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Impact of online public opinion regarding the Japanese nuclear wastewater incident on stock market based on the SOR model


  • The exposure of the Japanese nuclear wastewater incident has shaped online public opinion and has also caused a certain impact on stocks in aquaculture and feed industries. In order to explore the impact of network public opinion caused by public emergencies on relevant stocks, this paper uses the stimulus organism response(SOR) model to construct a framework model of the impact path of network public opinion on the financial stock market, and it uses emotional analysis, LDA and grounded theory methods to conduct empirical analysis. The study draws a new conclusion about the impact of online public opinion on the performance of relevant stocks in the context of the nuclear waste water incident in Japan. The positive change of media sentiment will lead to the decline of stock returns and the increase of volatility. The positive change of public sentiment will lead to the decline of stock returns in the current period and the increase of stock returns in the lag period. At the same time, we have proved that media attention, public opinion theme and prospect theory value have certain influences on stock performance in the context of the Japanese nuclear wastewater incident. The conclusion shows that after the public emergency, the government and investors need to pay attention to the changes of network public opinion caused by the event, so as to avoid the possible stock market risks.

    Citation: Wei Hong, Yiting Gu, Linhai Wu, Xujin Pu. Impact of online public opinion regarding the Japanese nuclear wastewater incident on stock market based on the SOR model[J]. Mathematical Biosciences and Engineering, 2023, 20(5): 9305-9326. doi: 10.3934/mbe.2023408

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  • The exposure of the Japanese nuclear wastewater incident has shaped online public opinion and has also caused a certain impact on stocks in aquaculture and feed industries. In order to explore the impact of network public opinion caused by public emergencies on relevant stocks, this paper uses the stimulus organism response(SOR) model to construct a framework model of the impact path of network public opinion on the financial stock market, and it uses emotional analysis, LDA and grounded theory methods to conduct empirical analysis. The study draws a new conclusion about the impact of online public opinion on the performance of relevant stocks in the context of the nuclear waste water incident in Japan. The positive change of media sentiment will lead to the decline of stock returns and the increase of volatility. The positive change of public sentiment will lead to the decline of stock returns in the current period and the increase of stock returns in the lag period. At the same time, we have proved that media attention, public opinion theme and prospect theory value have certain influences on stock performance in the context of the Japanese nuclear wastewater incident. The conclusion shows that after the public emergency, the government and investors need to pay attention to the changes of network public opinion caused by the event, so as to avoid the possible stock market risks.



    In a series of works [1,2], Kim and his collaborators introduced the following chemotaxis model

    {ut=(1w(uw)x)x,wt=dwxxκ(w)u, (1.1)

    where u is the bacterial density and w is the concentration of nutrient. d0 is the diffusion rate of nutrient. κ(w)0 is the consumption rate. Typical examples of κ(w) include κ(w)=wm with m0.

    System (1.1) is an alternative model to describe the propagation of traveling band of bacteria observed in the experiment of Adler [3]. Compared with the classical Keller-Segel system [4], this model is rigorously derived from the notion of "metric of food", and brings the theory of Riemannian geometry to the field of chemotaxis. Choi and Kim [1] have proved that system (1.1) with d=0 can generate traveling bands and traveling fronts under various assumptions on κ(w). They also generalized their results of [1] to the models with porous medium diffusion for the bacterial density, and showed that there exist compactly supported traveling waves for chemotaxis. Ahn, Choi and Yoo [5] proved the global existence of strong solutions of Cauchy problem of system (1.1) if the initial value of w has positive lower bound. Very recently, they [6] have generalized this result to the case where w, with infinite initial mass, can be zero at spatial infinity.

    In this paper, we are interested in the existence and stability of spiky patterns to system (1.1). We assume that the consumption rate is linear for simplicity, and write system (1.1) as

    {ut=(1w2(uxuwxw))x,wt=dwxxwu. (1.2)

    We shall consider the system in the half-space R+=[0,), with the following initial value

    (u,w)(x,0)=(u0(x),w0(x)), (1.3)

    and boundary conditions

    {(uxuwxw)(0,t)=0, w(0,t)=b,(u,w)(+,t)=(0,0). (1.4)

    where b>0 is a constant. That means we prescribe no-flux boundary condition for the bacterial density and saturated boundary condition for the oxygen. This kind of boundary conditions have also been used in a chemotaxis-fluid model to describe the formation of concentration patterns for aerobic bacteria observed in the experiment of [7].

    System (1.2) is actually a chemotaxis model with signal-suppressed motility. In other words, the diffusion rate of the bacterial density is monotonically decreasing as the concentration of the signal increases. There are several analytical works for the chemotaxis model of self-aggregation type with signal-suppressed motility in bounded domains. See [8,9] for the global existence of classical solutions if the motility function satisfies the power law, [10,11] for the existence of critical mass generating blowup if the motility is an exponential function, [12] for the formation of spiky patterns. In contrast, system (1.2) is of consumption type. That is the chemical signal w is consumed by the bacteria u. It turns out that such two types of chemotaxis model may exhibit different dynamics. Indeed, one can easily verify (following the argument of Proposition 2.1 of [13]) that if b=0 or w satisfies homogeneous Neumann boundary in the half space, then system (1.2) only has constant steady states, and no pattern exists. In other words, it is the nonhomogenous boundary condition that generates spiky patterns. Such phenomenon is quite different from the solution structures of chemotaxis model of self-aggregation type for which the intrinsic mechanics of chemotactic interaction generates spiky patterns (see [14,15]). Furthermore, Tao [16] showed that, under homogeneous Neumann boundary conditions in bounded domains, the multidimensional classical chemotaxis model of consumption type has a unique global bounded solutions under suitable assumptions on the initial data w0 and the chemotactic coefficient. In particular, the global existence or blow-up of solutions is independent of the initial data u0. This study indicates that the chemotaxis model of consumption type posses another different property from the one of self-aggregation type since the latter has the well-known critical mass on u0 for blow-up in dimension 2. This work subsequently led to various generalizations. Baghaei and Khelghati [17] improved the results of [16] to a larger set of w0 and chemotactic coefficient. Frassu and Viglialoro [18] further generalized the works of [16] to the models with indirect signal consumption. Recently, Li and Zhao [19] and Wang [20] proved that under homogeneous Neumann boundary conditions, the chemotaxis-consumption system with regular signal-dependent motility also has global bounded solutions under some assumptions on w0. It is worth mentioning that for the chemotaxis-consumption system with logarithmic sensitivity, Winkler [21,22] introduced the notion of renormalized solutions to handle the singularity in the study of global existence of large solutions.

    There are some studies on the dynamics of classical chemotaxis model of consumption type with nonhomogeneous boundary conditions. In the one dimensional case, Hong and Wang [23] studied the stability of steady state to the minimal model with Dirichlet boundary condition for the nutrient in bounded domains; Carrillo, Li and Wang [13] obtained the stability of steady state to the model with constant motility and logarithmic singular sensitivity in the half space. In the multidimensional case, Braukhoff and Lankeit [24] proved the existence and uniqueness of steady state to the minimal model with nonhomogeneous Robin boundary condition, while Lee, Wang and Yang [25] obtained similar results for the minimal model with Dirichlet boundary condition, and they further analyzed the boundary layer phenomena. Recently, Fuest, Lankeit and Mizukami [26] further showed the stability of steady state for the minimal parabolic-elliptic model on the base of the works on the steady state obtained in [24].

    One can observe from the boundary condition w(+,t)=0 that, in contrast with the models studied in the above mentioned works, system (1.2) is actually a chemotaxis model with singular sensitivity and singular motility. In this paper, we shall develop some new strategies to overcome analytical difficulties caused by the coupling of nonhomogenous boundary condition and singularities. And we obtain the following results:

    (1) system (1.2)–(1.4) admits a unique steady state (U,W), and Uλδ(x) as d0, where δ(x) is the Dirac function and λ is the initial bacterial density, i.e., λ=0u0(x)dx;

    (2) this spiky steady state (U,W) is asymptotically stable in the sense that if the initial data (u0,w0) is a small perturbation of (U,W) in some topology, then the solution (u,w) will converge to (U,W) time asymptotically.

    Following the argument of [13], one can easily show that result (1) holds. The aim of this paper is to show the nonlinear stability of steady state. The main difficulty of the problem is the presence of two types of singularities in the model: one is the logarithmic singularity of the sensitivity function, the other is the inverse square singularity of the signal-dependent motility. As in the arguments of [13,27,28,29], we relegate the former singularity by using the Cole-Hopf transformation to a nonlinear nonlocal term. However, this transformation is not powerful enough to settle the latter singularity. We shall develop new ideas to deal with the challenge of inverse square singularity of motility. Indeed, we first reformulate the problem in the perturbation variables using the method of anti-derivative, to classify the strength of singularity. Then we construct an appropriately approximate system, which retains some key structures of the original system, to establish the local well-posedness of the perturbation equations. In this step we will first prove that the approximate system is locally well-posed in a time interval [0,T], where T is independent of the artificial parameter ε; and then pass to the limit ε0+ by using the Aubin-Lions compactness lemma and a diagonal argument. Finally, to close the a priori estimate that is necessary to obtain the global well-posedness of the perturbation equations (or the asymptotic stability of steady state), we establish a new weighted elliptic estimate upon the weighted energy estimates where the weights are artfully chosen according to the nice structures of the equations.

    The paper is organized as follows. In Section 2, we present some elementary calculations and state the main results of this paper. In Section 3, we derive the perturbation equations, and establish the local well-posedness theory. Section 4 is devoted to the proof of nonlinear stability of the spiky steady state.

    In this section, we first show the existence of spiky steady state to the system (1.2)–(1.4). Then we present some elementary calculations and state the main results on the asymptotic stability of such spike profile.

    Owing to the zero-flux boundary condition for u, the mass of bacterial should be conserved. In other words,

    λ:=0u(x,t)dx=0u0(x)dx. (2.1)

    Thus, the steady state of (1.2) satisfies

    {(1W2(UxUWxW))x=0,dWxxWU=0,0U(x)dx=λ>0. (2.2)

    with boundary conditions

    (UxUWxW)(0)=0, W(0)=b, (U,W)(+)=(0,0). (2.3)

    Observe that when W>0, the steady state equations (2.2) and (2.3) is actually the m=χ=1 case of the chemotaxis model studied in [13]. Thus, according to Proposition 2.1 and Theorem 2.1 of [13], we have the following result.

    Proposition 2.1. (1) The system (2.2) and (2.3) has a unique smooth solution (U,W) satisfying U<0, W<0, and

    U(x)=λ26d(1+λ6dx)2, W(x)=b(1+λ6dx)2. (2.4)

    (2) U concentrates as a spike at x=0 as d0+, i.e.,

    U(x)λδ(x) in the sense of distribution as d0+.

    We next pay attention to the asymptotic stability of (U,W) to system (1.2)–(1.4). Because the chemical concentration w(x,t) has vacuum end state at x=+, there are two types of singularities in system (1.2): one is the singular sensitivity wxw, the other is the singular motility w2. To handle the former singularity, motivated by the works of [13,27,28,29], we employ the following Cole-Hopf type transformation

    v:=wxw, that is, (lnw)x=v, (2.5)

    which along with the boundary condition w(0,t)=b gives

    w(x,t)=bex0v(y,t)dy. (2.6)

    Hence we transform system (1.2) into a nonlocal system of viscous conservation laws as follows

    {ut=(w2(ux+uv))x,(x,t)R+×R+vt=dvxx(dv2u)x,(x,t)R+×R+w(x,t)=bex0v(y,t)dy,(u,v)(x,0)=(u0(x),v0(x)), (2.7)

    where v0=w0xw0. One may observe that the new system (2.7) still has singular motility near x=+ for the bacterial mass u. In this paper, we shall develop some novel ideas to solve this challenging problem.

    We next determine the boundary conditions of (2.7). The second equation of (1.2) gives (lnw)t=dvx+dv2u. Because b is a constant, for smooth solutions (lnw)t=0 at x=0, it then follows that

    dvx(dv2u)=0atx=0.

    Denote by (U,V)(x) the steady state of (2.7), where U(x) is given in (2.4). Then we have

    V(x)=WxW=λ3d(1+λ6dx)1.

    It is easy to see that

    V(x)0 as x.

    Because it is expected that v(x,t)V(x) as t, it is natural to impose v(+,t)=0. Therefore, the boundary conditions of (2.7) are

    {ux+uv=0, x=0dvx(dv2u)=0, x=0(u,v)(x,t)(0,0), x. (2.8)

    We also need some notation. Hk denotes the usual Sobolev space whose norm is abbreviated as f2k:=kj=0jxf2 with f:=fL2(R+), and Hkω is the weighted Sobolev space of measurable function f such that ωjxfL2(R+) with norm fω:=ωfL2(R+) and f2k,ω:=kj=0ωjxf2 for 0jk.

    We are now ready to state the main results.

    Theorem 2.1 (Local well-posedness). Let (U,V) be the steady state of (2.7) and (2.8). Assume that the initial perturbation around (U,V) satisfies ϕ0()=ψ0()=0, where

    (ϕ0,ψ0)(x)=x0(u0(y)U(y),v0(y)V(y))dy.

    Suppose that

    ϕ0H1(R+), ψ0L2(R+), ψ0xWL2(R+).

    Then there is a time T>0, such that the system (2.7) and (2.8) has a unique strong solution (u,v) on R+×(0,T), satisfying

    uUC([0,T];L2ω1)L2((0,T);H1ω2), vVC([0,T];L2ω2)L2((0,T);H1ω2),

    where ω1=U and ω2=1U.

    Theorem 2.2 (Global well-posedness). Assume that the conditions of Theorem 2.1 hold, and that there exists a constant δ0>0 such that,

    ψ02+ϕ02+ψ0x21,w2+ϕ0x2ω3+ϕ0xx2ω4δ0,

    where ω3=1U2 and ω4=1U3. Then the system (2.7) and (2.8) has a unique global solution (u,v)(x,t) satisfying

    {uUC([0,);H1)L2((0,);H2),vVC([0,);H1)L2((0,);H2). (2.9)

    Moreover, the following asymptotic convergence hold:

    supxR+|(u,v)(x,t)(U,V)(x)|0 as t+, (2.10)

    and

    u(,t)U()L1(R+)0 as t+. (2.11)

    Using the Cole-Hopf transformation (2.5), we transfer Theorem 2 to the original system (1.2)–(1.4).

    Theorem 2.3. Let (U,W) be the unique steady state of (1.2)–(1.4). Assume that the initial perturbation satisfies ϕ0()=ψ0()=0, where

    ϕ0(x)=x0(u0(y)U(y))dy, ψ0(x)=lnw0(x)+lnW(x).

    Suppose that there is a constant δ0>0 such that

    ψ02+ϕ02+ψ0x21,w2+ϕ0x2ω3+ϕ0xx2ω4δ0.

    Then the system (1.2)–(1.4) has a unique global solution (u,w)(x,t) satisfying

    {uUC([0,);H1)L2((0,);H2),wWC([0,);H1)L2((0,);H2).

    Moreover, we have the following asymptotic convergence:

    supxR+|(u,w)(x,t)(U,W)(x)|0 as t+,

    and

    (u,w)(,t)(U,W)()L1(R+)0 as t+.

    Remark 2.1. We provide both the pointwise convergence and L1 convergence for the solution. In contrast with the result of [6] where it is required infinite initial mass for w, our Theorem 2.3 implies that the chemical concentration w carries finite mass for all time.

    Remark 2.2. In view of its biological background, it is also interesting to study the stability of traveling waves to system (1.1). However, when we apply our argument to that problem, the perturbation equation involves several unfavorable terms which are sophisticated to estimate. We leave this problem for the future study.

    Remark 2.3. We shall remark that the steady state (U,W) obtained in Proposition 2.1 is a smooth solution of system (2.2) and (2.3), and it satisfies U(x)>0 and W(x)>0 for any x[0,+). In other words, U(x) only vanishes at the far field, and the singularity only happens at x=+. This fact enables us to take 1U as the key weight function, and derive the stability of steady state in specific weighted space.

    This section is devoted to proving Theorem 2.1, i.e., the local well-posedness of system (2.7) and (2.8). We first reformulate the problem in the perturbation variables using the method of anti-derivative. Because the perturbation system still has a singularity, we have to construct an appropriately approximate system. Then we prove that the approximate system is locally well-posed in a time interval [0,T] where T is independent of the artificial parameter ε. After establishing the uniqueness of solutions in weighted Sobolev space, we finally derive the local well-posedness of system (2.7) and (2.8) by the Aubin-Lions compactness lemma and a diagonal argument.

    The steady state (U,V) of system (2.7) and (2.8) satisfies

    {(W2(Ux+UV))x=0,dVxx(dV2U)x=0, (3.1)

    where the boundary conditions are given by

    (Ux+UV)(0)=(dVx(dV2U))(0)=0, (U,V)(+)=(0,0). (3.2)

    Integrating (3.1) in x, we have

    {Ux+UV=0,dVxdV2+U=0. (3.3)

    Observing that (u,v) satisfies the zero-flux boundary condition, the perturbation around (U,V) should have the conservation of mass. That is

    0(u(x,t)U(x),v(x,t)V(x))dx=0(u0(x)U(x),v0(x)V(x))dx=(0,0). (3.4)

    Then we could adopt the method of anti-derivative to decompose the solution (u,v) as

    (ϕ,ψ)(x,t)=x0(u(y,t)U(y),v(y,t)V(y))dy,

    which implies

    ϕx=uU, ψx=vV. (3.5)

    Substituting (3.5) into (2.7), integrating the equations in x, noting w=eψW, and using (3.1), we have

    {ϕt=W2e2ψ(ϕxx+ϕxψx+Uψx+Vϕx),ψt=dψxx2dVψxdψ2x+ϕx,

    which is equivalent to

    {W2ϕt=e2ψ(ϕxx+ϕxψx+Uψx+Vϕx),ψt=dψxx2dVψxdψ2x+ϕx. (3.6)

    The initial value of (ϕ,ψ) is given by

    (ϕ0,ψ0)(x):=(ϕ,ψ)(x,0)=x0(u0(y)U(y),v0(y)V(y))dy, (3.7)

    with

    (ϕ0,ψ0)(+)=(0,0), (3.8)

    and the boundary condition satisfies

    (ϕ,ψ)(0,t)=(0,0), (ϕ,ψ)(+,t)=(0,0). (3.9)

    We shall remark that the anti-derivative for v could remove the nonlocality of the problem, but it can not handle the singularity of the motility. Indeed, to overcome the difficulties caused by the singular motility (or degenerate relaxation), we construct an approximate system of (3.6) as

    {W2εϕt=e2ψ(ϕxx+ϕxψx+Uψx+Vεϕx),ψt=dψxx2dVεψxdψ2x+ϕx, (3.10)

    where ε>0 is a constant, Wε=W+ε and Vε=WWεV. Here we also approximate V by Vε so that system (3.10) retains the key structure of system (3.6):

    VεWε(1Wε)x=0. (3.11)

    Indeed, recalling that V=WxW, a direct calculation leads to

    VεWε(1Wε)x=VεWε+WεxW2ε=VW+WxW2ε=0.

    Employing the principle of contraction mapping (e.g., see [30]), one could easily get the local well-posedness for the approximate system on a time interval that may depend on ε.

    Proposition 3.1. Assume that the initial data (ϕ0,ψ0) satisfies

    ϕ0H1(R+), ψ0H1(R+).

    Then, there exists a constant T>0 depending on ε, ϕ0H1 and ψ0H1 such that the approximate system (3.10) with (3.7)–(3.9) has a unique local strong solution on R+×[0,T] satisfying

    (ϕ,ψ)C([0,T];H1)L2((0,T);H2).

    Proof.

    By Proposition 3.1, there exists a time T1>0 such that the system (3.10) with (3.7)–(3.9) has a unique solution (ϕ,ψ) on (0,T1) that satisfies ϕC([0,T];H1)L2((0,T);H2). Starting at T1, applying Proposition 3.1 again, we can extend the solution (ϕ,ψ) to another time T2=T1+t1, where t1>0 depends on ε, ϕ(T1)H1 and ψ(T1)H1. Continuing this procedure, we get two sequences {tj}j=1 and {Tj}j=1, where tj depends on ε, ϕ(Tj)H1 and ψ(Tj)H1, such that the solution (ϕ,ψ) exists on the time interval (0,Tj), and satisfies

    (ϕ,ψ)C([0,Tj];H1)L2((0,Tj);H2).

    Take the maximal existing time T as T=T1+j=1tj. Then the solution can be extended to (0,T) and satisfies

    (ϕ,ψ)C([0,T];H1)L2((0,T);H2),

    for any T(0,T). Clearly, if T<, then

    ¯limtT(ϕ(t)H1+ψ(t)H1)=. (3.12)

    However, one can not use Proposition 3.1 to derive the local well-posedness of system (3.6) by directly passing to the limit ε0, since the time interval [0,T] obtained in Proposition 3.1 depends on ε. In the following, we have to establish appropriate a priori estimates that are independent of ε.

    Proposition 3.2. Assume that (ϕ0,ψ0) satisfies

    ϕ0H1(R+), ψ0L2(R+), ψ0xWL2(R+). (3.13)

    Then there exists a constant T0>0 independent of ε, such that the approximate system (3.10) with (3.7)–(3.9) has a unique solution on R+×[0,T0], which satisfies

    supt[0,T0]0(Wεϕ2+ψ2+Wεϕ2x+ψ2xWε)dx20(Wεϕ20+ψ20+Wεϕ20x+ψ20xWε)dx, (3.14)

    and

    T000(ϕ2xWε+ϕ2xxWε+ψ2x+ψ2xxWε)C(T0). (3.15)

    Proof. Thanks to (3.12), it suffices to establish a priori estimate in the weighted Sobolev space that is independent of ε. To achieve this, we multiply the first equation of (3.10) by ϕWε, and integrate the resultant equation over (0,t)×(0,+) to get

    120Wεϕ2=t00e2ψ(VεWεϕϕx+ϕϕxxWε+ϕϕxψxWε+UWεϕψx)+120Wεϕ20=t00[VεWε(1Wε)x]e2ψϕϕxt00e2ψϕ2xWεt00e2ψϕϕxψxWε+t00UWεe2ψϕψx+120Wεϕ20. (3.16)

    By (3.11) we have

    120Wεϕ2+t00e2ψϕ2xWε=t00UWεe2ψϕψxt00e2ψϕϕxψxWε+120Wεϕ20.

    It follows from Young's inequality that

    |0UWεe2ψϕψxdx|Ce2ψL0Wεϕ2dx+Ce2ψL0ψ2xWεdx. (3.17)

    Using the inequality

    ϕ2(x,t)=2xϕϕx(y,t)dy2(0Wεϕ2)12(0ϕ2xWε)12, (3.18)

    we get

    |0e2ψϕϕxψxWεdx|CeψLϕL(R+)|0eψϕxψxWεdx|CeψLWεϕ12L2(R+)ϕxWε12L2(R+)eψϕxWεL2(R+)ψxWεL2(R+)Ce32ψLWεϕ12L2(R+)ψxWεL2(R+)eψϕxWε32L2(R+)Ce6ψLWεϕ2L2(R+)ψxWε4L2(R+)+18eψϕxWε2L2(R+)Ce6ψLWεϕ6L2(R+)+Ce6ψLψxWε6L2(R+)+18eψϕxWε2L2(R+). (3.19)

    Combining (3.17) and (3.19), one obtains

    120Wεϕ2+78t00e2ψϕ2xWεCt0e2ψL0(Wεϕ2+ψ2xWε)+Ct0e6ψL(0Wεϕ2dx+0ψ2xWεdx)3+120Wεϕ20. (3.20)

    Multiplying the second equation of (3.10) by ψ, we get

    120ψ2+dt00ψ2x=2dt00Vεψψx+t00ϕxψdt00ψψ2x+120ψ20. (3.21)

    By Young's inequality, we have

    |2d0Vεψψxdx|d40ψ2xdx+C0ψ2dx, (3.22)

    and

    |0ϕxψdx|Ce2ψL0ψ2dx+180e2ψϕ2xWεdx.

    Moreover, using (3.18) yields

    |d0ψψ2xdx|CψL(R+)0ψ2xdxCψ12L2(R+)ψx52L2Cψ2L2(R+)+Cψx103L2(R+). (3.23)

    Now substituting (3.22) and (3.23) into (3.21), we have

    120ψ2+d4t00ψ2xCt0(1+e2ψL)0ψ2+18t00e2ψϕ2xWε+Ct0(0ψ2xdx)53+120ψ20. (3.24)

    Combining (3.20) with (3.24) yields

    120(Wεϕ2+ψ2)dx+34t00e2ψϕ2xWεdxdτ+d4t00ψ2xdxdτCt0e2ψL0Wεϕ2dxdτ+Ct0e6ψL(0Wεϕ2dx)3dτ+Ct0(1+e2ψL)0ψ2dxdτ+Ct0e2ψL0ψ2xdxdτ+Ct0e6ψL(0ψ2xWεdx)3dτ+Ct0(0ψ2xdx)53dτ+120(Wεϕ20+ψ20)dx. (3.25)

    Multiplying the first equation of (3.10) by ϕxxWε, one gets

    120Wεϕ2x+t00e2ψϕ2xxWε=t00(e2ψVεWεϕxϕxx+e2ψϕxϕxxψxWε+e2ψUWεϕxxψx+Wxϕtϕx)+120Wεϕ20x=t00[e2ψ(VεWε+WxW2ε)ϕxϕxx+e2ψϕxϕxxψxWε+e2ψUWεϕxxψx]t00e2ψWxW2ε(Uϕxψx+ϕ2xψx+Vεϕ2x)+120Wεϕ20x, (3.26)

    where we have used the first equation of (3.10) in the second equality. As in (3.19),

    |0e2ψϕxϕxxψxWεdx|180e2ψϕ2xxWεdx+Ce6ψL(0Wεϕ2xdx)3+Ce6ψL(0ψ2xWεdx)3. (3.27)

    From Young's inequality, it follows that

    |0e2ψUWεϕxxψxdx|Ce2ψL0ψ2xdx+180e2ψϕ2xxWεdx, (3.28)
    |0e2ψWxW2εUϕxψxdx|Ce2ψL0Wεϕ2xdx+Ce2ψL0ψ2xWεdx, (3.29)

    and

    |0WxW2εe2ψϕ2xψxdx|CeψLϕxL(R+)0eψ|ϕxψx|WεdxCe32ψLWεϕx12L2(R+)eψϕxxWε12L2(R+)eψϕxWεL2(R+)ψxL2(R+)180e2ψϕ2xWεdx+Ce3ψLWεϕxL2(R+)eψϕxxWεL2(R+)ψx2L2(R+)180e2ψϕ2xWεdx+180e2ψϕ2xxWεdx+Ce6ψLWεϕx2L2(R+)ψx4L2(R+)180e2ψϕ2xWεdx+180e2ψϕ2xxWεdx+Ce6ψL(Wεϕx6L2+ψx6L2). (3.30)

    Substituting (3.27)–(3.30) into (3.26) leads to

    120Wεϕ2xdx+12t00e2ψϕ2xxWεdxdτCt00e2ψϕ2xWεdxdτ+Ct0e2ψL(0ψ2xdx+0Wεϕ2xdx)dτ+Ct0e6ψL(0Wεϕ2xdx+0ψ2xdx)3dτ+120Wεϕ20xdx. (3.31)

    Multiplying the second equation of (3.10) by ψxxWε, one gets

    120ψ2xWε+t00dψ2xxWε=t00(2dVεWεψxψxxϕxψxxWε+dψ2xψxxWε(1Wε)xψtψx)+120ψ20xWε. (3.32)

    By Young's inequality,

    |02dVεWεψxψxxdx|C0ψ2xWεdx+d80ψ2xxWεdx.

    Moreover, integration by parts leads to

    |0ϕxψxxWεdx|=|0ϕxxψxWεdx0(1Wε)xϕxψxdx+ϕxψxb+ε|x=0|180e2ψϕ2xWεdx+180e2ψϕ2xxWεdx+Ce2ψL0ψ2xWεdx+|ϕxψxb+ε|x=0|.

    The boundary term can be estimated as

    |ϕxψxb+ε|x=0|12(b+ε)ϕ2x|x=0+12(b+ε)ψ2x|x=0=1b+ε0ϕxϕxxdx1b+ε0ψxψxxdxδ0e2ψϕ2xxWεdx+Ce2ψL0Wεϕ2x+δ0ψ2xxWεdx+C0ψ2xWεdx,

    where δ is a small constant. It follows from (3.18) that

    |0dψ2xψxxWεdx|CψxL0ψxψxxWεdxCψx12L2(R+)ψxx12L2(R+)ψxWεL2(R+)ψxxWεL2(R+)CψxxWε32L2(R+)ψxWε32L2(R+)d80ψ2xxWεdx+C(0ψ2xWεdx)3.

    In view of the second equation of (3.10), it holds:

    0(1Wε)xψtψxdx=02dVε(1Wε)xψ2xdx0d(1Wε)xψxxψxdx0(1Wε)xϕxψxdx+0d(1Wε)xψ3xdx,

    where

    |02dVε(1Wε)xψ2xdx|C0ψ2xWεdx,
    |0d(1Wε)xψxxψxdx|C0ψ2xWεdx+d80ψ2xxWεdx,
    |0(1Wε)xϕxψxdx|180e2ψϕ2xWεdx+Ce2ψL0ψ2xWεdx,

    and by (3.18),

    |0d(1Wε)xψ3xdx|Cψx12L2(R+)ψxx12L2(R+)ψxWε2L2(R+)d80ψ2xxWεdx+C(0ψ2xWεdx)53.

    Then choosing δ1, by (3.31), we get

    120ψ2xWε+d4t00ψ2xxWεCt0(1+e2ψL)0ψ2xWε+14t00e2ψϕ2xxWε+Ct00e2ψϕ2xWε+Ct0[(0ψ2xWεdx)53+(0ψ2xWεdx)3]+Ct0e2ψL0Wεϕ2x+120ψ20xWε. (3.33)

    Combining (3.31) with (3.33), we have

    120(Wεϕ2x+ψ2xWε)+14t00(e2ψϕ2xxWε+dψ2xxWε)Ct00e2ψϕ2xWε+Ct0e6ψL(0Wεϕ2xdx)3+Ct0e2ψL0Wεϕ2x+Ct0(1+e6ψL)(0ψ2xWεdx)3+Ct0(1+e2ψL)0ψ2xWε+Ct0(0ψ2xWεdx)53+120(Wεϕ20x+ψ20xWε). (3.34)

    Multiplying (3.25) by K1 and combing the resultant inequality with (3.34), we have

    0(Wεϕ2+ψ2+Wεϕ2x+ψ2xWε)+t00(e2ψϕ2xWε+e2ψϕ2xxWε+ψ2x+ψ2xxWε)Ct0e2ψL0(Wεϕ2+Wεϕ2x)+Ct0e6ψL((0Wεϕ2)3+(0Wεϕ2x)3)+Ct0(1+e2ψL)0(ψ2+ψ2xWε)+Ct0(1+e6ψL)(0ψ2xWεdx)3+0(Wεϕ20+ψ20+Wεϕ20x+ψ20xWε), (3.35)

    which further gives

    0(Wεϕ2+ψ2+Wεϕ2x+ψ2xWε)dxCt0(1+[0(Wεϕ2+ψ2+Wεϕ2x+ψ2xWε)dx]3e6ψL)+0(Wεϕ20+ψ20+Wεϕ20x+ψ20xWε).

    Set H(t):=0(Wεϕ2+ψ2+Wεϕ2x+ψ2xWε)dx. Noting

    ez>1forz>0 andψ2L0(ψ2+ψ2x)dx,

    we are led to

    H(t)Ct0(H+1)3e6H+H0,

    where H0=0(Wεϕ20+ψ20+Wεϕ20x+ψ20xWε)dx. It is easy to verify that when ¯T0 satisfies

    2Ce12H0(H0+1)2¯T0min{12,H02}, (3.36)

    then

    H(t)2H0fort(0,¯T0). (3.37)

    Indeed, consider

    (H(t)+1)Mt0(H+1)3+(H0+1),

    where M=Ce12H0, then

    H(t)(H0+1)(12M(H0+1)2t)121fortsmall.

    Since (1x)12<1+x for x(0,12), it holds

    H(t)(H0+1)(1+2M(H0+1)2t)1=H0+2M(H0+1)2tH0+2M(H0+1)2t

    for t small. Thus, when we take ¯T0 satisfying (3.36), we have (3.37).

    If we take

    H1=0W(ϕ20+ϕ20x)dx+0(ϕ20+ϕ20x)dx+0(ψ2+ψ20xW)dx,H2=0W(ϕ20+ϕ20x)dx+0(ψ2+ψ20xW)dx,

    then H1>H0 and H2<H0. Now we take T0 satisfying

    2Ce12H1(H1+1)2T0=min{12,H22}.

    Clearly, T0 is independent of ε, and

    H(t)2H0fort(0,T0). (3.38)

    Thanks to Proposition 3.1, (3.13) and (3.38), for any 0<ε<1, system (3.10) with (3.7)–(3.9) has a unique solution (ϕ,ψ) on R+×(0,T0) satisfying (3.14). The other desired estimate (3.15) follows from (3.38) and an integration of (3.35) in t.

    Let us now study the local well-posedness of (3.6)–(3.9). We start with the uniqueness of the solutions.

    Proposition 3.3. Let (ϕ1,ψ1) and (ϕ2,ψ2) be two solutions of system (3.6)–(3.9) satisfying

    WϕiL((0,T);H1),ϕixWL2((0,T);H1),ψiL((0,T);L2),ψixWL((0,T);L2),ψixxWL2((0,T);L2),

    for i=1,2. Then (ϕ1,ψ1)(ϕ2,ψ2) on R+×[0,T].

    Proof. Define (ϕ,ψ) by

    ϕ=ϕ1ϕ2, ψ=ψ1ψ2.

    Then (ϕ,ψ) satisfies

    {W2ϕt=e2ψ1ϕxx+(ψ1x+V)e2ψ1ϕx+(e2ψ1e2ψ2)ϕ2xx+(e2ψ1e2ψ2)ϕ2xψ1x+e2ψ2ϕ2xψx+V(e2ψ1e2ψ2)ϕ2x+Ue2ψ1ψx+U(e2ψ1e2ψ2)ψ2x,ψt=dψxx(2dV+dψ1x+dψ2x)ψx+ϕx. (3.39)

    Multiplying the first equation of (3.39) by ϕW, and the second one by ψ, summing the resultant equations up, one gets after an integration by parts that

    120(Wϕ2+ψ2)+t00(e2ψ1ϕ2xW+dψ2x)=t00(e2ψ1e2ψ2)ϕ2xψ1xϕW+t00e2ψ2ϕ2xϕψxW+t00UWe2ψ1ϕψx+t00UW(e2ψ1e2ψ2)ψ2xϕ+t00VW(e2ψ1e2ψ2)ϕ2xϕ+t00ϕxψt00(2dV+dψ1x+dψ2x)ψxψt00(ψ1xWVW+(1W)x)e2ψ1ϕϕx+t00(e2ψ1e2ψ2)ϕ2xxϕW+120(Wϕ20+ψ20). (3.40)

    By Young's inequality, we have

    |0(e2ψ1e2ψ2)ϕ2xψ1xϕWdx|C0|ψϕϕ2xψ1x|WdxCϕLψLϕ2xWL2ψ1xWL2CWϕL2ϕxWL2ϕ2xWL2+CψL2ψxL2ϕ2xWL2C(Wϕ2L2+ψ2L2)ϕ2xW2L2+δ(ϕxW2L2+ψx2L2), (3.41)
    |0e2ψ2ϕ2xϕψxWdx|CϕLϕ2xWL2ψxWL2CWϕL2ϕxWL2+ψxW2L2ϕ2xW2L2δϕxW2L2+CWϕ2L2+CψxW2L2ϕ2xW2L2, (3.42)
    |0UWe2ψ1ϕψxdx|CψxW2L2+CWϕ2L2, (3.43)
    |0UW(e2ψ1e2ψ2)ψ2xϕdx|C0|ψψ2xϕ|dxCϕLψL2ψ2xL2CWϕ12L2ϕxW12L2ψL2C(ψ2L2+Wϕ2L2)+δϕxW2L2, (3.44)
    |0VW(e2ψ1e2ψ2)ϕ2xϕdx|C0|ψϕ2xϕ|WdxδϕxW2L2+CWϕ2L2+Cϕ2xW2L2ψ2L2, (3.45)

    and

    |0ϕxψdx0(2dV+dψ1x+dψ2x)ψxψdx|δψx2L2+δϕx2L2+Cψ2L2,

    where δ>0 is a small constant. A direct calculation yields VW+(1W)x=0, and then we get

    |0(ψ1xWVW+(1W)x)e2ψ1ϕϕx|=|0e2ψ1ψ1xϕϕxW|δϕxW2L2+CWϕ2L2. (3.46)

    Integration by parts leads to

    |0(e2ψ1e2ψ2)ϕ2xxϕW|=|0(e2ψ1e2ψ2)xϕϕ2xW+0(e2ψ1e2ψ2)ϕxϕ2xW+0(e2ψ1e2ψ2)(1W)xϕϕ2x|=|02e2ψ1ψxϕϕ2xW+20(e2ψ1e2ψ2)ψ2xϕϕ2xW+0(e2ψ1e2ψ2)ϕxϕ2xW+0(e2ψ1e2ψ2)(1W)xϕϕ2x|.

    As in (3.42),

    |02e2ψ1ψxϕϕ2xW|δϕxW2L2+CWϕ2L2+CψxW2L2ϕ2xW2L2. (3.47)

    As in (3.41),

    |02(e2ψ1e2ψ2)ψ2xϕϕ2xW|C(Wϕ2L2+ψ2L2)ϕ2xW2L2+δ(ϕxW2L2+ψx2L2),
    |0(e2ψ1e2ψ2)ϕxϕ2xW|CψLϕxWL2ϕ2xWL2C(ψ2L2+ψx2L2)ϕ2xW2L2+δϕxW2L2,

    and

    |0(e2ψ1e2ψ2)(1W)xϕϕ2x|CϕLψL2ϕ2xWL2δϕxW2L2+CWϕ2L2+Cψ2L2ϕ2xW2L2. (3.48)

    Now substituting (3.41)–(3.48) into (3.40), we arrive at

    120(Wϕ2+ψ2)dx+t00[(e2ψ1Cδ)ϕ2xW+(dCδ)ψ2x]dxCt0(Wϕ2L2+ψ2L2+ψxW2L2)(1+ϕ2xW2L2). (3.49)

    We next present the estimate for 0ψ2xWdx. Multiplying the second equation of (3.39) by ψxxW, we get

    120ψ2xW+t00dψ2xxW=t00(1W)xψxψt+t00ψxxW(2dV+dψ1x+dψ2x)ψx+t00ψxxϕxW. (3.50)

    Using the second equation of (3.39), we get

    0ψ2tC0(ψ2xx+ψ2x+(ψ21x+ψ22x)ψ2x+ϕ2x),

    and

    0(ψ21x+ψ22x)ψ2xCψx2LCψx2L2+ψxx2L2.

    Thus,

    |0(1W)xψxψt|C0ψ2xW+δ0ψ2tC0ψ2xW+δ0(ψ2xx+ϕ2x).

    Similarly,

    |0ψxxW(2dV+dψ1x+dψ2x)ψx|C0ψ2xW+δ0ψ2xxW,

    and

    0|ψxxϕx|Wd20ψ2xxW+C0ϕ2xW.

    Substituting these inequalities into (3.50), we get

    120ψ2xW+t00(d2Cδ)ψ2xxWCt00(ψ2xW+ϕ2xW). (3.51)

    Multiplying (3.49) by K1 and combing the resultant inequality with (3.51), we have

    0(Wϕ2+ψ2+ψ2xW)dxCt0(Wϕ2L2+ψ2L2+ψxW2L2)(1+ϕ2xW2L2).

    It then follows from the Gronwall's inequality that

    0(Wϕ2+ψ2W+ψ2xW)dx=0.

    Therefore, ϕ0 and ψ0. We complete the proof.

    We are now ready to prove the local existence of solutions to system (3.6)–(3.9).

    Proposition 3.4. Assume that (ϕ0,ψ0) satisfies

    ϕ0H1(R+), ψ0L2(R+), ψ0xWL2(R+).

    Then there exists a constant T>0, such that the system (3.6)–(3.9) has a unique solution (ϕ,ψ) on R+×(0,T), which satisfies

    supt[0,T]0(Wϕ2+ψ2+Wϕ2x+ψ2xW)dxC0(Wϕ20+ψ20+Wϕ20x+ψ20xW)dx, (3.52)

    where C is a constant independent of T.

    Proof. Owing to Proposition 3.2, there exists a constant T>0 independent of ε>0 such that the approximate system (3.10), subject to (3.7)–(3.9), has a unique solution (ϕϵ,ψϵ) satisfying

    supt[0,T]0(Wεϕ2ε+ψ2ε+Wεϕ2εx+ψ2εxWε)dx20(Wεϕ20+ψ20+Wεϕ20x+ψ20xWε)dxC0(ϕ20+ϕ20x+ψ20+ψ20xW), (3.53)

    where C is a constant independent of ε. Owing to (3.53) and (3.15), passing to the limit ε0+, applying the Banach-Alaoglu theorem and the diagonal argument, we know that there is a subsequence, still denoted by (ϕε,ψε), such that for any r(0,)

    ψεtψt  weakly in  L2((0,T);L2(0,r)),ϕεtϕt  weakly in  L2((0,T);L2(0,r)),ψεψ  weakly in  L2((0,T);H2(0,r)),ϕεϕ  weakly in  L2((0,T);H2(0,r)).

    Noting H2(0,r) and H1(0,r) compactly embed into H1(0,r) and L(0,r), respectively, for any r>0, we obtain from the Aubin-Lions compactness lemma that

    ψεψ  strongly in  L2((0,T);H1(0,r))C([0,T];L(0,r)),ϕεϕ  strongly in  L2((0,T);H1(0,r)).

    Observing that WεW and VεV in C[0,r], one can see that the nonlinear terms in (3.10), e2ψεϕεxψεx and dψ2εx converge strongly in L2((0,T);L2(0,r)) to e2ψϕxψx and dψ2x, respectively. Then one can take the limit as ε0 in (3.10) to derive that (ϕ,ψ) satisfies (3.6) in the sense of distribution. Moreover, it follows from the weakly lower semi-continuity of the norms, the first inequality of (3.53) that

    Wϕ2L((0,T);H1(0,r)+ψ2L((0,T);L2(0,r)+ψxW2L((0,T);L2(0,r)lim_ε0(Wεϕε2L((0,T);H1(0,r)+ψε2L((0,T);L2(0,r)+ψεxWε2L((0,T);L2(0,r))2lim_ε0(Wεϕ02L((0,T);H1(0,r)+ψ02L((0,T);L2(0,r)+ψ0xWε2L((0,T);L2(0,r))=2(Wϕ02L((0,T);H1(0,r)+ψ02L((0,T);L2(0,r)+ψ0xW2L((0,T);L2(0,r)).

    Therefore, (3.52) holds, and the proof is complete.

    Proof. [Proof of Theorem 2.1] It is a consequence of Propositions 3.3 and 3.4.

    In this section, we prove the global well-posedness of strong solutions to the system (3.6)–(3.9), which also implies the nonlinear stability of spiky steady state to the original chemotaxis system (1.2)–(1.4). We construct global solutions of system (3.6)–(3.9) in the more regular space:

    X(0,T):={(ϕ,ψ)(x,t)|ϕC([0,T];H2),ϕxC([0,T];L2ω3)L2((0,T);H2ω2),ϕxxC([0,T];L2ω4),ψC([0,T];H2),ψxC([0,T];H1ω2)L2((0,T);H2ω2).

    for T(0,+], where ω2=1U, ω3=1U2 and ω4=1U3. Set

    N2(t):=supτ[0,t](ϕ(,τ)2+ψ(,τ)2+ϕx(,τ)2ω3+ψx(,τ)21,ω2+ϕxx(,τ)2ω4).

    Since U(x)λ26d, the Sobolev embedding theorem implies

    supτ[0,t]{ϕ(,τ)L,ψ(,τ)L}N(t).

    Moreover, noting

    ψ2xU(x,t)=x(ψ2xU)xdx=x2ψxψxxUdxx(1U)xψ2xdxC0ψ2xUdx+C0ψ2xxUdx, (4.1)

    we have

    ψx(,t)ULCN(t). (4.2)

    Similarly,

    ϕ2xU(x,t)=x(ϕ2xU)xdx=x2ϕxϕxxUdxx(1U)xϕ2xdxC0ϕ2xUdx+C0ϕ2xxUdx, (4.3)

    which implies

    ϕx(,t)ULCN(t). (4.4)

    For system (3.6)–(3.9), we have the following results.

    Proposition 4.1. There exists a constant δ1>0, such that if N(0)δ1, then the system (3.6)–(3.9) has a unique global solution (ϕ,ψ)X(0,) satisfying

    ϕ(,τ)2+ψ(,τ)2+ϕx(,τ)2ω3+ψx(,τ)21,ω2+ϕxx(,τ)2ω4+t0(ϕx(τ)22,ω2+ψx(τ)22,ω2+ϕxx(τ)2ω3)dτCN2(0) (4.5)

    for any t[0,).

    Thanks to the local well-posedness established in Propositions 3.3 and 3.4, we only need to derive the following a priori estimates to prove Proposition 4.1.

    Proposition 4.2. Assume that the conditions of Proposition 4.1 hold, and that (ϕ,ψ)X(0,T) is a solution of system (3.6)–(3.9) for some constant T>0. Then there is a constant ε>0, independent of T, such that if N(t)ε for any 0<tT, then (ϕ,ψ) satisfies (4.5) for any 0tT.

    To establish the a priori estimate, we need the following Hardy inequality (see Lemma 3.4 of [13] for the proof).

    Lemma 4.1. (Hardy inequality) If fH10(0,), then for j1, it holds that

    0(1+kx)jf2(x)dx4(j+1)2k20(1+kx)j+2f2x(x)dx. (4.6)

    where k>0 is a constant.

    We start with the L2 estimate.

    Lemma 4.2. If N(t)1, then there exists a constant C>0 such that

    0(Uϕ2+ψ2)dx+t00(ϕ2+Uψ2)dxdτ+t00(ϕ2xU+ψ2x)dxdτC0(Uϕ20+ψ20)dx. (4.7)

    Proof. We rewrite (3.6) as

    {W2ϕt=ϕxx+Vϕx+Uψx+ϕxψx+(e2ψ1)(ϕxx+Vϕx+Uψx+ϕxψx),ψt=dψxx2dVψxdψ2x+ϕx. (4.8)

    Multiplying the first equation of (4.8) by ϕU, the second one by ψ, and integrating the resulting equations on (0,t)×(0,+), we have

    120(W2Uϕ2+ψ2)dx+t00ϕ2xUdxdτ+dt00ψ2xdxdτ+d0|Vx|ψ2xdxdτ=t0012[(1U)xx(VU)x]ϕ2dxdτ+t00ϕϕxψxUdxdτt00dψψ2xdxdτ+t00(e2ψ1)ϕU(ϕxx+ϕxψx+Uψx+Vϕx)dxdτ+120(W2Uϕ20+ψ20)dx. (4.9)

    By (2.4) and Hardy inequality, we get

    120ϕ2xUdx=03dλ2(1+λ6dx)2ϕ2xdx148d0ϕ2dx. (4.10)

    Owing to (3.3), it is easy to compute that

    (1U)xx(VU)x=0, (4.11)

    which gives

    012[(1U)xx(VU)x]ϕ2dx=0.

    By (4.2) and Young's inequality, we derive that

    |0ϕϕxψxUdx|CN(t)0|ϕϕx|UdxCN(t)0ϕ2xUdx+CN(t)0ϕ2dx. (4.12)

    Similarly, since ψ(,t)LN(t) and

    V=2U3d, (4.13)

    we have

    |0dψψ2xdx|dN(t)0ψ2xdx, (4.14)

    and

    0(e2ψ1)ϕU(ϕxx+ϕxψx+Uψx+Vϕx)=20e2ψψxϕϕxU0(e2ψ1)(ϕU)xϕx+0(e2ψ1)ϕU(ϕxψx+Uψx+Vϕx)Cψx(,t)UL0|ϕϕx|U+C0|ψ|(ϕ2xU+|Uxϕϕx|U2)+C01U|ϕψ(ϕxψx+Uψx+Vϕx)|CN(t)0(ϕ2+ϕ2xU+ψ2x), (4.15)

    where we have used the Taylor expansion

    |e2ψ1|=|2ψ+n=22nψnn!|2|ψ|+2|ψn=22n1ψn1n!|2|ψ|+2|ψ|n=22n1(12)n1C|ψ|. (4.16)

    Now substituting (4.10)–(4.15) into (4.9), noting Vx<0, and using Hardy inequality, we get

    0(Uϕ2+ψ2)dx+t00(ϕ2xU+ψ2x)dxdτ+t00(ϕ2+Uψ2)dxdτCN(t)t00(ϕ2+ϕ2xU+ψ2x)dxdτ+0(Uϕ20+ψ20)dx.

    Thus, we obtain (4.7) provided N(t)1.

    We next establish the H1 estimate.

    Lemma 4.3. If N(t)1, then the solution of (3.6)–(3.9) satisfies

    0(ϕ2x+ψ2xU)dx+t00(ϕ2xxU2+ψ2xxU)dxdτC0(ϕ20x+ψ20xU+Uϕ20+ψ20)dx. (4.17)

    Proof. Multiplying the first equation of (4.8) by ϕxxW2, we get

    120ϕ2x+t00ϕ2xxW2=t00UW2ϕxxψxt00VW2ϕxϕxxt00ϕxϕxxψxW2t00(e2ψ1)ϕxxW2(ϕxx+ϕxψx+Uψx+Vϕx)+120ϕ20x. (4.18)

    In view of (2.4), it holds

    U(x)=λ26dbW(x). (4.19)

    It then follows from Young's inequality that

    0UW2|ϕxxψx|dxC0|ϕxxψx|Wdx140ϕ2xxW2dx+C0ψ2xdx. (4.20)

    Moreover, by (4.13),

    |0VW2ϕxϕxxdx|120ϕ2xxW2dx+120V2W2ϕ2xdx120ϕ2xxW2dx+C0ϕ2xWdx. (4.21)

    Using (4.2), it is easy to see that

    |0ϕxϕxxψxW2dx|CN(t)0|ϕxϕxx|W32dxCN(t)0ϕ2xWdx+CN(t)0ϕ2xxW2dx. (4.22)

    By (4.16), the fact that ψ(,t)LN(t) and (4.2) again, one has

    0|(e2ψ1)ϕxxW2(ϕxx+ϕxψx+Uψx+Vϕx)|dxCN(t)0(ϕ2xW+ϕ2xxW2+ψ2x)dx. (4.23)

    Substituting (4.20)–(4.23) into (4.18), we get

    120ϕ2x+(14CN(t))t00ϕ2xxW2(C+CN(t))t00ϕ2xW+(C+CN(t))t00ψ2x+120ϕ20x.

    Thus, by (4.19) and Lemma 4.2, when N(t)1, we arrive at

    0ϕ2xdx+t00ϕ2xxU2dxdτ0ϕ20xdx+C0(Uϕ20+ψ20)dx. (4.24)

    Multiplying the second equation of (4.8) by ψxxU, we have

    120ψ2xU+dt00ψ2xxU=t00ϕxψxxU+2dt00VUψxψxxt00(1U)xψtψx+t00dUψ2xψxx+120ψ20xU. (4.25)

    By Young's inequality,

    |0ϕxψxxUdx|12d0ϕ2xUdx+d20ψ2xxUdx. (4.26)

    Moreover, (4.13) gives

    |2d0VUψxψxxdx|C01U|ψxψxx|dxC0ψ2xdx+d40ψ2xxUdx. (4.27)

    By (3.3) and (4.13),

    |(1U)x|=|Ux|U2=VU=23d1U, (4.28)

    which in combination with (4.2) leads to

    |0(1U)xψtψxdx|0|(1U)x(dψxx2dVψxdψ2x+ϕx)ψx|dxC0|ψxψxxU|dx+C0VUψ2xdx+C0|ψ3x|Udx+C0|ϕxψxU|dxC0ϕ2xUdx+(C+CN(t))0ψ2xdx+d80ψ2xxUdx, (4.29)

    and

    |0dUψ2xψxxdx|CN(t)0|ψxxψxU|dxCN(t)0ψ2xxUdx+CN(t)0ψ2xdx. (4.30)

    Now substituting (4.26)–(4.30) into (4.25), we derive that

    120ψ2xU+t00(d8CN(t))ψ2xxU(C+CN(t))t00ψ2x+Ct00ϕ2xU+120ψ20xU.

    Then by Lemma 4.2, when N(t)1, we have

    0ψ2xUdx+t00ψ2xxUdxdτ0ψ20xUdx+C0(Uϕ20+ψ20)dx. (4.31)

    Combining (4.31) and (4.24), we get (4.17).

    The H2 estimate is as follows.

    Lemma 4.4. If N(t)1, then it holds

    0(Uϕ2t+ψ2tU+ϕ2xxU+ψ2xxU)+t00(ϕ2txU+ψ2txU+ϕ2xxxU+ψ2xxxU)C0(Uϕ20+ψ20+ϕ20xU2+ψ20xU+ϕ20xxU3+ψ20xxU). (4.32)

    Proof. Differentiating the first equation of (4.8) with respect to t leads to

    W2ϕtt=ϕtxx+ϕtxψx+ϕxψtx+Uψtx+Vϕtx+(e2ψ1)(ϕtxx+ϕtxψx+ϕxψtx+Uψtx+Vϕtx)+2e2ψψt(ϕxx+ϕxψx+Uψx+Vϕx). (4.33)

    Multiplying (4.33) by ϕtU and integrating it in x and t, we get

    120W2Uϕ2t+t00ϕ2txU12t00[(1U)xx(VU)x]ϕ2tt00ϕtϕtxψxU+t00ϕtϕxψtxU+t00ϕtψtx+t00(e2ψ1)ϕtxxϕtU+t00(e2ψ1)(ϕtxψx+ϕxψtx+Uψtx+Vϕtx)ϕtU+t002e2ψψtϕtUϕxx+t002e2ψψtϕtU(ϕxψx+Uψx+Vϕx)+C0(ϕ20xxU3+ϕ20xU2+ψ20xU), (4.34)

    where we have used

    Uϕ2t|t=0C(ϕ20xxU3+ϕ20xU2+ψ20xU).

    By Young's inequality, Hardy's inequality, (4.2) and (4.4), we get

    |0ϕtϕtxψxUdx|N(t)20ϕ2txUdx+N(t)20ϕ2tdxCN(t)0ϕ2txUdx, (4.35)

    and

    |0ϕtϕxψtxUdx|CN(t)0ψ2txUdx+CN(t)0ϕ2tdxCN(t)0ψ2txUdx+CN(t)0ϕ2txUdx. (4.36)

    Moreover, integration by parts leads to

    |0ϕtψtxdx|=|0ψtϕtxdx|140ϕ2txUdx+C0ψ2tUdx. (4.37)

    Using (4.16), (4.2) and (4.4) again, a simple calculation yields

    |0(e2ψ1)ϕtxxϕtUdx|=|0[(e2ψ1)(ϕ2txU+ϕtxϕt(1U)x)+2e2ψψxϕtxϕtU]dx|CN(t)0(ϕ2txU+ϕ2t)dxCN(t)0ϕ2txUdx, (4.38)

    and

    |0(e2ψ1)(ϕtxψx+ϕxψtx+Uψtx+Vϕtx)ϕtUdx|CN(t)0|ϕtU(ϕtxψx+ϕxψtx+Uψtx+Vϕtx)|dxCN(t)0ϕ2txUdx+CN(t)0ψ2txUdx+CN(t)0ψ2tUdx. (4.39)

    Similarly,

    02e2ψϕtψtUϕxxdx=02(e2ψϕtψtU)xϕxdxCN(t)0(ϕ2txU+ψ2txU+ϕ2t+ψ2t)dxCN(t)0(ϕ2txU+ψ2txU)dx, (4.40)

    and

    |02e2ψϕtψtU(ϕxψx+Uψx+Vϕx)dx|CN(t)0(ϕ2txU+ψ2txU)dx. (4.41)

    Now substituting (4.35)–(4.41) into (4.34), we arrive at

    120Uϕ2t+(34CN(t))t00ϕ2txUCN(t)t00ψ2txU+Ct00ψ2tU+C0(ϕ20xxU3+ϕ20xU2+ψ20xU). (4.42)

    By (4.2), (3.6), and Lemmas 4.2 and 4.3, we estimate the last term of (4.42) as

    t00ψ2tUdxdτCt00(ψ2xxU+V2Uψ2x+1Uψ4x+ϕ2xU)dxdτCt00(ψ2xxU+(1+N2(t))ψ2x+ϕ2xU)dxdτC0(ϕ20x+ψ20xU+Uϕ20+ψ20)dx. (4.43)

    We next estimate t00ψ2txU. Differentiating the second equation of (3.6) with respect to t leads to

    ψtt=dψtxx2dVψtx2dψxψtx+ϕtx. (4.44)

    Multiplying (4.44) by ψtU, we have

    120ψ2tU+t00dψ2txU=t00ϕtxψtUt00d(1U)xψtxψtt002d1U(Vψtψtx+ψxψtψtx)+120ψ2tU|t=0. (4.45)

    Owing to Young's inequality, we have

    |0ϕtxψtUdx|120ϕ2txUdx+120ψ2tUdx (4.46)

    and

    |0d(1U)xψtxψtdx|C|0ψtxψtUdx|d40ψ2txUdx+C0ψ2tUdx. (4.47)

    By the boundedness of U(x) and V(x), we arrive that

    |02dVUψtψtxdx|C01U|ψtψtx|dxC0ψ2tUdx+d40ψ2txUdx. (4.48)

    Moreover, the fact that ψx(,t)LCN(t) leads to

    |02dψxψtψtxUdx|CN(t)0ψ2txUdx+CN(t)0ψ2tUdx. (4.49)

    Substituting (4.46)–(4.49) into (4.45) gives

    120ψ2tU+(d2CN(t))t00ψ2txU(C+CN(t))t00ψ2tU+12t00ϕ2txU+120ψ2tU|t=0. (4.50)

    Combing (4.42) and (4.50), by (4.43), we get

    0Uϕ2tdx+0ψ2tUdx+t00ϕ2txUdxdτ+t00ψ2txUdxdτC0(Uϕ20+ψ20+ϕ20xU2+ψ20xU+ϕ20xxU3+ψ20xxU)dx. (4.51)

    Squaring (3.6) and multiplying the resultant equations by 1U, owing to (4.51) and Lemma 4.3, we obtain

    0ϕ2xxUdxC0(W4Ue4ψϕ2t+ϕ2xψ2xU+Uψ2x+V2Uϕ2x)dxC0(Uϕ20+ψ20+ϕ20xU2+ψ20xU+ϕ20xxU3+ψ20xxU)dx, (4.52)

    and

    0ψ2xxUdxC0(ψ2tU+V2Uψ2x+ψ4xU+ϕ2xU)dxC0(Uϕ20+ψ20+ϕ20xU2+ψ20xU+ϕ20xxU3+ψ20xxU)dx. (4.53)

    Differentiating the first equation of (3.6) in x yields

    ϕxxx=ϕxxψxϕxψxxUxψxUψxxVxϕxVϕxx+W2ϕtx+2WWxϕt[(e2ψ1)(ϕxx+ϕxψx+Uψx+Vϕx)]x. (4.54)

    Squaring (4.54) and multiplying the resultant equations by 1U lead to

    t00ϕ2xxxUdxdτCt00(ϕ2xxψ2xU+ϕ2xψ2xxU+U2xUψ2x+Uψ2xx+V2xUϕ2x+V2Uϕ2xx+W4Uϕ2tx)+Ct00W2W2xUϕ2t+Ct001U|((e2ψ1)(ϕxx+ϕxψx+Uψx+Vϕx))x|2. (4.55)

    By (2.4), (4.51), the boundedness of U(x), and Hardy inequality, we have

    |t004W2W2xUϕ2tdxdτ|Ct00ϕ2txUdxdτC0(Uϕ20+ψ20+ϕ20xU2+ψ20xU+ϕ20xxU3+ψ20xxU)dx, (4.56)

    and

    t00W4Uϕ2txdxdτCt00ϕ2txUdxdτC0(Uϕ20+ψ20+ϕ20xU2+ψ20xU+ϕ20xxU3+ψ20xxU)dx. (4.57)

    Using (4.16) and Lemmas 4.2 and 4.3, with the fact that ψ(,t)LN(t), we get

    t001U|((e2ψ1)(ϕxx+ϕxψx+Uψx+Vϕx))x|2Ct00ψ2U|ϕxxx+ϕxxψx+ϕxψxx+Uxψx+Uψxx+Vxϕx+Vϕxx|2+Ct00e4ψU[ψx(ϕxx+ϕxψx+Uψx+Vϕx)]2CN(t)0(ϕ20x+ψ20xU+Uϕ20+ψ20)+CN(t)t00ϕ2xxxU. (4.58)

    Substituting (4.56)–(4.58) into (4.55), and using (4.52) and (4.53), when N(t)1, one gets

    t00ϕ2xxxUdxdτC0(Uϕ20+ψ20+ϕ20xU2+ψ20xU+ϕ20xxU3+ψ20xxU)dx. (4.59)

    Similarly, differentiating the second equation of (3.6) in x yields

    dψxxx=ψtx+2dVxψx+2dVψxx+2dψxψxxϕxx. (4.60)

    Then squaring (4.60), multiplying the resultant equations by 1U, using (4.51) and Lemmas 4.2 and 4.3, we arrive at

    t00ψ2xxxUdxdτCt00(ψ2txU+V2xUψ2x+V2Uψ2xx+ψ2xψ2xxU+ϕ2xxU)dxdτC0(Uϕ20+ψ20+ϕ20xU2+ψ20xU+ϕ20xxU3+ψ20xxU)dx. (4.61)

    From (4.59), (4.61) and (4.51)–(4.53), we get the desired (4.32).

    Notice that the estimate (4.32) requires that the initial data satisfies ϕ20xU2+ϕ20xxU3<. Hence, to guarantee the extension procedure works, we further need the following weighted elliptic estimate.

    Lemma 4.5. If N(t)1, we have

    ϕx(,t)U2L+ψx(,t)U2L+0ϕ2dx+0ϕ2xU2dx+0ϕ2xxU3dxC0(ϕ20+ψ20+ϕ20xU2+ψ20xU+ϕ20xxU3+ψ20xxU)dx. (4.62)

    Proof. By (4.1), (4.17) and (4.53), we have

    ψx(,t)U2LC0(Uϕ20+ψ20+ϕ20xU2+ψ20xU+ϕ20xxU3+ψ20xxU)dx. (4.63)

    We write the first equation of (4.8) as

    ϕxx+Vϕx=W2ϕtUψxϕxψx(e2ψ1)(ϕxx+ϕxψx+Uψx+Vϕx). (4.64)

    Multiplying (4.64) by ϕxU2V, noting

    0ϕxϕxxU2Vdx=U2V1ϕ2x|x=00(U2V1)xϕ2xdx,
    0(e2ψ1)U2Vϕxxϕxdx=0ϕ2xe2ψψxU2Vdx0ϕ2x2(e2ψ1)(1U2V)xdx(e2ψ1)2U2Vϕ2x|x=0,

    we get

    (e2ψ+1)2U2Vϕ2x|x=0+0(U2+(U2V1)x)ϕ2xdx=0W2U2Vϕxϕtdx+0ϕxψxUVdx+0ϕ2xψxU2Vdx0ϕ2xe2ψψxU2Vdx0ϕ2x2(e2ψ1)(1U2V)xdx+0(e2ψ1)U2Vϕx(ϕxψx+Uψx+Vϕx)dx. (4.65)

    A direct calculation by (3.3) gives

    U2+(U2V1)x=1U2VxU2V2>1U2.

    Thus,

    LHS of (4.65)0ϕ2xU2dx. (4.66)

    We next estimate the RHS of (4.65). By (4.2),

    |0ϕ2x2e2ψψxU2Vdx|+|0ϕ2x2(e2ψ1)(1U2V)xdx|CN(t)0ϕ2xU2dx.

    By (4.16),

    0(e2ψ1)U2Vϕx(ϕxψx+Uψx+Vϕx)dxCN(t)0(ϕ2xU2+|ϕxψx|UV)dx.

    Then we get from (4.13) and Young's inequality that

    RHS of (4.65)C0|ϕtϕx|U+C0|ϕxψx|UV+C0|ϕ2xψx|U2V+CN(t)0ϕ2xU2dx(12+CN(t))0ϕ2xU2dx+C0Uϕ2tdx+C0ψ2xUdx. (4.67)

    Now substituting (4.66) and (4.67) into (4.65), by Lemmas 4.3 and 4.4, we have

    0ϕ2xU2dxC0(Uϕ20+ψ20+ϕ20xU2+ψ20xU+ϕ20xxU3+ψ20xxU)dx, (4.68)

    which along with Hardy inequality gives

    0ϕ2dxC0(Uϕ20+ψ20+ϕ20xU2+ψ20xU+ϕ20xxU3+ψ20xxU)dx. (4.69)

    We next write the first equation of (3.6) as

    ϕxx=W2e2ψϕt(Vϕx+Uψx+ϕxψx).

    Squaring this equation and multiplying the resultant equation by 1U3, owing to (4.2) again, we obtain

    0ϕ2xxU3dxC0(Uϕ2t+V2U3ϕ2x+ψ2xU+ϕ2xψ2xU3)dxC(1+N(t))0ϕ2xU2dx+C0Uϕ2tdx+C0ψ2xUdx.

    Thus, by (4.68) and Lemma 4.4,

    0ϕ2xxU3dxC0(Uϕ20+ψ20+ϕ20xU2+ψ20xU+ϕ20xxU3+ψ20xxU)dx. (4.70)

    Using (4.3), (4.68) and (4.70), we get

    ϕx(,t)U2L0ϕ2xUdx+0ϕ2xxUdxC0(Uϕ20+ψ20+ϕ20xU2+ψ20xU+ϕ20xxU3+ψ20xxU)dx. (4.71)

    Therefore, (4.62) follows from (4.63) and (4.68)–(4.71). We complete the proof.

    Proof. [Proof of Proposition 4.2] It is a direct consequence of Lemmas 4.2–4.5.

    Proof. [Proof of Theorem 2.2] The a priori estimate (4.5) guarantees that if N(0) is small, then N(t) is small for all t>0. Thus, applying the standard extension argument, we obtain the global well-posedness of system (3.6)–(3.9) in X(0,). Owing to the transformation (3.5), system (2.7) and (2.8) has a unique global solution (u,v)(x,t) satisfying (2.9).

    We next prove the convergence (2.10). We first show that

    ϕx(,t)+ψx(,t)0as t. (4.72)

    It suffices to prove that ϕx(,t)2W1,1(0,) and ψx(,t)2W1,1(0,). By Lemma 4.2, we get

    00ϕ2xdxdtC00ϕ2xUdxdt<. (4.73)

    By Lemma 4.5, we have ψ(,t)LC and ψx(,t)WLC. In view of the first equation of (3.6), there exists a constant C such that

    |ddt0ϕ2xdx|=2|0ϕtxϕxdx|=2|0ϕxxϕtdx|=2|0W2ϕxxe2ψ(ϕxx+ϕxψx+Uψx+Vϕx)dx|Ceψ2L(0ϕ2xxW2dx+C(1+ψxW2L)0ϕ2xWdx+0ψ2xdx), (4.74)

    where we have used (4.13) and (4.19). Then integrating (4.74) with respect to t and using (4.5), we get

    0|ddt0ϕ2xdx|dt<,

    which, along with (4.73) leads to ϕx(,t)2W1,1(0,). Thus

    ϕx(,t)0ast. (4.75)

    Similarly, one has

    00ψ2xdxdt<. (4.76)

    Using the second equation of (3.6), there is a constant C>0 such that

    |ddt0ψ2xdx|=2|0ψxxψtdx|=2|0ψxx(dψxx2dVψxdψ2x+ϕx)dx|C(1+ψx2L)0ψ2xxdx+C0ψ2xdx+C0ϕ2xdx. (4.77)

    Then we integrate (4.77) with respect to t and make use of (4.5) to get

    0|ddt0ψ2xdx|dt<,

    which, along with (4.76) implies ψx(,t)2W1,1(0,). Thus

    ψx(,t)0ast. (4.78)

    (4.72) then follows from (4.75) and (4.78). By Cauchy-Schwarz inequality and (4.5), we get

    ϕ2x(x,t)=2xϕxϕxx(y,t)dy2(0ϕ2xdy)12(0ϕ2xxdy)12Cϕx(,t)0ast+.

    This implies

    supxR+|ϕx(x,t)|0ast+.

    Similarly, we have

    supxR+|ψx(x,t)|0ast+.

    Thus, (2.10) holds.

    Finally, we prove the L1 convergence. By Lemmas 4.2 and 4.4, we get

    00ϕ2xUC and 00ϕ2txUC. (4.79)

    A simple calculation gives

    |ddt0ϕ2xUdx|dt=2|0ϕtxϕxUdx|0ϕ2xUdx+0ϕ2txUdx. (4.80)

    Integrating (4.80) with respect to t and using (4.79), we obtain

    0|ddt0ϕ2xUdx|dt<,

    which, along with the first inequality (4.79) yields 0ϕ2x(x,t)UdxW1,1(0,). And then

    0ϕ2x(x,t)Udx0ast.

    Thus, from Hölder inequality and the fact that 0Udx<, it follows that

    0|ϕx(x,t)|dx(0ϕ2x(x,t)Udx)12(0Udx)120ast.

    This yields the convergence (2.11).

    Proof. [Proof of Theorem 2.3.] We just need to pass the results from v to w to complete the proof of Theorem 2.3. The transformation (3.5) and Theorem 2.2 give the regularity of wxwWxW.

    Next, we derive the results of wW. Let ξ:=wW. Owing to (2.5) and (3.5), it is easy to calculate that

    w(x,t)=bex0v(y,t)dy=bex0(ψx+V)dy=eψW.

    Thus, ξ=W(eψ1) and ξx=Wx(eψ1)Weψψx, which gives the regularity of wW.

    It is left to show the convergence. By Cauchy-Schwarz inequality and (4.5), we get

    ψ2(x,t)=2x0ψψx(y,t)dy2(0ψ2dy)12(0ψ2xdy)12Cψx(,t).

    Noting ψ(,t)LN(t)1, the Taylor expansion gives

    |1eψ|=|ψn=2(1)nψnn!|C|ψ|.

    Therefore, by (4.78), we get

    supxR+|ξ(x,t)|CsupxR+|ψ(x,t)|Cψx(,t)120 as t.

    For the L1 convergence, noting 0W(x)dx<, it follows from Hölder inequality and Hardy inequality that

    0|ξ(x,t)|dxC0W|ψ(x,t)|dxC(0W2dx)12(0W2ψ2(x,t)dx)12Cψx(,t)0 as t.

    We complete the proof of Theorem 2.3.

    We are concerned with the existence and stability of spiky patterns to the chemotaxis model (1.1) proposed by Kim and his collaborators [1,2]. This model was derived from the notion of "metric of food" which measures the amount of food. It avoids the mysterious assumption that the microscopic scale bacteria sense the macroscopic scale gradient of food. Moreover, this model also admits two types of traveling waves: traveling band and traveling front, under suitable assumptions on the consumption rates. Hence, it can be viewed as an alternative model to describe the propagation of traveling bands of bacteria observed in the experiment of Adler [3]. However, since the traveling wave of oxygen W vanishes at far field, one has to encounter the challenge of presence of two types of singularities in the study of stability of traveling waves. As the first step we investigate instead the stability of stationary waves to the model in the half space. In this case the model remains singular at the far field. We successfully find an effective strategy to handle the two types of singularities. In the following studies, we will apply the strategy of this paper to study the stability of traveling waves of the model by modifying some estimates.

    The potential biological application of our results is the explanation of formation of a plume pattern for aerobic bacteria observed in the experiment of [7], where the bacteria consume oxygen in a water drop. We conjecture that this plume pattern is a superposition of series of one dimensional spikes. However, owing to the lack of effective mathematical tools to handle the stability of biological patterns to a chemotaxis-fluid model, we consider a simplified fluid free chemotaxis model. And we expect that our argument is effective for more general chemotaxis models and even some chemotaxis-fluid models.

    The authors are grateful to the three referees for their insightful comments and suggestions, which lead to great improvements of our original manuscript. This work is supported by the Natural Science Foundation of Jilin Province (20210101144JC).

    The authors declare there is no conflict of interest.

    Proof. [Proof of Proposition 3.1] The local existence can be proved using the principle of contraction mapping. Set

    YT:={(f,g)|(f,g)L((0,T);H1),(fx,gx)L2((0,T);H1)}

    equipped with norm

    (f,g)YT:=(f,g)L((0,T);H1)+(fx,gx)L2((0,T);H1).

    Define a mapping Z: (ˆϕ,ˆψ)YTZ(ˆϕ,ˆψ) such that (ϕ,ψ)=Z(ˆϕ,ˆψ) is a solution of

    {W2εϕt=e2ψ(ϕxx+ˆϕxˆψx+Uˆψx+Vεˆϕx),ψt=dψxx2dVεˆψxdˆψx2+ˆϕx, (A1)

    with the initial and boundary conditions (3.7)–(3.9). Taking a ball

    BM,T:={(ˆϕ,ˆψ):(ˆϕ,ˆψ)YTM},

    where M is a constant to be determined later. We shall show that there are M and T such that (i) Z maps BM,T into itself; (ii) Z is a contraction in BM,T.

    We first show (i). According to the standard linear parabolic theory, for any (ˆϕ,ˆψ)YT, the second equation of (A1) has a unique strong solution ψ. Substituting ψ into the first equation, we obtain the existence of strong solution ϕ. Hence the mapping Z is well-defined.

    We next derive the estimates for (ϕ,ψ). Multiplying the second equation of (A1) by ψ gives

    120ψ2+dt00ψ2x12t00ψ2+Ct00(ˆψ2x+ˆϕ2x)+dt00ˆψ2x|ψ|+120ψ20,

    where

    d0ˆψ2x|ψ|dψL0ˆψ2xdM2ψ12L2ψx12L2d20ψ2x+0ψ2+CM4.

    Then choosing T12, we get

    sup0tT0ψ2dx+T00ψ2xdxdtC(M2+M4)T+20ψ20dx. (A2)

    Similarly, multiplying the second equation of (A1) by ψxx leads to

    sup0tT0ψ2xdx+T00ψ2xxdxdtCM2T+CM3T12(T00ˆψ2xx)12+0ψ20xdxCM2T+CM4T12+0ψ20xdx, (A3)

    where we have used

    0ˆψ4xM2ˆψx2LM2ˆψxL2ˆψxxL2M3ˆψxxL2.

    By (A2) and (A3), if we take M220(ψ20+ψ20x) and chose T small enough, then

    ψ2L((0,T);H1)M2. (A4)

    Multiplying the first equation of (A1) by ϕ gives

    120W2εϕ2+t00e2ψϕ2x=2t00e2ψψxϕxϕ+t00e2ψ(ˆϕxˆψx+Uˆψx+Vεˆϕx)ϕ+120W2εϕ20, (A5)

    where

    20e2ψ|ψxϕxϕ|2e2Me2ψϕx32L2ϕ12L2ψxL2120e2ψϕ2x+e4MM40ϕ2,
    0e2ψ|ˆϕxˆψxϕ|0ϕ2+e4Mˆϕx2Lˆψx2L20ϕ2+e4MM3ˆψxxL2,

    and

    0e2ψ|(Uˆψx+Vεˆϕx)ϕ|0ϕ2+e4MM2.

    If e4MM4Tε22, we get from (A5) that

    sup0tT0ϕ2+T00e2ψϕ2xCe4MM4T12ε2+Cε20ϕ20. (A6)

    Similarly, multiplying the first equation of (A1) by ϕxx gives

    120W2εϕ2x+12t00e2ψϕ2xxt00e2ψ|(ˆϕxˆψx+Uˆψx+Vεˆϕx)ϕxx|+2t00Wε|Wεxϕxϕt|+120W2εϕ20x

    where

    \begin{split} I\leq e^{2M}M^3\|\hat{\psi}_{xx}\|_{L^2}+e^{2M}M^2,\ II\leq\frac{1}{4}\int_0^{\infty}e^{2\psi}\phi_{xx}^2 +Ce^{2M}M^4+\int_0^{\infty}\phi_{x}^2.\end{split}

    Then choosing T\leq\frac{\varepsilon^{2}}{2} , we have

    \begin{equation} \begin{split} \sup\limits_{0\le t\le T}\int_0^{\infty}\phi_x^2 +\int_0^T\int_0^{\infty}e^{2\psi}\phi_{xx}^2\le C\varepsilon^{-2}e^{2M}M^4T^{\frac{1}{2}} +C\varepsilon^{-2}\int_0^{\infty}\phi_{0x}^2. \end{split} \end{equation} (A7)

    In view of (A2), (A3), (A6) and (A7), we choose M and T satisfying

    M = 4\int_0^{\infty}(\psi_{0}^2+\psi_{0x}^2) +4C\varepsilon^{-2}\int_0^{\infty}(\phi_{0}^2+\phi_{0x}^2)+1, \ 8C(M+M^3)T^{\frac{1}{2}}+8Ce^{4M}M^4\varepsilon^{-2}T^{\frac{1}{2}}\leq1,

    then \|(\phi, \psi)\|_{{\mathcal{Y}}_{T}}\le M , which verifies (i).

    We proceed to show (ii). For any (\hat{\phi}_1, \hat{\psi}_1) , (\hat{\phi}_2, \hat{\psi}_2)\in B_{M, T} , set (\phi_1, \psi_1) = \mathcal{Z}(\hat{\phi}_1, \hat{\psi}_1) , (\phi_2, \psi_2) = \mathcal{Z}(\hat{\phi}_2, \hat{\psi}_2) and (\bar{\phi}, \bar{\psi}): = (\phi_1, \psi_1) -(\phi_2, \psi_2) . Then (\bar{\phi}, \bar{\psi}) satisfies

    \begin{equation} \begin{cases} \begin{aligned} W_\varepsilon^2\bar{\phi}_t = &e^{2\psi_1}\bar{\phi}_{xx}+(e^{2\psi_1}-e^{2\psi_2})\phi_{2xx} +(e^{2\psi_1}-e^{2\psi_2})\hat{\phi}_{1x}\hat{\psi}_{1x}\\&+ e^{2\psi_2}(\hat{\phi}_{1x}-\hat{\phi}_{2x})\hat{\psi}_{1x} +e^{2\psi_2}\hat{\phi}_{2x}(\hat{\psi}_{1x}-\hat{\psi}_{2x})\\ &+U(e^{2\psi_1}-e^{2\psi_2})\hat{\psi}_{1x} +Ue^{2\hat{\psi}_2}(\hat{\psi}_{1x}-\hat{\psi}_{2x})\\& +V_\varepsilon(e^{2\psi_1}-e^{2\psi_2})\hat{\phi}_{1x} +V_\varepsilon e^{2\psi_2}(\hat{\phi}_{1x}-\hat{\phi}_{2x}),\\ \bar{\psi}_t = &d \bar{\psi}_{xx}-2d V_\varepsilon(\hat{\psi}_1-\hat{\psi}_2)_x -d(\hat{\psi}_1-\hat{\psi}_2)_x(\hat{\psi}_{1x}+\hat{\psi}_{2x}) +(\hat{\phi}_1-\hat{\phi}_2)_x \end{aligned} \end{cases} \end{equation} (A8)

    with zero initial-boundary conditions. Multiplying the second equation of (A8) by \bar{\psi} gives

    \int_0^{\infty}\bar{\psi}^2+d\int_0^T\int_0^{\infty}\bar{\psi}_x^2 \leq(C+M^2)\int_0^T\int_0^{\infty}|(\hat{\psi}_1-\hat{\psi}_2)_x|^2 +\int_0^T\int_0^{\infty}(\bar{\psi}^2 +C|(\hat{\phi}_1-\hat{\phi}_2)_x|^2).

    Thus, choosing T\leq\frac{1}{2} , we get

    \begin{split} \|\bar{\psi}\|_{L^{\infty}((0,T);L^2)}^2+\|\bar{\psi}_x\|_{L^2((0,T);L^2)}^2\leq& C(1+M^2)T\|(\hat{\psi}_1-\hat{\psi}_2)_x\|_{L^{\infty}((0,T);L^2)}^2\\& +CT\|(\hat{\phi}_1-\hat{\phi}_2)_x\|_{L^{\infty}((0,T);L^2)}^2. \end{split}

    Multiplying the second equation of (A8) by \bar{\psi}_{xx} , and noting

    \begin{split} \int_0^T\int_0^{\infty}|(\hat{\psi}_1+\hat{\psi}_2)_x|^2 |(\hat{\psi}_1-\hat{\psi}_2)_x|^2 &\leq M\|(\hat{\psi}_1-\hat{\psi}_2)_x\|_{L^{\infty}((0,T);L^2)}^2 \int_0^T\|(\hat{\psi}_1+\hat{\psi}_2)_{xx}\|_{L^2}\\ &\leq M^2T^{\frac{1}{2}}\|(\hat{\psi}_1-\hat{\psi}_2)_x\|_{L^{\infty}((0,T);L^2)}^2, \end{split}

    we get after choosing T\leq\frac{1}{2} that

    \begin{equation} \begin{split} &\|\bar{\psi}_x\|_{L^{\infty}((0,T);L^2)}^2+\|\bar{\psi}_{xx}\|_{L^2((0,T);L^2)}^2\\ &\leq CM^2T^{\frac{1}{2}}\|(\hat{\psi}_1-\hat{\psi}_2)_x\|_{L^{\infty}((0,T);L^2)}^2 +CT\|(\hat{\phi}_1-\hat{\phi}_2)_x\|_{L^{\infty}((0,T);L^2)}^2. \end{split} \end{equation} (A9)

    This also implies

    \begin{equation} \begin{split} \|\bar{\psi}\|_{L^{\infty}((0,T);L^\infty)}^2\leq CM^2T^{\frac{1}{2}}\|(\hat{\psi}_1-\hat{\psi}_2)_x\|_{L^{\infty}((0,T);L^2)}^2 +CT\|(\hat{\phi}_1-\hat{\phi}_2)_x\|_{L^{\infty}((0,T);L^2)}^2. \end{split} \end{equation} (A10)

    Multiplying the first equation of (A8) by \bar{\phi} , noting

    \begin{split} \int_0^T\int_0^\infty|(e^{2\psi_1}-e^{2\psi_2})\phi_{2xx}\bar{\phi}|&\leq \int_0^T\|\bar{\psi}\|_{L^{\infty}}\|\phi_{2xx}\|_{L^{2}}\|\bar{\phi}\|_{L^{2}}\\ &\leq T\|\bar{\phi}\|_{L^{\infty}((0,T);L^2)}^2 +M^2\|\bar{\psi}\|_{L^{\infty}((0,T);L^\infty)}^2,\end{split}

    we have

    \begin{split} &\varepsilon^2\|\bar{\phi}\|_{L^{\infty}((0,T);L^2)}^2 +\|\bar{\phi}_x\|_{L^{2}((0,T);L^2)}^2\\ &\leq C(M)T\left(\|\bar{\phi}\|_{L^{\infty}((0,T);L^2)}^2 +\|\bar{\psi}\|_{L^{\infty}((0,T);L^\infty)}^2\right)\\&\quad +C(M)T\left(\|(\hat{\psi}_1-\hat{\psi}_2)_x\|_{L^{\infty}((0,T);L^2)}^2 +\|(\hat{\phi}_1-\hat{\phi}_2)_x\|_{L^{\infty}((0,T);L^2)}^2\right).\end{split}

    Multiplying the first equation of (A8) by \bar{\phi}_{xx} , noting

    \begin{split} \int_0^T\int_0^\infty|(e^{2\psi_1}-e^{2\psi_2}) \phi_{2xx}\bar{\phi}_{xx}|&\leq \frac{1}{2}\int_0^T\int_0^\infty\bar{\phi}_{xx}^2+ C\int_0^Te^{6M}\|\bar{\psi}\|_{L^{2}}\|\bar{\psi}_x\|_{L^{2}} \|\phi_{2xx}\|_{L^{2}}^2\\ &\leq \frac{1}{2}\int_0^T\int_0^\infty\bar{\phi}_{xx}^2+ C(M)\|\bar{\psi}\|_{L^{\infty}((0,T);L^2)} \|\bar{\psi}_x\|_{L^{\infty}((0,T);L^2)},\end{split}

    and

    \begin{split} &\int_0^T\int_0^\infty e^{2\psi_2}(\hat{\phi}_{1}-\hat{\phi}_{2})_x\hat{\psi}_{1x}\bar{\phi}_{xx}\\ &\leq \frac{1}{2}\int_0^T\int_0^\infty\bar{\phi}_{xx}^2+ C(M)\|(\hat{\phi}_1-\hat{\phi}_2)_x\|_{L^{\infty}((0,T);L^2)}^2\int_0^T \|\hat{\psi}_{1xx}(\cdot,t)\|_{L^{2}}\\ &\leq \frac{1}{2}\int_0^T\int_0^\infty\bar{\phi}_{xx}^2+ C(M)T^{\frac{1}{2}}\|(\hat{\phi}_1-\hat{\phi}_2)_x\|_{L^{\infty}((0,T);L^2)}^2, \end{split}

    we have

    \begin{split} &\varepsilon^2\|\bar{\phi}_x\|_{L^{\infty}((0,T);L^2)}^2 +\|\bar{\phi}_{xx}\|_{L^{2}((0,T);L^2)}^2\\& \leq C(M)\|\bar{\psi}\|_{L^{\infty}((0,T);L^2)} \|\bar{\psi}_x\|_{L^{\infty}((0,T);L^2)}\\&\quad +C(M)T\left(\|(\hat{\psi}_1-\hat{\psi}_2)_x\|_{L^{\infty}((0,T);L^2)}^2 +\|(\hat{\phi}_1-\hat{\phi}_2)_x\|_{L^{\infty}((0,T);L^2)}^2\right).\end{split}

    Therefore, owing to (A9)–(A10), we can take T small enough to derive

    \begin{equation} \|(\bar{\phi},\bar{\psi})\|_{\mathcal{Y}_T}\le \frac{1}{2} \|(\hat{\phi}_1-\hat{\phi}_2, \hat{\psi}_1-\hat{\psi}_2)\|_{\mathcal{Y}_T}, \end{equation} (A11)

    which verifies (ii).

    Now we apply the contraction mapping principle to obtain that system (3.10) has a solution. The uniqueness follows from a similar argument as (A11) and the Gronwall's inequality.



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