Analytical Properties for the Fifth Order Camassa-Holm (FOCH) Model

Mingxuan Zhu; Lu Cao; Zaihong Jiang; Zhijun Qiao

doi:10.2991/jnmp.k.210519.001

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Volume 28, Issue 3, September 2021, Pages 321 - 336

Analytical Properties for the Fifth Order Camassa-Holm (FOCH) Model

Authors

Mingxuan Zhu¹, Lu Cao², Zaihong Jiang²^,, Zhijun Qiao³^{, *}^,

¹School of Mathematical Sciences, Qufu Normal University, Qufu 273100, P. R. China

²Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, P. R. China

³School of Mathematical and Statistical Science, University of Texas Rio Grande Valley, Edinburg, TX 78539, USA

^*Corresponding author. Email: zhijun.qiao@utrgv.edu

Corresponding Author

Zhijun Qiao

Received 21 March 2021, Accepted 6 May 2021, Available Online 16 June 2021.

DOI: 10.2991/jnmp.k.210519.001 How to use a DOI?
Keywords: The Fifth order Camassa-Holm (FOCH) model; global existence; infinite propagation speed; long time behavior
Abstract: This paper devotes to present analysis work on the fifth order Camassa-Holm (FOCH) model which recently proposed by Liu and Qiao. Firstly, we establish the local and global existence of the solution to the FOCH model. Secondly, we study the property of the infinite propagation speed. Finally, we discuss the long time behavior of the support of momentum density with a compactly supported initial data.
Copyright: © 2021 The Authors. Published by Atlantis Press B.V.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

In this paper, we consider the following fifth order Camassa-Holm (FOCH) model [29]:

{mt+umx+buxm=0,t>0,x∈ℝ,m=(1-α2∂x2)(1-β2∂x2)u,t>0,x∈ℝ, (1.1)

where b ∈ ℝ is a constant, α, β ∈ ℝ are two parameters, α ≠ β, αβ ≠ 0. Without loss of generality, we only consider the case α > 0, β > 0. When α < 0, β < 0, one can get the similar results by using the corresponding absolute values |α| and |β| instead of α, β.

In what follows, we present some mathematical results related to the topic of this paper. Liu and Qiao [29] obtained some interesting solutions including explicit single pseudo-peakons, two-peakon, and N-peakon solutions. Detailed dynamical interactions for two-pseudo-peakons and three-pseudo-peakons were also investigated in their paper with numerical simulations. There have been extensive studies on high order Camassa-Holm type equations in the mathematics physics fields [5,16,17,20,22,31,38–41]. For the case α = β = 1, on the circle, McLachlan and Zhang [31] established the local well-posedness of the solution in H^s with s>72, it was shown that system (1.1) with α = β = 1 does’t admit finite time blow-up solutions. Tang and Liu [38] proved that the Cauchy problem for this equation is locally well-posed in the critical Besov space B2,17/2 or Bp,rs(1≤p,r≤+∞ and s>max{3+1p,72}). The peakon-like solution and ill-posedness was also studied in [38]. For the case b = 2, m = u − u_xx + u_xxxx, by using the Kato’s theory, the local well-posedness [39] was studied in the Sobolev space H^s with s>92. Ding [16,17] investigated the stationary solution, generality mild traveling solutions and conservative solution. Coclite, Holden and Karlsen [5] established the existence of global weak solutions. They also presented some invariant spaces under the action of the equation. In [20], the infinite propagation speed was considered for the case m = 4u − 5u_xx + u_xxxx. They also proved asymptotic behavior of the solution under the condition that the initial data decays exponentially and algebraically.

When β = 0 (or α = 0), it means m = u − α²u_xx. The Camassa-Holm equation, the Degasperis-Procesi equation, and the Holm-Staley b-family equations are the special cases of equation (1.1) with b = 2, b = 3 and b ∈ ℝ, respectively. These equations arise at various levels of approximation in shallow water theory, and possess a physics background with shallow water propagation, the bi-Hamiltonian structure, Lax pair, and explicit solutions including classical soliton, cuspon, and peakon solutions.

In 1993, Camassa and Holm [3] derived an integrable shallow water equation with peaked solitons, which was called the Camassa-Holm equation. In 1999, Degasperis and Procesi [15] extended the Camassa-Holm equation to a new water wave equation (Degasperis-Procesi equation). Both Camassa-Holm equation and Degasperis-Procesi equation have attracted much attention. They are completely integrable [11,12,14,35]. Infinitely many conservation laws have been shown in [14,27,37]. For the Camassa-Holm equation, in [9,28], They proved the local well-posedness for the initial datum in H^s with s > 3/2. There were many works to study the blow-up phenomenon, such as [8–10,24,28,32]. McKean [32] (See also [24] for a simple proof) proved that if and only if some portion of the positive part of y₀ = u₀ − u₀_xx lies to the left of some portion of its negative part, then the Camassa-Holm equation blow-up in finite time. The hierarchy properties, related finite-dimensional constrained flows, and algebro-geometric solutions of the Camassa-Holm equation were proposed in [34]. In [1], they studied the global conservative solution for the Camassa-Holm equation. Global dissipative solution have been shown in [2]. Constantin and Strauss [13] studied the orbital stability of the peakons. Himonas and his collaborators [21] obtained the persistence properties and unique continuation of solutions of the Camassa-Holm equation. In [25], the authors deduced the limit of the support of momentum density as t goes to +∞. In [4,6,7,23,26,30,33,35,36,42], they have investigated some mathematical properties for the Degasperis-Procesi equation. For the Holm-Staley b-family equation, mathematical studies have also been presented in [18,19,43].

The paper is organized as follows. In Section 2, we establish the local well-posedness and blow up scenario for the FOCH model. Conditions for global existence are found in Section 3. In Section 4, we establish the property of the infinite propagation speed for the FOCH model. In Section 5, we discuss the long time behavior for the support of momentum density of the FOCH model.

2. LOCAL WELL-POSEDNESS AND BLOW UP SCENARIO

Similar to the Camassa-Holm equation [9], we can establish the following local well-posedness theorem for the FOCH model (1.1).

Theorem 2.1.

Let u₀ ∈ H^s(ℝ) with s>72. Then there exist a T > 0 depending on ǁu₀ǁ_H^s, such that the FOCH model (1.1) has a unique solution

u∈C([0,T);Hs(ℝ))∩C1([0,T);Hs-1(ℝ)).

Morever, the map u₀ ∈ H^s → u ∈ C([0, T); H^s(ℝ)) ∩ C¹([0, T); H^s⁻¹(ℝ)) is continuous.

The proof is similar to that of Theorem 2.1 in [9,39]. To make the paper concise, we would like to omit the detail proof here. The maximum value of T in Theorem 2.1 is called the lifespan of the solution, in general. If T < ∞, that is

limt→T-∥u∥Hs=∞,

we say the solution blows up in finite time.

Before going to the blow up scenario, we have the following Lemma.

Lemma 2.2.

As m=(1-α2∂x2)(1-β2∂x2)u, then

u=p*m, p=α2α2-β2p1-β2α2-β2p2,

where p1=12αe-|x|α, p2=12βe-|x|β, α ≠ β, α > 0, β > 0.

Proof. Taking fourier transform to m=(1-α2∂x2)(1-β2∂x2)u, we have

m^=(1+α2ξ2)(1+β2ξ2)u^.

Notice that when f(x) = e⁻^a^|^x^|, a > 0 then

f^(ξ)=2aξ2+a2.

It follows that

u^(ξ)=11+α2ξ2⋅11+β2ξ2⋅m^(ξ)=p^1(ξ)⋅p^2(ξ)⋅m^(ξ),

where p1=12αe-|x|α, p2=12βe-|x|β. Then,

u(x)=ℱ-1(p^1(ξ)⋅p^2(ξ)⋅m^(ξ))=p1*p2*m(x).

Let p = p₁ ∗ p₂, we have

p(x)=14αβ∫ℝe-|x-y|αe-|y|βdy=12(α2-β2)[αe-|x|α-βe-|x|β]=α2(α2-β2)p1-β2(α2-β2)p2.

By Lemma 2.2, we can rewrite u(x, t) as

u=[α2(α2-β2)p1-β2(α2-β2)p2]*m=α2(α2-β2)[e-xα∫-∞xeξαm(ξ,t)dξ+exα∫x+∞e-ξαm(ξ,t)dξ] -β2(α2-β2)[e-xβ∫-∞xeξβm(ξ,t)dξ+exβ∫x+∞e-ξβm(ξ,t)dξ]. (2.1)

Differentiating u with respect to x, we have

ux=12(α2-β2)[-e-xα∫-∞xeξαm(ξ,t)dξ+exα∫x+∞e-ξαm(ξ,t)dξ] +12(α2-β2)[e-xβ∫-∞xeξβm(ξ,t)dξ-exβ∫x+∞e-ξβm(ξ,t)dξ].

Then, we present the precise blow-up scenario.

Theorem 2.3.

Assume that u₀ ∈ H⁴(ℝ) and let T be the maximal existence time of the solution u(x, t) to equation (1.1), α ≠ β, α > 0, β > 0 with the initial data u₀(x).

(1).
If b>12, then the corresponding solution of the FOCH model (1.1) blows up in finite time if and only if
limt→Tinfx∈ℝ{ux(x,t)}=-∞.
(2).
If b<12, then the corresponding solution of the FOCH model (1.1) blows up in finite time if and only if
limt→Tsupx∈ℝ{ux(x,t)}=+∞.

Proof. By direct calculation, we have

∥m∥L22=∫ℝ[u-(α2+β2)uxx+α2β2uxxxx]2dx=∫ℝu2+(α2+β2)2uxx2-2(α2+β2)uuxx+α4β4uxxxx2+2α2β2uuxxxx-2(α2+β2)α2β2uxxuxxxxdx=∫ℝu2+(α2+β2)2uxx2+2(α2+β2)ux2+α4β4uxxxx2+2α2β2uxx2+2(α2+β2)α2β2uxxx2dx.

Hence

c∥u∥H42≤∥m∥L22≤C∥u∥H42,

where c and C are positive constants depending on α and β. If b>12, direct calculation we have

ddt∫ℝm2dx=(1-2b)∫ℝuxm2dx≤(1-2b)infx∈ℝ{ux(x,t)}∫ℝm2dx.

If

infx∈ℝ{ux(x,t)}≥-M,

then

ddt∫ℝm2dx≤-(1-2b)M∫ℝm2dx.

By using the Gronwall inequality,

∥m∥L22=∫ℝm2dx≤e-(1-2b)M∫ℝm02dx=e-(1-2b)M∥m0∥L22.

Therefore the H⁴-norm of the solution is bounded on [0, T).

On the other hand,

u=α2(α2-β2)p1*m-β2(α2-β2)p2*m=α2(α2-β2)∫ℝp1(x-ξ)m(ξ)dξ-β2(α2-β2)∫ℝp2(x-ξ)m(ξ)dξ.

By the Sobolev’s embedding ‖u_x‖_∞ ≤ ‖u‖_H⁴, it tells us if H⁴-norm of the solution is bounded, then the L^∞-norm of the first derivative is bounded.

By the same argument, we can get the similar result for b<12. So, we omit the details and complete the proof of Theorem 2.3.

3. GLOBAL EXISTENCE

In this section, we study the global existence. Before going to our main results, we give the particle trajectory as

{qt=u(q,t), 0<t<T,x∈ℝ,q(x,0)=x, x∈ℝ, (3.1)

where T is the lifespan of the solution. Taking derivative (3.1) with respect to x, we obtain

dqtdx=qtx=ux(q,t)qx, t∈(0,T).

Therefore

{qx=exp{∫0tux(q,s)ds}, 0<t<T, x∈ℝ,qx(x,0)=1, x∈ℝ,

which is always positive before the blow-up time. Therefore, the function q(x, t) is an increasing diffeomorphism of the line before blow-up. In fact, direct calculation yields

ddt(m(q)qxb)=[mt(q)+u(q,t)mx(q)+bux(q,t)m(q)]qxb=0.

Hence, we have the following identity

m(q)qxb=m0(x), 0<t<T,x∈ℝ. (3.2)

Theorem 3.1.

Assume that u₀ ∈ H⁴(ℝ), α ≠ β, α > 0, β > 0, if b=12 or b = 2, then the corresponding solution of FOCH model (1.1) will exist globally in time.

Remark 3.1.

If α = 0 or β = 0, system (1.1) reduce to the well-known b-family equation. The global existence for b=12 and Theorem 3.2 can be reduce to the results for b-family equation [18]. The global existence for b = 2 is the new discovery compared to the b-family equation.

Proof. Let

E(t)=∫ℝu2+(α2+β2)ux2+α2β2uxx2dx.

Differentiating E(t), we have

ddtE(t)=∫ℝ2uut+2(α2+β2)uxuxt+2α2β2uxxuxxtdx=∫ℝ2uut-2(α2+β2)uuxxt+2α2β2uuxxxxtdx=2∫ℝumtdx=(b-2)∫ℝu2mxdx.

It yields that E(t) = E(0) when b = 2. By the Sobolev’s imbedding, we have

∥ux∥L∞≤∥u∥H22≤CE(t)=CE(0).

The global existence for b = 2 is completed by Theorem 2.3. Applying m on (1.1) and integration by parts, we obtain

ddt∫ℝm2dx=-2∫ℝbuxm2+mmxudx=-2∫ℝbuxm2-m22uxdx=(1-2b)∫ℝuxm2dx.

If b=12, then ddt∫ℝm2dx=0. Hence,

∥u∥H42≤∥m∥L22=∥m0∥L22.

It follows that the corresponding solution of FOCH model (1.1) exists globally when b=12.

Theorem 3.2.

Supposing that u₀ ∈ H⁴, α ≠ β, α > 0, β > 0, m0=(1-α2∂x2)(1-β2∂x2)u0 does not change sign. Then the corresponding solution to (1.1) exists globally.

Proof. We can assume that m₀ ≥ 0. It is sufficient to prove u_x is bounded for all t. In fact,

ux=12(α2-β2)[exα∫x+∞e-ξαm(ξ,t)dξ-e-xα∫-∞xeξαm(ξ,t)dξ] +12(α2-β2)[e-xβ∫-∞xeξβm(ξ,t)dξ-exβ∫x+∞e-ξβm(ξ,t)dξ].

If m₀ ≥ 0, α > β > 0, then

ux=12(α2-β2)[exα∫x+∞e-ξαm(ξ,t)dξ-e-xα∫-∞xeξαm(ξ,t)dξ] +12(α2-β2)[e-xβ∫-∞xeξβm(ξ,t)dξ-exβ∫x+∞e-ξβm(ξ,t)dξ]≤12(α2-β2)[exα∫x∞e-ξαm(ξ,t)dξ+e-xβ∫-∞xeξβm(ξ,t)dξ]≤12(α2-β2)[exαe-xα∫x∞m(ξ,t)dξ+e-xβexβ∫-∞xm(ξ,t)dξ]≤12(α2-β2)[∫ℝm(ξ,t)dξ+∫ℝm(ξ,t)dξ]=1(α2-β2)∫ℝm0(ξ,t)dξ

and

ux=12(α2-β2)[exα∫x∞e-ξαm(ξ,t)dξ-e-xα∫-∞xeξαm(ξ,t)dξ] +12(α2-β2)[e-xβ∫-∞xeξβm(ξ,t)dξ-exβ∫x+∞e-ξβm(ξ,t)dξ]≥12(α2-β2)[-e-xα∫-∞xeξαm(ξ,t)dξ-exβ∫x+∞e-ξβm(ξ,t)dξ]≥-1(α2-β2)∫ℝm(ξ,t)dξ=-1(α2-β2)∫ℝm0(ξ,t)dξ.

That is

|ux|≤1(α2-β2)∫ℝm0(ξ,t)dξ.

If m₀ ≥ 0, 0 < α < β, then

ux=12(α2-β2)[exα∫x∞e-ξαm(ξ,t)dξ-e-xα∫-∞xeξαm(ξ,t)dξ] +12(α2-β2)[e-xβ∫-∞xeξβm(ξ,t)dξ-exβ∫x+∞e-ξβm(ξ,t)dξ]≤12(α2-β2)[-e-xα∫-∞xeξαm(ξ,t)dξ-exβ∫x+∞e-ξβm(ξ,t)dξ]≤12(α2-β2)[-e-xαexα∫-∞xm(ξ,t)dξ-exβe-xβ∫x∞m(ξ,t)dξ]≤12(β2-α2)[∫ℝm(ξ,t)dξ+∫ℝm(ξ,t)dξ]=1(β2-α2)∫ℝm0(ξ,t)dξ

and

ux=12(α2-β2)[exα∫x∞e-ξαm(ξ,t)dξ-e-xα∫-∞xeξαm(ξ,t)dξ] +12(α2-β2)[e-xβ∫-∞xeξβm(ξ,t)dξ-exβ∫x+∞e-ξβm(ξ,t)dξ]≥12(α2-β2)[exα∫x∞e-ξαm(ξ,t)dξ+e-xβ∫-∞xeξβm(ξ,t)dξ]≥-1(β2-α2)∫ℝm(ξ,t)dξ=-1(β2-α2)∫ℝm0(ξ,t)dξ.

That is

|ux|≤1(β2-α2)∫ℝm0(ξ,t)dξ.

When m₀ ≤ 0, via the similar approach that is used above, we could also obtain the global existence result. So, we omit the details and complete the proof of Theorem 3.2.

4. INFINITE PROPAGATION SPEED

The main theorem reads as follows:

Theorem 4.1.

Assume that the initial datum u₀(x) ∈ H⁴(ℝ) is compactly supported in [a, c], then for t ∈ (0, T), the corresponding solution u(x, t) to the FOCH model (1.1) α ≠ β, α > 0, β > 0 has the following property:

u(x,t)={α2(α2-β2)e-xαE1(t)-β2(α2-β2)e-xβE2(t),as x>q(c,t),α2(α2-β2)exαF1(t)-β2(α2-β2)exβF2(t),as x<q(a,t),

where

E1(t)=∫ℝexαm(x,t)dx, F1(t)=∫ℝe-xαm(x,t)dx,

and

E2(t)=∫ℝexβm(x,t)dx, F2(t)=∫ℝe-xβm(x,t)dx,

denote continuous nonvanishing functions.

Furthermore, if α > 0, 0<β≤32α, 0≤b≤min{3-2β2α2,53}, E₁(t) is strictly increasing function, while F₁(t) is strictly decreasing function.

Similarly, if β > 0, 0<α≤32β, 0≤b≤min{3-2α2β2,53}, E₂(t) is strictly increasing function, while F₂(t) is strictly decreasing function.

Remark 4.1.

Theorem 4.1 implies that the strong solution u(x, t) doesn’t have compact x-support for any t > 0 in its lifespan, although the corresponding u₀(x) is compactly supported.

Proof. Since u₀(x) has a compact support in the interval [a, c], so does m0(x)=(1-α2∂x2)(1-β2∂x2)u0(x). Equation (3.2) tells us that m(x)=(1-α2∂x2)(1-β2∂x2)u(x) is compactly supported in the interval [q(a, t), q(c, t)] well-defined in its lifespan. Hence the following functions are well-defined

E1(t)=∫ℝexαm(x,t)dx, F1(t)=∫ℝe-xαm(x,t)dx,E2(t)=∫ℝexβm(x,t)dx, F2(t)=∫ℝe-xβm(x,t)dx.

Using (3.2),

m(q(x,t),t)≡0, x<a or x>c,

we know

u(x,t)=(α2(α2-β2)p1-β2(α2-β2)p2)*m(x,t)=α2(α2-β2)∫ℝe-|x-ξ|αm(ξ)dξ-β2(α2-β2)∫ℝe-|x-ξ|βm(ξ)dξ=α2(α2-β2)∫q(a,t)q(c,t)e-|x-ξ|αm(ξ)dξ-β2(α2-β2)∫q(a,t)q(c,t)e-|x-ξ|βm(ξ)dξ.

Then, for x > q(c, t), we have

u(x,t)=α2(α2-β2)∫q(a,t)q(c,t)e-x-ξαm(ξ)dξ-β2(α2-β2)∫q(a,t)q(c,t)e-x-ξβm(ξ)dξ=α2(α2-β2)e-xα∫q(a,t)q(c,t)eξαm(ξ)dξ-β2(α2-β2)e-xβ∫q(a,t)q(c,t)eξβm(ξ)dξ=α2(α2-β2)e-xαE1(t)-β2(α2-β2)e-xβE2(t). (4.1)

Similarly, when x < q(a, t), we have

u(x,t)=α2(α2-β2)∫q(a,t)q(c,t)ex-ξαm(ξ)dξ-β2(α2-β2)∫q(a,t)q(c,t)ex-ξβm(ξ)dξ=α2(α2-β2)exα∫q(a,t)q(c,t)e-ξαm(ξ)dξ-β2(α2-β2)exβ∫q(a,t)q(c,t)e-ξβm(ξ)dξ=α2(α2-β2)exαF1(t)-β2(α2-β2)exβF2(t). (4.2)

On the other hand,

dE1(t)dt=∫ℝeξαmt(ξ,t)dξ.

It is easy to get

mt=-mx-bmux=[(α2+β2)uxxx-α2β2uxxxxx-ux]u-b[u-(α2+β2)uxx+α2β2uxxxx]ux=(α2+β2)uxxxu-α2β2uxxxxxu-(b+1)uux+b(α2+β2)uxxux-bα2β2uxxxxux. (4.3)

Taking (4.3) into dE1(t)dt, we obtain

dE1(t)dt=∫ℝexαmtdx=∫ℝexα[(α2+β2)uxxxu-α2β2uxxxxxu-(b+1)uux+b(α2+β2)uxxux-bα2β2uxxxxux]dx=(α2+β2)∫ℝexαuxxxudx-α2β2∫ℝexαuxxxxxudx-(b+1)∫ℝexαuuxdx +b(α2+β2)∫ℝexαuxxuxdx-bα2β2∫ℝexαuxxxxuxdx=∑i=15Ii. (4.4)

I₁-I₅ can be estimated as follows:

I1=(α2+β2)∫ℝexαuxxxudx=-α2+β22α3∫ℝexαu2dx+3(α2+β2)2α∫ℝexαux2dx, (4.5)

I2=-α2β2∫ℝexαuxxxxxudx=β22α3∫ℝexαu2dx-5β22α∫ℝexαux2dx+5αβ22∫ℝexαuxx2dx, (4.6)

I3=-(b+1)∫ℝexαuuxdx=b+12α∫ℝexαu2dx, (4.7)

I4=b(α2+β2)∫ℝexαuxxuxdx=-b(α2+β2)2α∫ℝexαux2dx, (4.8)

I5=-bα2β2∫ℝexαuxxxxuxdx=bβ22α∫ℝexαux2dx-3bαβ22∫ℝexαuxx2dx. (4.9)

Combining (4.5)–(4.9) to (4.4), we have

dE1(t)dt=∫ℝexαmtdx=b2α∫ℝexαu2dx-(b-3)α2+2β22α∫ℝexαux2dx+(5-3b)αβ22∫ℝexαuxx2dx. (4.10)

For α > 0, 0<β≤32α, 0≤b≤min{3-2β2α2,53}, from (4.10), E₁(t) is strictly increasing for nontrivial solution.

Similary,

dF1(t)dt=∫ℝe-xαmtdx=-b2α∫ℝe-xαu2dx+(b-3)α2+2β22α∫ℝe-xαux2dx-(5-3b)αβ22∫ℝe-xαuxx2dx. (4.11)

For α > 0, 0<β≤32α, 0≤b≤min{3-2β2α2,53}, from (4.11), F₁(t) is strictly decreasing for nontrivial solution.

dE2(t)dt=∫ℝexβmtdx=b2β∫ℝexβu2dx-(b-3)β2+2α22β∫ℝexβux2dx+(5-3b)βα22∫ℝexβuxx2dx. (4.12)

For β > 0, 0<α≤32β, 0≤b≤min{3-2α2β2,53}, from (4.12), E₂(t) is strictly increasing for nontrivial solution.

dF2(t)dt=∫ℝe-xβmtdx=-b2β∫ℝe-xβu2dx+(b-3)β2+2α22β∫ℝe-xβux2dx-(5-3b)βα22∫ℝe-xβuxx2dx. (4.13)

For β > 0, 0<α≤32β, 0≤b≤min{3-2α2β2,53}, from (4.13), F₂(t) is strictly decreasing for nontrivial solution.

This complete the proof of Theorem 4.1.

Remark 4.2.

Let

u′(x,t)={α2(α2-β2)e-xαE1(t),as x>q(c,t),α2(α2-β2)exαF1(t),as x<q(a,t), u″(x,t)={β2(α2-β2)e-xβE2(t),as x>q(c,t),β2(α2-β2)exβF2(t),as x<q(a,t),

We rewrite u = u′− u″, as consequences of (4.1) and (4.2), we have

u′(x,t)=-αux′(x,t)=α2uxx′(x,t)=α2(α2-β2)e-xαE1(t),as x>q(c,t),u′(x,t)=αux′(x,t)=α2uxx′(x,t)=α2(α2-β2)exαF1(t),as x<q(a,t).

and

u″(x,t)=-βux″(x,t)=β2uxx″(x,t)=β2(α2-β2)e-xβE2(t),as x>q(c,t),u″(x,t)=βux″(x,t)=β2uxx″(x,t)=β2(α2-β2)exβF2(t),as x<q(a,t).

Theorem 4.2.

Suppose the initial value u₀(x) ∈ H⁴(ℝ), m0=(1-α2∂x2)(1-β2∂x2)u0, α > β > 0 m₀ doesn’t change sign on ℝ and u₀ has compact support in the interval [a, c]. Then for t ∈ (0, T), the corresponding solution u(x, t) of equation (1.1) satisfies

12(α+β)e-xα|E1(t)|≤u(x,t)≤α2(α2-β2)e-xα|E1(t)|,as x>q(c,t),12(α+β)exα|F1(t)|≤u(x,t)≤α2(α2-β2)exα|F1(t)|,as x<q(a,t).

where

E1(t)=∫ℝeξαm(ξ,t)dξ, F1(t)=∫ℝe-ξαm(ξ,t)dξ,

denote continuous nonvanishing functions.

Remark 4.3.

We assume α > β > 0 to get the above conclusion in Theorem 4.2, because the position of α, β is symmetric, then β > α > 0, we have results similar to the above conclusions about E2(t)=∫ℝeξβm(ξ,t)dξ, F2(t)=∫ℝe-ξβm(ξ,t)dξ,

12(α+β)e-xβ|E2(t)|≤u(x,t)≤β2(β2-α2)e-xβ|E2(t)|,as x>q(c,t),12(α+β)exβ|F2(t)|≤u(x,t)≤β2(β2-α2)exβ|F2(t)|,as x<q(a,t).

Theorem 4.2 can be seem as a generalization of the result in [20]. Comparing with Theorem 4.1, it show more detailed estimation by adding the additional condition on m₀.

Proof. If u₀ has a compact support set [a, c], then the corresponding m₀ also has a corresponding compact support set [a, c]. It is known from (3.2) that m has the same compact support set [q(a, t), q(c, t)]. We define

u1=(1-βα)⋅α2(α2-β2)∫ℝe-|x-ξ|αmdξ=12(α+β)∫ℝe-|x-ξ|αmdξ,u2=α2(α2-β2)∫ℝe-|x-ξ|αmdξ. (4.14)

According to E1(t)=∫ℝeξαm(ξ,t)dξ,F1(t)=∫ℝe-ξαm(ξ,t)dξ, then

u1(x,t)=12(α+β)e-xαE1(t),u2(x,t)=α2(α2-β2)e-xαE1(t), as x>q(c,t),u1(x,t)=12(α+β)exαF1(t),u2(x,t)=α2(α2-β2)exαF1(t), as x<q(a,t).

According to (4.1) and (4.14), we obtain

u2(x,t)-u(x,t)=β2(α2-β2)∫ℝe-|x-ξ|βmdξ,u(x,t)-u1(x,t)=β2(α2-β2)∫ℝ(e-|x-ξ|α-e-|x-ξ|β)mdξ.

Then, we obtain

{u1(x,t)≤u(x,t)≤u2(x,t),m0≥0,u2(x,t)≤u(x,t)≤u1(x,t),m0≤0.

If m₀ ≥ 0,

{12(α+β)e-xαE1(t)≤u(x,t)≤α2(α2-β2)e-xαE1(t),as x>q(c,t),12(α+β)exαF1(t)≤u(x,t)≤α2(α2-β2)exαF1(t),as x<q(a,t). (4.15)

If m₀ ≤ 0,

{α2(α2-β2)e-xαE1(t)≤u(x,t)≤12(α+β)e-xαE1(t),as x>q(c,t),α2(α2-β2)exαF1(t)≤u(x,t)≤12(α+β)exαF1(t),as x<q(a,t). (4.16)

The proof of Theorem 4.2 is finished.

5. LONG TIME BEHAVIOR FOR THE SUPPORT OF MOMENTUM DENSITY

After the global existence of solution is established, we will discuss the long time behavior for the support of momentum density of the FOCH model. Now, we give the lemma and main theorem as follows:

Lemma 5.1.

Let α > β > 0, Assume the initial value u₀ ≢ 0 has a compact supported set [a, c].

(1).
If m₀(x) ≥ 0(≢ 0), x ∈ [a, c], then we have
limt→+∞F1(t)=0.
(2).
If m₀(x) ≤ 0(≢ 0), x ∈ [a, c], then we have
limt→+∞E1(t)=0.

Remark 5.1.

By the same argument, we can get a similar conclusion for β > α > 0. If m₀(x) ≥ 0(≢ 0), x ∈ [a, c], then we have

limt→+∞F2(t)=0.

If m₀(x) ≤ 0(≢ 0), x ∈ [a, c], then we have

limt→+∞E2(t)=0.

Proof. (1) For m₀(x) > 0, from (3.2), we have E₁(t) > 0, F₁(t) > 0, E₂(t) > 0, F₂(t) > 0, for all t ≥ 0. As F₁(t) > 0, we claim that

limt→+∞F1(t)=0.

Otherwise, there is a constant ∊₀ > 0, for any T > 0, there will exist a t > T, such that F₁(t) ≥ ∊₀.

For any d < a, from (4.15) we have

ddtq(d,t)=u(q(d,t),t)≥12(α+β)eq(d,t)αF1(t)≥12(α+β)eq(d,t)α∊0.

It follows that

e-q(d,t)α≤-∊02α(α+β)t+e-dαα.

Taking T=2(α+β)∊0e-dα, however, when t = T + 1,

-∊02α(α+β)t+e-dαα<0,

This is the contradiction. So our claim is right.

(2). For m₀(x) < 0, from (3.2), we have E₁(t) < 0, F₁(t) < 0, E₂(t) < 0, F₂(t) < 0, for all t ≥ 0. As F₁(t) > 0, we claim that

limt→+∞E1(t)=0.

Otherwise, there is a constant ∊₀ > 0, for any T > 0, for any T > 0, there will exist a t > T, such that E₁(t) ≤ −∊₀.

For any h > c, from (4.16) we have

ddtq(h,t)=u(q(h,t),t)≤12(α+β)e-q(h,t)αE1(t)≤-∊02(α+β)e-q(h,t)α.

It follows that

eq(h,t)α≤-∊02α(α+β)t+e-dαα.

Taking T=2(α+β)∊0e-dα, however, when t = T + 1,

-∊02α(α+β)t+e-dαα<0,

This is the contradiction. So our claim is right.

Theorem 5.2.

If b > 1, α > β > 0, and suppose that m0(x)∈L1b and u₀(x) has a compact supported set [a, c].

(1).
If m₀(x) ≥ 0(≢ 0), x ∈ [a, c], then we have
eq(c,t)α(b-1)-eq(a,t)α(b-1)→+∞, as t→+∞. (5.1)
(2).
If m₀(x) ≤ 0(≢ 0), x ∈ [a, c], then we have
e-q(a,t)α(b-1)-e-q(c,t)α(b-1)→+∞, as t→+∞. (5.2)

Remark 5.2.

For the case β > α > 0, by using the properties of E₂ and F₂ in Remark 5.1, one can replace α with β in (5.1) and (5.2).

Proof. (1) By (3.2) and direct calculation, we have

(∫ac(m0)1bdx)b=(∫ac(m(q,t)qxb)1bdx)b=(∫ac(m(q,t))1bqxdx)b=(∫q(a,t)q(c,t)(m(ξ,t))1bdξ)b≤(∫q(a,t)q(c,t)m(ξ,t)e-ξαdξ)[∫q(a,t)q(c,t)eξα(b-1)dξ]b-1=F1(t)[α(b-1)(eq(c,t)α(b-1)-eq(a,t)α(b-1))](b-1).

It follows

[α(b-1)(eq(c,t)α(b-1)-eq(a,t)(b-1)α)](b-1)≥(∫ac(m0)1bdx)bF1(t).

Using the limit

limt→+∞F1=0,

we can get

eq(c,t)α(b-1)-eq(a,t)α(b-1)→+∞, as t→+∞.

(2). Direct calculation, we have

(∫ac(-m0)1bdx)b=(∫ac(-m0)1bdx)b=(∫ac(-m(q,t)qxb)1bdx)b=(∫ac(-m(q,t))1bqxdx)b=(∫q(a,t)q(c,t)(-m(ξ,t))1bdξ)b≤(∫q(a,t)q(c,t)(-m(ξ,t)eξα)dξ)[∫q(a,t)q(c,t)e-ξα(b-1)dξ]b-1=-E1(t)[α(b-1)(e-q(a,t)α(b-1)-e-q(c,t)α(b-1))]b-1.

It follows

[α(b-1)(e-q(a,t)α(b-1)-e-q(c,t)α(b-1))]b-1≥(∫ac(-m0)1bdx)b-E1(t).

Using the limit

limt→+∞E1=0,

we can obtain

e-q(a,t)α(b-1)-e-q(c,t)α(b-1)→+∞, as t→+∞.

Theorem 5.3.

If b = 1, suppose that m₀(x) ∈ L₁ and u₀(x) has a compact supported set [a, c].

(1).
If m₀(x) ≥ 0(≢ 0), x ∈ [a, c], then we have
q(c,t)→+∞, as t→+∞.
(2).
If m₀(x) ≤ 0(≢ 0), x ∈ [a, c], then we have
q(a,t)→-∞, as t→+∞.

Proof. We only present the proof for α > β > 0. The case β > α > 0 can be proved by the same argument. (1) As m₀(x) ≥ 0, for any t ≥ 0, we have F₁(t)> 0. According to Lemma 5.1, we know

limt→+∞F1(t)=0.

Direct calculation, we have

∫acm0dx=∫acm(q,t)qxdx≤eq(c,t)α∫q(a,t)q(c,t)m(ξ,t)e-ξαdξ=eq(c,t)αF1(t).

It follows

eq(c,t)α≥∫acm0dxF1(t)→+∞, as t→+∞,

then we can get

q(c,t)→+∞, as t→+∞.

(2). As m₀(x) ≤ 0, for any t ≥ 0, we have E₁(t) < 0. According to Lemma 5.1, we know

limt→+∞E1(t)=0.

Direct calculation, we have

∫ac(-m0)dx=∫ac(-m(q,t)qx)dx≤e-q(a,t)α∫q(a,t)q(c,t)(-m(ξ,t))eξαdξ=-e-q(a,t)αE1(t).

It follows

e-q(a,t)α≥∫ac(-m0)dx-E1(t)→+∞, as t→+∞,

then we can get

q(a,t)→-∞, as t→+∞.

Theorem 5.4.

If 0 < b < 1, α > β > 0, m0(x)∈L1b or b = 0, m₀ ∈ L_∞. Suppose that u₀(x) has a compact supported set [a, c].

(1).
If m₀(x) ≥ 0(≢ 0) for x ∈ [a, c], then we have
eq(c,t)α-eq(a,t)α→+∞, as t→+∞. (5.3)
(2).
If m₀(x) ≤ 0(≢ 0) for x ∈ [a, c], then we have
e-q(a,t)α-e-q(c,t)α→+∞, as t→+∞. (5.4)

Remark 5.3.

For the case β > α > 0, by using the properties of E₂ and F₂ in Remark 5.1, one can replace α with β in (5.3) and (5.4).

Proof. (1). For m₀(x) ≥ 0, we have F₁(t) > 0 for all t ≥ 0. From Lemma 5.1, we know

limt→+∞F1(t)=0.

According to the conservation law

∫ℝmdx=∫ℝm0dx, ∫ℝm1bdx=∫ℝm01bdx.If 0<b<1 and {γ+ηb=1,2γ+η=1,0<γ,η<1.⇒{0<η=22-b-1<1,0<γ=1+1b-2<1.

By direct calculation, we obtain

∫ℝm0dx=∫ℝmdx=∫ℝm(q,t)qxdx=[∫ac(me-qαqx)γ(m1bqx)η(eqαqx)γdx]≤(∫acme-qαqxdx)γ(∫acm1bqxdx)η(∫aceqαqxdx)γ=(∫q(a,t)q(c,t)me-ξαdξ)γ(∫ℝm1bdξ)η(∫q(a,t)q(c,t)eξαdξ)γ=(F1(t))γ(∫ℝm1bdξ)η(∫q(a,t)q(c,t)eξαdξ)γ=(F1(t))γ(∫ℝm01bdξ)η(αeq(c,t)α-αeq(a,t)α)γ.

It follows

(αeq(c,t)α-αeq(a,t)α)γ≥∫ℝm0dx(F1(t))γ(∫ℝm01bdξ)η→+∞,

then we can obtain

eq(c,t)α-eq(a,t)α→+∞, as t→+∞.

If b = 0, we can obtain

∫ℝm0dx=limb→0∫ℝmdx=∫ℝm(q,t)qxdx=limb→0[∫ac(me-qαqx)γ(m1bqx)η(eqαqx)γdx]≤limb→0(∫acme-qαqxdx)γ(∫acm1bqxdx)η(∫aceqαqxdx)γ=limb→0(∫q(a,t)q(c,t)me-ξαdξ)γ(∫ℝm1bdξ)η(∫q(a,t)q(c,t)eξαdξ)γ=limb→0(F1(t))γ(∫ℝm1bdξ)η(∫q(a,t)q(c,t)eξαdξ)γ=limb→0(F1(t))γ(∫ℝm01bdξ)η(αeq(c,t)α-αeq(a,t)α)γ.

It follows

(αeq(c,t)α-αeq(a,t)α)γ≥∫ℝm0dx(F1(t))γ(limb→0∫ℝm01bdξ)η→+∞,

then we can obtain

eq(c,t)α-eq(a,t)α→+∞, as t→+∞.

(2). For m₀(x) ≤ 0, we have E₁(t) < 0 for all t ≥ 0. From Lemma 5.1, we know

limt→+∞E1(t)=0.

Similarly, according to the conservation law

∫ℝmdx=∫ℝm0dx, ∫ℝm1bdx=∫ℝm01bdx.If 0<b<1 and {γ+ηb=1,2γ+η=1,0<γ,η<1.⇒{0<η=22-b-1<1,0<γ=1+1b-2<1.

By direct calculation, we obtain

-∫ℝm0dx=-∫ℝmdx=-∫ℝm(q,t)qxdx=[∫ac(-meqαqx)γ((-m)1bqx)η(e-qαqx)γdx]≤(∫ac-meqαqxdx)γ(∫ac(-m)1bqxdx)η(∫ace-qαqxdx)γ=(∫q(a,t)q(c,t)-meξαdξ)γ(∫ℝ(-m)1bdξ)η(∫q(a,t)q(c,t)e-ξαdξ)γ=(-E1(t))γ(∫ℝm1bdξ)η(∫q(a,t)q(c,t)e-ξαdξ)γ=(-E1(t))γ(∫ℝm01bdξ)η(αe-q(a,t)α-αe-q(c,t)α)γ.

It follows

(αe-q(a,t)α-αe-q(c,t)α)γ≥-∫ℝm0dx(-E1(t))γ(∫ℝ(-m0)1bdξ)η→+∞,

then we can obtain

e-q(a,t)α-e-q(c,t)α→+∞, as t→+∞.

If b = 0, we can obtain

-∫ℝm0dx=-limb→0∫ℝmdx=-∫ℝm(q,t)qxdx=limb→0[∫ac(-meqαqx)γ((-m)1bqx)η(e-qαqx)γdx]≤limb→0(∫ac-meqαqxdx)γ(∫ac(-m)1bqxdx)η(∫ace-qαqxdx)γ=limb→0(∫q(a,t)q(c,t)-meξαdξ)γ(∫ℝ(-m)1bdξ)η(∫q(a,t)q(c,t)e-ξαdξ)γ=limb→0(-E1(t))γ(∫ℝm1bdξ)η(∫q(a,t)q(c,t)e-ξαdξ)γ=limb→0(-E1(t))γ(∫ℝm01bdξ)η(αe-q(a,t)α-αe-q(c,t)α)γ.

It follows

(αe-q(a,t)α-αe-q(c,t)α)γ≥-∫ℝm0dx(-E1(t))γ(limb→0∫ℝ(-m0)1bdξ)η→+∞,

then we can obtain

e-q(a,t)α-e-q(c,t)α→+∞, as t→+∞.

6. CONCLUSION

We have considered the FOCH model α ≠ β, α > 0, β > 0. When α, β is negative, one can get the same results by taking absolute value |α| and |β|. This model is highly related to the classical Camassa-Holm equation, the Degasperis-Procesi equation and the Holm-Staley b-family equation. We have studied some mathematical property, such as global existence, infinite propagation speed and long time behavior of the support of momentum density. Another highly related equation is (1.1) with α = β. Due to (1.1) with α = β doesn’t have the structure in Lemma 2.2 and (2.1), some results in this manuscript may can’t been realized for α = β.

CONFLICTS OF INTEREST

The authors declare they have no conflicts of interest.

ACKNOWLEDGEMENTS

This work has partially been supported by the National Natural Science Foundation of China (No. 12071439 and No. 11971475) and the 2019 - 2020 Hunan overseas distinguished professorship project (No. 2019014). The third author Jiang was partially supported by NSFC (grant No. 11101376, 11671309, 11671364) and ZJNSF (grant No. LY19A010016). The fourth author Qiao thanks the UT President’s Endowed Professorship (Project # 450000123) for its partial support.

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Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 28 - 3
Pages: 321 - 336
Publication Date: 2021/06/16
ISSN (Online): 1776-0852
ISSN (Print): 1402-9251
DOI: 10.2991/jnmp.k.210519.001 How to use a DOI?
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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ris enw bib

TY  - JOUR
AU  - Mingxuan Zhu
AU  - Lu Cao
AU  - Zaihong Jiang
AU  - Zhijun Qiao
PY  - 2021
DA  - 2021/06/16
TI  - Analytical Properties for the Fifth Order Camassa-Holm (FOCH) Model
JO  - Journal of Nonlinear Mathematical Physics
SP  - 321
EP  - 336
VL  - 28
IS  - 3
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.k.210519.001
DO  - 10.2991/jnmp.k.210519.001
ID  - Zhu2021
ER  -

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