# Well-Posedness, Blow-Up Phenomena, and Asymptotic Profile for a Weakly Dissipative Modified Two-Component Camassa-Holm Equation

Authors: Yongsheng Mi; Chunlai Mu
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We study the Cauchy problem of a weakly dissipative modified two-component Camassa-Holm equation. We firstly establish the local well-posedness result. Then we present a precise blow-up scenario. Moreover, we obtain several blow-up results and the blow-up rate of strong solutions. Finally, we consider the asymptotic behavior of solutions.


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## Body

## 1. Introduction

In this paper, we consider the Cauchy problem of the following weakly dissipative modified two-component Camassa-Holm system:(1)mt+umx+2mux+ρρ¯x+λm=0,t>0,x∈ℝ,ρt+(ρu)x+λρ=0,t>0,x∈ℝ,m(0,x)=m0(x),x∈ℝ,ρ(0,x)=ρ0(x),x∈ℝ,
where m=(1-∂x2)u,ρ=(1-∂x2)(ρ¯-ρ¯0), and λ is a nonnegative dissipative parameter.The Camassa-Holm equation [1] has been recently extended to a two-component integrable system (CH2)
(2)mt+umx+2mux=ρρx,t>0,x∈ℝ,ρt+(ρu)x=0,t>0,x∈ℝ,
with m=u-uxx, which is a model for wave motion on shallow water, where u(t,x) describes the horizontal velocity of the fluid and ρ(t,x) is in connection with the horizontal deviation of the surface from equilibrium, all measured in dimensionless units. Moreover, u and ρ satisfy the boundary conditions: u→0 and ρ→1 as |x|→∞. The system can be identified with the first negative flow of the AKNS hierarchy and possesses the interesting peakon and multikink solutions [2]. Moreover, it is connected with the time-dependent Schrödinger spectral problem [2]. Popowicz [3] observes that the system is related to the bosonic sector of an N=2 supersymmetric extension of the classical Camassa-Holm equation. Equation (2) with ρ≡0 becomes the Camassa-Holm equation, which has global conservative solutions [4] and dissipative solutions [5]. For other methods to handle the problems relating to various dynamic properties of the Camassa-Holm equation and other shallow water equations, the reader is referred to [6–8] and the references therein.Since the system was derived physically by Constantin and Ivanov [9] in the context of shallow water theory (also by Chen et al. in [2] and Falqui in [10]), many researchers have paid extensive attention to it. In [11], Escher et al. establish the local well-posedness and present the precise blow-up scenarios and several blow-up results of strong solutions to (2) on the line. In [9], Constantin and Ivanov investigate the global existence and blow-up phenomena of strong solutions of (2) on the line. Later, Guan and Yin [12] obtain a new global existence result for strong solutions to (2) and get several blow-up results, which improve the recent results in [9]. Recently, they study the global existence of weak solutions to (2) [13]. In [14], Henry studies the infinite propagation speed for (2). Gui and Liu [15] establish the local well-posedness for (2) in a range of the Besov spaces; they also derive a wave-breaking mechanism for strong solutions. Mustafa [16] gives a simple proof of existence for the smooth travelling waves for (2). Hu and Yin [17, 18] study the blow-up phenomena and the global existence of (2) on the circle.Recently, the CH2 system was generalized into the following modified two-component Camassa-Holm (MCH2) system:(3)mt+umx+2mux=-gρρ¯x,t>0,x∈ℝ,ρt+(ρu)x=0,t>0,x∈ℝ,
where m=(1-∂x2)u,ρ=(1-∂x2)(ρ¯-ρ¯0),u denotes the velocity field, ρ¯0 is taken to be a constant, and g is the downward constant acceleration of gravity in applications to shallow water waves. This MCH2 system does admit peaked solutions in the velocity and average density; we refer this to [19] for details. There, the authors analytically identified the steepening mechanism that allows the singular solutions to emerge from smooth spatially confined initial data. They found that wave breaking in the fluid velocity does not imply singularity in the pointwise density ρ at the point of vertical slope. Some other recent works can be found in [20–31]. We find that the MCH2 system is expressed in terms of an averaged or filtered density ρ¯ in analogy to the relation between momentum and velocity by setting ρ=(1-∂x2)(ρ¯-ρ¯0), but it may not be integrable unlike the CH2 system. The important point here is that MCH2 has the following conservation law ∫R‍(u2+ux2+ρ2+ρx2)dx, which plays a crucial role in the study of (3), noting that, for the CH2 system, we cannot obtain the conservation of H1 norm.In general, it is quite difficult to avoid energy dissipation mechanisms in the real world. Ghidaglia [32] studies the long-time behavior of solutions to the weakly dissipative KdV equation as a finite-dimensional dynamical system. Recently, Hu and Yin [33] study the blow-up and blow-up rate of solutions to a weakly dissipative periodic rod equation. In [34, 35], Hu considered global existence and blow-up phenomena for a weakly dissipative two-component Camassa-Holm system on the circle and on the line. However, (1) on the line (nonperiodic case) has not been studied yet. The aim of this paper is to study the blow-up phenomena and asymptotic profile of the strong solutions to (1). We find that asymptotic profile of solutions to the weakly dissipative modified two-component periodic Camassa-Holm system (1) is similar to that of the modified two-component Camassa-Holm system (3), such as the local well-posedness and the blow-up scenario. In addition, we also find that the blow-up rate of (1) is not affected by the weakly dissipative term, but the occurrence of blow-up of (1) is affected by the dissipative parameter.This paper is organized as follows. In Section2, we present some notations and establish the local well-posedness for system (1) by applying Kato's semigroup approach to nonlinear hyperbolic evolution equations. In Section 3, we prove a precise blow-up scenario result. In Section 4, we present the blow-up results for strong solutions to (1) provided that the initial data satisfy appropriate conditions and we derive a blow-up rate estimate result. Finally, we consider the asymptotic behavior of solutions.


## 2. Local Well-Posedness

We now provide the framework in which we will reformulate system (1). With m=u-uxx,ρ=γ-γxx, and γ=ρ¯-ρ¯0, we can rewrite (1) as follows:
(4)mt+umx+2mux+ργx+λy=0,t>0,x∈ℝ,ρt+(ρu)x+λρ=0,t>0,x∈ℝ,y(0,x)=u0(x)-u0,xx(x),x∈ℝ,ρ(0,x)=γ0(x)-γ0,xx(x),x∈ℝ.
Note that if p(x):=(1/2)e-|x|,x∈R, then (1-∂x2)-1f=p*f for all f∈L2(R),p*y=u, and p*ρ=γ. Here, we denote by * the convolution. Using this identity, we can rewrite (5) as follows:
(5)ut+uux=-∂xp*(u2+12ux2+12γ2-12γx2)-λu,t>0,x∈ℝ,γt+uγx=-p*((uxγx)x+uxγ)-λγ,t>0,x∈ℝ,u(0,x)=u0(x),x∈ℝ,γ(0,x)=γ0(x),x∈ℝ,
or we can write it in the equivalent form
(6)ut+uux=-∂x(1-∂x2)-1(u2+12ux2+12γ2-12γx2)-λu,t>0,x∈ℝ,γt+uγx=-∂x(1-∂x2)-1(uxγx)-(1-∂x2)-1uxγ-λγ,t>0,x∈ℝ,u(0,x)=u0(x),x∈ℝ,γ(0,x)=γ0(x),x∈ℝ.
The local well-posedness of the Cauchy problem (5) in Sobolev spaces Hs with s>5/2 can be obtained by applying Kato's theorem [23, 36]. As a result, we have the following well-posedness result.Theorem 1.
Givenz0=z(x,0)=(u0,γ0)∈Hs×Hs, s>5/2, then there exist a maximal T=T(∥z0∥Hs×Hs)>0 and a unique solution z=(u,γ) to (3) (or (6)) such that
(7)z=(·,z0)∈C([0,T];Hs×Hs)∩C1([0,T];Hs-1×Hs-1).
Moreover, the solution depends continuously on the initial data; that is, the mapping u0→u(·,u0):Hs→C([0,T];Hs(ℝ))∩C1([0,T];Hs-1(ℝ)) is continuous.


## 3. The Precise Blow-Up Scenario

In this section, we present the precise blow-up scenarios for solutions to (6).Theorem 2.
Letz0=(u,γ)∈Hs×Hs,s>5/2, be given and assume that T is the maximal existence time of the corresponding solution z=(u,γ) to (6) with initial data z0; if there exists M>0 such that
(8)∥ux(t,·)∥L∞+∥γ(t,·)∥L∞+∥γx(t,·)∥L∞≤M,t∈[0,T),
then the Hs×Hs norm of z(t,·) does not blow up on [0,T).The proof of the theorem is similar to the proof of Theorem 3 in [22]; we omit it here.Consider the following differential equation:(9)dq(x,t)dt=u(q(x,t),t),t∈[0,T),q(0,t)=x,x∈ℝ.
By applying the classical results on the theory of ordinary differential equations, we may derive the following properties of the solution q of (9), which are crucial in the proof of global existence and blow-up of solutions.Lemma 3 (see [23]).
Letu0∈Hs,s≥5/2, and let T be the maximal existence time of the corresponding solution u(t,x) to (9). Then (9) has a unique solution q∈C1([0,T)×ℝ,ℝ). Moreover, the map q(t,·) is an increasing diffeomorphism of ℝ with
(10)qx(x,t)=exp(∫0t‍ux(q(x,s),s)ds)>0,qx(x,0)=1,x∈R,0≤t<T.Lemma 4.
Letz0=(u0,γ0)∈Hs×Hs with s>5/2, and let T>0 be the maximal existence time of the corresponding solution z∈(u,γ) to (5). Then, one has
(11)ρ(t,q(x,t))qx=ρ0e-λt,∀(t,x)∈[0,T)×R.
Moreover, if there exists M1>0, such that ux(t,x)≥-M1 for all (t,x)∈[0,T)×R, then
(12)∥ρ(t,·)∥L∞=∥ρ(t,q(t,x))∥L∞≤eM1T∥ρ0(·)∥L∞,∀t∈[0,T).Proof.
Differentiating the left-hand side of (6) with respect to t, in view of (9) and the second equation of (5), we have
(13)ddt(ρ(t,q(x,t))qx)=ρtqx+ρxqtqx+ρqxt=-λρqx.
Solving the equation, we get (11).
By Lemma3, in view of (11) and the assumption of the lemma, we obtain
(14)∥ρ(t,·)∥L∞=∥ρ(t,q(t,·))∥∞=∥e-λt-∫0t‍ux(s,·)dxρ0(·)∥L∞≤eM1T∥ρ(·)∥L∞,∀t∈[0,T).
The following result is proved only with regard tor=3, since we can obtain the same conclusion for the general case r>5/2 by using Theorem 1 and a simple density argument.We now present a precise blow-up scenario for strong solutions to (5).Theorem 5.
Lety0=(u0,γ0)∈Hs×Hs, s>5/2, and let T be the maximal existence of the corresponding solution z=(u,γ) to (6). Then, the solution blows up in finite time if and only if
(15)liminft→T,x∈ℝux(t,x)=-∞
or
(16)limsupt→T{∥γx(t,·)∥L∞}=+∞.Proof.
Multiplying the first equation in (5) by m=u-uxx and integrating by parts, we obtain
(17)ddt∫ℝ‍m2dx=2∫ℝ‍mmtdx=2∫ℝ‍m(-umx-2mux-ργx)dx-2λ∫ℝ‍m2dx=-3∫ℝ‍m2uxdx-2∫ℝ‍mργxdx-2λ∫ℝ‍m2dx.
Repeating the same procedure to the second equation in (5), we get
(18)ddt∫ℝ‍ρ2dx=-∫ℝ‍ρ2ux-2λ∫ℝ‍ρ2dx.
A combination of (17) and (18) yields
(19)ddt∫ℝ‍(m2+ρ2)dx=-3∫ℝ‍m2uxdx-2∫ℝ‍mργxdx-∫ℝ‍ρ2ux-2λ∫ℝ‍(m2+ρ2)dx.
Differentiating the first equation in (5) with respect to x, multiplying by mx=ux-uxxx, and then integrating over ℝ, we obtain
(20)ddt∫ℝ‍mx2dx=-5∫ℝ‍mx2uxdx+2∫ℝ‍m2uxdx-2∫ℝ‍mxρxγxdx-2∫ℝ‍mxργxxdx-2λ∫ℝ‍mx2dx.
Similarly,
(21)ddt∫ℝ‍ρx2dx=-3∫ℝ‍ρx2uxdx+∫ℝ‍ρ2uxxxdx-2λ∫ℝ‍ρx2dx.
A combination of (17)–(21) yields
(22)ddt∫ℝ‍(m2+ρ2+mx2+ρx2)dx=-∫ℝ‍m2uxdx-5∫ℝ‍mx2uxdx-2∫ℝ‍mργxdx-2∫ℝ‍mxρxγxdx-2λ∫ℝ‍(m2+ρ2)dx-2∫ℝ‍mxργxxdx-∫ℝ‍ρ2uxdx-3∫ℝ‍ρx2uxdx+∫ℝ‍ρ2uxxxdx-2λ∫ℝ‍(mx2+ρx2)dx=-∫ℝ‍m2uxdx-5∫ℝ‍mx2uxdx-∫ℝ‍ρ2uxdx-3∫ℝ‍ρx2uxdx-2λ∫ℝ‍(m2+ρ2)dx+∫ℝ‍ρ2uxxxdx-2∫ℝ‍mργxdx-2∫ℝ‍mxρxγxdx-2∫ℝ‍mxργxxdx-2λ∫ℝ‍(mx2+ρx2)dx.
Assume that there exist M1>0 and M2>0 such that ux(t,x)≥M1 and ∥γx(t,·)∥L∞≤M2 for all (t,x)∈[0,T)×R; then it follows from Lemma 4 that
(23)∥ρ(t,·)∥L∞≤eM1T∥ρ0(·)∥L∞.
Therefore,
(24)ddt∫ℝ‍(m2+ρ2+mx2+ρx2)dx≤(5M1+2λ)∫ℝ‍(m2+ρ2+mx2+ρx2)dx+(M2+eM1T∥ρ0(·)∥L∞)×∫ℝ‍(m2+ρ2+mx2+ρx2+uxxx2+γxx2)dx≤(5M1+2λ)∫ℝ‍(m2+ρ2+mx2+ρx2)dx+2(M2+eM1T∥ρ0(·)∥L∞)×∫ℝ‍(m2+ρ2+mx2+ρx2)dx≤(5M1+2λ)+2(M2+eM1T∥ρ0(·)∥L∞)×∫ℝ‍(m2+ρ2+mx2+ρx2)dx.
The previous discussion shows that if there exist M1>0 and M2>0 such that ux(t,x)≥M1 and ∥γx(t,·)∥≤M2 for all (t,x)∈[0,T)×ℝ, then there exist two positive constants K and k such that the following estimate holds:
(25)∥u(t,·)∥Hs2+∥v(t,·)∥Hs2≤Kekt,t∈[0,T).
This inequality, Sobolev's embedding theorem, and Theorem2 guarantee that the solution does not blow up in finite time.
On the other hand, we see that, if(26)liminft→T,x∈ℝux(t,x)=-∞orlimsupt→T{∥γx(t,·)∥L∞}=+∞,
then, by Sobolev's embedding theorem, the solution will blow up in finite time. This completes the proof of the theorem.


## 4. Blow-Up Results and Blow-Up Rate Estimate

In this section, we investigate the blow-up phenomena of strong solutions to (6). We now present the first blow-up result.Lemma 6.
Letz0=(u0,γ0)∈Hs×Hs,s>5/2, and let T be the maximal existence time of the solution z=(u,γ) to (6) with the initial data z0. Then, for all t∈[0,T), one has
(27)∥u(t,·)∥H12+∥γ(t,·)∥H12=e-2λt(∥u0∥H12+∥γ0∥H12).
Moreover,
(28)∥u(t,·)∥L∞≤22∥u(t,·)∥H1≤22(∥u0∥H12+∥γ0∥H12)1/2,∥γ(t,·)∥L∞≤22∥γ(t,·)∥H1≤22(∥u0∥H12+∥γ0∥H12)1/2.Proof.
Denote(29)f(u,γ)=u2+12ux2+12γ2-12γx2,g=g(u,γ)=(uxγx)x+uxγ.
In view of the identity -∂x2p*f=f-p*f, we can obtain, from (6),
(30)utx=-ux2-uuxx+f-p*f,vtx=-uxγx-uγxx-∂xp*g.
Therefore, an integration by parts yields
(31)12ddt(∥u∥H12+∥γ∥H12)=∫ℝ‍(uut+uxutx+γγt+γxγtx)dx=∫ℝ‍u(-uux-∂x2p*f-λu)+ux(-ux2-uuxx+f-p*f-λux)+γ(-uγx-λγ)+γx(-uγx-uγxx-∂xp*g-λγx)dx=∫ℝ‍[-12ux3+ux(u2+12ux2+12γ2-12γx2)-uγγx-γ(uxxγx+uxγ)-uγxγx2-uγxγxx-λ(u2+ux2+γ2+γx2)12]dx=-λ∫ℝ‍(u2+ux2+γ2+γx2)dx.
Thus, the statement of the conservation law follows. The remaining part of this lemma can be easily deduced from the conservation law. The proof of the lemma is complete.Lemma 7 (see [37]).
LetT>0 and v∈C1([0,T);H2). Then, for every t∈[0,T), there exists at least one point ξ∈R with
(32)m(t):=infx∈R[vx(t,x)]=vx(t,ξ(t)).
The function m(t) is absolutely continuous on (0,T) with
(33)dmdt=vtx(t,ξ(t)),a.e.,on(0,T).Theorem 8.
Letz0=(u0,γ0)∈Hs×Hs,s>5/2, and let T be the maximal existence time of the solution z=(u,γ) to the (6) with the initial data z0. If there exists some x0∈ℝ such that
(34)u0′(x0)<-λ-[λ2+(∥u0∥H12+∥γ0∥H12)]1/2,
then the existence time T is finite and the slope of u tends to negative infinity as t goes to T while u remains uniformly bounded on [0,T].Proof.
As mentioned earlier, here we only need to show that the previous theorem holds fors=3. Differentiating the first equation of (6) with respect to x, in view of ∂x2p*f=p*f-f, we have
(35)utx+uuxx=-12ux2+u2+12γ2-12γx2-p*(u2+12ux2+12γ2-12γx2)-λux.
Note that
(36)dux(t,q(t,x))dt=uxt(t,q(t,x))+uxx(t,q(t,x))qt(t,x)=uxt(t,q(t,x))+u(t,q(tx))uxx(t,q(t,x)).
We know that p*(u2+(1/2)ux2)≥(1/2)u2and
(37)∥p*γx2∥L∞≤∥p∥L∞∥γx2∥L1=12∥γx2∥L1.
By (35) and (36) and the previous estimates, we deduce that
(38)dux(t,q(t,x))dt≤-12ux2(t,q(t,x))+12u2(t,q(t,x))+14γ2(t,q(t,x))+34p*(γx2)(t,q(t,x))-λux≤-12ux2(t,q(t,x))+12u2(t,q(t,x))+14γ2(t,q(t,x))+38∥γx∥L1-λux≤-12ux2(t,q(t,x))+14∥u∥H12+18∥γ∥H12+38∥γx2∥L1-λux≤-12ux2(t,q(t,x))+12(∥u∥H12+∥γ∥H12)-λux=-12ux2(t,q(t,x))-λux+12(∥u0∥H12+∥γ0∥H12),
in view of Lemma 6. Take
(39)K=22(∥u0∥H12+∥γ0∥H12)1/2
and define g(t)=ux(t,q(t,x0)). It then follows from (38) that on [0,T),
(40)g′(t)≤-12g2(t)-λg+K2=-12(g(t)+λ+λ2+2K2)×(g(t)+λ-λ2+2K2).
Note that if g(0)≤-λ-λ2+2K2, then g(t)≤-λ-λ2+2K2, forallt∈[0,T). Therefore, we can solve the previous inequality to obtain
(41)g(0)+λ+λ2+2K2g(0)+λ-λ2+2K2eλ2+2K2t-1≤2λ2+2K2g(t)+λ-λ2+2K2.
Due to 0<(g(0)+λ+λ2+2K2)/(g(0)+λ-λ2+2K2)<1, then there exists T, and 0<T<(1/λ2+2K2)ln((g(0)+λ+λ2+2K2)/(g(0)+λ-λ2+2K2)), such that limt→Tg(t)=-∞. Applying Theorem 5, the solution z does not exist globally in time.Next, we give a blow-up result ifu0 and γ0 are odd.Theorem 9.
Letz0=(u0,γ0)∈Hs×Hs,s>5/2, and let T be the maximal existence time of the solution z=(u,γ) to (6) with the initial data z0. If u0 and γ0 are odd, and furthermore
(42)u0′(x0)<-λ-[λ2+12(∥u0∥H12+∥γ0∥H12)]1/2,
then T is finite and ux(t,0)→-∞ as t goes to T.Proof.
As mentioned earlier, here we only need to show that the previous theorem holds fors=3. Note that (6) is the invariant under the transformation(u,x)→(-u,-x) and (γ,x)→(-γ,-x). Thus, we deduce that if u0(x) and γ0(x) are odd, then u(t,x) and γ(t,x) are odd for any t∈[0,T). By continuity with respect to x of z and zxx, we have
(43)u(t,0)=uxx(t,0)=γ(t,0)=γxx(t,0)=0,∀t∈[0,T).
Hence, in view of (35) and Lemma 6, we obtain
(44)utx(t,0)=-12ux2(t,0)-12γx2-p*(u2+12ux+12γ2-12γx2)(t,0)-λux≤-12ux2(t,0)+12p*γx2(t,0)-λux≤-12ux2(t,0)-λux+14(∥u0∥H12+∥γ0∥H12).
Take
(45)K=12(∥u0∥H12+∥γ0∥H12)1/2
and define g(t)=ux(t,q(t,x0)). It then follows from (38) that on [0,T),
(46)g′(t)≤-12g2(t)-λg+K2=-12(g(t)+λ+λ2+2K2)×(g(t)+λ-λ2+2K2).
Note that if g(0)≤-λ-λ2+2K2, then g(t)≤-λ-λ2+2K2, for all t∈[0,T). Therefore, we can solve the previous inequality to obtain
(47)g(0)+λ+λ2+2K2g(0)+λ-λ2+2K2eλ2+2K2t-1≤2λ2+2K2g(t)+λ-λ2+2K2.
Due to 0<(g(0)+λ+λ2+2K2)/(g(0)+λ-λ2+2K2)<1, then there exists T, and 0<T<(1/λ2+2K2)ln((g(0)+λ+λ2+2K2)/(g(0)+λ-λ2+2K2)), such that limt→Tg(t)=-∞. Applying Theorem 5, the solution z does not exist globally in time.Next, we give more insight into the blow-up rate for the wave-breaking solutions to (6).Theorem 10.
Letz0=(u0,γ0)∈Hs×Hs,s≥5/2,z=(u,γ) be the corresponding solution to (6) with initial data z0 and satisfy ∥γx(t,x)∥L∞≤M,for all (t,x)∈[0,T)×R, and T be the maximal existence time of the solution. Then let one has
(48)limt→T(infx∈ℝ(ux(t,x)(T-t)))=-2.Proof.
By Lemma6, we get the uniform bound of u. Set m(t)=infx∈ℝux(t,x). By the proof of Theorem 8 (or Theorem 9), we find a constant K>0 such that
(49)|g′(t)+12g(t)+λg(t)≤K|,
where K depends only on ∥u0∥H1 and ∥γ0∥H1. It follows that
(50)-K-12λ2≤g′(t)+12(g(t)+λ)2≤K+12λ2,a.e.,on(0,T).
Choose ϵ∈(0,1/2). Since liminfr→T(y(t)+λ)=-∞ by Theorem 5, there is some t0∈(0,T) with g(t0)+λ<0 and (g(t0)+λ)2>(K+(1/2)λ2)/ϵ. Let us first prove that
(51)(g(t)+λ)2>1ϵ(K+12λ2),t∈[t0,T).
Since g is locally Lipschitz, there is some δ>0 such that
(52)(g(t)+λ)2>1ϵ(K+12λ2),t∈(t0,t0+δ).
Note thatg is locally Lipschitz and therefore absolutely continuous. Integrating the previous relation on (t0,t0+δ) yields that
(53)g(t0+δ)+λ≤g(t0)+λ<0.
It follows from the previous inequality that
(54)(g(t0+δ)+λ)2≥(g(t0)+λ)2>1ϵ(K+12λ2).
By (50)-(51), we infer that
(55)12-ϵ≤-g′(t)(m+λ)2≤12+ϵ,a.e.,on(0,T).
For t∈(t0,T), integrating (55) on  (t,T) to get
(56)(12-ϵ)(T-t)≤-1g(t)+λ≤(12+ϵ)(T-t),t∈(t0,T).
Since g(t)+λ<0 on [t0,T), it follows that
(57)1(1/2)+ϵ≤-(g(t)+λ)(T-t)≤1(1/2)+ϵ,t∈(t0,T).
By the arbitrariness of ϵ∈(0,1/2), the statement of the theorem follows.


## 5. Asymptotic Profile

In this section, we focus on the persistence property of the solution to (6) in L∞-space. Precisely, we give an asymptotic description on how the solutions behave under the initial values possess algebraic decay at infinity. Recently, the asymptotic behavior for the celebrated Camassa-Holm equation was investigated in [38]. We notice that in [39], the authors showed that the solution of the Camassa-Holm equation and its first-order spatial derivative retain exponential decay at infinity as their initial values behave. After all, the exponential decay of initial value is a faster way; this motivates us to establish the decay rate of solution if its initial value decays algebraically. We show that the strong solution of (6) corresponding to initial data with a slower algebraically decaying way will keep this behavior in the x-variable at infinity in its lifespan. In order to achieve our result, we first recall the following lemma.Lemma 11 (see [40]).
For a functionΦN(X) defined next, there exists a constant Cθ which only depends on θ∈(0,1], such that for any positive integer N≥2(58)ΦN(x)∫R‍e-|x-y|1ΦN(y)dy≤Cθ,
where
(59)ΦN(x)={1,x≤1,xθ,x∈(1,N),Nθ,X≥N.Theorem 12.
Assume thatX0(x)=(u0(x),γ0(x))T∈Hs×Hs with s>5/2 satisfies that for some θ∈(0,1](60)|X0(x)|,|X0x(x)|~O(x-θ)asx↑∞.
Then, the corresponding strong solution X(x)=(u(x),γ(x))T∈C([0,T);Hs×Hs) to (6) satisfies that
(61)|X(x)|,|Xx(x)|~O(x-θ)asx↑∞,
uniformly in the time interval [0,T).Notation.
One has(62)|f(x)|~O(|g(x)|)asx↑∞iflimx→∞|f(x)||g(x)|=L,
whereL is a nonnegative constant. In order to shorten the presentation in the sequel, we introduce
(63)F(u,γ)=u2+12ux2+12γ2-12γx2,H(u,γ)=(uxγx)x+uxγ.Proof.
The first step is devoted to giving estimates on∥u(x,t)∥∞ and ∥γ(x,t)∥∞, where ∥·∥p is the standard Lp(ℝ) norm.
Multiplying the first equation of (6) by u2n-1 with n∈Z+ and integrating both sides with respect to x variable, we obtain
(64)∫ℝ‍u2n-1utdx+∫ℝ‍u2n-1uuxdx+∫ℝ‍u2n-1∂x(G*F)dx+λ∫ℝ‍u2ndx=0.
The first term in (64) is
(65)∫ℝ‍u2n-1utdx=12nddt∥u∥2n2n=∥u∥2n2n-1ddt∥u∥2n;
for the second term of (64), we have
(66)∫ℝ‍u2n-1uuxdx≤∥ux∥∞∥u∥2n2n.
It follows from the Höder inequality that
(67)∫ℝ‍u2n-1∂x(G*F(u))dx≤∥u∥2n2n-1∥∂x(G*F)∥2m.
Therefore,
(68)ddt∥u∥2n≤(∥ux∥∞+λ)∥u∥2n+∥∂x(G*F)∥2n.
Similarly, for the estimate of ∥γ(x)∥∞, we have by multiplying γ2n-1 and integration
(69)∫ℝ‍γ2n-1γtdx+∫ℝ‍γ2n-1uγxdx+∫ℝ‍γ2n-1(G*H)dx+λ∫ℝ‍γ2n-1dx=0,ddt∥γ∥2n≤(∥γx∥∞+λ)∥u∥2n+∥G*H∥2n.
By the Sobolev embedding theorem, there exists a constant M>0 such that
(70)ddt(∥γ∥2n+∥u∥2n)≤M(∥γ∥2n+∥u∥2n)+∥∂x(G*H)∥2n+∥G*H∥2n.
In view of Gronwall's inequality, we have the estimate
(71)∥γ∥2n+∥u∥2n≤eMt(∫0t‍∥γ0∥2n+∥u0∥2n+∫0t‍(∥∂x(G*H)∥2n+∥G*H∥2n)dτ).
We can take limits as N goes to infinity to obtain
(72)∥γ∥∞+∥u∥∞≤eMt(∫0t‍∥γ0∥∞+∥u0∥∞+∫0t‍(∥∂x(G*H)∥∞+∥G*H∥∞)dτ).
The second step is to establish estimates for∥ux∥∞ and ∥γx∥∞ by using the same method as previously mentioned. Differentiating the first equation of (6) with respect to x produces the following equation:
(73)uxt+ux2+uuxx+∂x2(G*F)+λux=0.
Multiplying (73) by ux2n-1, and then integrating by parts, one obtains
(74)∫ℝ‍ux2n-1uxtdx+∫ℝ‍ux2n+1dx-12n∫ℝ‍ux2nux+∫ℝ‍ux2n-1∂x2(G*F)dx+λ∫ℝ‍ux2n=0.
Similarly, one can obtain
(75)∥ux∥∞≤eMt(∥u0x∥∞+∫0t‍∥∂x2(G*F)∥∞dτ).
For the second equation of (6), we may get
(76)∫ℝ‍γx2n-1γxtdx+∫ℝ‍uxγx2ndx+∫ℝ‍uγxxγx2n-1dx+∫ℝ‍γx2n-1∂x(G*H)dx+λ∫ℝ‍γx2n=0,ddt∥γx∥2n≤(∥ux∥∞+λ)∥γx∥2n+∥γxx∥∞∥u∥2n+∥∂x(G*H)∥2n,
and by Gronwall's inequality
(77)∥γx∥∞≤eMt(∥γ0x∥∞+∫0t‍(∥u∥∞+∥∂x(G*H)∥∞)dτ).
In order to arrive at our result, we introduce a weighted continuous function which is independent on t as follows:
(78)ΦN(x)={1,x≤1,xθ,x∈(1,N),Nθ,X≥N,
where θ∈(0,1],N∈Z+,N>2. It is trivial that
(79)0≤ΦN′(x)≤ΦN(x),a.e.,x∈R,
where the derivative is with respect to the variable x. From the first equation of (6) and (73), we have
(80)ΦNut+ΦNuux+ΦN∂x(G*F)+λΦNu=0,(81)ΦNuxt+ΦNux2+ΦNuuxx+ΦN∂x2(G*F)+λΦNux=0.
Next, in order to obtain the estimates on ∥uΦN∥∞ and ∥uxΦN∥∞, we apply a similar technique that was used before for ∥u(x,t)∥∞ and ∥ux(x,t)∥∞ step by step to (80) and (81). For (81), we need to eliminate the term with the second-order derivative in order to attain the estimate for uxΦN. Using integration by parts, we obtain
(82)|∫ℝ‍(uxΦN)2n-1ΦNuuxxdx|=|∫R‍(uxΦN)2n-1u((ΦNux)x-ΦN′ux)dx|=|∫ℝ‍u((uxΦN)2n2n)xdx-∫ℝ‍u(uxΦN)2n-1ΦN′uxdx|≤2(∥ux∥∞+∥u∥∞)∥uxΦN∥2n2n,
where (79) is used directly. Therefore, with these preparations, it holds that
(83)∥uΦN∥∞+∥uxΦN∥∞≤e2Mt(∥u0ΦN∥∞+∥u0xΦN∥∞)+e2Mt∫0t‍(∥ΦN∂x(G*H)∥∞+∥ΦN∂x2(G*F)∥∞)dτ.
For the second equation of (6), we have
(84)ΦNγt+ΦNuγx+ΦNG*H+λγ=0,ΦNγxt+ΦNuxγx+ΦNuγxx+ΦN∂xG*H+λγx=0.
Therefore,
(85)ddt∥γΦN∥2n≤∥u∥∞∥ΦNγ∥2n+12n∥ux∥∞∥γΦN∥2n+∥ΦNG*H∥2n,ddt∥γxΦN∥2n≤3(∥u∥∞+∥ux∥∞)∥γxΦN∥2n+∥ΦN∂x(G*H)∥2n.
Then by (85), we obtain
(86)∥γΦN∥∞+∥γxΦN∥∞≤e3Mt(∥γ0ΦN∥∞+∥γ0xΦN∥∞)+e3Mt∫0t‍(∥ΦN∂x(G*H)∥∞+∥ΦN∂x2(G*F)∥∞)dτ.
On the other hand, for a suitable function f, one obtains, due to Lemma 11,
(87)|ΦN∂x(G*f2)|≤ΦN(x)∫R‍e-|x-y|1ΦN(y)ΦN(y)f(y)f(y)dy≤∥fΦN∥∞∥f∥∞ΦN(x)∫R‍e-|x-y|1ΦN(y)dy≤Cθ∥fΦN∥∞∥f∥∞.
Similarly,
(88)|ΦN∂x2(G*f2)|≤Cθ∥fΦN∥∞∥f∥∞.
Note that there are two additional quantities in (86) to be dealt with. Let us estimate ∥ΦN(G*H)∥∞ first. One has
(89)|ΦN(G*H)|=12|ΦN(x)∫R‍e-|x-y|((uxγx)x+uxγ)dy|≤Cθ∥uxΦN∥∞(∥γ∥∞+∥γx∥∞).
It is similar to the remaining ∥ΦN∂x(G*H)∥∞. Then, combining (87)–(89) with (83) and (86), it follows that there exists a constant C~=C~(M,T)>0 such that
(90)Γ(t)≤C~Γ(0)+C~∫0t‍ζ(τ)Γ(τ)dτ≤C~(Γ(0)+∫0t‍Γ(τ)dτ),
where
(91)Γ(t)=∥uΦN∥∞+uxΦN∥∞+γΦN∥∞+γxΦN∥∞,ζ(t)=∥u∥∞+∥ux∥∞+∥γ∥∞+∥γx∥∞
are introduced just for simplicity. Next for any N∈Z+,t∈[0,T], and x>0, we have, by Gronwall's inequality,
(92)Γ(t)≤C~0Γ(0)≤C~0(∥u0(x)xθ∥∞+∥u0x(x)xθ∥∞+∥γ0(x)xθ∥∞+∥γ0x(x)xθ∥∞).
Finally, passing limit as N goes to infinity in the previous inequality, we obtain
(93)∥u(x,t)xθ∥∞+∥ux(x,t)xθ∥∞+∥γ(x,t)xθ∥∞+∥γx(x,t)xθ∥∞≤∥u0(x)xθ∥∞+∥u0x(x)xθ∥∞+∥γ0(x)xθ∥∞+∥γ0(x)xθ∥∞.
We complete the proof.



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*Source: 547261-2013-09-11.xml*