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Research article

Modeling the effects of trait-mediated dispersal on coexistence of mutualists

  • Even though mutualistic interactions are ubiquitous in nature, we are still far from making good predictions about the fate of mutualistic communities under threats such as habitat fragmentation and climate change. Fragmentation often causes declines in abundance of a species due to increased susceptibility to edge effects between remnant habitat patches and lower quality "matrix" surrounding these focal patches. It has been argued that ecological communities are replete with trait-mediated indirect effects, and that these effects may sometimes contribute more to the dynamics of a population than direct density-mediated effects, e.g., lowering an organism's fitness through competitive interactions. Although some studies have focused on trait-mediated behavior such as trait-mediated dispersal, in which an organism changes its dispersal patterns due to the presence of another species, they have been mostly limited to predator-prey systems–little is known regarding their effect on other interaction systems such as mutualism. Here, we explore consequences of fragmentation and trait-mediated dispersal on coexistence of a system of two mutualists by employing a model built upon the reaction diffusion framework. To distinguish between trait-mediated dispersal and density-mediated effects, we isolate effects of trait-mediated dispersal on the mutualistic system by excluding any direct density-mediated effects in the model. Our results demonstrate that fragmentation and trait-mediated dispersal can have important impacts on coexistence of mutualists. Specifically, one species can be better able to invade and persist than the other and be crucial to the success of the other species in the patch. Matrix quality degradation can also bring about a complete reversal of the role of which species is supporting the other's persistence in the patch, even as the patch size remains constant. As most mutualistic relationships are identified based on density-mediated effects, such an effect may be easily overlooked.

    Citation: James T. Cronin, Jerome Goddard II, Amila Muthunayake, Ratnasingham Shivaji. Modeling the effects of trait-mediated dispersal on coexistence of mutualists[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 7838-7861. doi: 10.3934/mbe.2020399

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  • Even though mutualistic interactions are ubiquitous in nature, we are still far from making good predictions about the fate of mutualistic communities under threats such as habitat fragmentation and climate change. Fragmentation often causes declines in abundance of a species due to increased susceptibility to edge effects between remnant habitat patches and lower quality "matrix" surrounding these focal patches. It has been argued that ecological communities are replete with trait-mediated indirect effects, and that these effects may sometimes contribute more to the dynamics of a population than direct density-mediated effects, e.g., lowering an organism's fitness through competitive interactions. Although some studies have focused on trait-mediated behavior such as trait-mediated dispersal, in which an organism changes its dispersal patterns due to the presence of another species, they have been mostly limited to predator-prey systems–little is known regarding their effect on other interaction systems such as mutualism. Here, we explore consequences of fragmentation and trait-mediated dispersal on coexistence of a system of two mutualists by employing a model built upon the reaction diffusion framework. To distinguish between trait-mediated dispersal and density-mediated effects, we isolate effects of trait-mediated dispersal on the mutualistic system by excluding any direct density-mediated effects in the model. Our results demonstrate that fragmentation and trait-mediated dispersal can have important impacts on coexistence of mutualists. Specifically, one species can be better able to invade and persist than the other and be crucial to the success of the other species in the patch. Matrix quality degradation can also bring about a complete reversal of the role of which species is supporting the other's persistence in the patch, even as the patch size remains constant. As most mutualistic relationships are identified based on density-mediated effects, such an effect may be easily overlooked.


    In the present work, we shall consider a chemotaxis-haptotaxis model

    {ut=Δuχ(uv)ξ(uw)+μu(1uw),τvt=Δvv+u,wt=vw+ηu(1uw), (1.1)

    where χ and ξ are positive parameters. In the model (1.1), u represents the density of cancer cell, v and w denote the density of matrix degrading enzymes (MDEs) and the extracellular matrix (ECM) with the positive sensitivity χ, ξ, respectively. Such an important extension of chemotaxis to a more complex cell migration mechanism has been proposed by Chaplain and Lolas [3] to describe the cancer cell invasion of tissue. In that process, cancer invasion is associated with the degradation of ECM, which is degraded by MDEs secreted by cancer cells. Besides random motion, the migration of invasive cells is oriented both by a chemotaxis mechanism and by a haptotaxis mechanism.

    In the past ten more years, the global solvability, boundedness and asymptotic behavior for the corresponding no-flux or homogeneous Neumann boundary-initial value problem in bounded domain and its numerous variants have been widely investigated for certain smooth initial data. For the full parabolic system of (1.1), Pang and Wang [4] studied the global boundedness of classical solution in the case τ=1 in 2D domains, and the global solvability also was established for three dimension. When η=0 and τ=1, Tao and Wang [5] proved the existence and uniqueness of global classical solution for any χ>0 in 1D intervals and for small χμ>0 in 2D domains, and Tao [6] improved the results for any μ>0 in two dimension; Cao [7] proved for small χμ>0, the model (1.1) processes a global and bounded classical solution in 3D domains.

    When τ=0, the second equation of (1.1) becomes an elliptic function. In the case of η>0, Tao and Winkler [8] proved the global existence of classical solutions in 2D domains for any μ>0. In the case of η=0, the global existence and boundedness for this simplified model under the condition of μ>(N2)+Nχ in any N-D domains in [9]. Moreover, the stabilization of solutions with on-flux boundary conditions was discussed in [10]. For the explosion phenomenon, Xiang [11] proved that (1.1) possess a striking feature of finite-time blow-up for N3 with μ=η=τ=0; the blow-up results for two dimension was discussed in [2] with wt=vw+ηw(1w) and μ=0.

    When χ=0, the system (1.1) becomes a haptotaxis-only system. The local existence and uniqueness of classical solutions was proved in [12]. In [13,14,15], the authors respectively established the global existence, the uniform-in-time boundedness of classical solutions and the asymptotic behavior. Very recently, Xiang[11] showed that the pure haptotaxis term cannot induce blow-up and pattern for N3 or τ=0 in the case of μ=η=0.

    Without considering the effect of the haptotaxis term in (1.1), we may have the extensively-studied Keller-Segel system, which was proposed in [16] to describe the collective behavior of cells under the influence of chemotaxis

    {tu=Δuχ(uv),τtv=Δuλv+u (1.2)

    with u and v denoting the cell density and chemosignal concentration, respectively. There have been a lot of results in the past years (see [17,18,19,20,21], for instance). Here we only mention some global existence and blow-up results in two dimensional space. For the parabolic-elliptic case of (1.2) with λ=0, 8πχ was proved to be the mass threshold in two dimension in [22,23,24] (see also [25,26] for related results in the bounded domain); namely, the chemotactic collapse (blowup) should occur if and only if u0L1 is greater than 8πχ. If u0L1<8πχ, the existence of free-energy solutions were improved in [22]. Furthermore, the asymptotic behavior was given by a unique self-similar profile of the system (see also [27] for radially symmetric results concerning self-similar behavior). For the results in the threshold 8πχ, we refer readers for [28,29,30] for more details. For the parabolic-elliptic model in higher dimensions (N3) in (1.2), the solvability results were discussed in [31,32,33,34] with small data in critical spaces like LN2(RN),LN2w(RN),MN2(RN), i.e., those which are scale-invariant under the natural scaling. Blowing up solutions to the parabolic-elliptic model of (1.2) in dimension N3 have been studied in [35,36,37,38].

    In the case τ=1, Calvez and Corrias [1] showed that under hypotheses u0ln(1+|x|2)L1(R2) and u0lnu0L1(R2), any solution exists globally in time if u0L1<8πχ. In [39], the extra assumptions on u0 were removed, while the condition on mass was restricted to u0L1<4πχ. The value 4πχ appeared since a Brezis-Merle type inequality played an essential role there. These results were improved in [40,41] to global existence of all solutions with u0L1<8πχ by two different method. Furthermore the global existence of solutions was also obtained under some condition on u0 in the critical case u0L1=8πχ in[40]. The blow-up results of the parabolic-parabolic case in the whole space were discussed in [42,43] with the second equation was replaced by tv=Δu+u.

    However, the global solvability and explosion phenomenon of chemotaxis-haptotaxis model in the whole space have never been touched. Here we consider the global solvability of a simplified model of (1.1)

    {ut=Δuχ(uv)ξ(uw),xR2, t>0,vt=Δvv+u,xR2, t>0,wt=vw,xR2, t>0,u(x,0)=u0(x),  v(x,0)=v0(x),  w(x,0)=w0(x),xR2. (1.3)

    Main results. We assume that the initial data satisfies the following assumptions:

    (u0,v0,w0)H2(R2)×H3(R2)×H3(R2) and u0,v0,w0 are nonnegative, (1.4)
    u0L1(R2,ln(1+|x|2)dx) and u0lnu0L1(R2) (1.5)

    and

    Δw0L(R2) and w0L(R2). (1.6)

    Theorem 1.1. Let χ>0, ξ>0 and the initial data (u0,v0,w0) satisfy (1.4)–(1.6). If m:=u0L1<8πχ, then the corresponding chemotaxis-haptotaxis system (1.3) possesses a unique global-in-time, nonnegative and strong solution (u,v,w) fulfilling that for any T<

    (u,v,w)C(0,T;H2(R2)×H3(R2)×H3(R2)).

    Remark 1.1. Our theorem extends the previous results in two aspects. First, our result agrees with that in [1] by setting w=0, which proved that if u0L1<8πχ, then the Cauchy problem of the system (1.2) admits a global solution. Secondly, our theorem extends Theorem 1.1 in [2], where the authors proved that 4πχ is the critical mass of the system (1.3) in bounded domains, implying the negligibility of haptotaxis on global existence.

    We obtain the critical mass value using the energy method in [1,22]. The energy functional:

    F(u,v,w)(t)=R2ulnuχR2uvξR2uw+χ2R2(v2+|v|2),t(0,Tmax) (1.7)

    as shown in [2] comes out to be the key ingredient leading to the global existence of solutions under the smallness condition for the mass. Under the assumption

    u0L1<8πχ (1.8)

    and (1.5), we can derive an integral-type Gronwall inequality for F(t). As a result, we can get a priori estimate for the R2ulnu, which is the key step to establish the global existence of solutions to the system (1.3).

    The rest of this paper is organized as follows. In Section 2, we prove local-in-time existence of the solution, and obtain the blow-up criteria for the solution. In Section 3, we give the proof of the Theorem 1.1.

    In the following, (u)+ and (u) will denote the positive and negative part of u as usual, while Lp:=Lp(R2).

    We now establish the local existence and uniqueness of strong solutions to system (1.3). Our strategy is first to construct an iteration scheme for (1.3) to obtain the approximate solutions and then to derive uniform bounds for the approximate solutions to pass the limit.

    Lemma 2.1. Let χ>0, ξ>0 and u00. Then, there exists a maximal existence time Tmax>0, such that, if the initial data (u0,v0,w0) satisfy (1.4), then there exists a unique solution (u,v,w) of (1.3) satisfying for any T<Tmax, and

    (u,v,w)C(0,T;H2(R2)×H3(R2)×H3(R2)). (2.1)

    Furthermore, u, v and w are all nonnegative.

    Proof. To obtain the local solution, we follow similar procedures of an iterative scheme developed in [45,46]. We construct the solution sequence (uj,vj,wj)j0 by iteratively solving the Cauchy problems of the following system

    {tuj+1=Δuj+1χ(uj+1vj)ξ(uj+1wj),xR2, t>0,tvj+1=Δvj+1vj+1+uj,xR2, t>0,twj+1=vj+1wj+1,xR2, t>0,u(x,0)=u0(x),  v(x,0)=v0(x),  w(x,0)=w0(x),xR2. (2.2)

    We first set (u0(x,t),v0(x,t),w0(x,t))=(u0(x),v0(x),w0(x)). We point out that the system is decouple, then by the linear parabolic equations theory in [44,Theorem Ⅲ.5.2], we can obtain the unique solution u1,v1V1,122([0,T]×R2), then we get w1C1([0,T],H1(R2)) by directly solving the ordinary equation. Similarly, we define (uj,vj,wj) iteratively.

    In the following, we shall prove the convergence of the iterative sequences {uj,vj,wj}j1 in C(0,T;X) with X:= H2×H3×H3 for some small T>0. To obtain the uniform estimates, we may use the standard mollifying procedure. However, since the procedure is lengthy, we omit the details, like in the proofs of Theorem 1.1 in [45] and Theorem 2.1 in [46].

    Uniform estimates: We will use the induction argument to show that the iterative sequences {uj,vj,wj}j1 are in C(0,T;X) with X:= H2×H3×H3 for some small T>0, which means that there exists a constant R>0 such that, for any j, the following inequality holds for a small time interval

    sup0tT(ujH2+vjH3+wjH3)R, (2.3)

    where R=2{u0H2+v0H3+w0H3}+8. Due to the definition of R, the case j=0 is obvious. Then, we need to show that (2.3) is also true for j+1. This will be done by establishing the energy estimate for (uj+1,vj+1,wj+1). First, we begin with the estimate of vj+1.

    (ⅰ) Estimates of vj+1. Taking the L2 inner product of the second equation of (2.2) with vj+1, integrating by parts and using Young's inequality, we have

    12ddtvj+1(t)2L2+vj+12L2=R2(vj+1)2+R2vj+1uj12vj+12L2+12uj2L2. (2.4)

    To show the H1 estimate of vj+1, we will multiply the second equation of (2.2) by tvj+1, integrating by parts and then obtain

    12ddtvj+1(t)2L2+tvj+12L2=R2vj+1tvj+1+R2ujtvj+112tvj+12L2+vj+12L2+uj2L2. (2.5)

    For the H2 estimate of vj+1, by Young's inequality, we have

    12ddt2vj+1(t)2L2+Δvj+12L2=R2(Δvj+1)2+R2Δvj+1Δuj12Δvj+12L2+12Δuj2L2. (2.6)

    Similarly, integrating by parts, it is clear that for all t(0,T)

    ddt3vj+1(t)2L2=2R23vj+13(Δvj+1vj+1+uj)=2R2|4vj+1|22R2|3vj+1|22R24vj+12uj4vj+12L223vj+12L2+2uj2L2,

    togethering with (2.3)–(2.6) and adjusting the coefficients carefully, we can find a positive constant α such that

     ddtvj+1(t)2H3+α(vj+12H4+tvj+12L2)c1(vj+12H3+uj2H2) (2.7)

    with c1>0. Here after ci(i=2,3...) denotes the constant independent of R. Integrating on (0,t), we can obtain for all t(0,T)

    vj+1(t)2H3+αt0(vj+1(s)2H4+tvj+1(s)2L2)ec1Tv02H3+ec1TTsupt(0,T)uj(t)2H2ec1Tv02H3+ec1TTR22v02H3+2, (2.8)

    by choosing T>0 small enough to satisfy ec1T<2 and TR2<1.

    (ⅱ) The estimate of wj+1. In fact, the third component of the above solution of (2.2) can be expressed explicitly in terms of vj+1. This leads to the representation formulae

    wj+1(x,t)=w0(x)et0vj+1(x,s)ds, (2.9)
    wj+1(x,t)=w0(x)et0vj+1(x,s)dsw0(x)et0vj+1(x,s)dst0vj+1(x,s)ds (2.10)

    as well as

    Δwj+1(x,t)=Δw0(x)et0vj+1(x,s)ds2et0vj+1(x,s)dsw0(x)t0vj+1(x,s)ds+w0(x)et0vj+1(x,s)ds|t0vj+1(x,s)ds|2w0(x)et0vj+1(x,s)dst0Δvj+1(x,s)ds. (2.11)

    From (2.9), we can easily get for t(0,T)

    wj+1Lpw0Lp,p(1,]. (2.12)

    From (2.10), by (2.8), the definition of R and the following inequality

     t0f(x,s)dsLp={R2|t0f(x,s)ds|pdx}1p{tp1R2t0|f|pdsdx}1ptsups(0,t)f(s)Lp,for all p(1,), (2.13)

    we can obtain

    wj+1L2w0L2+w0et0vj+1dst0vj+1dsL2w0L2+w0Lt0vj+1dsL2w0L2+w0LTsupt(0,T)vj+1L2w0L2+c2R2Tw0L2+1 (2.14)

    by setting T small enough to satisfy c2R2T<1.

    Similarly, by the embedding H2W1,4 and (2.13), we can obtain from (2.11) for all t(0,T)

    Δwj+1L2Δw0L2+2w0t0vj+1dsL2+w0|t0vj+1ds|2L2 +w0t0Δvj+1dsL2Δw0L2+2w0L4t0vj+1dsL4+w0L|t0vj+1ds|2L2 +w0Lt0Δvj+1dsL2Δw0L2+2w0L4Tsupt(0,T)vj+1L4+w0LT2supt(0,T)vj+12L2 +w0LTsupt(0,T)Δvj+1L2Δw0L2+c3R2T+c3R3T2Δw0L2+2 (2.15)

    by setting T small enough to satisfy c3R2T<1 and c3R3T2<1.

    Now we deduce the L2 norm of 3wj+1. According to the equation of w and Hölder inequality, we can easily get for all t(0,T)

    ddt3wj+1(t)2L2=2R23wj+13wj+1t=2R23wj+13(vj+1wj+1)23wj+1L23(vj+1wj+1)L2c43wj+1L23wj+1vj+1+32wj+1vj+1+32vj+1wj+1+3vj+1wj+1L2c4{3wj+12L2vj+1L+3wj+1L22wj+1L4vj+1L4+3wj+1L22vj+1L2wj+1L+3wj+1L23vj+1L2wj+1L}. (2.16)

    By Galiardo-Nirenberg inequality, we have

    2wj+1L4c53wj+134L2wj+114L,wj+1Lc53wj+112L2wj+112L.

    Together with Young's inequality, (2.8), (2.12) and (2.16), we can get

    ddt3wj+1(t)2L2c6(vj+12H3+1)3wj+12L2+c6wj+12Lc7R23wj+12L2+c6w02Lc7R23wj+12L2+c7R2. (2.17)

    Then, we can deduce from Gronwall's inequality that for all t(0,T)

    3wj+12L2ec7R2T33w02L2+ec7R2T3c7R2T23w02L2+2 (2.18)

    by setting T small enough to satisfy ec7R2T3<2 and c7R2T<1.

    Combining with (2.12)–(2.15) and (2.18), we can see that

    wj+1H32w0H3+5. (2.19)

    (ⅲ) Estimates of uj+1. Taking the L2 inner product of the equation of uj+1 in (2.2), integrating by part we obtain

    12ddtuj+1(t)2L2+uj+12L2=χR2uj+1vjuj+1+ξR2uj+1wjuj+1uj+1L2vjLuj+1L2+uj+1L2wjLuj+1L2. (2.20)

    By (2.8), (2.19) and (2.20) and the embedding H3W1,, we can see for all t(0,T)

    12ddtuj+1(t)2L2+uj+12L2c8R2uj+12L2+12uj+12L2. (2.21)

    Now we turn to show the L2-estimate of uj+1. Multiplying Δuj+1 to both sides of the first equation of (2.3) and integrating by parts, we obtain for all t(0,T)

    12ddtuj+1(t)2L2+Δuj+12L2=χR2Δuj+1(uj+1vj)+ξR2Δui+1(ui+1wj)=I1+I2. (2.22)

    By Hölder inequality and Young's inequality, it yields that

    I1c9Δuj+1L2(uj+1vj)L2c10Δuj+1L2{uj+1H1vjH2}14Δuj+12L2+c210vj2H2uj+12H1.

    Applying the similar procedure to I2, we can obtain

    I214Δuj+12L2+c210wj2H2uj+12H1,

    which entails that for all t(0,T)

    12ddtuj+1(t)2L2+12Δuj+12L2c11R2uj+12H1. (2.23)

    Similar as (2.16), we can get

    ddt2uj+1(t)2L2=2R22uj+12uj+1t=2R22uj+1{2(Δuj+1χ(uj+1vj)ξ(uj+1wj))}=2R22uj+12Δuj+12χR22uj+13(uj+1vj)2ξR22ui+13(uj+1wj)=23uj+12L2+2χR23uj+12(uj+1vj)+2ξR23ui+12(ui+1wj)3uj+12L2+c12{2(uj+1vj)2L2+2(ui+1wj)2L2}3uj+12L2+c13R2uj+12H2. (2.24)

    Together with (2.21), (2.23) and (2.24), and adjusting the coefficients carefully, we can find a positive constant β such that

    ddtuj+1(t)2H2+βuj+12H3c14R2uj+12H2, (2.25)

    which implies from Gronwall's inequality that

    uj+12H2+βt0uj+12H3ec14R2Tu02H22u02H2,for all t(0,T), (2.26)

    by choosing T small enough to satisfy ec14R2T<2.

    Combining (2.8), (2.19) and (2.26), we can get for all t(0,T)

    uj+1H2+vj+1H2+wj+1H32{u0H1+v0H2+w0H3}+7R, (2.27)

    by the definition of R.

    Convergence: The derivation of the relevant estimates of uj+1uj, vj+1vj and wj+1wj are similar to the ones of uj+1, vj+1 and wj+1, so we omit the details. For simplicity, we denote δfj+1:=fj+1fj. Subtracting the j-th equations from the (j+1)-th equations, we have the following equations for δuj+1,δvj+1 and δwj+1 :

    {tδuj+1=Δδuj+1χ(δuj+1vj)χ(ujδvj)ξ(δuj+1wj)ξ(ujδwj),tδvj+1=Δδvj+1δvj+1+δuj,[1mm]tδwj+1=vj+1δwj+1δvj+1wj+1. (2.28)

    (ⅰ) Estimates of δvj+1. Using the same procedure as proving (2.8), we can obtain that for all t(0,T)

    δvj+1(t)2H3+αt0(δvj+1(s)2H4+tδvj+1(s)2L2)dsec15Tc15Tsup0tTδuj(t)2H2. (2.29)

    (ⅱ) Estimates of δwj+1. According to the third equation of (2.28), we have for all t(0,T)

    δwj+1(t)=t0etsvj+1(τ)dτwj+1(s)δvj+1(s)ds. (2.30)

    Using the same procedure as proving (2.19) entails that for all t(0,T)

    δwj+1L2supt(0,T)wj+1Lsupt(0,T)δvj+1L2Tc16RTsupt(0,T)δvj+1L2,
    δwj+1L2t0wj+1(s)δvj+1(s)dsL2+t0wj+1(s)δvj+1(s)dsL2+t0wj+1δvj+1tsvj+1dτdsL2c17RTsupt(0,T)δvj+1L2+c17RTsupt(0,T)δvj+1L2+c17R2T2supt(0,T)δvj+1L2c18(R2+R)Tsupt(0,T)δvj+1H1,Δδwj+1L2=t0Δ{etsvj+1dτwj+1δvj+1}dsL2Tsupt(0,T)Δ{etsvj+1dτwj+1δvj+1}L2Tsupt(0,T){(vj+12H2T2+vj+1H2T)wj+1H2δvj+1H2}c19(R3+R2)Tsupt(0,T)δvj+1H2

    and

    3δwj+12L2ec20R2TTsupt(0,T)δvj+1H3,

    which imply that for all t(0,T)

    δwj+1H3c21(R3+R2+R)Tsup0tTδuj(t)H2. (2.31)

    by setting T<1.

    (ⅲ) Estimates of δuj+1. Using the same procedure as proving (2.26) entails that for all t(0,T)

    sup0tTδuj+12H2+βt0δuj+12H3ec22RTc22RTsup0tT(δvj2H3+δwj2H2). (2.32)

    Combining with (2.29), (2.31) and (2.32), we can obtain for all t(0,T)

    sup0tT(δuj+1H2+δvj+1H3+δwj+1H3)ec23(R3+R2+R)Tc23(R3+R2+R)Tsup0tT(δujH2+δvjH3+δwjH3). (2.33)

    Taking T>0 small enough, we can find a constant r(0,1) such that

    sup0tT(δuj+1H2+δvj+1H3+δwj+1H3)rsup0tT(δujH2+δvjH3+δwjH3) (2.34)

    for any j1 and t(0,T). From the above inequality, we find that (uj,vj,wj) is a Cauchy sequence in the Banach space C(0,T;X) for some small T>0, and thus its corresponding limit denoted by (u,v,w) definitely exists in the same space.

    Uniqueness: If (ˉu,ˉv,ˉw) is another local-in-time solution of system (1.3), (˜u,˜v,˜w):=(uˉu,vˉv,wˉw) solves

    {t˜u=Δ˜uχ(˜uˉv)χ(u˜v)ξ(˜uˉw)ξ(u˜w),xR2, 0<tT,t˜v=Δ˜v˜v+˜u,xR2, 0<tT,t˜w=v˜w˜vˉw,xR2, 0<tT,˜u(x,0)=˜v(x,0)=˜w(x,0)=0,xR2,

    where T is any time before the maximal time of existence. Following a same procedure as (2.34), we can deduce that ˜u=˜v=˜w=0, which implies the uniqueness of the local solution.

    Nonnegativity: The nonnegativity of wj can be easily obtained by (2.9) and the nonnegativity of w0. We will use the induction argument to show that uj and vj are nonnegative for all j>0. We assume that uj and vj are nonnegative. If we apply the maximum principle to the second equation of (2.2), we find vj+1 is nonnegative (uj is nonnegative). Then we turn to deal with uj+1. Let us decompose uj+1=uj+1+uj+1, where uj+1+={uj+1uj+100uj+1<0anduj+1={uj+1uj+100uj+1>0. Now multiplying the negative part uj+1 on both sides of the first equation of (2.2) and integrating over [0,t]×R2, we can get

    t0R2tuj+1uj+1dxds=t0(uj+1)2L2ds+χt0R2uj+1vjuj+1+ξR2uj+1wjuj+1Ct0(uj+1)2L2(vj2L+wj2L)+12(uj+1)2L2ds

    by Young's inequality and the fact the weak derivative of uj+1 is uj+1 if uj+1<0, otherwise zero. Since uj+1, tuj+1L2(0,T;L2(R2)), we can have

    t0R2tuj+1(uj+1)dxds=12((uj+1)(0)L2(uj+1)(0)L2),

    together with the above inequality, it holds that

    (uj+1)(t)2L2(uj+1)(0)2L2exp(Ct0(vj2L+wj2L)ds).

    Due to the fact uj+1(0) is nonnegative, we can deduce that uj+1 is nonnegative. This completes the proof of Lemma 2.1.

    Remark 2.1. Since the above choice of T depends only on u0H2(R2), v0H3(R2) and w0H3(R2), it is clear by a standard argument that (u,v,w) can be extended up to some Tmax. If Tmax< in Lemma 2.1, then

    lim suptTmax(u(t)H2(R2)+v(t)H3(R2)+w(t)H3(R2))=. (2.35)

    In order to show the H2×H3×H3-boundedness of (u,v,w), it suffices to estimate a suitable Lp-norm of u, with some large, but finite p.

    Lemma 2.2. Suppose that χ,ξ>0 and the initial data (u0,v0,w0) satisfy all the assumptions presented in Lemma 2.1. Then for every K>0 there is C>0 such that whenever (u,v,w)C(0,T;H2(R2)×H3(R2)×H3(R2)) solves (1.3) for some T>0 and q0>2 satisfies

    u(,t)Lq0K,for allt(0,T), (2.36)

    then

    u(t)H2(R2)+v(t)H3(R2)+w(t)H3(R2)Cfor allt(0,T). (2.37)

    Proof. Firstly, we suppose that for some q0>2 and K>0

    u(t)Lq0(R2)K,for all t(0,T). (2.38)

    By the Duhamel principle, we represent u and v of the following integral equations

    u(t)=etΔu0χt0e(tτ)Δ(uv)(τ)dτξt0e(tτ)Δ(uw)(τ)dτ (2.39)

    and

    v(t)=et(Δ+1)v0+t0e(tτ)(Δ+1)u(τ)dτ. (2.40)

    where etΔf(x)=R2G(xy,t)f(y)dy and

    G(x,t)=Gt(x):=1(4πt)exp(|x|24t)

    is the Gaussian heat kernel. The following well-known LpLq estimates of the heat semigroup play an important role in the proofs [47,48]. For 1pq and fLq(R2), we have

    etΔfLp(4πt)(1q1p)fLq,etΔfLpC3t12(1q1p)fLq,

    where C3 is a constant depending on p and q. Then, according to (2.39), we can see that for q0>2 and all t(0,T)

    u(t)Lu0L+χt0(tτ)121q0uvLq0dτ+ξt0(tτ)121q0uwLq0dτu0L+Kχt0(tτ)121q0vLdτ+Kξt0(tτ)121q0wLdτ. (2.41)

    From (2.40) and the above LpLq estimates of the heat semigroup, we have

    v(t)Lqv0Lq+t0e(tτ)(tτ)(1q01q)u(t)Lq0dτC4,q(1,] (2.42)

    and by the embedding H3(R2)W1,(R2)

    v(t)Lv0L+t0e(tτ)(tτ)121q0u(t)Lq0dτC5, (2.43)

    where C4 and C5 depend on v0H3 and K in (2.38).

    According to the equation of w, we can see that for some C6=C6(w0H3,v0H2,K,T)>0 and all t(0,T)

    w(t)Lw0et0v(s)dsL+w0et0v(s)dst0v(s)dsLw0H3+w0Lsupt(0,T)v(t)LTC6 (2.44)

    by the embedding H3(R2)W1,(R2). Inserting (2.43) and (2.44) into (2.41), this yields for all t(0,T)

    u(t)Lu0L+KC5χt0(tτ)121q0dτ+KC6ξt0(tτ)121q0dτC7, (2.45)

    where C7 depends on u0H2, v0H2, w0H3, K and T.

    Integrating by parts and by Young's inequality, we can obtain from the second equation of (1.3) that for all t(0,T)

    12ddttv(t)2L2+tv2L2=R2(tv)2+R2tvtu12tv2L2+12tu2L2 (2.46)

    and

    12ddtv(t)2L2+Δv2L2=R2vΔv+R2uΔvv2L2+12u2L2+12Δv2L2. (2.47)

    Similarly, according to the first equation of (1.3), (2.43) and (2.44), we have for all t\in(0, T)

    \begin{align} \frac{1}{2} \frac{d}{d t}\left\|u(t)\right\|_{L^{2}}^{2}+\left\|\nabla u\right\|_{L^{2}}^{2}& = \chi\int_{\mathbb{R}^2}u\nabla v\cdot \nabla u+\xi\int_{\mathbb{R}^2}u\nabla w\cdot \nabla u \\&\leq\frac{\chi^2}{2}\left\|u \nabla v\right\|_{L^{2}}^{2}+\frac{\xi^2}{2}\left\|u \nabla w\right\|_{L^{2}}^{2}+\frac{1}{2}\left\|\nabla u\right\|_{L^{2}}^{2} \\&\leq\frac{\chi^2C_5^2}{2}\left\|u\right\|_{L^{2}}^{2}+\frac{\xi^2C_6^2}{2}\left\|u \right\|_{L^{2}}^{2}+\frac{1}{2}\left\|\nabla u\right\|_{L^{2}}^{2} \end{align} (2.48)

    and by (2.27) for some \theta > 0

    \begin{align} \frac{1}{2} \frac{d}{d t}\left\|\partial_{t} u(t)\right\|_{L^{2}}^{2}+\left\|\nabla \partial_{t} u\right\|_{L^{2}}^{2} = &\chi\int_{\mathbb{R}^2}\partial_{t}u\nabla v\cdot \nabla \partial_{t}u+\chi\int_{\mathbb{R}^2}u\partial_{t}\nabla v\cdot \nabla \partial_{t}u \\&+\xi\int_{\mathbb{R}^2}\partial_{t}u\nabla w\cdot \nabla \partial_{t}u+\xi\int_{\mathbb{R}^2}u\nabla \partial_{t}w\cdot \nabla \partial_{t}u \\ \leq &C_8\left(\|\nabla v\|_{L^{\infty}}+\|\nabla w\|_{L^{\infty}}\right)\left(\theta\left\|\partial_{t} u\right\|_{L^{2}}^{2}+\frac{1}{\theta}\left\|\nabla\partial_{t} u\right\|_{L^{2}}^{2}\right) \\&+C_8\|u\|_{L^{\infty}}\left(\theta\left\|\nabla\partial_{t} v\right\|_{L^{2}}^{2}+\theta\left\|\nabla\partial_{t} w\right\|_{L^{2}}^{2}+\frac{2}{\theta}\left\|\nabla\partial_{t} u\right\|_{L^{2}}^{2}\right) \\ \leq& C_8(C_5+C_6)\theta\left\|\partial_{t} u\right\|_{L^{2}}^{2}+\frac{C_8(C_5+C_6+2C_7)}{\theta}\left\|\nabla\partial_{t} u\right\|_{L^{2}}^{2} \\&+\theta C_8C_7\left(\left\|\nabla\partial_{t} v\right\|_{L^{2}}^{2}+\left\|\nabla\partial_{t} w\right\|_{L^{2}}^{2}\right). \end{align} (2.49)

    Now we turn to estimate the last term of the right side of (2.49). According to the third equation of (1.3), (2.42) and (2.43), we obtain for some C_9 > 0

    \begin{align} \left\|\nabla\partial_{t} w\right\|_{L^{2}}^{2}&\leq \left\|\nabla v\right\|_{L^{2}}^{2}\left\| w\right\|_{L^{\infty}}^{2}+\left\|\nabla w\right\|_{L^{2}}^{2}\left\| v\right\|_{L^{\infty}}^{2}\\&\leq \left\| w_0\right\|_{L^{\infty}}^{2}\left\|\nabla v\right\|_{L^{2}}^{2}+C_4^2\left(\|\nabla w_0\|_{L^{2}}^{2}+T^2\sup\limits_{t\in(0,T)}\|\nabla v\|_{L^{\infty}}^2\left\| w_0\right\|_{L^{2}}^2\right)\\&\leq C_9\left\|\nabla v\right\|_{L^{2}}^{2}+C_9. \end{align} (2.50)

    Combining with (2.46)–(2.50) and setting \theta > 0 to satisfy \frac{C_8(C_5+C_6+2C_7)}{\theta} < \frac{1}{2} , we can obtain such Gronwall-type inequality

    \begin{align} &\frac{d}{d t}\left\{\left\|\partial_{t}v(t)\right\|_{L^{2}}^{2}+\left\|\nabla v(t)\right\|_{L^{2}}^{2}+\left\|u(t)\right\|_{L^{2}}^{2}+\left\|\partial_{t} u(t)\right\|_{L^{2}}^{2}\right\} \\&+C_{10} \left(\left\|\nabla\partial_{t} v\right\|_{L^{2}}^{2}+\left\|\nabla u\right\|_{L^{2}}^{2}+\left\|\nabla v\right\|_{L^{2}}^{2}+\left\|\Delta v\right\|_{L^{2}}^{2}+\left\|\nabla \partial_{t} u\right\|_{L^{2}}^{2}\right) \\ \leq &C_{10}\left(\left\|\partial_{t}v\right\|_{L^{2}}^{2}+\left\|\nabla v\right\|_{L^{2}}^{2}+\left\|u\right\|_{L^{2}}^{2}+\left\|\partial_{t} u\right\|_{L^{2}}^{2}+1\right),\quad \text{for all } t\in(0,T), \end{align} (2.51)

    then by direct integration, we can have for some C_{11} = C_{11}(\|w_0\|_{H^{3}}, \| v_0\|_{H^{2}}, \|u_0\|_{H^{2}}, K, T) > 0

    \begin{eqnarray} &\ &\int_0^t \left(\left\|\nabla\partial_{t} v\right\|_{L^{2}}^{2}+\left\|\nabla u\right\|_{L^{2}}^{2}+\left\|\nabla v\right\|_{L^{2}}^{2}+\left\|\Delta v\right\|_{L^{2}}^{2}+\left\|\nabla \partial_{t} u\right\|_{L^{2}}^{2}\right)\\&\ &+\left\{\left\|\partial_{t}v\right\|_{L^{2}}^{2}+\left\|\nabla v\right\|_{L^{2}}^{2}+\left\|u\right\|_{L^{2}}^{2}+\left\|\partial_{t} u\right\|_{L^{2}}^{2}\right\}\leq C_{11} ,\quad \text{for all } t\in(0,T). \end{eqnarray} (2.52)

    By (2.42) and (2.52) and the second equation of v in (1.3), we have

    \begin{eqnarray} \left\|\Delta v\right\|_{L^{2}}\leq \left\|\partial_{t} v\right\|_{L^{2}}+\left\| v\right\|_{L^{2}}+\left\|u\right\|_{L^{2}}\leq C_{12} ,\quad \text{for all } t\in(0,T), \end{eqnarray} (2.53)

    where C_{12} = C_{12}(\|w_0\|_{H^{3}}, \| v_0\|_{H^{2}}, \|u_0\|_{H^{2}}, K, T) . Hence by the equation of w and Young's inequality, we obtain for some C_{13} = C_{13}(\|w_0\|_{H^{3}}, \| v_0\|_{H^{2}}, \|u_0\|_{H^{2}}, K, T) > 0

    \begin{align} \left\|\Delta w\right\|_{L^{2}}\leq& \left\|\Delta w_0\right\|_{L^{2}}+2\left\|\nabla w_0\right\|_{L^{4}}\left\|\int_0^t \nabla v\right\|_{L^{4}}+\left\|w_0\right\|_{L^{\infty}}\left\|\left|\int_0^t \nabla v\right|^2\right\|_{L^{2}}+\left\|w_0\right\|_{L^{\infty}}\left\|\int_0^t\Delta v\right\|_{L^{2}} \\ \leq& \left\|\Delta w_0\right\|_{L^{2}}+2\left\|\nabla w_0\right\|_{L^{4}}\sup\limits_{t\in(0,T)}\left\|\nabla v\right\|_{L^{4}}T \\&+\left\|w_0\right\|_{L^{\infty}}\sup\limits_{t\in(0,T)}\left\|\nabla v\right\|_{L^{4}}^2T+\left\|w_0\right\|_{L^{\infty}}\sup\limits_{t\in(0,T)}\left\|\Delta v\right\|_{L^{2}}T \\ \leq&C_{13},\quad \text{for all } t\in(0,T). \end{align} (2.54)

    By (2.42) and (2.43) and the embedding W^{2, 2}(\mathbb{R}^2)\hookrightarrow W^{1, 4}(\mathbb{R}^2) , we can see

    \begin{align} \frac{d}{d t}\left\|\nabla^{3} w(t)\right\|_{L^{2}}^{2} &\leqslant C \left\{\left\|\nabla^{3} w\right\|_{L^{2}}^{2}\|v\|_{L^{\infty}}+\left\|\nabla^{3} w\right\|_{L^{2}}\left\|\nabla^{2} w\right\|_{L^{4}}\|\nabla v\|_{L^{4}}\right.\\ &\quad\left.+\left\|\nabla^{3} w\right\|_{L^{2}}\left\|\nabla^{2} v\right\|_{L^{2}}\|\nabla w\|_{L^{\infty}}+\left\|\nabla^{3} w\right\|_{L^{2}}\|w\|_{L^{\infty}}\left\|\nabla^{3} v\right\|_{L^{2}}\right\} \\&\leq C_{14}\left\|\nabla^{3} w\right\|_{L^{2}}^{2}+C_{14}\left\|\nabla^{3} v\right\|_{L^{2}}^{2}+C_{14},\quad \text{for all } t\in(0,T). \end{align} (2.55)

    Integrating on (0, t) , we have

    \begin{eqnarray} \begin{aligned} \left\|\nabla^{3} w\right\|_{L^{2}}^{2} \leq C_{15}\left\|\nabla^{3} w_0\right\|_{L^{2}}^{2}+ C_{15}\int_0^t \left\|\nabla^{3} v\right\|_{L^{2}}^{2}+C_{15},\quad \text{for all } t\in(0,T). \end{aligned} \end{eqnarray} (2.56)

    Now we turn to estimate the second integral of the right side of (2.56). Applying \nabla to the second equation of (1.3) and rewriting the equation as \nabla\Delta v = \nabla v_t+\nabla v-\nabla u , then by (2.52) we have that

    \begin{align} \int_0^t \left\|\nabla^{3} v\right\|_{L^{2}}^{2} &\leq \int_0^t \left\|\nabla v_t\right\|_{L^{2}}^{2}+ \int_0^t \left\|\nabla v\right\|_{L^{2}}^{2}+\int_0^t \left\|\nabla u\right\|_{L^{2}}^{2}\\&\leq C_{11},\quad \text{for all } t\in(0,T). \end{align} (2.57)

    Inserting (2.57) into (2.56), we can obtain that for some C_{16} > 0

    \begin{eqnarray} \begin{aligned} \left\|\nabla^{3} w\right\|_{L^{2}} \leq C_{16},\quad \text{for all } t\in(0,T). \end{aligned} \end{eqnarray} (2.58)

    Now we deduce the L^2 -norm of \nabla u and \nabla^2 u . We multiply the first equation of (1.3) by -\Delta u , integrate by parts and then obtain

    \begin{align} \frac{1}{2} \frac{d}{d t}\|\nabla u(t)\|_{L^2}^2+\|\Delta u\|_{L^2}^2 = & \chi \int_{\mathbb{R}^2} \Delta u \nabla \cdot(u \nabla v)+\xi \int_{\mathbb{R}^2} \Delta u \cdot \nabla(u \nabla w) \\ = & \chi \int_{\mathbb{R}^2} \Delta u \nabla u \cdot \nabla v+\chi \int_{\mathbb{R}^2} u\Delta u \Delta v \\&+\xi \int_{\mathbb{R}^2} \Delta u\nabla u \cdot \nabla w+\xi \int_{\mathbb{R}^2} u\Delta u \Delta w \\ = &I_1+I_2+I_3+I_4. \end{align} (2.59)

    Then by (2.42), (2.45) and (2.52), Hölder's inequality and Young's inequality, we have

    \begin{aligned} I_1+I_2 & \leqslant \chi\|\nabla v\|_{L^{\infty}}\|\Delta u\|_{L^2}\|\nabla u\|_{L^2}+\chi\|\Delta v\|_{L^2}\|\Delta u\|_{L^2}\|u\|_{L^{\infty}} \\ & \leqslant \frac{1}{4}\|\Delta u\|_{L^2}^2+C_{17}\|\nabla u\|_{L^2}^2+C_{17}. \end{aligned}

    Similarly, according to (2.43), (2.45) and (2.56), we can obtain

    \begin{aligned} I_3+I_4 & \leqslant \chi\|\nabla v\|_{L^{\infty}}\|\Delta u\|_{L^2}\|\nabla u\|_{L^2}+\chi\|\Delta v\|_{L^2}\|\Delta u\|_{L^2}\|u\|_{L^{\infty}} \\ & \leqslant \frac{1}{4}\|\Delta u\|_{L^2}^2+C_{17}\|\nabla u\|_{L^2}^2+C_{18}. \end{aligned}

    Then we have

    \frac{1}{2} \frac{d}{d t}\|\nabla u(t)\|_{L^2}^2+\frac{1}{2}\|\Delta u\|_{L^2}^2\leq C_{19}\|\nabla u\|_{L^2}^2+C_{19}.

    Integrating on (0, t) , we have for some C_{20} > 0

    \begin{eqnarray} \left\|\nabla u\right\|_{L^{2}}\leq C_{20},\quad \text{for all } t\in(0,T). \end{eqnarray} (2.60)

    Rewriting the first equation of (1.3) as \Delta u = u_t+\chi\nabla\cdot(u\nabla v)+\xi\nabla\cdot(u\nabla w) , and by (2.42)–(2.44), (2.53) and (2.54), we have for some C_{21} > 0

    \begin{align} \left\|\Delta u\right\|_{L^{2}} \leq& \left\|\partial_{t} u\right\|_{L^{2}}+\chi\left\|\nabla u\cdot\nabla v\right\|_{L^{2}}+\chi\left\|u\Delta v\right\|_{L^{2}}+\xi\left\|\nabla u\cdot\nabla w\right\|_{L^{2}}+\xi\left\| u\Delta w\right\|_{L^{2}} \\ \leq&\left\|\partial_{t} u\right\|_{L^{2}}+\chi\left\|\nabla u\right\|_{L^{2}}\left\|\nabla v\right\|_{L^{\infty}}+\chi\left\|\Delta v\right\|_{L^{2}}\left\|u\right\|_{L^{\infty}} \\&+\xi\left\|\nabla u\right\|_{L^{2}}\left\|\nabla w\right\|_{L^{\infty}}+\xi\left\|\Delta w\right\|_{L^{2}}\left\|u\right\|_{L^{\infty}} \\ \leq&C_{21}. \end{align} (2.61)

    For the L^2 -norm of \nabla^3 v , integrating by parts, we deduce that

    \begin{aligned} \frac{d}{d t} \left\|\nabla^3 v(t)\right\|_{L^{2}}^2 & = 2 \int_{\mathbb{R}^2} \nabla^3 v \nabla^3(\Delta v-v+u) \\ & = -2 \int_{\mathbb{R}^2}\left|\nabla^4 v\right|^2-2 \int_{\mathbb{R}^2}\left|\nabla^3 v\right|^2+2 \int_{\mathbb{R}^2} \nabla^3 v \nabla^3 u \\ &\leq-\left\|\nabla^4 v\right\|_{L^{2}}^2-2\left\|\nabla^3 v\right\|_{L^{2}}^2+\left\|\nabla^2 u\right\|_{L^{2}}^2, \end{aligned}

    then by (2.61) and Gronwall's inequality, we can see that for all t\in(0, T)

    \begin{eqnarray} \left\|\nabla^3 v(t)\right\|_{L^{2}}\leq C_{22}. \end{eqnarray} (2.62)

    Putting (2.52)–(2.54), (2.58) and (2.60)–(2.62) together, we conclude that for some C > 0

    \begin{eqnarray} \|u(t)\|_{H^2(\mathbb{R}^2)}+\|v(t)\|_{H^3(\mathbb{R}^2)}+\|w(t)\|_{H^3(\mathbb{R}^2)}\leq C,\quad \forall t\in(0,T), \end{eqnarray} (2.63)

    which completes the proof.

    As a preparation, we first state some results concerning the system which will be used in the proof of Theorem 1.1.

    Lemma 3.1. The local-in-time classical solution (u, v, w) of system (1.3) satisfies

    \begin{eqnarray} \|u(t)\|_{L^{1}} = \left\|u_{0}\right\|_{L^{1}}: = M, \quad \forall t \in\left(0, T_{\max}\right) \end{eqnarray} (3.1)

    and

    \begin{eqnarray} \|v(t)\|_{L^{1}} = \left\|u_{0}\right\|_{L^{1}}+\left(\left\|v_{0}\right\|_{L^{1}}-\left\|u_{0}\right\|_{L^{1}}\right) e^{-t}, \quad \forall t \in\left(0, T_{\max}\right). \end{eqnarray} (3.2)

    Proof. Integrating the first and second equation of (1.3) on \mathbb{R}^2 , we can obtain

    \frac{d}{d t} \int_{\mathbb{R}^2} u = \int_{\mathbb{R}^2} \Delta u-\chi\int_{\mathbb{R}^2} \nabla \cdot(u \nabla v)-\xi\int_{\mathbb{R}^2} \nabla \cdot(u \nabla w) = 0

    and

    \frac{d}{d t} \int_{\mathbb{R}^2} v = \int_{\mathbb{R}^2} \Delta v-\int_{\mathbb{R}^2} v+\int_{\mathbb{R}^2} u = -\int_{\mathbb{R}^2} v+\int_{\mathbb{R}^2} u,

    which can easily yield (3.1) and (3.2).

    The following energy

    F(t) = \int_{\mathbb{R}^2} u \ln u-\chi \int_{\mathbb{R}^2} u v-\xi \int_{\mathbb{R}^2} u w+\frac{\chi}{2} \int_{\mathbb{R}^2}\left(v^{2}+|\nabla v|^{2}\right)

    plays a key role in the proof. The main idea of the proof is similar to the strategy introduced in [2].

    Lemma 3.2. Assume that (1.4) and (1.5) holds. Let (u, v, w) be the local-in-time classical solution of system (1.3). Then F(t) satisfies

    \begin{eqnarray} F(t)+\chi\int_0^t\int_{\mathbb{R}^{2}} v_{t}^{2}+\int_0^t\int_{\mathbb{R}^{2}} u|\nabla(\ln u-\chi v-\xi w)|^{2} = F(0)+\xi \int_0^t\int_{\mathbb{R}^{2}} u v w, \quad \forall t \in\left(0, T_{\max}\right). \end{eqnarray} (3.3)

    Proof. We use the same ideas as in the proofs of [45,Theorem 1.3], [46,Lemma 3.1] and [1,Theorem 3.2]. The equation of u can be written as u_t = \nabla \cdot(u \nabla(\ln u-\chi v-\xi w)) . Multiplying by \ln u-\chi v-\xi w and integrating by parts, we obtain

    \begin{align} -\int_{\mathbb{R}^{2}} u|\nabla(\ln u-\chi v-\xi w)|^{2} & = \int_{\mathbb{R}^{2}} u_{t}(\ln u-\chi v-\xi w) \\ & = \frac{d}{d t} \int_{\mathbb{R}^{2}}(u \ln u-\chi u v-\xi u w)+\chi \int_{\mathbb{R}^{2}} u v_{t}+\xi \int_{\mathbb{R}^{2}} u w_{t}. \end{align} (3.4)

    Substituting the second and third equation of (1.3) into (3.4) and integrating by parts, we have

    \begin{align} &-\int_{\mathbb{R}^{2}} u|\nabla(\ln u-\chi v-\xi w)|^{2} \\ = &\frac{d}{d t} \int_{\mathbb{R}^{2}}(u \ln u-\chi u v-\xi u w)+\chi \int_{\mathbb{R}^{2}} \left( v_{t}-\Delta v+v\right) v_{t}+\xi \int_{\mathbb{R}^{2}} uvw \\ = & \frac{d}{d t} \int_{\mathbb{R}^{2}}(u \ln u-\chi u v-\xi u w)-\chi\int_{\mathbb{R}^{2}} v_{t}^{2}+\frac{\chi}{2} \frac{d}{d t} \int_{\mathbb{R}^{2}}\left(v^{2}+|\nabla v|^{2}\right)+\xi \int_{\mathbb{R}^{2}} uvw, \end{align} (3.5)

    which, upon being integrated from 0 to t , yields simply that (3.3). We give some lemmas to deal with the term \int_{\mathbb{R}^{2}} u\ln u in (1.7).

    Lemma 3.3. ([1,Lemma 2.1]) Let \psi be any function such that e^{\psi} \in L^{1}\left(\mathbb{R}^{2}\right) and denote \bar{u} = M e^{\psi}\left(\int_{\mathbb{R}^{2}} e^{\psi} d x\right)^{-1} with M a positive arbitrary constant. Let E: L_{+}^{1}\left(\mathbb{R}^{2}\right) \rightarrow \mathbb{R} \cup\{\infty\} be the entropy functional

    \begin{eqnarray*} E(u ; \psi) = \int_{\mathbb{R}^{2}}\left(u \ln u-u \psi\right) d x \end{eqnarray*}

    and let RE: L_{+}^{1}\left(\mathbb{R}^{2}\right) \rightarrow \mathbb{R} \cup\{\infty\} defined by

    RE(u \mid \bar{u}) = \int_{\mathbb{R}^{2}} u \ln(\frac{u}{\bar{u}}) d x

    be the relative (to \bar{u} ) entropy.

    Then E(u; \psi) and RE(u\mid\bar{u}) are finite or infinite in the same time and for all u in the set \mathcal{U} = \left\{u \in L_{+}^{1}\left(\mathbb{R}^{2}\right), \int_{\mathbb{R}^{2}} u(x) d x = M\right\} and it holds true that

    E(u ; \psi)-E(\bar{u};\psi) = RE(u\mid\bar{u}) \geq 0 .

    Next, we give a Moser-Trudinger-Onofri inequality.

    Lemma 3.4. ([1,Lemma 3.1]) Let H be defined as H(x) = \frac{1}{\pi} \frac{1}{\left(1+|x|^{2}\right)^{2}} . Then

    \begin{eqnarray} \int_{\mathbb{R}^{2}} e^{\varphi(x)} H(x) d x \leq \exp \left\{\int_{\mathbb{R}^{2}} \varphi(x) H(x) d x+\frac{1}{16 \pi} \int_{\mathbb{R}^{2}}|\nabla \varphi(x)|^{2} d x\right\}, \end{eqnarray} (3.6)

    for all functions \varphi \in L^{1}\left(\mathbb{R}^{2}, H(x) d x\right) such that |\nabla \varphi(x)| \in L^{2}\left(\mathbb{R}^{2}, d x\right) .

    Lemma 3.5. ([1,Lemma 2.4]) Let \psi be any function such that e^\psi \in L^1\left(\mathbb{R}^2\right) , and let f be a non-negative function such that \left(f \bf{1}_{\{f \leq 1\}}\right) \in L^1\left(\mathbb{R}^2\right) \cap L^1\left(\mathbb{R}^2, |\psi(x)| d x\right) . Then there exists a constant C such that

    \int_{\mathbb{R}^2} f(x)(\ln f(x))_{-} d x \leq C-\int_{\{f \leq 1\}} f(x) \psi(x) d x.

    With the help of Lemma 3.2–3.5, we now use the subcritical mass condition (1.8) to derive a Gronwall-type inequality and to get a time-dependent bound for \|(u \ln u)(t)\|_{L^{1}} .

    Lemma 3.6. Under the subcritical mass condition (1.8) and (1.5), there exists C = C\left(u_{0}, v_{0}, w_{0}\right) > 0 such that

    \begin{eqnarray} \|(u \ln u)(t)\|_{L^{1}}+\|v(t)\|_{H^{1}}^{2} \leq C e^{\frac{\xi K}{\gamma} t}, \quad \forall t \in\left(0, T_{\max}\right), \end{eqnarray} (3.7)

    where K > 0 and \gamma are defined by (3.8) and (3.10) below, respectively.

    Proof. According to the third equation of (1.3), we have for all t\in (0, T)

    \begin{eqnarray} \|w\|_{L^{\infty}} \leq \|w_{0}\|_{L^{\infty}}: = K, \end{eqnarray} (3.8)

    then we apply the estimate of (3.3) to find that

    \begin{eqnarray} F(t)+\int_0^t\int_{\mathbb{R}^{2} } u|\nabla(\ln u-\chi v-\xi w)|^{2}\leq F(0)+\xi K \int_0^t \int_{\mathbb{R}^{2} } u v, \quad \forall t \in\left(0, T_{\max}\right). \end{eqnarray} (3.9)

    For our later purpose, since M < \frac{8\pi}{\chi} , we first choose some positive constant \gamma > 0 small enough to satisfy

    \begin{eqnarray} \chi-\frac{M(\chi+\gamma)^2}{8\pi} > 0, \end{eqnarray} (3.10)

    then by the definition of F(t) in (1.7), we use (3.1) and (3.8) to deduce that

    \begin{eqnarray} F(t)& = &\int_{\mathbb{R}^2} u \ln u-\chi \int_{\mathbb{R}^2} u v-\xi \int_{\mathbb{R}^2} u w+\frac{\chi}{2} \int_{\mathbb{R}^2}\left(v^{2}+|\nabla v|^{2}\right) \\&\geq&\int_{\mathbb{R}^2} u \ln u-(\chi+\gamma) \int_{\mathbb{R}^2} u v-\xi KM+\frac{\chi}{2} \int_{\mathbb{R}^2}\left(v^{2}+|\nabla v|^{2}\right)+ \gamma\int_{\mathbb{R}^2} u v. \end{eqnarray} (3.11)

    Similar as the calculation shown in [1], we set \bar{u}(x, t) = M e^{(\chi+\gamma) v(x, t)} H(x)\left(\int_{\mathbb{R}^{2}} e^{(\chi+\gamma) v(x, t)} H(x) d x\right)^{-1} , where H(x) is defined in Lemma 3.4. Then, we can apply the Entropy Lemma 3.3 with \psi = (\chi+\gamma) v+\ln H to obtain

    \begin{align} E(u ;(\chi+\gamma)v+\ln H) & \geq E(\bar{u} ;(\chi+\gamma)v+\ln H) \\ & = M \ln M-M \ln \left(\int_{\mathbb{R}^{2}} e^{(\chi+\gamma) v(x, t)} H(x) d x\right). \end{align} (3.12)

    Furthermore, applying Lemma 3.4 with \varphi = (\chi+\gamma) v to the last term in the right hand side of (3.12), we have that

    \begin{align} E(u ;(\chi+\gamma)v+\ln H) & = \int_{\mathbb{R}^2} u \ln u-(\chi+\gamma) \int_{\mathbb{R}^2} u v-\int_{\mathbb{R}^2} u \ln H\\ &\geq M \ln M-M \ln \left(\int_{\mathbb{R}^{2}} e^{(\chi+\gamma) v(x, t)} H(x) d x\right)\\ &\geq M \ln M-M(\chi+\gamma)\int_{\mathbb{R}^{2}} vH-\frac{M(\chi+\gamma)^2}{16\pi}\int_{\mathbb{R}^{2}}|\nabla v|^{2}. \end{align} (3.13)

    Then by Young's inequality, we have M(\chi+\gamma)\int_{\mathbb{R}^{2}} vH\leq \frac{M(\chi+\gamma)^2}{16\pi}\int_{\mathbb{R}^{2}}v^{2}+ 4M\pi \int_{\mathbb{R}^{2}}H^2 . Together with (3.13) and the fact \int_{\mathbb{R}^{2}} H^2(x) dx = \frac{1}{3\pi} , we can easily obtain

    \begin{align} &\int_{\mathbb{R}^2} u \ln u-(\chi+\gamma) \int_{\mathbb{R}^2} u v-\int_{\mathbb{R}^2} u \ln H\\ \geq &M \ln M-M(\chi+\gamma)\int_{\mathbb{R}^{2}} vH-\frac{M(\chi+\gamma)^2}{16\pi}\int_{\mathbb{R}^{2}}|\nabla v|^{2}\\ \geq &M \ln M-\frac{M(\chi+\gamma)^2}{16\pi}\int_{\mathbb{R}^{2}}v^{2}-\frac{M(\chi+\gamma)^2}{16\pi}\int_{\mathbb{R}^{2}}|\nabla v|^{2}- \frac{4}{3}M. \end{align} (3.14)

    Substituting (3.14) into (3.11), we have

    \begin{align} F(t)&\geq\int_{\mathbb{R}^2} u \ln u-(\chi+\gamma) \int_{\mathbb{R}^2} u v+\frac{\chi}{2} \int_{\mathbb{R}^2}\left(v^{2}+|\nabla v|^{2}\right)+\gamma\int_{\mathbb{R}^2} u v-\xi KM \\&\geq M \ln M+\int_{\mathbb{R}^2} u \ln H+\left(\frac{\chi}{2}-\frac{M(\chi+\gamma)^2}{16\pi}\right)\int_{\mathbb{R}^2}\left(v^{2}+|\nabla v|^{2}\right)+ \gamma\int_{\mathbb{R}^2} u v-(\xi K+ \frac{4}{3})M \\&\geq M \ln M+\int_{\mathbb{R}^2} u \ln H+\gamma\int_{\mathbb{R}^2} u v-(\xi K+ \frac{4}{3})M \end{align} (3.15)

    by (3.10). Now we turn to estimate the second term on the right side of (3.15). We set \phi(x) = \ln (1+|x|^2) , then we can obtain by Young's inequality

    \begin{align*} &\ \frac{d}{dt}\int_{\mathbb{R}^2} u\phi = \int_{\mathbb{R}^2} u_t\phi = \int_{\mathbb{R}^2} u \nabla \phi\cdot\nabla(\ln u-\chi v-\xi w)\nonumber\\ \leq&\int_{\mathbb{R}^2}u|\nabla \phi|^{2}+\frac{1}{4}\int_{\mathbb{R}^2} u|\nabla(\ln u-\chi v-\xi w)|^{2},\quad \text{for all}\ t\in(0,T_{\max}). \end{align*}

    By the fact |\nabla \phi(x)| = \left|\frac{2x}{1+|x|^2}\right|\leq1 , we have

    \begin{align*} \frac{d}{dt}\int_{\mathbb{R}^2} u\phi\leq \int_{\mathbb{R}^2}u+\frac{1}{4}\int_{\mathbb{R}^2} u|\nabla(\ln u-\chi v-\xi w)|^{2}\leq M+\frac{1}{4}\int_{\mathbb{R}^2} u|\nabla(\ln u-\chi v-\xi w)|^{2}, \end{align*}

    upon being integrated from 0 to t , which yields simply that for all t\in(0, T_{\max})

    \begin{eqnarray} \int_{\mathbb{R}^2} u\ln (1+|x|^2)\leq\int_{\mathbb{R}^2} u_0\ln (1+|x|^2)+Mt+\frac{1}{4}\int_0^t\int_{\mathbb{R}^2} u|\nabla(\ln u-\chi v-\xi w)|^{2}. \end{eqnarray} (3.16)

    By the definition of H(x) , we have for all t\in(0, T_{\max})

    \begin{align} \int_{\mathbb{R}^2} u \ln H& = -2\int_{\mathbb{R}^2}u\ln (1+|x|^2)- M\ln \pi \\&\geq-2\int_{\mathbb{R}^2} u_0\ln (1+|x|^2)-2Mt-\frac{1}{2}\int_0^t\int_{\mathbb{R}^2} u|\nabla(\ln u-\chi v-\xi w)|^{2}- M\ln \pi. \end{align} (3.17)

    Substituting (3.15) and (3.17) into (3.9), we have for all t\in(0, T_{\max})

    \begin{eqnarray} \gamma\int_{\mathbb{R}^2} u v\leq \xi K \int_0^t \int_{\mathbb{R}^{2} } u v+2Mt+F(0)+2\int_{\mathbb{R}^2}u_0\ln (1+|x|^2)+(\ln \pi+\xi K+ \frac{4}{3}-\ln M )M. \end{eqnarray} (3.18)

    From (1.4), we have assumed for convenience that u_{0} \ln u_{0} and u_0\ln (1+|x|^2) belongs to L^{1}(\mathbb{R}^2) for convenience. Then we conclude an integral-type Gronwall inequality as follows

    \begin{eqnarray} \gamma\int_{\mathbb{R}^2} u v\leq \xi K \int_0^t \int_{\mathbb{R}^{2} } u v+2Mt+C_1, \quad \forall t \in\left(0, T_{\max}\right), \end{eqnarray} (3.19)

    where C_1 = F(0)+2\int_{\mathbb{R}^2}u_0\ln (1+|x|^2)+(\ln \pi+\xi K+ \frac{4}{3}-\ln M)M is a finite number. Solving the integral-type Gronwall inequality (3.19) via integrating factor method, we infer that for some C_2 > 0

    \int_{\mathbb{R}^2} u v+\int_{0}^{t} \int_{\mathbb{R}^2} u v \leq C_{2} e^{\frac{\xi K}{\gamma} t}, \quad \forall t \in\left(0, T_{\max}\right) .

    Then by (3.9), one can simply deduce that F(t) grows no great than exponentially as well:

    \begin{eqnarray} F(t) \leq C_{3} e^{\frac{\xi K}{\gamma} t}, \quad \forall t \in\left(0, T_{\max}\right). \end{eqnarray} (3.20)

    Similarly, this along with (1.7) shows that for some C_4 > 0

    \begin{eqnarray} \int_{\mathbb{R}^2} u\ln u+\int_{\mathbb{R}^2} v^{2}+\int_{\mathbb{R}^2}|\nabla v|^{2} \leq C_{4} e^{\frac{\xi K}{\gamma} t}, \quad \forall t \in\left(0, T_{\max}\right). \end{eqnarray} (3.21)

    According to Lemma 3.5 with \psi = -(1+\delta) \ln \left(1+|x|^{2}\right) , for arbitrary \delta > 0 in order to have e^{-(1+\delta) \ln \left(1+|x|^{2}\right)} \in L^{1}\left(\mathbb{R}^{2}\right) , we have for all t \in(0, T_{\max})

    \begin{align} &\int_{\mathbb{R}^{2}} u(\ln u)_{-} d x\\ \leq & (1+\delta) \int_{\mathbb{R}^{2}} u \ln \left(1+|x|^{2}\right) d x+C_5 \\ \leq & (1+\delta) \left\{\int_{\mathbb{R}^2} u_0\ln (1+|x|^2)+Mt+\frac{1}{4}\int_0^t\int_{\mathbb{R}^2} u|\nabla(\ln u-\chi v-\xi w)|^{2}\right\}+C_5\\ \leq &\frac{1+\delta}{4}\left\{F(0)-F(t)+ \xi K \int_0^t \int_{\mathbb{R}^{2} } u v\right\}+M(1+\delta)t+C_6\\ \leq &C_{7}e^{\frac{\xi K}{\gamma} t} \end{align} (3.22)

    for some C_i > 0 (i = 5, 6, 7) . Finally, the identity

    \begin{eqnarray} \int_{\mathbb{R}^{2}}|u \ln u| d x = \int_{\mathbb{R}^{2}} u \ln u d x+2 \int_{\mathbb{R}^{2}} u(\ln u)_{-} d x \end{eqnarray} (3.23)

    gives that \|(u \ln u)(t)\|_{L^{1}}\leq C_{8}e^{\frac{\xi K}{\gamma} t} for some C_8 > 0 . Together with (3.21), this easily yield (3.7).

    Next, we wish to raise the regularity of u based on the local L^{1} -boundedness of u \ln u . In particular, for subcritical mass M , we have \int_{\mathbb{R}^{2}}(u(x, t)-k)_{+} d x \leq M for any k > 0 , while for k > 1 we have for all t \in(0, T_{\max})

    \begin{align} \int_{\mathbb{R}^{2}}(u(x, t)-k)_{+} d x & \leq \frac{1}{\ln k} \int_{\mathbb{R}^{2}}(u(x, t)-k)_{+} \ln u(x, t) d x \\ & \leq \frac{1}{\ln k} \int_{\mathbb{R}^{2}} u(x, t)(\ln u(x, t))_{+} d x\leq \frac{C e^{\frac{\xi K}{\gamma} t}}{\ln k}. \end{align} (3.24)

    Lemma 3.7. Under the condition (1.5) and (1.8), for any T \in(0, T_{\max}) , there exists C(T) > 0 such that the local solution (u, v, w) of (1.1) verifies that for any p\geq2

    \begin{eqnarray} \int_{\mathbb{R}^{2}} u^{p}(x, t) d x \leq C(T),\ \forall t \in\left(0, T\right], \end{eqnarray} (3.25)

    where C(T) = 2^p\bar{C}(T)+(2k)^{p-1}M with k and \bar{C}(T) respectively given by (3.37) and (3.40) below, which are finite for any T > 0 .

    Proof. Let k > 0 , to be chosen later. We derive a non-linear differential inequality for the quantity Y_{p}(t): = \int_{\mathbb{R}^{2}}(u(x, t)-k)_{+}^{p} d x , which guarantees that the L^{p} -norm of u remains finite.

    Multiplying the equation of u in (1.3) by p(u-k)_{+}^{p-1} yields, using integration by parts,

    \begin{align} &\frac{d}{d t} \int_{\mathbb{R}^{2}}(u-k)_{+}^{p} d x \\ = &-4 \frac{(p-1)}{p} \int_{\mathbb{R}^{2}}\left|\nabla(u-k)_{+}^{\frac{p}{2}}\right|^{2} d x -(p-1) \chi\int_{\mathbb{R}^{2}}(u-k)_{+}^{p} \Delta v d x-p k \chi\int_{\mathbb{R}^{2}}(u-k)_{+}^{p-1} \Delta v d x \\ & -(p-1)\xi \int_{\mathbb{R}^{2}}(u-k)_{+}^{p} \Delta w d x-p k \xi\int_{\mathbb{R}^{2}}(u-k)_{+}^{p-1} \Delta w d x \\ = &I_1+I_2+I_3+I_4+I_5. \end{align} (3.26)

    Now using the equation of v in (1.3) and the nonnegativity of v , one obtains

    \begin{eqnarray} I_2& = &-(p-1)\chi \int_{\mathbb{R}^{2}}(u-k)_{+}^{p} \Delta v dx \\& = &(p-1)\chi \int_{\mathbb{R}^{2}}(u-k)_{+}^{p}(-v_t-v+u)dx \\&\leq&-(p-1)\chi \int_{\mathbb{R}^{2}}(u-k)_{+}^{p}v_t+(p-1)\chi \int_{\mathbb{R}^{2}}(u-k)_{+}^{p+1}dx+(p-1)k\chi \int_{\mathbb{R}^{2}}(u-k)_{+}^{p}dx \end{eqnarray} (3.27)

    and

    \begin{eqnarray} I_3& = &-p k \chi \int_{\mathbb{R}^{2}}(u-k)_{+}^{p-1} \Delta vdx\\& = &pk\chi \int_{\mathbb{R}^{2}}(u-k)_{+}^{p-1}(-v_t-v+u)dx\\&\leq&-pk \chi \int_{\mathbb{R}^{2}}(u-k)_{+}^{p-1}v_tdx+pk\chi \int_{\mathbb{R}^{2}}(u-k)_{+}^{p}dx+pk^2\chi \int_{\mathbb{R}^{2}}(u-k)_{+}^{p-1}dx. \end{eqnarray} (3.28)

    Using Gagliardo-Nirenberg inequality \int_{\mathbb{R}^{2}} f^{4}(x) d x \leq C \int_{\mathbb{R}^{2}} f^{2}(x) d x \int_{\mathbb{R}^{2}}|\nabla f(x)|^{2} d x with f = (u-k)_{+}^{\frac{p}{2}} and Hölder inequality, we obtain for \varepsilon > 0

    \begin{align} \left|\int_{\mathbb{R}^{2}}(u-k)_{+}^{p} v_t d x\right| & \leq\left(\int_{\mathbb{R}^{2}}(u-k)_{+}^{2 p} d x\right)^{\frac{1}{2}}\left\|v_t\right\|_{L^{2}} \\ & \leq C\left(\int_{\mathbb{R}^{2}}(u-k)_{+}^{p} d x\right)^{\frac{1}{2}}\left(\int_{\mathbb{R}^{2}}\left|\nabla(u-k)_{+}^{\frac{p}{2}}\right|^{2} d x\right)^{1 / 2}\left\|v_t\right\|_{L^{2}} \\ & \leq C(p)\varepsilon\left\| v_t\right\|_{L^{2}}^{2} \int_{\mathbb{R}^{2}}(u-k)_{+}^{p} d x+\frac{2}{\varepsilon p} \int_{\mathbb{R}^{2}}\left|\nabla(u-k)_{+}^{\frac{p}{2}}\right|^{2} d x. \end{align} (3.29)

    Similarly, we have, for p \geq \frac{3}{2}

    \begin{align} \left|\int_{\mathbb{R}^{2}}(u-k)_{+}^{p-1} v_t d x\right| \leq & \left(\int_{\mathbb{R}^{2}}(u-k)_{+}^{2(p-1)} d x\right)^{\frac{1}{2}}\left\| v_t\right\|_{L^{2}} \\ \leq& \left(C(M, p)+C(p) \int_{\mathbb{R}^{2}}(u-k)_{+}^{2 p} d x\right)^{\frac{1}{2}}\left\| v_t\right\|_{L^{2}} \\ \leq & C(M, p)\left\| v_t\right\|_{L^{2}}+ C(p)\varepsilon\left\| v_t\right\|_{L^{2}}^{2} \int_{\mathbb{R}^{2}}(u-k)_{+}^{p} d x \\ &+\frac{p-1}{\varepsilon p^2 k} \int_{\mathbb{R}^{2}}\left|\nabla(u-k)_{+}^{\frac{p}{2}}\right|^{2} d x. \end{align} (3.30)

    Then we can see that

    \begin{eqnarray} I_2+I_3&\leq &(p-1)\int_{\mathbb{R}^{2}}(u-k)_{+}^{p+1}dx+\frac{(p-1)}{p} \int_{\mathbb{R}^{2}}\left|\nabla(u-k)_{+}^{\frac{p}{2}}\right|^{2} d x \\&\ &+C(p,\chi)(k+1)\left\|v_{t} \right\|_{L^{2}}^2 \int_{\mathbb{R}^{2}}(u-k)_{+}^{p} d x+ C(M,p,\chi)k\left\|v_{t} \right\|_{L^{2}} \\&\ &+(2p-1)k\chi\int_{\mathbb{R}^{2}}(u-k)_{+}^{p} d x+pk^2\chi\int_{\mathbb{R}^{2}}(u-k)_{+}^{p-1} d x \end{eqnarray} (3.31)

    by setting \varepsilon = 4\chi . According to the equation of w and v and (3.8), one obtains for all t\in(0, T)

    \begin{align} -\Delta w(x, t) = & -\Delta w_{0}(x) \mathrm{e}^{-\int_{0}^{t} v(x, s) \mathrm{d} s}+2 \mathrm{e}^{-\int_{0}^{t} v(x, s) \mathrm{d} s} \nabla w_{0}(x) \cdot \int_{0}^{t} \nabla v(x, s) \mathrm{d} s \\&-w_{0}(x) \mathrm{e}^{-\int_{0}^{t} v(x, s) \mathrm{d} s} \left|\int_{0}^{t} \nabla v(x, s) \mathrm{d} s\right|^{2}+w_{0}(x) \mathrm{e}^{-\int_{0}^{t} v(x, s) \mathrm{d} s} \int_{0}^{t} \Delta v(x, s) \mathrm{d} s \\ \leq&\|\Delta w_{0}\|_{L^{\infty}}-\mathrm{e}^{-\int_{0}^{t} v(x, s) \mathrm{d} s}\left(\sqrt{w_0}\int_{0}^{t} \nabla v(x, s) \mathrm{d} s-\frac{\nabla w_{0}}{\sqrt{w_0}} \right)^2+\mathrm{e}^{-\int_{0}^{t} v(x, s)ds}\frac{\left|\nabla w_{0}\right|^2}{w_0} \\&+w_{0}(x) \mathrm{e}^{-\int_{0}^{t} v(x, s) \mathrm{d} s} \int_{0}^{t} \left(v_s(x,s)+v-u\right) \mathrm{d} s. \end{align} (3.32)

    Here to estimate the last integral of the right side of (3.32) we first note (1.7) guarantees that

    \begin{aligned} w_{0}(x) \mathrm{e}^{-\int_{0}^{t} v(x, s) \mathrm{d} s} \int_{0}^{t} \left(v_s(x,s)+v-u\right) \mathrm{d}s&\leq \|w_{0}\|_{L^{\infty}}\mathrm{e}^{-\int_{0}^{t} v(x, s) \mathrm{d} s}\left[v(x,t)-v_0+\int_{0}^{t} v(x, s)\mathrm{d} s\right]\\&\leq \|w_{0}\|_{L^{\infty}}v+\frac{\|w_{0}\|_{L^{\infty}}}{\mathrm{e}}, \quad \forall t \in\left(0, T\right) \end{aligned}

    by the nonnegativity of w_0 and v_0 and the fact \mathrm{e}^{-x}x\leq\frac{1}{\mathrm{e}} for all x > 0 . Substituting (3.8) and (3.32) into (3.26), we have

    \begin{align*} I_4+I_5 = &-(p-1)\xi \int_{\mathbb{R}^{2}}(u-k)_{+}^{p} \Delta w d x-p k\xi \int_{\mathbb{R}^{2}}(u-k)_{+}^{p-1} \Delta w d x\nonumber\\ \leq&(p-1)K\xi \int_{\mathbb{R}^{2}}(u-k)_{+}^{p}v d x+ (p-1)K_1 \xi\int_{\mathbb{R}^{2}}(u-k)_{+}^{p} dx\nonumber\\&+p k K\xi\int_{\mathbb{R}^{2}}(u-k)_{+}^{p-1}v d x+p k K_1 \xi \int_{\mathbb{R}^{2}}(u-k)_{+}^{p-1}d x, \end{align*}

    where K_1 = \|\Delta w_{0}\|_{L^{\infty}}+4\|\nabla \sqrt{w_{0}}\|_{L^{\infty}}^2+\frac{K}{\mathrm{e}} . Applying similar procedure as (3.29) and (3.30) to \int_{\mathbb{R}^{2}}(u-k)_{+}^{p} vdx and \int_{\mathbb{R}^{2}}(u-k)_{+}^{p-1} vdx , this yields

    \begin{align} I_4+I_5\leq&\frac{(p-1)}{p} \int_{\mathbb{R}^{2}}\left|\nabla(u-k)_{+}^{\frac{p}{2}}\right|^{2} d x+C(p,K,\xi)k\left\| v\right\|_{L^{2}}^{2} \int_{\mathbb{R}^{2}}(u-k)_{+}^{p} d x+ C(M, p,K,\xi)k\left\| v\right\|_{L^{2}}\\&+(p-1)K_1 \xi \int_{\mathbb{R}^{2}}(u-k)_{+}^{p} dx+p k K_1\xi\int_{\mathbb{R}^{2}}(u-k)_{+}^{p-1}d x \end{align} (3.33)

    by setting \varepsilon = 2K\xi . Combining (3.26), (3.31) and (3.33), we have for all t\in(0, T)

    \begin{align} &\frac{d}{d t} \int_{\mathbb{R}^{2}}(u-k)_{+}^{p} d x\\ \leq&-2\frac{(p-1)}{p} \int_{\mathbb{R}^{2}}\left|\nabla(u-k)_{+}^{\frac{p}{2}}\right|^{2} d x+(p-1) \int_{\mathbb{R}^{2}}(u-k)_{+}^{p+1} dx \\&+[(2p-1)k\chi+(p-1)K_1\xi]\int_{\mathbb{R}^{2}}(u-k)_{+}^{p} dx+(pk^2\chi+pkK_1\xi)\int_{\mathbb{R}^{2}}(u-k)_{+}^{p-1} dx \\&+C(p,K,\chi,\xi)(k+1)\left(\left\|\partial_{t} v\right\|_{L^{2}}^2+\left\| v\right\|_{L^{2}}^2\right) \int_{\mathbb{R}^{2}}(u-k)_{+}^{p} d x+ C(M,p,K,\chi,\xi)k\left(\left\|\partial_{t} v\right\|_{L^{2}}+\left\| v\right\|_{L^{2}}\right). \end{align} (3.34)

    Next, we estimate the nonlinear and negative contribution -2\frac{(p-1)}{p} \int_{\mathbb{R}^{2}}|\nabla(u-k)_{+}^{\frac{p}{2}}|^{2} d x in terms of \int_{\mathbb{R}^{2}}(u-k)_{+}^{p+1} dx , with the help of the Sobolev's inequality \|f\|_{L^2}^2\leq c_1 \|\nabla f\|_{L^1}^2 . Indeed, by (3.24),

    \begin{align} \int_{\mathbb{R}^{2}}(u-k)_{+}^{p+1} d x & = \int_{\mathbb{R}^{2}}\left((u-k)_{+}^{\frac{(p+1)}{2}}\right)^{2} d x \leq c_1\left(\int_{\mathbb{R}^{2}}\left|\nabla(u-k)_{+}^{\frac{(p+1)}{2}}\right| d x\right)^{2} \\ & = C(p)\left(\int_{\mathbb{R}^{2}}(u-k)_{+}^{\frac{1}{2}}\left|\nabla(u-k)_{+}^{\frac{p}{2}}\right| d x\right)^{2} \\ & \leq C(p) \int_{\mathbb{R}^{2}}(u-k)_{+} d x \int_{\mathbb{R}^{2}}\left|\nabla(u-k)_{+}^{p / 2}\right|^{2} d x \\ & \leq C(p) \frac{C e^{\frac{\xi K}{\gamma} T}}{\ln k} \int_{\mathbb{R}^{2}}\left|\nabla(u-k)_{+}^{p / 2}\right|^{2} d x, \quad \forall 0 < t \leq T. \end{align} (3.35)

    Moreover, since for p \geq 2 it holds true that

    \begin{equation} \int_{\mathbb{R}^{2}}(u-k)_{+}^{p-1} d x \leq \int_{\mathbb{R}^{2}}(u-k)_{+} d x+\int_{\mathbb{R}^{2}}(u-k)_{+}^{p} d x. \end{equation} (3.36)

    Inserting (3.35) and (3.36) into (3.34) gives for p \geq 2 and 0 < t \leq T that

    \begin{equation*} \begin{aligned} &\frac{d}{d t} \int_{\mathbb{R}^{2}}(u-k)_{+}^{p} d x \\& \leq(p-1)\left(1-\frac{2\ln k}{p C(p)C e^{\frac{\xi K}{\gamma} T}}\right) \int_{\mathbb{R}^{2}}(u-k)_{+}^{p+1} d x \\ &\quad+C(p,K,\chi,\xi)k\left(1+\left\|\partial_{t} v\right\|_{L^{2}}^2+\left\| v\right\|_{L^{2}}^2\right) \int_{\mathbb{R}^{2}}(u-k)_{+}^{p} d x+ C(M,p,K,\chi,\xi)k\left(\left\|\partial_{t} v\right\|_{L^{2}}+\left\| v\right\|_{L^{2}}+1\right). \end{aligned} \end{equation*}

    For any fixed p we can choose k = k(p, T) sufficiently large such that

    \begin{equation} \delta: = \frac{2\ln k}{p C(p)C e^{\frac{\xi K}{\gamma} T}}-1 > 0, \end{equation} (3.37)

    namely, k = \exp\left(\frac{(1+\delta)p C(p)C e^{\frac{\xi K}{\gamma} T}}{2}\right) . For such a k , using the interpolation

    \begin{aligned} \int_{\mathbb{R}^{2}}(u-k)_{+}^{p} d x & \leq\left(\int_{\mathbb{R}^{2}}(u-k)_{+} d x\right)^{\frac{1}{p}}\left(\int_{\mathbb{R}^{2}}(u-k)_{+}^{p+1} d x\right)^{\left(1-\frac{1}{p}\right)} \\ & \leq M^{\frac{1}{p}}\left(\int_{\mathbb{R}^{2}}(u-k)_{+}^{p+1} d x\right)^{\left(1-\frac{1}{p}\right)}, \end{aligned}

    we end up with the following differential inequality for Y_{p}(t) , p\geq2 fixed and 0 < t \leq T

    \begin{align} \frac{d}{d t} Y_{p}(t)& \leq-(p-1) M^{-\frac{1}{p-1}} \delta Y_{p}^{\beta}(t)+c_{2}(p,K,\chi,\xi)k\left(1+\left\|\partial_{t} v\right\|_{L^{2}}^2+\left\| v\right\|_{L^{2}}^2\right) Y_{p}(t) \\&\quad+c_{3}(M,p,K,\chi,\xi)k\left(1+\left\|\partial_{t} v\right\|_{L^{2}}^2+\left\| v\right\|_{L^{2}}^2\right), \end{align} (3.38)

    where \beta = \frac{p}{p-1} > 1 . Let us write the differential inequality (3.38) as follows for simplicity:

    \begin{equation} \frac{d}{d t} Y_{p}(t) \leq-\tilde{C} Y_{p}^{\beta}(t)+g(t) Y_{p}(t)+g(t), \quad 0 < t \leq T, \end{equation} (3.39)

    where g(t) = \bar{C}(M, p, K, \chi, \xi)k\left(1+\left\|\partial_{t} v\right\|_{L^{2}}^2+\left\| v\right\|_{L^{2}}^2\right) and \tilde{C} = (p-1) M^{-\frac{1}{p-1}} \delta > 0 . According to (3.7), (3.9) and (3.20), we can see that g(t)\leq \bar{C}(M, p, K, \chi, \xi)ke^{\frac{\xi Kt}{\gamma}} . Then by comparison inequality, we show that there exists a constant \bar{C}(T) such that for all t\in(0, T)

    \begin{align} Y_{p}(t) &\leq Y_{p}(0)\exp\left(\int_0^t g(s)ds\right)+\int_0^tg(\tau)\exp\left(\int_\tau^t g(s)ds\right)d\tau \\&\leq Y_{p}(0)\bar{C}(M, p, K,\chi,\xi) k e^{\frac{\xi K T}{\gamma}}T+\bar{C}(M, p, K,\chi,\xi) k e^{\frac{\xi K T}{\gamma}} e^{\bar{C}(M, p, K,\chi,\xi) k e^{\frac{\xi K T}{\gamma}}} T: = \bar{C}(T). \end{align} (3.40)

    It is sufficient to observe that for any k > 0

    \begin{align} \int_{\mathbb{R}^{2}} u^{p}(x, t) d x & = \int_{\{u \leq 2 k\}} u^{p}(x, t) d x+\int_{\{u > 2 k\}} u^{p}(x, t) d x \\ & \leq(2 k)^{p-1} M+2^{p} \int_{\{u > 2 k\}}(u(x, t)-k)^{p} d x \\ & \leq(2 k)^{p-1} M+2^{p} \int_{\mathbb{R}^{2}}(u(x, t)-k)_{+}^{p} d x, \end{align} (3.41)

    where the inequality x^{p} \leq 2^{p}(x-k)^{p} , for x \geq 2 k , has been used. Therefore, (3.25) follows for any p \geq 2 by (3.40) and (3.41) choosing k = k(p, T) sufficiently large such that (3.37) holds true.

    Proof of Theorem 1.1. According to the local L^p- boundedness of Lemma 3.7 and Lemma 2.2 we must have the local H^2\times H^3\times H^3 -boundedness of (u, v, w) , which contracts the extensibility criteria in (2.35). Then we must obtain that T_{\max} = \infty , that is, the strong solution (u, v, w) of (1.3) exists globally in time and is locally bounded as in (2.2).

    The authors convey sincere gratitude to the anonymous referees for their careful reading of this manuscript and valuable comments which greatly improved the exposition of the paper. The authors are supported in part by National Natural Science Foundation of China (No. 12271092, No. 11671079).

    The authors declare there is no conflict of interest.



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