Citation: Changgui Gu, Ping Wang, Tongfeng Weng, Huijie Yang, Jos Rohling. Heterogeneity of neuronal properties determines the collective behavior of the neurons in the suprachiasmatic nucleus[J]. Mathematical Biosciences and Engineering, 2019, 16(4): 1893-1913. doi: 10.3934/mbe.2019092
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In the present work, we shall consider a chemotaxis-haptotaxis model
{ut=Δu−χ∇⋅(u∇v)−ξ∇⋅(u∇w)+μu(1−u−w),τvt=Δv−v+u,wt=−vw+ηu(1−u−w), | (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>(N−2)+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 N≥3 with μ=η=τ=0; the blow-up results for two dimension was discussed in [2] with wt=−vw+ηw(1−w) 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 N≤3 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=Δu−χ∇⋅(u∇v),τ∂tv=Δu−λv+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 ‖u0‖L1 is greater than 8πχ. If ‖u0‖L1<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 (N≥3) 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 N≥3 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 u0lnu0∈L1(R2), any solution exists globally in time if ‖u0‖L1<8πχ. In [39], the extra assumptions on u0 were removed, while the condition on mass was restricted to ‖u0‖L1<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 ‖u0‖L1<8πχ by two different method. Furthermore the global existence of solutions was also obtained under some condition on u0 in the critical case ‖u0‖L1=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=Δu−χ∇⋅(u∇v)−ξ∇⋅(u∇w),x∈R2, t>0,vt=Δv−v+u,x∈R2, t>0,wt=−vw,x∈R2, t>0,u(x,0)=u0(x), v(x,0)=v0(x), w(x,0)=w0(x),x∈R2. | (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) |
u0∈L1(R2,ln(1+|x|2)dx) and u0lnu0∈L1(R2) | (1.5) |
and
Δw0∈L∞(R2) and ∇√w0∈L∞(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:=‖u0‖L1<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 ‖u0‖L1<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)=∫R2ulnu−χ∫R2uv−ξ∫R2uw+χ2∫R2(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
‖u0‖L1<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 u0≥0. 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)j≥0 by iteratively solving the Cauchy problems of the following system
{∂tuj+1=Δuj+1−χ∇⋅(uj+1∇vj)−ξ∇⋅(uj+1∇wj),x∈R2, t>0,∂tvj+1=Δvj+1−vj+1+uj,x∈R2, t>0,∂twj+1=−vj+1wj+1,x∈R2, t>0,u(x,0)=u0(x), v(x,0)=v0(x), w(x,0)=w0(x),x∈R2. | (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,v1∈V1,122([0,T]×R2), then we get w1∈C1([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}j≥1 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}j≥1 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
sup0≤t≤T(‖uj‖H2+‖vj‖H3+‖wj‖H3)≤R, | (2.3) |
where R=2{‖u0‖H2+‖v0‖H3+‖w0‖H3}+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
12ddt‖vj+1(t)‖2L2+‖∇vj+1‖2L2=−∫R2(vj+1)2+∫R2vj+1uj≤−12‖vj+1‖2L2+12‖uj‖2L2. | (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
12ddt‖∇vj+1(t)‖2L2+‖∂tvj+1‖2L2=−∫R2vj+1∂tvj+1+∫R2uj∂tvj+1≤12‖∂tvj+1‖2L2+‖vj+1‖2L2+‖uj‖2L2. | (2.5) |
For the H2 estimate of vj+1, by Young's inequality, we have
12ddt‖∇2vj+1(t)‖2L2+‖∇Δvj+1‖2L2=−∫R2(Δvj+1)2+∫R2Δvj+1Δuj≤−12‖Δvj+1‖2L2+12‖Δuj‖2L2. | (2.6) |
Similarly, integrating by parts, it is clear that for all t∈(0,T)
ddt‖∇3vj+1(t)‖2L2=2∫R2∇3vj+1∇3(Δvj+1−vj+1+uj)=−2∫R2|∇4vj+1|2−2∫R2|∇3vj+1|2−2∫R2∇4vj+1∇2uj≤−‖∇4vj+1‖2L2−2‖∇3vj+1‖2L2+‖∇2uj‖2L2, |
togethering with (2.3)–(2.6) and adjusting the coefficients carefully, we can find a positive constant α such that
ddt‖vj+1(t)‖2H3+α(‖vj+1‖2H4+‖∂tvj+1‖2L2)≤c1(‖vj+1‖2H3+‖uj‖2H2) | (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+α∫t0(‖vj+1(s)‖2H4+‖∂tvj+1(s)‖2L2)≤ec1T‖v0‖2H3+ec1TTsupt∈(0,T)‖uj(t)‖2H2≤ec1T‖v0‖2H3+ec1TTR2≤2‖v0‖2H3+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)e−∫t0vj+1(x,s)ds, | (2.9) |
∇wj+1(x,t)=∇w0(x)e−∫t0vj+1(x,s)ds−w0(x)e−∫t0vj+1(x,s)ds∫t0∇vj+1(x,s)ds | (2.10) |
as well as
Δwj+1(x,t)=Δw0(x)e−∫t0vj+1(x,s)ds−2e−∫t0vj+1(x,s)ds∇w0(x)⋅∫t0∇vj+1(x,s)ds+w0(x)e−∫t0vj+1(x,s)ds|∫t0∇vj+1(x,s)ds|2−w0(x)e−∫t0vj+1(x,s)ds∫t0Δvj+1(x,s)ds. | (2.11) |
From (2.9), we can easily get for t∈(0,T)
‖wj+1‖Lp≤‖w0‖Lp,∀p∈(1,∞]. | (2.12) |
From (2.10), by (2.8), the definition of R and the following inequality
‖∫t0f(x,s)ds‖Lp={∫R2|∫t0f(x,s)ds|pdx}1p≤{tp−1∫R2∫t0|f|pdsdx}1p≤tsups∈(0,t)‖f(s)‖Lp,for all p∈(1,∞), | (2.13) |
we can obtain
‖∇wj+1‖L2≤‖∇w0‖L2+‖w0e−∫t0vj+1ds∫t0∇vj+1ds‖L2≤‖∇w0‖L2+‖w0‖L∞‖∫t0∇vj+1ds‖L2≤‖∇w0‖L2+‖w0‖L∞Tsupt∈(0,T)‖∇vj+1‖L2≤‖∇w0‖L2+c2R2T≤‖∇w0‖L2+1 | (2.14) |
by setting T small enough to satisfy c2R2T<1.
Similarly, by the embedding H2↪W1,4 and (2.13), we can obtain from (2.11) for all t∈(0,T)
‖Δwj+1‖L2≤‖Δw0‖L2+2‖∇w0∫t0∇vj+1ds‖L2+‖w0|∫t0∇vj+1ds|2‖L2 +‖w0∫t0Δvj+1ds‖L2≤‖Δw0‖L2+2‖∇w0‖L4‖∫t0∇vj+1ds‖L4+‖w0‖L∞‖|∫t0∇vj+1ds|2‖L2 +‖w0‖L∞‖∫t0Δvj+1ds‖L2≤‖Δw0‖L2+2‖∇w0‖L4Tsupt∈(0,T)‖∇vj+1‖L4+‖w0‖L∞T2supt∈(0,T)‖∇vj+1‖2L2 +‖w0‖L∞Tsupt∈(0,T)‖Δvj+1‖L2≤‖Δw0‖L2+c3R2T+c3R3T2≤‖Δw0‖L2+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)
ddt‖∇3wj+1(t)‖2L2=2∫R2∇3wj+1∇3wj+1t=−2∫R2∇3wj+1∇3(vj+1wj+1)≤2‖∇3wj+1‖L2‖∇3(vj+1wj+1)‖L2≤c4‖∇3wj+1‖L2‖∇3wj+1vj+1+3∇2wj+1∇vj+1+3∇2vj+1∇wj+1+∇3vj+1wj+1‖L2≤c4{‖∇3wj+1‖2L2‖vj+1‖L∞+‖∇3wj+1‖L2‖∇2wj+1‖L4‖∇vj+1‖L4+‖∇3wj+1‖L2‖∇2vj+1‖L2‖∇wj+1‖L∞+‖∇3wj+1‖L2‖∇3vj+1‖L2‖wj+1‖L∞}. | (2.16) |
By Galiardo-Nirenberg inequality, we have
‖∇2wj+1‖L4≤c5‖∇3wj+1‖34L2‖wj+1‖14L∞,‖∇wj+1‖L∞≤c5‖∇3wj+1‖12L2‖wj+1‖12L∞. |
Together with Young's inequality, (2.8), (2.12) and (2.16), we can get
ddt‖∇3wj+1(t)‖2L2≤c6(‖vj+1‖2H3+1)‖∇3wj+1‖2L2+c6‖∇wj+1‖2L∞≤c7R2‖∇3wj+1‖2L2+c6‖∇w0‖2L∞≤c7R2‖∇3wj+1‖2L2+c7R2. | (2.17) |
Then, we can deduce from Gronwall's inequality that for all t∈(0,T)
‖∇3wj+1‖2L2≤ec7R2T3‖∇3w0‖2L2+ec7R2T3c7R2T≤2‖∇3w0‖2L2+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+1‖H3≤2‖w0‖H3+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
12ddt‖uj+1(t)‖2L2+‖∇uj+1‖2L2=χ∫R2uj+1∇vj⋅∇uj+1+ξ∫R2uj+1∇wj⋅∇uj+1≤‖uj+1‖L2‖∇vj‖L∞‖∇uj+1‖L2+‖uj+1‖L2‖∇wj‖L∞‖∇uj+1‖L2. | (2.20) |
By (2.8), (2.19) and (2.20) and the embedding H3↪W1,∞, we can see for all t∈(0,T)
12ddt‖uj+1(t)‖2L2+‖∇uj+1‖2L2≤c8R2‖uj+1‖2L2+12‖∇uj+1‖2L2. | (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)
12ddt‖∇uj+1(t)‖2L2+‖Δuj+1‖2L2=χ∫R2Δuj+1∇⋅(uj+1∇vj)+ξ∫R2Δui+1∇⋅(ui+1∇wj)=I1+I2. | (2.22) |
By Hölder inequality and Young's inequality, it yields that
I1≤c9‖Δuj+1‖L2‖∇(uj+1∇vj)‖L2≤c10‖Δuj+1‖L2{‖uj+1‖H1‖vj‖H2}≤14‖Δuj+1‖2L2+c210‖vj‖2H2‖uj+1‖2H1. |
Applying the similar procedure to I2, we can obtain
I2≤14‖Δuj+1‖2L2+c210‖wj‖2H2‖uj+1‖2H1, |
which entails that for all t∈(0,T)
12ddt‖∇uj+1(t)‖2L2+12‖Δuj+1‖2L2≤c11R2‖uj+1‖2H1. | (2.23) |
Similar as (2.16), we can get
ddt‖∇2uj+1(t)‖2L2=2∫R2∇2uj+1∇2uj+1t=2∫R2∇2uj+1{∇2(Δuj+1−χ∇⋅(uj+1∇vj)−ξ∇⋅(uj+1∇wj))}=2∫R2∇2uj+1∇2Δuj+1−2χ∫R2∇2uj+1∇3(uj+1∇vj)−2ξ∫R2∇2ui+1∇3(uj+1∇wj)=−2‖∇3uj+1‖2L2+2χ∫R2∇3uj+1∇2(uj+1∇vj)+2ξ∫R2∇3ui+1∇2(ui+1∇wj)≤−‖∇3uj+1‖2L2+c12{‖∇2(uj+1∇vj)‖2L2+‖∇2(ui+1∇wj)‖2L2}≤−‖∇3uj+1‖2L2+c13R2‖uj+1‖2H2. | (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
ddt‖uj+1(t)‖2H2+β‖uj+1‖2H3≤c14R2‖uj+1‖2H2, | (2.25) |
which implies from Gronwall's inequality that
‖uj+1‖2H2+β∫t0‖uj+1‖2H3⩽ | (2.26) |
by choosing T small enough to satisfy e^{c_{14}R^2T} < 2 .
Combining (2.8), (2.19) and (2.26), we can get for all t\in(0, T)
\begin{eqnarray} \left\|u^{j+1}\right\|_{H^{2}}+\left\|v^{j+1}\right\|_{H^{2}}+\left\|w^{j+1}\right\|_{H^{3}}\leq 2\left\{\left\|u_0\right\|_{H^1}+\left\|v_0\right\|_{H^2}+\left\|w_0\right\|_{H^3}\right\}+7\leq R, \end{eqnarray} | (2.27) |
by the definition of R .
Convergence: The derivation of the relevant estimates of u^{j+1}-u^j , v^{j+1}-v^j and w^{j+1}-w^j are similar to the ones of u^{j+1} , v^{j+1} and w^{j+1} , so we omit the details. For simplicity, we denote \delta f^{j+1}: = f^{j+1}-f^j . Subtracting the j -th equations from the (j+1) -th equations, we have the following equations for \delta u^{j+1}, \delta v^{j+1} and \delta w^{j+1} :
\begin{eqnarray} \left\{\begin{array}{l} \partial_{t} \delta u^{j+1} = \Delta\delta u^{j+1}-\chi\nabla \cdot\left(\delta u^{j+1} \nabla v^{j}\right)-\chi\nabla \cdot\left( u^{j} \nabla\delta v^{j}\right) -\xi\nabla \cdot\left(\delta u^{j+1} \nabla w^{j}\right)-\xi\nabla \cdot\left(u^{j} \nabla\delta w^{j}\right), \\ \partial_{t} \delta v^{j+1} = \Delta\delta v^{j+1}-\delta v^{j+1}+\delta u^{j},\\ [1mm] \partial_{t}\delta w^{j+1} = -v^{j+1}\delta w^{j+1}-\delta v^{j+1} w^{j+1}. \end{array}\right. \end{eqnarray} | (2.28) |
(ⅰ) Estimates of \delta v^{j+1} . Using the same procedure as proving (2.8), we can obtain that for all t \in (0, T)
\begin{eqnarray} \left\|\delta v^{j+1}(t)\right\|_{H^3}^2+\alpha\int_0^t\left(\left\|\delta v^{j+1}(s)\right\|_{H^{4}}^{2}+\left\|\partial_{t}\delta v^{j+1}(s)\right\|_{L^{2}}^{2} \right)ds \leq e^{ c_{15}T} c_{15}T \sup \limits_{0 \leq t \leq T}\left\|\delta u^{j}(t)\right\|_{H^2}^2. \end{eqnarray} | (2.29) |
(ⅱ) Estimates of \delta w^{j+1} . According to the third equation of (2.28), we have for all t \in (0, T)
\begin{equation} \delta w^{j+1}(t) = -\int_0^t e^{-\int_s^t v^{j+1}(\tau) d \tau} w^{j+1}(s) \delta v^{j+1}(s) d s. \end{equation} | (2.30) |
Using the same procedure as proving (2.19) entails that for all t \in (0, T)
\left\|\delta w^{j+1}\right\|_{L^2} \leqslant \sup\limits_{t \in (0, T)}\left\|w^{j+1}\right\|_{L^{\infty}} \sup \limits_{t \in(0, T)}\left\|\delta v^{j+1}\right\|_{L^2} T \leqslant c_{16}R T \sup\limits _{t \in (0, T)}\left\|\delta v^{j+1}\right\|_{L^2}, |
\begin{eqnarray*} \begin{aligned} \left\|\nabla \delta w^{j+1}\right\|_{L^2} \leqslant &\left\|\int_0^t \nabla w^{j+1}(s)\delta v^{j+1}(s) d s\right\|_{L^2}+\left\|\int_0^t w^{j+1}(s) \nabla \delta v^{j+1}(s) d s\right\|_{L^2} \\ &+\left\|\int_0^t w^{j+1} \delta v^{j+1} \int_s^t \nabla v^{j+1} d \tau d s\right\|_{L^2} \\ \leqslant & c_{17}R T\sup \limits_{t \in (0, T)}\left\|\delta v^{j+1}\right\|_{L^2}+c_{17}R T \sup \limits_{t \in (0, T)}\left\|\nabla \delta v^{j+1}\right\|_{L^2}+c_{17}R^2 T^2\sup \limits_{t \in (0, T)}\left\|\delta v^{j+1}\right\|_{L^2} \\ \leqslant & c_{18}(R^2+R) T\sup \limits_{t \in (0, T)}\left\|\delta v^{j+1}\right\|_{H^{1}},\\ \left\|\Delta \delta w^{j+1}\right\|_{L^2} = &\left\|-\int_0^t \Delta\left\{e^{-\int_s^{t} v^{j+1} d \tau} w^{j+1} \delta v^{j+1}\right\} d s\right\|_{L^2}\\ \leqslant& T \sup \limits_{t \in (0, T)}\left\|\Delta\left\{e^{-\int_s^{t} v^{j+1} d \tau} w^{j+1} \delta v^{j+1}\right\}\right\|_{L^2}\\ \leqslant &T \sup \limits_{t \in (0, T)}\Big\{\left(\left\| v^{j+1}\right\|_{H^2}^2T^2+\left\|v^{j+1}\right\|_{H^2}T\right)\left\|w^{j+1}\right\|_{H^2}\left\|\delta v^{j+1}\right\|_{H^2}\Big\}\\ \leqslant &c_{19}(R^3+R^2) T \sup \limits_{t \in (0, T)}\left\|\delta v^{j+1}\right\|_{H^2} \end{aligned} \end{eqnarray*} |
and
\left\|\nabla^3 \delta w^{j+1}\right\|_{L^2}^2 \leqslant e^{c_{20}R^2T}T\sup \limits_{t \in (0, T)} \left\|\delta v^{j+1}\right\|_{H^3}, |
which imply that for all t \in (0, T)
\begin{eqnarray} \left\| \delta w^{j+1}\right\|_{H^3}\leq c_{21}(R^3+R^2+R)T \sup \limits_{0 \leq t \leq T}\left\|\delta u^{j}(t)\right\|_{H^2}. \end{eqnarray} | (2.31) |
by setting T < 1 .
(ⅲ) Estimates of \delta u^{j+1} . Using the same procedure as proving (2.26) entails that for all t \in (0, T)
\begin{eqnarray} \sup \limits_{0 \leq t \leq T}\left\|\delta u^{j+1}\right\|_{H^2}^2+\beta\int_0^t \left\|\delta u^{j+1}\right\|_{H^3}^2 \leqslant e^{c_{22}RT}c_{22}RT\sup \limits_{0 \leq t \leq T}\left(\left\|\delta v^j\right\|_{H^3}^2+\left\|\delta w^j\right\|_{H^2}^2\right). \end{eqnarray} | (2.32) |
Combining with (2.29), (2.31) and (2.32), we can obtain for all t \in (0, T)
\begin{align} &\sup\limits_{0 \leq t \leq T}\left(\left\|\delta u^{j+1}\right\|_{H^{2}}+\left\|\delta v^{j+1}\right\|_{H^3}+\left\|\delta w^{j+1}\right\|_{H^3}\right) \\ \leq& e^{c_{23}(R^3+R^2+R)T}c_{23}(R^3+R^2+R)T \sup\limits _{0 \leq t \leq T}\left(\left\|\delta u^j\right\|_{H^{2}}+\left\|\delta v^j\right\|_{H^3}+\left\|\delta w^j\right\|_{H^3}\right). \end{align} | (2.33) |
Taking T > 0 small enough, we can find a constant r \in(0, 1) such that
\begin{equation} \sup\limits _{0 \leq t \leq T}\left(\left\|\delta u^{j+1}\right\|_{H^{2}}+\left\|\delta v^{j+1}\right\|_{H^3}+\left\|\delta w^{j+1}\right\|_{H^3}\right) \leq r \sup \limits_{0 \leq t \leq T}\left(\left\|\delta u^j\right\|_{H^{2}}+\left\|\delta v^j\right\|_{H^3}+\left\|\delta w^j\right\|_{H^3}\right) \end{equation} | (2.34) |
for any j \geq 1 and t \in (0, T) . From the above inequality, we find that \left(u^j, v^j, w^j\right) is a Cauchy sequence in the Banach space C\left(0, T; X\right) for some small T > 0 , and thus its corresponding limit denoted by \left(u, v, w\right) definitely exists in the same space.
Uniqueness: If (\bar{u}, \bar{v}, \bar{w}) is another local-in-time solution of system (1.3), (\tilde{u}, \tilde{v}, \tilde{w}): = (u-\bar{u}, v-\bar{v}, w-\bar{w}) solves
\begin{eqnarray*} \left\{\begin{array}{lll} \partial_{t} \tilde{u} = \Delta \tilde{u}-\chi\nabla \cdot\left(\tilde{u} \nabla \bar{v}\right)-\chi\nabla \cdot\left( u \nabla\tilde{v}\right) -\xi\nabla \cdot\left(\tilde{u} \nabla\bar{w}\right)-\xi\nabla \cdot\left(u \nabla \tilde{w}\right), &{} x\in\mathbb{R}^2,\ 0 < t\leq T,\\ \partial_{t} \tilde{v} = \Delta\tilde{v}-\tilde{v}+\tilde{u},&{}x\in\mathbb{R}^2,\ 0 < t\leq T,\\ \partial_{t}\tilde{w} = -v\tilde{w}-\tilde{v}\bar{w},&{}x\in\mathbb{R}^2,\ 0 < t\leq T,\\ \tilde{u}(x,0) = \tilde{v}(x,0) = \tilde{w}(x,0) = 0,&{} x\in\mathbb{R}^2, \end{array}\right. \end{eqnarray*} |
where T is any time before the maximal time of existence. Following a same procedure as (2.34), we can deduce that \tilde{u} = \tilde{v} = \tilde{w} = 0 , which implies the uniqueness of the local solution.
Nonnegativity: The nonnegativity of w^j can be easily obtained by (2.9) and the nonnegativity of w_0 . We will use the induction argument to show that u^j and v^j are nonnegative for all j > 0 . We assume that u^j and v^j are nonnegative. If we apply the maximum principle to the second equation of (2.2), we find v^{j+1} is nonnegative ( u^j is nonnegative). Then we turn to deal with u^{j+1} . Let us decompose u^{j+1} = u_{+}^{j+1}-u_{-}^{j+1} , where u_{+}^{j+1} = \left\{\begin{array}{ll}u^{j+1} & u^{j+1} \geq 0 \\ 0 & u^{j+1} < 0\end{array}\quad\text{and} \quad u_{-}^{j+1} = \left\{\begin{array}{cc}-u^{j+1} &u^{j+1} \leq 0 \\ 0 & u^{j+1} > 0\end{array}\right.\right. . Now multiplying the negative part u_{-}^{j+1} on both sides of the first equation of (2.2) and integrating over [0, t]\times{\mathbb{R}^2} , we can get
\begin{aligned} &\int_0^t \int_{\mathbb{R}^2} \partial_t u^{j+1}u_{-}^{j+1} d x d s\\ = &-\int_0^t\left\|\nabla\left(u_{-}^{j+1}\right)\right\|_{L^2}^2 d s+ \chi\int_0^t\int_{\mathbb{R}^2} u^{j+1}\nabla v^{j}\cdot\nabla u_{-}^{j+1}+\xi\int_{\mathbb{R}^2} u^{j+1}\nabla w^{j}\cdot\nabla u_{-}^{j+1} \\ \leq& C \int_0^t\left\|\left(u^{j+1}\right)_{-}\right\|_{L^2}^2\left(\left\|\nabla v^j\right\|_{L^{\infty}}^2+\left\|\nabla w^j\right\|_{L^{\infty}}^2\right)+\frac{1}{2}\left\|\nabla\left(u_{-}^{j+1}\right)\right\|_{L^2}^2 d s \end{aligned} |
by Young's inequality and the fact the weak derivative of u_{-}^{j+1} is -\nabla u^{j+1} if u_{-}^{j+1} < 0 , otherwise zero. Since u_{-}^{j+1}, \ \partial_t u^{j+1} \in L^2\left(0, T; L^2\left(\mathbb{R}^2\right)\right) , we can have
\int_0^t \int_{\mathbb{R}^2} \partial_t u^{j+1}\left(u^{j+1}\right)_{-} d x d s = \frac{1}{2}\left(\left\|\left(u_{-}^{j+1}\right)(0)\right\|_{L^2}-\left\|\left(u_{-}^{j+1}\right)(0)\right\|_{L^2}\right), |
together with the above inequality, it holds that
\left\|\left(u_{-}^{j+1}\right)(t)\right\|_{L^2}^2 \leq\left\|\left(u_{-}^{j+1}\right)(0)\right\|_{L^2}^2 \exp \left(C \int_0^t\left(\left\|\nabla v^j\right\|_{L^{\infty}}^2+\left\|\nabla w^j\right\|_{L^{\infty}}^2\right) d s\right). |
Due to the fact u_{-}^{j+1}(0) is nonnegative, we can deduce that u^{j+1} is nonnegative. This completes the proof of Lemma 2.1.
Remark 2.1. Since the above choice of T depends only on \|u_0\|_{H^2(\mathbb{R}^2)} , \|v_0\|_{H^3(\mathbb{R}^2)} and \|w_0\|_{H^3(\mathbb{R}^2)} , it is clear by a standard argument that (u, v, w) can be extended up to some T_{\max}\leq\infty . If T_{\max} < \infty in Lemma 2.1, then
\begin{eqnarray} \limsup _{t \rightarrow T_{\max}}\left(\|u(t)\|_{H^2(\mathbb{R}^2)}+\|v(t)\|_{H^3(\mathbb{R}^2)}+\|w(t)\|_{H^3(\mathbb{R}^2)}\right) = \infty. \end{eqnarray} | (2.35) |
In order to show the H^2\times H^3\times H^3 -boundedness of (u, v, w) , it suffices to estimate a suitable L^p -norm of u , with some large, but finite p .
Lemma 2.2. Suppose that \chi, \xi > 0 and the initial data (u_0, v_0, w_0) satisfy all the assumptions presented in Lemma 2.1. Then for every K > 0 there is C > 0 such that whenever (u, v, w) \in C\left(0, T;H^2(\mathbb{R}^2)\times H^3(\mathbb{R}^2)\times H^3(\mathbb{R}^2)\right) solves (1.3) for some T > 0 and q_0 > 2 satisfies
\begin{eqnarray} \|u(\cdot, t)\|_{L^{q_0}} \leq K, \mathit{\text{for all}} t \in(0, T), \end{eqnarray} | (2.36) |
then
\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 \mathit{\text{for all}} t \in(0, T). \end{eqnarray} | (2.37) |
Proof. Firstly, we suppose that for some q_0 > 2 and K > 0
\begin{eqnarray} \|u(t)\|_{L^{q_0}(\mathbb{R}^2)}\leq K, \quad \text{for all } t\in(0,T). \end{eqnarray} | (2.38) |
By the Duhamel principle, we represent u and v of the following integral equations
\begin{eqnarray} u(t) = e^{t \Delta} u_{0}-\chi\int_{0}^{t} e^{(t-\tau) \Delta} \nabla\cdot(u \nabla v)(\tau) d \tau-\xi\int_{0}^{t} e^{(t-\tau) \Delta} \nabla\cdot(u \nabla w)(\tau) d \tau \end{eqnarray} | (2.39) |
and
\begin{eqnarray} v(t) = e^{-t(-\Delta+1)} v_{0}+\int_{0}^{t} e^{-(t-\tau)(-\Delta+1)} u(\tau) d \tau. \end{eqnarray} | (2.40) |
where e^{t \Delta} f(x) = \int_{\mathbb{R}^{2}} G(x-y, t)f(y)dy and
G(x, t) = G_t(x): = \frac{1}{(4 \pi t)} \exp \left(-\frac{|x|^2}{4 t}\right) |
is the Gaussian heat kernel. The following well-known L^{p}-L^{q} estimates of the heat semigroup play an important role in the proofs [47,48]. For 1 \leqslant p \leqslant q \leqslant \infty and f \in L^{q}\left(\mathbb{R}^{2}\right) , we have
\begin{aligned} &\left\|e^{t \Delta} f\right\|_{L^p} \leqslant(4 \pi t)^{-\left(\frac{1}{q}-\frac{1}{p}\right)}\|f\|_{L^q}, \\ &\left\|\nabla e^{t \Delta} f\right\|_{L^p} \leqslant C_3 t^{-\frac{1}{2}-\left(\frac{1}{q}-\frac{1}{p}\right)}\|f\|_{L^q}, \end{aligned} |
where C_3 is a constant depending on p and q . Then, according to (2.39), we can see that for q_0 > 2 and all t\in(0, T)
\begin{align} \|u(t)\|_{L^{\infty}}&\leq \|u_0\|_{L^{\infty}}+\chi\int_{0}^{t}(t-\tau)^{-\frac{1}{2}-\frac{1}{q_0}}\|u\cdot\nabla v\|_{L^{q_0}}d \tau+\xi\int_{0}^{t}(t-\tau)^{-\frac{1}{2}-\frac{1}{q_0}}\|u\cdot\nabla w\|_{L^{q_0}}d \tau \\&\leq\|u_0\|_{L^{\infty}}+K\chi\int_{0}^{t}(t-\tau)^{-\frac{1}{2}-\frac{1}{q_0}}\|\nabla v\|_{L^{\infty}}d \tau+K\xi\int_{0}^{t}(t-\tau)^{-\frac{1}{2}-\frac{1}{q_0}}\|\nabla w\|_{L^{\infty}}d \tau. \end{align} | (2.41) |
From (2.40) and the above L^{p}-L^{q} estimates of the heat semigroup, we have
\begin{eqnarray} \|v(t)\|_{L^{q}}\leq \|v_0\|_{L^{q}}+\int_{0}^{t}e^{-(t-\tau)}(t-\tau)^{-(\frac{1}{q_0}-\frac{1}{q})}\|u(t)\|_{L^{q_0}}d \tau\leq C_4,\forall q\in(1,\infty] \end{eqnarray} | (2.42) |
and by the embedding H^3(\mathbb{R}^2)\hookrightarrow W^{1, \infty}(\mathbb{R}^2)
\begin{eqnarray} \|\nabla v(t)\|_{L^{\infty}}\leq \|\nabla v_0\|_{L^{\infty}}+\int_{0}^{t}e^{-(t-\tau)}(t-\tau)^{-\frac{1}{2}-\frac{1}{q_0}}\|u(t)\|_{L^{q_0}}d \tau\leq C_5, \end{eqnarray} | (2.43) |
where C_4 and C_5 depend on \| v_0\|_{H^{3}} and K in (2.38).
According to the equation of w , we can see that for some C_6 = C_6(\|w_0\|_{H^{3}}, \| v_0\|_{H^{2}}, K, T) > 0 and all t\in(0, T)
\begin{align} \|\nabla w(t)\|_{L^{\infty}}&\leq \left\|\nabla w_0e^{-\int_0^tv(s) ds}\right\|_{L^{\infty}}+\left\|w_0e^{-\int_0^tv(s) ds}\int_0^t \nabla v(s) ds\right\|_{L^{\infty}} \\&\leq \|w_0\|_{H^{3}}+\|w_0\|_{L^{\infty}}\sup\limits_{t\in(0,T)}\|\nabla v(t)\|_{L^{\infty}}T\leq C_6 \end{align} | (2.44) |
by the embedding H^3(\mathbb{R}^2)\hookrightarrow W^{1, \infty}(\mathbb{R}^2) . Inserting (2.43) and (2.44) into (2.41), this yields for all t\in(0, T)
\begin{eqnarray} \|u(t)\|_{L^{\infty}}\leq\|u_0\|_{L^{\infty}}+KC_5\chi\int_{0}^{t}(t-\tau)^{-\frac{1}{2}-\frac{1}{q_0}}d \tau+KC_6\xi\int_{0}^{t}(t-\tau)^{-\frac{1}{2}-\frac{1}{q_0}}d \tau\leq C_7, \end{eqnarray} | (2.45) |
where C_7 depends on \|u_0\|_{H^{2}} , \| v_0\|_{H^{2}} , \|w_0\|_{H^{3}} , 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\in(0, T)
\begin{align} \frac{1}{2} \frac{d}{d t}\left\|\partial_{t}v(t)\right\|_{L^{2}}^{2}+\left\|\nabla\partial_{t} v\right\|_{L^{2}}^{2}& = -\int_{\mathbb{R}^2} (\partial_{t}v)^{2}+\int_{\mathbb{R}^2} \partial_{t}v\partial_{t}u \\&\leq-\frac{1}{2}\left\|\partial_{t}v\right\|_{L^{2}}^{2}+\frac{1}{2}\left\|\partial_{t}u\right\|_{L^{2}}^{2} \end{align} | (2.46) |
and
\begin{align} \frac{1}{2} \frac{d}{d t}\left\|\nabla v(t)\right\|_{L^{2}}^{2}+\left\|\Delta v\right\|_{L^{2}}^{2}& = \int_{\mathbb{R}^2} v \Delta v+\int_{\mathbb{R}^2} u \Delta v\\&\leq-\left\|\nabla v\right\|_{L^{2}}^{2}+\frac{1}{2}\left\|u\right\|_{L^{2}}^{2}+\frac{1}{2}\left\|\Delta v\right\|_{L^{2}}^{2}. \end{align} | (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.
[1] | D. K. Welsh, J. S. Takahashi and S. A. Kay, Suprachiasmatic nucleus: cell autonomy and network properties, Annu. Rev. Physiol., 72 (2010), 551–577. |
[2] | G. Zerbini, T. Kantermann and M. Merrow, Strategies to decrease social jetlag: reducing evening blue light advances sleep and melatonin, Eur. J. Neurosci., (2018). |
[3] | A. Slomski, Melatonin Improves Sleep in Patients With Circadian Disruption, JAMA, 320 (2018), 749. |
[4] | H. Daido, Why circadian rhythms are circadian: competitive population dynamics of biological oscillators, Phys. Rev. Lett., 87 (2001), 048101. |
[5] | C. A. Czeisler, J. F. Duffy, T. L. Shanahan, et al., Stability, precision, and near-24-hour period of the human circadian pacemaker, Science, 284 (1999), 2177–2181. |
[6] | J. A. Mohawk, C. B. Green and J. S. Takahashi, Central and peripheral circadian clocks in mammals, Annu. Rev. Neurosci., 35 (2012), 445–462. |
[7] | D. K. Welsh, D. E. Logothetis, M. Meister, et al., Individual neurons dissociated from rat suprachiasmatic nucleus express independently phased circadian firing rhythms, Neuron., 14 (1995), 697–706. |
[8] | S. Honma, W. Nakamura, T. Shirakawa, et al., Diversity in the circadian periods of single neurons of the rat suprachiasmatic nucleus depends on nuclear structure and intrinsic period, Neurosci. Lett., 358 (2004), 173–176. |
[9] | S. M. Reppert and D. R. Weaver, Molecular analysis of mammalian circadian rhythms, Annu. Rev. Physiol., 63 (2001), 647–676. |
[10] | E. D. Herzog, S. J. Aton, R. Numano, et al. Temporal precision in the mammalian circadian system: a reliable clock from less reliable neurons, J. Biol. Rhythms., 19 (2004), 35–46. |
[11] | D. M. Berson, Strange vision: ganglion cells as circadian photoreceptors, Trends Neurosci., 26 (2003), 314–320. |
[12] | S. Usui, Y. Takahashi, T. Okazaki, Range of entrainment of rat circadian rhythms to sinusoidal light-intensity cycles, Am. J. Physiol. Regul. Integr. Comp. Physiol., 278 (2000), R1148–1156. |
[13] | U. Abraham, A. E. Granada, P. O. Westermark, et al., Coupling governs entrainment range of circadian clocks, Mol. Syst. Biol., 6 (2010): 438. |
[14] | R. Refinetti, (2006) Circadian Physiology (2nd edn.). Boca Raton, Florida: CRC Press. |
[15] | H. O. de la Iglesia, T. Cambras, W. J. Schwartz, et al., Forced desynchronization of dual circadian oscillators within the rat suprachiasmatic nucleus, Curr. Biol., 14 (2004), 796–800. |
[16] | A. B. Webb, N. Angelo, J. E. Huettner, et al., Intrinsic, nondeterministic circadian rhythm generation in identified mammalian neurons, Proc. Natl. Acad. Sci. USA, 106 (2009), 16493–16498. |
[17] | P. O. Westermark, D. K. Welsh., H. Okamura, et al., Quantification of circadian rhythms in single cells, PLoS Comput. Biol., 5 (2009), e1000580. |
[18] | T. Hamada, J. LeSauter, J. M.Venuti, et al., Expression of Period genes: rhythmic and nonrhythmic compartments of the suprachiasmatic nucleus pacemaker, J. Neurosci., 21 (2001), 7742–7750. |
[19] | A. B. Webb, S. R. Taylor, K. A. Thoroughman, et al., Weakly circadian cells improve resynchrony, PLoS Comput. Biol., 8 (2012), e1002787. |
[20] | J. H. Rohling, H. T. vanderLeest, S. Michel, et al., Phase resetting of the mammalian circadian clock relies on a rapid shift of a small population of pacemaker neurons, PLoS One, 6 (2011), e25437. |
[21] | H. Dardente, V. J. Poirel, P. Klosen, et al., Per and neuropeptide expression in the rat suprachiasmatic nuclei: compartmentalization and differential cellular induction by light, Brain Res., 958 (2002), 261–271. |
[22] | H. T. vanderLeest, J. H. Rohling, S. Michel, et al., Phase shifting capacity of the circadian pacemaker determined by the SCN neuronal network organization, PLoS One 4 (2009), e4976. |
[23] | M. D. Schwartz, C. Wotus, T. Liu, et al., Dissociation of circadian and light inhibition of melatonin release through forced desynchronization in the rat, Proc. Natl. Acad. Sci. U S A, 106 (2009), 17540–17545. |
[24] | T. Noguchi, K. Watanabe, A. Ogura, et al., The clock in the dorsal suprachiasmatic nucleus runs faster than that in the ventral, Eur. J. Neurosci., 20 (2004), 3199–3202. |
[25] | S. Yamaguchi, H. Isejima, T. Matsuo, et al., Synchronization of cellular clocks in the suprachiasmatic nucleus, Science, 302 (2003), 1408–1412. |
[26] | H. Ohta, S. Yamazaki and D.G.McMahon, Constant light desynchronizes mammalian clock neurons, Nat. Neurosci., 8 (2005), 267–269. |
[27] | S. J. Aton, C. S. Colwell, A. J. Harmar, et al., Vasoactive intestinal polypeptide mediates circadian rhythmicity and synchrony in mammalian clock neurons, Nat. Neurosci., 8 (2005), 476–483. |
[28] | L. P. Morin, SCN organization reconsidered, J. Biol. Rhythms., 22 (2007), 3–13. |
[29] | H. Albus, M. J. Vansteensel, S. Michel, et al., A GABAergic mechanism is necessary for coupling dissociable ventral and dorsal regional oscillators within the circadian clock, Curr. Biol., 15 (2005), 886–893. |
[30] | J. A. Evans, T. L. Leise, O. Castanon-Cervantes, et al., Dynamic interactions mediated by nonredundant signaling mechanisms couple circadian clock neurons, Neuron., 80 (2013), 973–983. |
[31] | S. R. Taylor, T. J. Wang, D. Granados-Fuentes, et al., Resynchronization Dynamics Reveal that the Ventral Entrains the Dorsal Suprachiasmatic Nucleus, J. Biol. Rhythms., 32 (2017), 35–47. |
[32] | J. C. Leloup and A. Goldbeter, Toward a detailed computational model for the mammalian circadian clock, Proc. Natl. Acad. Sci. U S A, 100 (2003), 7051–7056. |
[33] | H. Dardente and N. Cermakian, Molecular circadian rhythms in central and peripheral clocks in mammals, Chronobiol. Int., 24 (2007), 195–213. |
[34] | S. Becker-Weimann, J. Wolf, H. Herzel, et al., Modeling feedback loops of the Mammalian circadian oscillator, Biophys. J., 87 (2004), 3023–3034. |
[35] | D. Gonze, S. Bernard, C. Waltermann, et al., Spontaneous synchronization of coupled circadian oscillators, Biophys. J., 89 (2005), 120–129. |
[36] | A. T. Winfree, (2001) The geometry of biological time. New York: Springer-Verlag. |
[37] | S. Bernard, D. Gonze, B. Cajavec, et al., Synchronization-induced rhythmicity of circadian oscillators in the suprachiasmatic nucleus, PLoS Comput. Biol., 3 (2007), e68. |
[38] | J. C. Locke, P. O. Westermark, A. Kramer, et al., Global parameter search reveals design principles of the mammalian circadian clock, BMC Syst. Biol., 2 (2008), 22. |
[39] | T. Hirota and Y. Fukada, Resetting mechanism of central and peripheral circadian clocks in mammals, Zoolog. Sci., 21 (2004), 359–368. |
[40] | C. Liu, D. R. Weaver, S. H. Strogatz, et al., Cellular construction of a circadian clock: period determination in the suprachiasmatic nuclei, Cell, 91 (1997), 855–860. |
[41] | B. Ananthasubramaniam, E. D. Herzog and H. Herzel, Timing of neuropeptide coupling determines synchrony and entrainment in the mammalian circadian clock, PLoS Comput. Biol., 10 (2014), e1003565. |
[42] | Z. Lu, K. Klein-Cardena, S. Lee, et al., Resynchronization of circadian oscillators and the east-west asymmetry of jet-lag,. Chaos, 26 (2016), 094811. |
[43] | C. Schmal, J. Myung, H. Herzel, et al., A theoretical study on seasonality, Front. Neurol., 6 (2015), 94. |
[44] | C. Vasalou, E. D. Herzog and M. A. Henson, Small-World Network Models of Intercellular Coupling Predict Enhanced Synchronization in the Suprachiasmatic Nucleus, J. Biol. Rhythms., 24 (2009), 243–254. |
[45] | C. Vasalou and M. A. Henson, A multicellular model for differential regulation of circadian signals in the core and shell regions of the suprachiasmatic nucleus, J. Theor. Biol., 288 (2011), 44–56. |
[46] | C. Bodenstein, M. Gosak, S. Schuster, et al., Modeling the seasonal adaptation of circadian clocks by changes in the network structure of the suprachiasmatic nucleus, PLoS Comput. Biol., 8 (2012), e1002697. |
[47] | H. Kori and A. S. Mikhailov, Entrainment of randomly coupled oscillator networks by a pacemaker, Phys. Rev. Lett., 93 (2004), 254101. |
[48] | H. Kori and A. S. Mikhailov, Strong effects of network architecture in the entrainment of coupled oscillator systems, Phys. Rev. E. Stat. Nonlin. Soft. Matter. Phys., 74 (2006), 066115. |
[49] | S. Varadarajan, M. Tajiri, R. Jain, et al., Connectome of the Suprachiasmatic Nucleus: New Evidence of the Core-Shell Relationship, eNeuro., 5 (2018). |
[50] | L. P. Morin and C. N. Allen, The circadian visual system, 2005, Brain Res. Rev., 51 (2006), 1–60. |
[51] | J. Hannibal, M. Moller, O. P. Ottersen, et al., PACAP and glutamate are co-stored in the retinohypothalamic tract, J. Comp. Neurol., 418 (2000), 147–155. |
[52] | D. A. Golombek, R. E. Rosenstein, Physiology of circadian entrainment, Physiol. Rev., 90 (2010), 1063–1102. |
[53] | C. Gu, A. Ramkisoensing, Z. Liu, et al., The proportion of light-responsive neurons determines the limit cycle properties of the suprachiasmatic nucleus, J. Biol. Rhythms., 29 (2014), 16–27. |
[54] | C. Gu, Z. Liu, W. J. Schwartz, et al, Photic desynchronization of two subgroups of circadian oscillators in a network model of the suprachiasmatic nucleus with dispersed coupling strengths, PLoS One, 7 (2012), e36900. |
[55] | C. Gu, H. Yang, J. H. Meijer, et al., Dependence of the entrainment on the ratio of amplitudes between two subgroups in the suprachiasmatic nucleus, Phys. Rev. E., 97 (2018), 062215. |
[56] | J. H. Meijer and W. J. Rietveld, Neurophysiology of the suprachiasmatic circadian pacemaker in rodents, Physiol. Rev., 69 (1989), 671–707. |
[57] | C. Gu, J. Xu, Z. Liu, et al., Entrainment range of nonidentical circadian oscillators by a light-dark cycle, Phys. Rev. E. Stat. Nonlin Soft. Matter. Phys., 88 (2013), 022702. |
[58] | C. Gu and H. Yang, The asymmetry of the entrainment range induced by the difference in intrinsic frequencies between two subgroups within the suprachiasmatic nucleus, Chaos 27 (2017), 063115. |
[59] | C. Gu, X. Liang, H. Yang, et al., Heterogeneity induces rhythms of weakly coupled circadian neurons, Sci. Rep., 6 (2016), 21412. |
[60] | C. Gu and H. Yang, Differences in intrinsic amplitudes of neuronal oscillators improve synchronization in the suprachiasmatic nucleus, Chaos, 27 (2017), 093108. |
[61] | C. Gu, H. Yang and Z. Ruan, Entrainment range of the suprachiasmatic nucleus affected by the difference in the neuronal amplitudes between the light-sensitive and light-insensitive regions, Phys. Rev. E., 95 (2017), 042409. |
[62] | C. Gu, M. Tang, J. H. Rohling, et al., The effects of non-self-sustained oscillators on the en-trainment ability of the suprachiasmatic nucleus, Sci. Rep., 6 (2016), 37661. |
[63] | C. Gu, H. Yang and J. H. Rohling, Dissociation between two subgroups of the suprachiasmatic nucleus affected by the number of damped oscillated neurons, Phys. Rev. E., 95 (2017), 032302. |
[64] | C. Gu, J. Wang and Z. Liu, Free-running period of neurons in the suprachiasmatic nucleus: Its dependence on the distribution of neuronal coupling strengths, Phys Rev E Stat Nonlin Soft Matter. Phys., 80 (2009), 030904. |
[65] | C. Gu, J. H. Rohling, X. Liang, et al. Impact of dispersed coupling strength on the free running periods of circadian rhythms, Phys. Rev. E., 93 (2016), 032414. |
[66] | C. Gu, M. Tang and H.Yang, The synchronization of neuronal oscillators determined by the directed network structure of the suprachiasmatic nucleus under different photoperiods, Sci. Rep., 6 (2016), 28878. |
[67] | H. T. VanderLeest, T. Houben, S. Michel, et al., Seasonal encoding by the circadian pacemaker of the SCN, Curr. Biol., 17 (2007), 468–473. |
[68] | C. Gu and H. Yang, The circadian rhythm induced by the heterogeneous network structure of the suprachiasmatic nucleus, Chaos, 26 (2016), 053112. |
[69] | C. G. Gu, P. Wang and H. J. Yang, Entrainment range affected by the heterogeneity in the amplitude relaxation rate of suprachiasmatic nucleus neurons, Chin. Phys. B, 28 (2019), 018701. |
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