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Global existence and propagation speed for a Degasperis-Procesi equation with both dissipation and dispersion

  • 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

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  • 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:

    {ututxx+k(uuxx)x+4uux+λ(uuxx)=3uxuxx+uuxxx,t>0,xR,u(0,x)=u0(x),xR, (1)

    where supp u0[a,b] is a compactly supported initial data, kR is an arbitrary dispersion coefficient, and λ>0 is a dissipative parameter.

    When k=0 and λ=0, (1) becomes the Degasperis-Procesi equation [3]

    ututxx+4uux=3uxuxx+uuxxx,t>0,xR,

    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 k=0 and λ>0 in (1), then it becomes the weakly dissipative Degasperis-Procesi equation [5,6,18,19]

    vtvtxx+4vvx+λ(vvxx)=3vxvxx+vvxxx,t>0,xR, (2)

    which has been analyzed in several papers: Local well-posedness of (2) in Hs,s>3/2, waw proved in [5]. Existence and uniqueness of global weak solutions of (2) was established in [6]. Further global existence, blow-up results, persistence properties, and propagation speed were derived in [18,5]. Lenells and Wunsch [12] point out that the weakly dissipative the Degasperis procesi equation (2) are equivalent to their non-dissipative counterparts up to a simple change of variables. More precisely, if u(t,x) and v(t,x) are related by

    v(t,x)=eλtu(1eλtλ,x),

    then v(t,x) satisfies the weakly dissipative Degasperis-Procesi equation (2) if and only if u(t,x) satisfies the respective non-dissipative equation. It was observed already in [18] that the properties of (2) are similar to the properties of non-dissipative Degasperi-Procesi equation restricted to a finite time interval. It was also noted that there are considerable differences between theses equations with respect to their long time behavior. However, the work done above was only involved in dissipative terms. It is found [13,15,20]that such kinds of considerable differnces were investigated for the different model (nonlinear wave equations). Recently, Novruzov and Hagverdiyev [17] analyzed the behavior of solutions to the dissipative Camassa-Holm equation with arbitrary dispersion coefficient.

    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 (k=0 and λ=0) and (1) (k0 and λ>0), some behaviors of solutions to the dissipative Degasperis-Procesi equation (1) with arbitrary dispersion are similar to the ones of Degasperis-Procesi equation(k=0 and λ=0), such as, the local well-posedness and the blow-up scenario. However, the dissipative term λ(uuxx) and the dispersive term k(uxuxxx) in (1) do have impacts on the global existence and the propagation speed of its solutions, which are shown below in Theorem 3.3 and Theorem 4.1, respectively. In particular, the propagation speed is seriously affected by both the dissipative parameter λ and the dispersion coefficient k.

    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 Hs(R), s>3/2, we briefly collect the needed results from [21] in order to pursue our goal.

    With m:=uuxx, The equation (1) takes the following form of a quasi-linear evolution equation of hyperbolic type:

    {mt+(u+k)mx+3uxm+λm=0,t>0,xR,m(0,x)=u0(x)u0,xx(x),xR. (3)

    Note that if p(x):12e|x|, xR, then (12x)1f=pf for all fL2(R) and pm=u. Using this identity, we can rewrite equation (3) as follows:

    {ut+(u+k)ux+xp(32u2)+λu=0,t>0,xR,u(0,x)=u0(x),xR. (4)

    The local well-posedness of the Cauchy problem (4) with initial data u0Hs(R), s>3/2, and λ=0 can be obtained by applying the Kato theorem [11]. It is easy to see that the same result holds for the Cauchy problem (4). As a result, we have the following well-posedness result.

    Lemma 2.1. [19] Given u0Hs(R),s>3/2, there exists a maximal time T=T(u0,k,λ)>0 and a unique solution u to initial-value problem (4), such that

    u=u(,u0)C([0,T);Hs(R))C1([0,T);Hs1(R)).

    Moreover, the solution depends continuously on the initial data, i.e. the mapping u0u(,u0):Hs(R)C([0,T);Hs(R))C1([0,T);Hs1(R)) is continuous and the maximal time of existence T>0 can be chosen to be independent of s.

    Consider the following differential equation:

    {φt=u(t,φ(t,x))+k,t[0,T),φ(0,x)=x,xR, (5)

    where u(t,x) is the corresponding strong solution to (1). Applying classical results in the theory of ordinary differential equations, one can obtain the following two results on φ which are crucial in the proof of global existence and blow-up solutions.

    Lemma 2.2. [19] Let u0Hs(R),s>3/2, and let T>0 be the maximal existence time of the corresponding solution u to equation (4).Then equation (5) has a unique solution φC1([0,T)×R,R). Moreover, the map φ(t,) is an increasing diffeomorphism of R with

    φx(t,x)=exp(t0ux(τ,φ(τ,x))dτ)>0,(t,x)[0,T)×R.

    Lemma 2.3. Let u0Hs(R),s>3/2, and let T>0 be the maximal existence time of the corresponding solution u to equation (4). Then we have

    m(t,φ(t,x))φ3x(t,x)=m0(x)eλt. (6)

    Proof. Differentiating the left-hand side of equation (6) with respect to t, in view of the relations (5) and (3), we obtain

    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 L-norm of the strong solution by using conservation law. This enables us to guarantee that the solution exists globally.

    Lemma 3.1. If u0Hs(R),s>3/2, then as long as the solution u(t,x) given by Lemma 2.1 exists, we have

    m(t,x)v(t,x)dx=e2λtm(0,x)v(0,x)dx, (7)

    where m(t,x)=u(t,x)uxx(t,x) and v(t,x)=(42x)1u. Moreover, we have

    u(t)2L24e2λtu02L2. (8)

    Proof. Applying Lemma 2.1 and a simple density argument, we only need to show that this lemma with some s>3/2. Thus we take s=3 in the proof. Let T>0 be the maximal time of existence of the solution u to equation (4) with initial data u0H3(R) such that uC([0,T);H3(R))C1([0,T);H2(R)), which is guaranteed by the local well-posedness Lemma 2.1. Applying the operator (12x) on the both sides of equation (4), we have

    mt+(12x)x(12u2+ku)+x(32u2)+λm=0.

    Multiplying the above equation by v(t,x) and integrating by parts with respect to x, in view of 4vvxx=u, we obtain

    vmtdx=v(12x)x(12u2+ku)dx32vx(u2)dxλvmdx=vx(12x)(12u2+ku)dx+32vxu2dxλvmdx=2vxu2dx12vxxxu2dx+kvx(12x)(4vvxx)dxλvmdx=12uxu2dx+kvx(4v5vxx+vxxxx)dxλvmdx=λvmdx.

    Since

    12ddtmvdx=12mtvdx+12mvtdx=mtvdx,

    it follows that

    12ddtmvdx=λvmdx,

    which implies the desired conserved quantity. In view of the above conservation law, it then follows that

    u(t,)2L2=ˆu(t,)2L241+ξ24+ξ2|ˆu(t,ξ)|2dξ=4(ˆm(t),ˆv(t))=4(m(t),v(t))=4(m0,v0)e2λt=4(ˆm0,ˆv0)e2λt4e2λt1+ξ24+ξ2|ˆu0(ξ)|2dξ4e2λtˆu02L2=4e2λtu02L2.

    This completes the proof of Lemma 3.1.

    The following important estimate can be obtained by Lemma 3.1.

    Lemma 3.2. Assume u0Hs(R),s>3/2. Let T>0 be the maximal existence time of the solution u to the equation (4) guaranteed by Lemma 2.1. Then we have

    u(t,x)Leλt(3λu02L2+u0L).

    Proof. Applying Lemma 2.1 and a simple density argument, it suffices to consider s=3. Let T>0 be the maximal time of existence of the solution u to (4) with the initial value u0H3(R) such that uC([0,T);H3(R))C1([0,T);H2(R)), which is guaranteed by Lemma 2.1. By (4), we obtain

    ut+(u+k)ux=3p(uux)λu. (9)

    Note that

    3p(uux)=32e|xη|uuηdη=32xex+ηuuηdη32xexηuuηdη=34xe|xη|u2dη34xe|xη|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 φ=φ(t,x) is the C1 solution to (5). It then follows from (9) and (10) that

    34φ(t,x)e|φ(t,x)η|u2dηdu(t,φ(t,x))dt+λu(t,φ(t,x))34φ(t,x)e|φ(t,x)η|u2dη.

    It thus transpires that

    |du(t,φ(t,x))dt+λu(t,φ(t,x))|34e|φ(t,x)η|u2dη34u2(t,η)dη.

    In view of Lemma 3.1, we have

    3e2λtu02L2du(t,φ(t,x))dt+λu(t,φ(t,x))3e2λtu02L2.

    Integrating the above inequality with respect to t<T on [0,t] yields

    3λ(1eλt)u02L2eλtu(t,φ(t,x))u0(x)3λ(1eλt)u02L2.

    Thus,

    |u(t,φ(t,x))|u(t,φ(t,x))Leλt(3λ(1eλt)u02L2+u0L). (11)

    Using the Sobolev embedding to ensure the uniform boundedness of ux(τ,η) for (τ,η)[0,t]×R with t[0,T), in view of Lemma 2.3, we obtain for every t[0,T) a constant κ(t)>0 such that

    eκ(t)φx(t,x)eκ(t),xR.

    We now deduce from the above equation that the function φ(t,) is strictly increasing on R with limx±φ(t,x)=± as long as t[0,T). Thus, by (11) we can obtain

    u(t,x)L=u(t,φ(t,x))Leλt(3λ(1eλt)u02L2+u0L). (12)

    This completes the proof of Lemma 3.2.

    We now present the global existence result.

    Theorem 3.3. Assume u0Hs(R),s>3/2. If m0=u0u0,xx satisfying m0L2<4λ5, then the corresponding solution u(t,x) to equation (4) exists globally.

    Proof. We only assume s=3 to prove the above theorem. Let T>0 be the maximal time of existence of the solution u to equation (4) with initial data u0H3(R). Multiplying equation (3) by m and using integration by parts with respect to x, we have

    ddt12m2dx=kmmxdxλm2dx3m2uxdxummxdx=λm2dx52m2uxdx.

    Again, multiplying the above equality by 2e2λt, we get

    e2λtddtm2dx+2e2λtλm2dx=5e2λtm2uxdx.

    Therefore,

    ddt(e2λtm2dx)=5e2λtm2uxdx. (13)

    On the other hand, note that if p(x)=12e|x|, then (12x)1f=pf for all fL2(R) and u=pm. From this relation, we find

    uxLpxL2mL212mL2. (14)

    Using (14), we from (13) obtain

    ddt(e2λtm2dx)52e2λt(m2dx)32=52eλt(e2λtm2dx)32.

    From the above inequality, we easily derive that

    ddt(e2λtm2dx)1254eλt.

    Integrating the above inequality with respect to t yields

    (e2λtm2dx)12(m20dx)1254λ(eλt1).

    Thus,

    (e2λtm2dx)12(m20dx)1254λ.

    We deduce from the above inequality that

    1(m20dx)1254λ(e2λtm2dx)12,

    which implies

    mL21eλt(m01L254λ). (15)

    Using (14), (15), and the condition of Theorem, we have

    uxL12mL2<mL21eλt(m01L254λ).

    The above inequality and Lemma 2.2 imply T=. This proves that the solution u exists globally in time. This completes the proof of Theorem 3.3.

    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 m0 in the point x0.

    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 x-support. Motivated by recently work [9,16], the purpose of this section is to give a more detailed description on the corresponding strong solution u(t,x) to (4) in its lifespan with initial data u0(x) being compactly supported.

    Theorem 4.1. Assume that for some T>0 and s>3/2,uC([0,T);Hs(R)) is a strong solution of (1). If u0(x)=u(0,x) has compact support in [a,b], then for any t[0,T], we have

    u(t,x)={E+(t)ex,xφ(t,b),E(t)ex,xφ(t,a), (16)

    where E+ and E are continuous non-vanishing functions with E+(0)=E(0)=0 and E+(t)>0 for t(0,T) is a strictly increasing function, while E(t)<0 for t(0,T) is a strictly decreasing function. Furthermore, we get

    E+(t)c1(λ,u0L2,u0L,b)e(kλ)tand|E(t)|c2(λ,u0L2,u0L,a)e(k+λ)t.

    Proof. From Lemma 2.3, we see that

    m(t,x)=(I2x)u(t,x)=(I2x)u0(φ1(t,x))eλt(xφ(t,φ1(t,x)))3. (17)

    In addition, u0 has a compact support in x in the interval [a,b] for any t[0,T]. Therefore so does m(t,) in the interval [φ(t,a),φ(t,b)]. By the relation u=pm with p(x)=12e|x|,xR, we have

    u(t,x)=12exxeηm(t,η)dη+12exxeηm(t,η)dη, (18)

    and

    ux(t,x)=12exxeηm(t,η)dη+12exxeηm(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)eηm(t,η)dη,

    one has from (17) that

    u(t,x)=12ex(φ(t,a)+φ(t,b)φ(t,a)+φ(t,b))eηm(t,η)dη+12exxeηm(t,η)dη=12exf+(t),xφ(t,b). (20)

    In the same way, we obtain

    u(t,x)=12exf(t),xφ(t,a).

    It then from (19) and (20) follows that

    u(t,x)=ux(t,x)=uxx(t,x)=12exf+(t),xφ(t,b), (21)

    and similarly,

    u(t,x)=ux(t,x)=uxx(t,x)=12exf(t),xφ(t,a). (22)

    By the definition of f+(t), we get

    f+(0)=eηm0(η)dη=eη(u0u0,xx)(η)dη=eηu0dη+eηu0,x(η)dη=0.

    Since m(t,) is supported on compact interval [φ(t,a),φ(t,b)], for fixed t we have

    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+(u2u2x)x+λm+kmx)dη=mueη|+eηmudη(u2u2x)eη|+eη(u2u2x)dηλeηmdηkeηmxdη=eη(u2+u2x)dη+eη(u2u2x)dηuuxeη|+12u2eη|12eηu2dη(λk)eηmdη=32eηu2dη(λk)eηmdη.

    Therefore,

    df+(t)dt+(λk)f+(t)32eηu2dη.

    Multiplying the above inequality by e(λk)t, we obtain

    d(f+(t)e(λk)t)dt>0,

    so that f+(t)>0 for any t>0 is a strictly increasing function. Next, along the curve φ(t,b), simple calculations yields the following estimation

    φ(t,b)=t0u(τ,φ(τ,0))dτ+kt+bt0eλτ(3λu02L2+u0L)dτ+kt+b=(eλtλ1λ)(3λu02L2+u0L)+kt+b1λ(3λu02L2+u0L)+kt+b.

    It then follows from (21) that

    u(t,φ(t,b))=12eφ(t,b)f+(t)12e1λ(3λu02L2+u0L)+|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)Leλt(3λu02L2+u0L). (24)

    Combining (23) with (24), we deduce that

    f+(t)2(3λu02L2+u0L)e1λ(3λu02L2+u0L)|b|+(kλ)t:=c1(λ,u0L2,u0L,b)e(kλ)t.

    In an analogous way, we can easy to verify that

    f(0)=0

    and

    df(t)dt+(λ+k)f(t)32eηu2dη.

    Following the similar argument of the function f+(t) and curve φ(t,b), it is found that

    f(t)e(λ+k)t0,
    f(t)<0,

    and

    φ(t,a)1λ(3λu02L2+u0L)+kt+a. (25)

    Combining (22) with (25), in view of Lemma 3.2, we obtain

    |f(t)|2(3λu02L2+u0L)e1λ(3λu02L2+u0L)|a|(k+λ)t:=c2(λ,u0L2,u0L,a)e(k+λ)t.

    By taking E±(t)=12f±(t), this completes the proof of Theorem 4.1.

    We would like to thank anonymous referee for his useful comments.



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