In this paper, we consider the dissipative Degasperis-Procesi equation with arbitrary dispersion coefficient and compactly supported initial data. We establish the simple condition on the initial data which lead to guarantee that the solution exists globally. We also investigate the propagation speed for the equation under the initial data is compactly supported.
Citation: Guenbo Hwang, Byungsoo Moon. Global existence and propagation speed for a Degasperis-Procesi equation with both dissipation and dispersion[J]. Electronic Research Archive, 2020, 28(1): 15-25. doi: 10.3934/era.2020002
[1] | Guenbo Hwang, Byungsoo Moon . Global existence and propagation speed for a Degasperis-Procesi equation with both dissipation and dispersion. Electronic Research Archive, 2020, 28(1): 15-25. doi: 10.3934/era.2020002 |
[2] | Jun Meng, Shaoyong Lai . $ L^1 $ local stability to a nonlinear shallow water wave model. Electronic Research Archive, 2024, 32(9): 5409-5423. doi: 10.3934/era.2024251 |
[3] | Lynnyngs K. Arruda . Multi-shockpeakons for the stochastic Degasperis-Procesi equation. Electronic Research Archive, 2022, 30(6): 2303-2320. doi: 10.3934/era.2022117 |
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[6] | Xu Liu, Jun Zhou . Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity. Electronic Research Archive, 2020, 28(2): 599-625. doi: 10.3934/era.2020032 |
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[10] | Yang Cao, Qiuting Zhao . Initial boundary value problem of a class of mixed pseudo-parabolic Kirchhoff equations. Electronic Research Archive, 2021, 29(6): 3833-3851. doi: 10.3934/era.2021064 |
In this paper, we consider the dissipative Degasperis-Procesi equation with arbitrary dispersion coefficient and compactly supported initial data. We establish the simple condition on the initial data which lead to guarantee that the solution exists globally. We also investigate the propagation speed for the equation under the initial data is compactly supported.
In the present paper, we study the Cauchy problem to the dissipative Degasperis-Procesi equation with arbitrary dispersion term and compactly supported initial data:
{ut−utxx+k(u−uxx)x+4uux+λ(u−uxx)=3uxuxx+uuxxx,t>0,x∈R,u(0,x)=u0(x),x∈R, | (1) |
where supp
When
ut−utxx+4uux=3uxuxx+uuxxx,t>0,x∈R, |
which arises in the shallow-water medium-amplitude regime [2,10], introduced to capture stronger nonlinear effects that will allow for breaking waves, since the latter are not modeled by the shallow-water small-amplitude regime characteristic for the KdV equation. In this regime only the Camass-Holm equation [1] and the Degasperis-Procesi equation [4] arise as integrable model equations, with the same accuracy of approximation to the governing equations for water waves. It has bi-Hamiltonian structure and an infinite sequence of conserved quantities, and admits exact peakon solutions which are analogous to the Camassa-Holm peakons. After the Degasperis-Procesi equation was derived, many papers were devoted to its study, cf. [3,8,12,14,21] and the citations therein.
If
vt−vtxx+4vvx+λ(v−vxx)=3vxvxx+vvxxx,t>0,x∈R, | (2) |
which has been analyzed in several papers: Local well-posedness of (2) in
v(t,x)=e−λtu(1−e−λtλ,x), |
then
In the present paper, we discuss the global existence and the propagation speed of strong solutions to the equation (1). Our result shows that in comparison between the Degasperis-Procesi equation (
Our paper is organized as follows. In Section 2, we provide some preliminary materials which are crucial in the proof of our result. We gave the global existence result in Section 3. We study the propagation speed of strong solutions to the equation (1) under the condition that the initial data has compact support in Section 4.
Since we shall also use a priori estimates and further properties of solutions in
With
{mt+(u+k)mx+3uxm+λm=0,t>0,x∈R,m(0,x)=u0(x)−u0,xx(x),x∈R. | (3) |
Note that if
{ut+(u+k)ux+∂xp∗(32u2)+λu=0,t>0,x∈R,u(0,x)=u0(x),x∈R. | (4) |
The local well-posedness of the Cauchy problem (4) with initial data
Lemma 2.1. [19] Given
u=u(⋅,u0)∈C([0,T);Hs(R))∩C1([0,T);Hs−1(R)). |
Moreover, the solution depends continuously on the initial data, i.e. the mapping
Consider the following differential equation:
{φt=u(t,φ(t,x))+k,t∈[0,T),φ(0,x)=x,x∈R, | (5) |
where
Lemma 2.2. [19] Let
φx(t,x)=exp(∫t0ux(τ,φ(τ,x))dτ)>0,∀(t,x)∈[0,T)×R. |
Lemma 2.3. Let
m(t,φ(t,x))φ3x(t,x)=m0(x)e−λt. | (6) |
Proof. Differentiating the left-hand side of equation (6) with respect to
ddt{m(t,φ(t,x))φ3x(t,x)}=(mt(t,φ)+mx(t,φ)φt(t,x))φ3x(t,x)+3m(t,φ)φ2x(t,x)φxt(t,x)=[mt(t,φ)+(u(t,φ)+k)mx(t,φ)+3m(t,φ)ux(t,φ)]φ3x(t,x)=−λm(t,φ(t,x))φ3x(t,x), |
which completes the proof of the Lemma 2.3.
In this section, we will derive a conservation law for strong solutions to equation (4). we then establish a priori estimate for the
Lemma 3.1. If
∫∞−∞m(t,x)v(t,x)dx=e−2λt∫∞−∞m(0,x)v(0,x)dx, | (7) |
where
‖u(t)‖2L2≤4e−2λt‖u0‖2L2. | (8) |
Proof. Applying Lemma 2.1 and a simple density argument, we only need to show that this lemma with some
mt+(1−∂2x)∂x(12u2+ku)+∂x(32u2)+λm=0. |
Multiplying the above equation by
∫∞−∞vmtdx=−∫∞−∞v(1−∂2x)∂x(12u2+ku)dx−32∫∞−∞v∂x(u2)dx−λ∫∞−∞vmdx=∫∞−∞vx(1−∂2x)(12u2+ku)dx+32∫∞−∞vxu2dx−λ∫∞−∞vmdx=2∫∞−∞vxu2dx−12∫∞−∞vxxxu2dx+k∫∞−∞vx(1−∂2x)(4v−vxx)dx−λ∫∞−∞vmdx=12∫∞−∞uxu2dx+k∫∞−∞vx(4v−5vxx+vxxxx)dx−λ∫∞−∞vmdx=−λ∫∞−∞vmdx. |
Since
12ddt∫∞−∞mvdx=12∫∞−∞mtvdx+12∫∞−∞mvtdx=∫∞−∞mtvdx, |
it follows that
12ddt∫∞−∞mvdx=−λ∫∞−∞vmdx, |
which implies the desired conserved quantity. In view of the above conservation law, it then follows that
‖u(t,⋅)‖2L2=‖ˆu(t,⋅)‖2L2≤4∫∞−∞1+ξ24+ξ2|ˆu(t,ξ)|2dξ=4(ˆm(t),ˆv(t))=4(m(t),v(t))=4(m0,v0)e−2λt=4(ˆm0,ˆv0)e−2λt≤4e−2λt∫∞−∞1+ξ24+ξ2|ˆu0(ξ)|2dξ≤4e−2λt‖ˆu0‖2L2=4e−2λt‖u0‖2L2. |
This completes the proof of Lemma 3.1.
The following important estimate can be obtained by Lemma 3.1.
Lemma 3.2. Assume
‖u(t,x)‖L∞≤e−λt(3λ‖u0‖2L2+‖u0‖L∞). |
Proof. Applying Lemma 2.1 and a simple density argument, it suffices to consider
ut+(u+k)ux=−3p∗(uux)−λu. | (9) |
Note that
−3p∗(uux)=−32∫∞−∞e−|x−η|uuηdη=−32∫x−∞e−x+ηuuηdη−32∫∞xex−ηuuηdη=34∫x−∞e−|x−η|u2dη−34∫∞xe−|x−η|u2dη. | (10) |
In view of (5), we have
d(u(t,φ(t,x))dt=ut(t,φ(t,x))+ux(t,φ(t,x))φt(t,x)=(ut+(u+k)ux)(t,φ(t,x)), |
where
−34∫∞φ(t,x)e−|φ(t,x)−η|u2dη≤du(t,φ(t,x))dt+λu(t,φ(t,x))≤34∫φ(t,x)−∞e−|φ(t,x)−η|u2dη. |
It thus transpires that
|du(t,φ(t,x))dt+λu(t,φ(t,x))|≤34∫∞−∞e−|φ(t,x)−η|u2dη≤34∫∞−∞u2(t,η)dη. |
In view of Lemma 3.1, we have
−3e−2λt‖u0‖2L2≤du(t,φ(t,x))dt+λu(t,φ(t,x))≤3e−2λt‖u0‖2L2. |
Integrating the above inequality with respect to
−3λ(1−e−λt)‖u0‖2L2≤eλtu(t,φ(t,x))−u0(x)≤3λ(1−e−λt)‖u0‖2L2. |
Thus,
|u(t,φ(t,x))|≤‖u(t,φ(t,x))‖L∞≤e−λt(3λ(1−e−λt)‖u0‖2L2+‖u0‖L∞). | (11) |
Using the Sobolev embedding to ensure the uniform boundedness of
e−κ(t)≤φx(t,x)≤eκ(t),x∈R. |
We now deduce from the above equation that the function
‖u(t,x)‖L∞=‖u(t,φ(t,x))‖L∞≤e−λt(3λ(1−e−λt)‖u0‖2L2+‖u0‖L∞). | (12) |
This completes the proof of Lemma 3.2.
We now present the global existence result.
Theorem 3.3. Assume
Proof. We only assume
ddt12∫∞−∞m2dx=−k∫∞−∞mmxdx−λ∫∞−∞m2dx−3∫∞−∞m2uxdx−∫∞−∞ummxdx=−λ∫∞−∞m2dx−52∫∞−∞m2uxdx. |
Again, multiplying the above equality by
e2λtddt∫∞−∞m2dx+2e2λtλ∫∞−∞m2dx=−5e2λt∫∞−∞m2uxdx. |
Therefore,
ddt(e2λt∫∞−∞m2dx)=5e2λt∫∞−∞m2uxdx. | (13) |
On the other hand, note that if
‖ux‖L∞≤‖px‖L2‖m‖L2≤12‖m‖L2. | (14) |
Using (14), we from (13) obtain
ddt(e2λt∫∞−∞m2dx)≤52e2λt(∫∞−∞m2dx)32=52e−λt(e2λt∫∞−∞m2dx)32. |
From the above inequality, we easily derive that
ddt(e2λt∫∞−∞m2dx)−12≥−54e−λt. |
Integrating the above inequality with respect to
(e2λt∫∞−∞m2dx)−12−(∫∞−∞m20dx)−12≥54λ(e−λt−1). |
Thus,
(e2λt∫∞−∞m2dx)−12≥(∫∞−∞m20dx)−12−54λ. |
We deduce from the above inequality that
1(∫∞−∞m20dx)−12−54λ≥(e2λt∫∞−∞m2dx)12, |
which implies
‖m‖L2≤1eλt(‖m0‖−1L2−54λ). | (15) |
Using (14), (15), and the condition of Theorem, we have
‖ux‖L∞≤12‖m‖L2<‖m‖L2≤1eλt(‖m0‖−1L2−54λ). |
The above inequality and Lemma 2.2 imply
Remark 1. Theorem 3.3 demonstrate the difference of the previous global existence result for the Degasperis-Procesi equation. The obtained condition is simple and convenient since we do not use a condition of the sign of the potential
Recently, the results of the infinite propagation speed for the Camassa-Holm equation and the Degasperis-Procesi equation was extensively established [7,22]. Infinite propagation speed means that they loose instantly the property of having compact
Theorem 4.1. Assume that for some
u(t,x)={E+(t)e−x,x≥φ(t,b),E−(t)ex,x≤φ(t,a), | (16) |
where
E+(t)≤c1(λ,‖u0‖L2,‖u0‖L∞,b)e(k−λ)tand|E−(t)|≤c2(λ,‖u0‖L2,‖u0‖L∞,a)e−(k+λ)t. |
Proof. From Lemma 2.3, we see that
m(t,x)=(I−∂2x)u(t,x)=(I−∂2x)u0(φ−1(t,x))e−λt(∂xφ(t,φ−1(t,x)))3. | (17) |
In addition,
u(t,x)=12e−x∫x−∞eηm(t,η)dη+12ex∫∞xe−ηm(t,η)dη, | (18) |
and
ux(t,x)=−12e−x∫x−∞eηm(t,η)dη+12ex∫∞xe−ηm(t,η)dη. | (19) |
Moreover, defining
f+(t)=∫φ(t,b)φ(t,a)eηm(t,η)dη,f−(t)=∫φ(t,b)φ(t,a)e−ηm(t,η)dη, |
one has from (17) that
u(t,x)=12e−x(∫φ(t,a)−∞+∫φ(t,b)φ(t,a)+∫∞φ(t,b))eηm(t,η)dη+12ex∫∞xe−ηm(t,η)dη=12e−xf+(t),x≥φ(t,b). | (20) |
In the same way, we obtain
u(t,x)=12exf−(t),x≤φ(t,a). |
It then from (19) and (20) follows that
u(t,x)=−ux(t,x)=uxx(t,x)=12e−xf+(t),x≥φ(t,b), | (21) |
and similarly,
u(t,x)=ux(t,x)=uxx(t,x)=12exf−(t),x≤φ(t,a). | (22) |
By the definition of
f+(0)=∫∞−∞eηm0(η)dη=∫∞−∞eη(u0−u0,xx)(η)dη=∫∞−∞eηu0dη+∫∞−∞eηu0,x(η)dη=0. |
Since
df+dt=∫φ(t,b)φ(t,a)eηmt(t,η)dη=∫∞−∞eηmt(t,η)dη. |
Next, integration by parts, (18), (19) and equation using equation (3) yield the following identities
df+(t)dt=∫∞−∞eηmt(t,η)dη=−∫∞−∞eη((mu)x+(u2−u2x)x+λm+kmx)dη=−mueη|∞−∞+∫∞−∞eηmudη−(u2−u2x)eη|∞−∞+∫∞−∞eη(u2−u2x)dη−λ∫∞−∞eηmdη−k∫∞−∞eηmxdη=∫∞−∞eη(u2+u2x)dη+∫∞−∞eη(u2−u2x)dη−uuxeη|∞−∞+12u2eη|∞−∞−12∫∞−∞eηu2dη−(λ−k)∫∞−∞eηmdη=32∫∞−∞eηu2dη−(λ−k)∫∞−∞eηmdη. |
Therefore,
df+(t)dt+(λ−k)f+(t)≥32∫∞−∞eηu2dη. |
Multiplying the above inequality by
d(f+(t)e(λ−k)t)dt>0, |
so that
φ(t,b)=∫t0u(τ,φ(τ,0))dτ+kt+b≤∫t0e−λτ(3λ‖u0‖2L2+‖u0‖L∞)dτ+kt+b=−(e−λtλ−1λ)(3λ‖u0‖2L2+‖u0‖L∞)+kt+b≤1λ(3λ‖u0‖2L2+‖u0‖L∞)+kt+b. |
It then follows from (21) that
u(t,φ(t,b))=12e−φ(t,b)f+(t)≥12e−1λ(3λ‖u0‖2L2+‖u0‖L∞)+|b|−ktf+(t). | (23) |
On the other hand, by Lemma 3.2 we know that
u(t,φ(t,b))≤‖u(t,φ(t,b)‖L∞≤e−λt(3λ‖u0‖2L2+‖u0‖L∞). | (24) |
Combining (23) with (24), we deduce that
f+(t)≤2(3λ‖u0‖2L2+‖u0‖L∞)e1λ(3λ‖u0‖2L2+‖u0‖L∞)−|b|+(k−λ)t:=c1(λ,‖u0‖L2,‖u0‖L∞,b)e(k−λ)t. |
In an analogous way, we can easy to verify that
f−(0)=0 |
and
df−(t)dt+(λ+k)f−(t)≤−32∫∞−∞e−ηu2dη. |
Following the similar argument of the function
f−(t)e(λ+k)t≤0, |
f−(t)<0, |
and
φ(t,a)≥−1λ(3λ‖u0‖2L2+‖u0‖L∞)+kt+a. | (25) |
Combining (22) with (25), in view of Lemma 3.2, we obtain
|f−(t)|≤2(3λ‖u0‖2L2+‖u0‖L∞)e1λ(3λ‖u0‖2L2+‖u0‖L∞)−|a|−(k+λ)t:=c2(λ,‖u0‖L2,‖u0‖L∞,a)e−(k+λ)t. |
By taking
We would like to thank anonymous referee for his useful comments.
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