b(p+l-1)}{2^{s+4}c_{19}\xi(p-1)} in (3.43). Since s > \gamma_{1}\gamma_{2}+\theta, we can derive from (3.11)-(3.14) that

    \begin{align} \int_{\Omega}(u+1)^{p+\theta-1}v_{1}^{\gamma_{1}} \leq \frac{b(p+\theta-1)}{2^{s+4}\chi(p-1)}\int_{\Omega}(u+1)^{p+s-1} +c_{11}\int_{\Omega}v_{1}^{\frac{(p+s-1)\gamma_{1}}{s-\theta}} \end{align} (3.44)

    for all t\in(0, T_{\max}), with c_{11} = \left(\frac{2^{s+4}\chi(p-1)}{b(p+\theta-1)}\right)^{\frac{p+\theta-1}{s-\theta}} > 0. Due to s-\theta > \gamma_{1}\gamma_{2}, from Young's inequality and Lemma 2.2, we can obtain

    \begin{align} \int_{\Omega}v_{1}^{\frac{(p+s-1)\gamma_{1}}{s-\theta}} \leq \frac{b(p+\theta-1)}{2^{s+4}\chi \eta_{4}(p-1)c_{11}}\int_{\Omega}v_{1}^{\frac{p+s-1}{\gamma_{2}}}+c_{12} \leq \frac{b(p+\theta-1)}{2^{s+4}\chi(p-1)c_{11}}\int_{\Omega}(u+1)^{p+s-1}+c_{13} \end{align} (3.45)

    for all t\in(0, T_{\max}), with c_{12} = \left(\frac{2^{s+4}\chi \eta_{4}(p-1)c_{11}}{b(p+\theta-1)}\right) ^{\frac{\gamma_{1}\gamma_{2}}{s-\theta-\gamma_{1}\gamma_{2}}}|\Omega| and c_{13} = c_{12}+c_{3}. In view of Young's inequality, it is easy to get

    \begin{align} \int_{\Omega}(u+1)^{p+\theta-1}v \leq\frac{b(p+\theta-1)}{2^{s+4}\chi(p-1)}\int_{\Omega}(u+1)^{p+s-1} +c_{14}\int_{\Omega}v^{\frac{p+s-1}{s-\theta}} \end{align} (3.46)

    for all t\in (0, T_{\max}), with c_{14} = \left(\frac{2^{s+4}\chi(p-1)}{b(p+\theta-1)}\right) ^{\frac{p+\theta-1}{s-\theta}} > 0. Due to s-\theta > \gamma_{1}\gamma_{2}, thus we use Lemma 2.2 with \tau = \frac{p+s-1}{s-\theta} and Young's inequality to obtain

    \begin{align} \int_{\Omega}v^{\frac{p+s-1}{s-\theta}} \leq \eta_{3}\eta_{4}\int_{\Omega}(u+1) ^{\frac{\gamma_{1}\gamma_{2}(p+s-1)}{s-\theta}}+c_{4} \leq \frac{b(p+\theta-1)}{2^{s+4}\chi(p-1)c_{14}}\int_{\Omega}(u+1)^{p+s-1}+c_{15} \end{align} (3.47)

    for all t\in (0, T_{\max}), with c_{15} = (\eta_{3}\eta_{4})^{\frac{s-\theta}{s-\theta-\gamma_{1}\gamma_{2}}}\cdot\left(\frac{2^{s+4}\chi(p-1)c_{14}}{b(p+\theta-1)}\right) ^{\frac{\gamma_{1}\gamma_{2}}{s-\theta-\gamma_{1}\gamma_{2}}}+c_{4}. For s > 1, we get from (3.19) that

    \begin{align} \int_{\Omega}(u+1)^{p}\leq c_{21}\int_{\Omega}(u+1)^{p+s-1}+c_{22} \end{align} (3.48)

    for all t\in (0, T_{\max}), where c_{21} = \frac{b}{2^{s+2}(m+1)} and c_{22} = \left(\frac{2^{s+2}(m+1)}{b}\right)^{\frac{p}{s-1}}|\Omega|. Collecting (3.40)–(3.48), we can infer from (3.10) that

    \begin{align} \frac{1}{p}\frac{d}{dt}\int_{\Omega}(u+1)^{p}+\int_{\Omega}(u+1)^{p} \leq-\frac{b}{2^{s+2}}\int_{\Omega}(u+1)^{p+s-1}+c_{26} \end{align} (3.49)

    for all t\in (0, T_{\max}), where c_{26} = \big(c_{11}c_{13}+c_{14}c_{15}\big)\cdot\frac{\chi(p-1)}{p+\theta-1}+ \big(c_{1}c_{16}+c_{2}c_{19}\big)\cdot\frac{\xi(p-1)}{p+l-1}+c_{22}(m+1). Hence, we can conclude (3.1) by using the ODE comparison principle. Thus, we complete the proof of Lemma 3.1.

    Now, we are in a position to prove Theorem 1.1.

    Proof of Theorem 1.1 Let \Omega\subset\mathbb{R}^n(n\geq1) be a bounded domain with smooth boundary and the parameters fulfill \xi, \chi, \gamma_{2}, \gamma_{4} > 0 and \gamma_{1}, \gamma_{3}\geq1. Assume that the nonlinear functions \varphi, \psi and f satisfy the conditions (1.5) and (1.6) with a, b > 0, s > 1 and \theta, l\in \mathbb{R}. According to Lemma 3.1, for any p > \max\{1, 1-\theta, 1-l, n\gamma_{1}\gamma_{2}, n\gamma_{3}\gamma_{4}, \gamma_{1}\gamma_{2}-s+1, \gamma_{3}\gamma_{4}-s+1\}, there exists c_{27} > 0 such that \|u\|_{L^{p}(\Omega)}\leq c_{27} for all t\in(0, T_{\max}). We deal with the second, the third, the fourth and the fifth equations in system (1.4) by elliptic L^{p}- estimate to obtain

    \begin{align} \|v(\cdot, t)\|_{W^{2, \frac{p}{\gamma_{1}\gamma_{2}}}(\Omega)} +\|v_{1}(\cdot, t)\|_{W^{2, \frac{p}{\gamma_{2}}}(\Omega)} +\|w(\cdot, t)\|_{W^{2, \frac{p}{\gamma_{3}\gamma_{4}}}(\Omega)} +\|w_{1}(\cdot, t)\|_{W^{2, \frac{p}{\gamma_{4}}}(\Omega)} \leq c_{28} \end{align} (3.50)

    for all t\in(0, T_{\max}), with some c_{28} > 0. Applying the Sobolev imbedding theorem, we can infer that

    \begin{align} \|v(\cdot, t)\|_{W^{1, \infty}} +\|v_{1}(\cdot, t)\|_{W^{1, \infty}} +\|w(\cdot, t)\|_{W^{1, \infty}} +\|w_{1}(\cdot, t)\|_{W^{1, \infty}} \leq c_{29} \end{align} (3.51)

    for all t\in(0, T_{\max}), with some c_{29} > 0. In view of Moser iteration [41, Lemma A.1], there exists c_{30} > 0 such that

    \|u(\cdot, t)\|_{L^{ \infty}(\Omega)}\leq c_{30}

    for all t\in(0, T_{\max}), which combining with Lemma 2.1 implies that T_{\max} = \infty. The proof of Theorem 1.1 is complete.

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

    We would like to thank the anonymous referees for many useful comments and suggestions that greatly improve the work. This work was partially supported by NSFC Grant NO. 12271466, Scientific and Technological Key Projects of Henan Province NO. 232102310227, NO. 222102320425 and Nanhu Scholars Program for Young Scholars of XYNU NO. 2020017.

    The authors declare there is no conflict of interest.



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