# Wave Breaking and Global Existence for the Generalized Periodic Camassa-Holm Equation with the Weak Dissipation

Authors: Ying Zhang; Congming Peng
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

In this paper, a family of the weakly dissipative periodic Camassa-Holm type equation cubic and quartic nonlinearities is considered. The precise blow-up scenarios of strong solutions and several conditions on the initial data to guarantee blow-up of the induced solutions are described in detail. Finally, we establish a sufficient condition for global solutions.


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## Body

## 1. Introduction

In this paper, we are concerned with the periodic Cauchy problem of the generalized Camassa-Holm (CH) type equation the weak dissipation as follows:(1)ut−utxx+3uux+λu−uxx=2uxuxx+uuxxx+αux+βu2ux+γu3ux+Γuxxx,ut,x+1=ut,x,t≥0,x∈ℝ,u0,x=u0x,x∈ℝ,where α,β,γ,λ, and Γ are arbitrary constants.Whenλ=α=β=γ=Γ=0, the generalized CH type equation in (1) recovers the well-known integrable Camassa-Holm equation [1]
(2)ut−utxx+3uux=2uxuxx+uuxx,which was derived by Camassa and Holm for shallow water waves [2] and is named after them. In fact, the CH equation was originally derived by Fokas and Fuchssiteiner [1] as a bi-Hamiltonian generalization of KdV. It is completely integrable and has infinitely many conservation laws [2–4]. In the recent years, the CH equation has caught a lot of attention from different perspectives, such as well-posedness, blow-up phenomena of solutions, and their stability. For example, the local well-posedness and the global strong solutions for certain class of initial data were studied [4, 5]. Existence and uniqueness results for classical solution of the periodic CH equation were established in [6]. The blow-up phenomena of the periodic CH equation were also investigated in a number of papers (see [2, 4–8] and references therein). Orbital stability of the peakons for the CH equation was studied by Lenells in [9, 10].Whenλ=β=γ=0 and αΓ≠0, we have the Dullin-Gottwald-Holm (DGH) equation [11, 12]. And if α=β=γ=Γ=0 and λ>0, it reduces to the weakly dissipative CH equation, whereas if β=γ=0 and λ>0,αΓ≠0, it reduces to the weakly dissipative DGH equation. The well-posedness and blow-up phenomena of solutions of the Cauchy problem for the weakly dissipative DGH equation were studied, see, for example, [13, 14]. The similar research of the dissipative DGH equation higher-order nonlinearities and arbitrary coefficients and the other related models was also discussed in [15–19]. For some new and important developments for searching for solving numerical solutions for some PDE, the reader is referred to [20, 21] and references therein.Ifλ=0, the first equation of (1) becomes the equation
(3)ut−utxx+3uux=2uxuxx+uuxxx+αux+βu2ux+γu3ux+Γuxxx,which is related to the following physically relevant model:
(4)ut−utxx+3uux=2uxuxx+uuxxx−cux+β0βuxxx−ω1α2u2ux−ω2α3u3ux,with(5)c=1+Ω2−Ω,α=c21+c2,β0=cc4+6c2−16c2+12,β=3c4+8c2−16c2+12,ω1=−3cc2−1c2−22c2+13,ω2=c2−12c2−28c2−12c2+15,satisfying c⟶1,β⟶5/12,β0⟶1/4,ω1,ω2⟶0, and α⟶1/2 when Ω⟶0. Here, Ω is a parameter related to the rotational frequency due to Coriolis effect which is typically a manifestation of rotation when Newton’s laws are applied to model physical phenomena on Earth’s surface. The Cauchy problem of Equations (3) and (4) has been studied in ref. [16, 22–26].Whenλ>0, the first equation of (1) can be seen as a weakly dissipative Camassa-Holm (CH) type equation. For the weakly dissipative CH type equations: the local well-posedness, global existence, and blow-up phenomena of the Cauchy problem of the weakly dissipative CH equation
(6)ut−utxx+3uux+λu−uxx=2uxuxx+uuxxx,on the line [27] and on the circle [27] were studied. They found that the behaviors of Equation (44) are similar to the CH equation in a finite interval of time, such as the local well-posedness and the blow-up phenomena, and that there are considerable differences between them in their long time behaviors. Thereafter, a new global existence result and a new blow-up result for strong solutions to the equation certain profiles were presented in [28]. The obtained results improve considerably the previous results. Later on, a new blow-up result for positive strong solutions of (6) was presented by Novruzov [29]. In particular, they used a condition where the initial data u0x and its derivative are not simultaneously involved and the parameter λ is not bounded from above. The well-posedness and wave-breaking phenomena to the weakly dissipative CH equation quadratic and cubic nonlinearities
(7)ut−utxx+3u2ux+λu−uxx=2uxuxx+uuxxx,were considered by Freire et al. [30]. The novelty of their work is the method of group analysis was applied in order to construct conserved currents, and therefore, the conserved quantities were established as an extremely natural consequence of them. Subsequently, the periodic Cauchy problem of Equation (7) was considered by Ji and Zhou [31] and their local well-posedness was established via Kato’s theory [32]; then, a sufficient condition on the initial data to guarantee the wave breaking was given and the global existence of solutions was given finally. Recently, Freire [33] considered the Cauchy problem of the weakly dissipative CH Equation (1). In their paper, some time-dependent energy functionals of solutions were proved, then the existence of wave-breaking phenomena was investigated, and necessary conditions for its existence were also obtained.In general, it is difficulty to avoid energy dissipation mechanics in a real world. So it is reasonable to investigate the model with energy dissipation in propagation of nonlinear waves, see [34, 35] and references therein. Inspired by the previous work, the aim of the paper is to investigate whether the periodic Cauchy problem of the equation in (1) has the similar remarkable properties as that on the entire line. The outline of the paper is as follows. In Section 2, we obtain the local well-posedness and wave-breaking criterion. In Section 3, a blow-up scenario for strong solutions is described and some sufficient conditions for wave breaking of strong solutions in finite time are established. Furthermore, the blow-up rate of blow-up solutions of (1) is also derived. In Section 4, a sufficient condition for global solutions is provided.


### 1.1. Notations

Throughout this paper, we identity all spaces of periodic functions with function spaces over the unit circleS in ℝ2, i.e., S=ℝ/ℤ. The norm of the Sobolev space HsS, s∈ℝ, by ⋅Hs. Since all space of functions are over S, for simplicity, we drop S in our notations of function spaces if there is no ambiguity.


## 1.1. Notations

Throughout this paper, we identity all spaces of periodic functions with function spaces over the unit circleS in ℝ2, i.e., S=ℝ/ℤ. The norm of the Sobolev space HsS, s∈ℝ, by ⋅Hs. Since all space of functions are over S, for simplicity, we drop S in our notations of function spaces if there is no ambiguity.


## 2. Preliminaries and Local Well-Posedness

Let us introduce the subject of investigation of this paper. In order to establish the local well-posedness result by Kato’ theorem. Rewrite problem (1) as follows:
(8)mt+u+Γmx+2uxm+λm−∂xhu=0,t>0,x∈ℝ,ut,x+1=ut,x,t≥0,x∈ℝ,u0,x=u0x,x∈ℝ,where
(9)m=u−uxx,hu≔α+Γu+β3u3+γ4u4.DenoteGx≔coshx−x−1/2/2sinh1/2,x∈ℝ. Then 1−∂x2−1f=Λ−2f=G∗f for all f∈L2S. Our system (8) can be written in the following transport type:
(10)ut+u+Γux=∂xG∗hu−∂xG∗u2+ux22−λu,t>0,x∈ℝ,ut,x+1=ut,x,t≥0,x∈ℝ,u0,x=u0x,x∈ℝ,or equivalently,(11)ut+u+Γux=∂xΛ−2hu−∂xΛ−2u2+ux22−λu,t>0,x∈ℝ,ut,x+1=ut,x,t≥0,x∈ℝ,u0,x=u0x,x∈ℝ.We begin by presenting the local well-posedness result for the periodic Cauchy problem (11). Concerning the generalized CH equation in (1) is suitable for applying Kato’s theory [32]; one may follow the similar argument as in [30, 33] to obtain the following theorem.Theorem 1.
Givenu0∈HsS with s>3/2. Then, there exists a maximal time T=Tu0>0 and unique solution u to (1) satisfying the initial condition u0,x=u0x such that
(12)u·,u0∈C0,T;HsS∩C10,T;Hs−1S.
Moreover, the solution depends continuously on the initial data, in the sense that the mappingu0⟶u·,u0: HsS⟶C0,T;HsS∩C10,T;Hs−1S is continuous and T does not depend on S.Remark 2.
The equation in (1), for λ>0, can be seen as the weakly dissipative periodic Camassa-Holm type equation. However, no matter the value of parameter λ, the problem (1) is locally well-posed as shown in Theorem 1.Lemma 3. (see [8]).
(i)
For everyf∈H1S, we have(13)maxx∈0,1f2x=e+12e−1fH1S2,where the constant e+1/2e−1 is sharp.
(ii)
For everyf∈H3S, we have(14)maxx∈0,1f2x≤cfH1S2,with the best possible constant c lying within the range 1,13/12. Moreover, the best constant c is e+1/2e−1.
The next theorem will establish the time-dependent conserved quantities of solutions to problem (1), which is crucial in the investigation of wave-breaking phenomena and global existence of solutions.Theorem 4.
Assume thatut,x be the solution to problem (1) with the initial data u0x∈HsS,s>3/2 such that ut,x and its derivatives up to second order go to 0 as x⟶0 and x⟶1. Let
(15)H0t≔∫Sut,xdx,H1t≔12∫Su2t,x+ux2t,xdx.
Then, for anyt∈0,T, we have
(16)H0t=e−λtH00,H1t=e−2λtH10.Proof.
Integrating the first equation of system (1) by parts, in view of the periodicity of u, we have
(17)ddt∫Sut,xdx+λ∫Sut,xdx=0.
Then integrating (17) with respect to t over 0,t implies
(18)H0t=∫Sut,xdx=e−λt∫Su0xdx=e−λtH00.
For the proof of the other conserved quantity, multiplying both sides of the first equation in (1) by 2u, we have
(19)2uut−2uutxx+6u2ux+2λu2−uuxx=4uuxuxx+2u2uxxx+2αuux+2βu3ux+2γu4ux+2Γuuxxx.
Integrating (19) with respect to t over 0,t yields
(20)ddt∫Su2+ux2dx+2λ∫Su2+ux2dx=−2∫S∂∂xu3−u2uxx−uutx+Γ2ux2−Γuuxx−α2u2−β4u4−γ5u5−λuuxdx=0,
i.e.,H1t=e−2λtH10, which completes the proof of the theorem.It is worth mentioning that ifλ>0, the results of Theorem 1 implies
(21)H0t≤H00,H1t≤H10,which will be of great relevance in our investigation of wave breaking. Combining this observation with the Sobolev Embedding Theorem, we give the following remark.Remark 5.
Ifu0x∈HsS,s>3/2, and λ>0, then the H1-norm of the corresponding solution of system (1) is bounded above by u0H1S.Once we have presented conditions for having locally well-posed solutions, a natural question is whether (1) admits wave breaking, which can be assured by the following blow-up criterion. In the following proof, we only prove the case when s=3, since the same conclusion for general case s>3/2 can be obtained by using denseness.Theorem 6.
Letu0x∈Hs,s>3/2 be given and assume that T>0 is the maximal existence time of the corresponding solution ut,x to problem (1). Then, T is finite if and only if
liminft⟶Tinfx∈Suxt,x=−∞.Proof.
Note thatm=u−uxx, a simple computation yields
(22)mL22=∫Su−uxx2dx=∫Su2+2ux2+uxx2dx,mxL22=∫Sux−uxxx2dx=∫Sux2+2uxx2+uxxx2dx.
Therefore, we can conclude that(23)uH32≤mH12≤3uH32.
Multiplying2m to both sides of the first equation in (8), we get
(24)2mmt+2u+Γmmx+4uxm2+2λm2−2m∂xhu=0.
Integrating (24) with respect to x over S yields
(25)ddt∫Sm2dx=−∫S2u+Γmmx+4uxm2+2λm2−2m∂xhudx=−∫S3uxm2+2λm2−2m∂xhudx,where we used the relations
(26)∫S2mmxdx=∫Sdm2=0,∫S2ummxdx=−∫Suxm2dx=0.
Ifu0∈H4, differentiating the first equation in (8) with respect to x, we obtain
(27)mtx+3uxmx+u+Γmxx+2uxxm+λmx−∂x2hu=0.
Multiplying2mx to both sides of Equation (27), then integrating the result with respect to x over S, it follows that
(28)ddt∫Smx2dx=−6∫Suxmx2dx−2∫Su+Γmxmxxdx−4∫Suxxmmxdx−2λ∫Smx2dx+2∫Smx∂x2hudx.
Since that(29)2∫Su+Γmxmxxdx=−∫Suxmx2dx,4∫Suxxmmx=−2∫Suxm2dx,
we have(30)ddt∫Smx2dx=−5∫Suxmx2dx+2∫Suxm2dx−2λ∫Smx2dx+2∫Smx∂x2hudx,with
(31)∂x2hu=α+Γuxx+2βuux2+βu2uxx+3γu3uxx,
which yields that(32)ddt∫Smx2dx=−5∫Suxmx2dx+2∫Suxm2dx−2λ∫Smx2dx+2∫Smx∂x2hudx.
Then, by approximatingu0 in H3 by function u0n∈H4n≥1, (32) also holds for u0∈H3. In fact, let un be the solution to problem (1) with the initial data u0n. By local well-posedness theorem, we know that un∈C0,Tn;H4∩C10,Tn;H3 for n≥1,unt,x⟶ut,x in H3 and Tn⟶T as n⟶∞. Since u0nx∈H4, we have
(33)ddt∫Smxn2dx=−5∫Suxnmxn2dx+2∫Suxnmn2dx−2λ∫Smxn2dx+2∫Smxn∂x2hundx.
Sinceun⟶u in H3 as n⟶∞, it follows that uxn⟶ux,uxxn⟶uxx in L∞ as n⟶∞. Meanwhile, mn⟶m in H1 and mxn⟶mx in L2 as n⟶∞. Letting n⟶∞ in (33), it follows that (32) holds for u0x∈H3.
Thus, we get(34)ddt∫Sm2+mx2dx=−5∫Suxmx2dx−∫Suxm2dx−2λ∫Sm2+mx2dx+2∫Sm∂xhu+mx∂x2hudx.
Note that(35)2∫Sm∂xhu+mx∂x2hudx=2∫SmΛ2∂xhudx≤mL22+Λ2∂xhuL22≤cmH12.
Here, we have used the facts that(36)Λ2∂xhuL2=∂xhuH2≤huH3,huH3≤cuH3=cΛ−2uH1=cmH1.
Therefore, (34) is reduced to
(37)ddt∫Sm2+mx2dx=−5∫Suxmx2dx−∫Suxm2dx−2λ∫Sm2+mx2dx+cmH12.
If there exists some positive constantM>0 such that ux>−M, then we have
(38)ddt∫Sm2+mx2dx≤5M∫Smx2dx+M∫Sm2dx−2λ∫Sm2+mx2dx+cmH12≤5M−2λ+cmH12.
By using Gronwall’s inequality, we get(39)mH12≤e5M−2λ+ctm0H12≤e5M−2λ+cTm0H12.
Therefore,(40)uH32≤mH12≤e5M−2λ+cTm0H12≤3e5M−2λ+cTu0H12,
which implies theH3-norm of the solution ut,x to (1) does not blow up in finite time.
On the other hand, if(41)liminft⟶Tinfx∈Suxt,x=−∞,
then solutionut,x to (1) will blow up in finite time. This completes the proof of the theorem.Remark 7.
Similar to the case of nonperiodic, we prove the following result: Letu0x∈Hs,s>3/2 be given and λ>0. And u be the corresponding solution ut,x to problem (1). Assume that ux>−k for some positive constant k. Then, we obtain that uHsS≤eσtu0HsS, for a certain positive constant σ.


## 3. Blow-up Solutions and Blow-up Rate

In this section, we establish some sufficient conditions for the breaking of waves for the initial-value problem (1). To this end, we need the following lemma.Lemma 8.
LetT>0 and vt,x∈C10,T;H2 be a given function. Then, for any t∈0,T, there exists at least one point ξt∈S such that
(42)mt=infx∈Svxt,x=vxt,ξt,
and the functionmt is almost everywhere differentiable in 0,T, with
(43)m′t=vtxt,ξt,a.e.on0,T.Now we are in position to state the following that provide a case that wave breaks in finite time.Theorem 9.
Letu0∈Hs with s>3/2 and λ>0. Assume k≔maxα+Γ,β/3,γ/4 and U0≔maxu0H1,u0H14. If there exists some x0∈S such that
(44)u0′x0<−λ−λ2+21+32·e+1e−11+kU0.
Then, the corresponding solutionut,x to (1) blows up in finite time in the following sense: there exists a T1 with
(45)0<T1<1λ2+21+3/2·e+1/e−11+kU0lnm0+λ−λ2+21+3/2·e+1/e−11+kU0m0+λ+λ2+21+3/2·e+1/e−11+kU0,
such that(46)liminft↑T1infx∈Suxt,x=−∞.Proof.
. Differentiating the first equation in (11) with respect to x, we get
(47)uxt+ux22+u+Γuxx+λux=u2−Λ−2u2+ux2−hu+Λ−2hu,where we have used the relation ∂x2Λ−2=Λ−2−1.
According to Lemma8 and the local well-posedness theorem, there is at least one point ξt∈S satisfying mt=infx∈Suxt,x=uxt,ξt. Hence,
(48)uxxt,ξt=0,a.e.t∈0,T.
Thus, we get(49)m′t=−12m2t−λmt+ft,ξt,where ft,ξt is given by
(50)ft,ξt=u2t,ξt−G∗u2+ux22t,ξt−hut,ξt+G∗hut,ξt.
Note that(51)u2t,ξt≤e+12e−1uH12≤e+12e−1u0H12,G∗u2+ux2L∞≤GL∞u2+ux2L1≤e+12e−1uH12≤e+12e−1u0H12,G∗hu≤GL∞huL∞≤e+12e−1huH1,
then(52)G∗hu−hu≤1+e+12e−1huH1.
LetU0≔maxu0H1,u0H14, it follows from (9) that
(53)huH1≤kmaxu0H1,u0H14≤kU0,with k≔maxα+Γ,β/3,γ/4 is a positive constant. Therefore, we obtain
(54)f≤e+1e−1u0H12+1+e+12e−1kU0≤1+32·e+1e−11+kU0.
Combining (49) and (54), we have
(55)m′t≤−12m2t−λmt+1+3e+12e−11+kU0≐−12m2t−λmt+c=−12mt+λ+λ2+2cmt+λ−λ2+2ca.e.on0,T,
with(56)c=1+32·e+1e−11+kU0.
According to assumption (44), we have m0<−λ−λ2+2c, hence m′0<0. Considering the continuity of mt with respect to t, we can obtain that for any t∈0,T, m′t<0 and mt<−λ−λ2+2c.
Then by solving the inequality (54), it follows that
(57)m0+λ+λ2+2cm0+λ−λ2+2ceλ2+2ct≤mt+λ+λ2+2cmt+λ−λ2+2c,
then(58)m0+λ+λ2+2cm0+λ−λ2+2ceλ2+2ct−1≤2λ2+2cmt+λ−λ2+2c≤0.
Notice that0<m0+λ+λ2+2c/m0+λ−λ2+2c<1, there exists
(59)0<T1≤1λ2+2clnm0+λ−λ2+2cm0+λ+λ2+2c
such that(60)liminft↑T1mt=−∞,
which demonstrates that the solutionut,x blows up at a time 0<T≤T1.Theorem 10.
IfT<∞ is the blow-up time of the solution to (1) with initial data u0∈Hs,s>3/2 satisfying the assumption of Theorem 9. Then,
(61)liminft↑Tinfx∈Suxt,xT−t=−2.Proof.
From (49) and (54), we know that
(62)−c≤m′t+12m2t+λmt≤c,a.e.on0,T,therefore,
(63)−c−12λ2≤m′t+12mt+λ2≤c+12λ2,a.e.on0,T.
Choose0<ε<1/2. Since liminft↑Tmt=∞, we get liminft↑Tmt+λ=−∞; there is some point t0∈0,T such that
(64)mt0+λ<0,mt0+λ2>1εc+12λ2.
Sincemt is absolutely continuous on 0,T. By the above differential inequality, it follows that mt is strictly decreasing on t0,T and hence
(65)mt+λ2>1εc+12λ2,t∈t0,T.
Combining (63) and (63), we get
(66)−12−ε≤m′tmt+λ2≤−12+ε,a.e.ont∈t0,T.Since ε∈0,1/2 is arbitrary, the above relation implies
(67)liminft↑Tmt+λT−t=−2,namely, liminft↑Tmt+λT−t=−2. This implies liminft↑Tinfx∈Suxt,xT−t=−2 by in view of the definition of mt.


## 4. Global Existence

In this section, we turn our attention to existence of the global solution of system (1). We begin with the related results as follows.Lemma 11.
Letu∈C10,T,Hs be the solution of problem (1) with u0∈Hs,s>3/2, and T>0 be the maximal time of existence. Then, the problem
(68)qtt,x=ut,q+Γ,t,x∈0,T×S,q0,x=x,x∈S.
has a unique solutionqt,x∈C10,T×ℝ,ℝ and qt,· is an increasing diffeomorphism of the line with
(69)qxt,x=exp∫0tuxs,qs,xds>0,t,x∈0,T×ℝ.
Furthermore,(70)mt,qt,xqx2t,x=m0xe−λt+∫0teλs−tqx2s,x∂xhus,qds,where q is the solution of the problem (72) and hu is the function given in (44).Proof.
The proof of Lemma11 is similar as for the classic CH Equation (2), see ref. [36] for details.Theorem 12.
Letu0∈Hs,s>3/2, and m0x=u0x−u0′′x. Assume that
m0x≥0 on S or m0x≥0 on S does not change sign
(71)sgnm0=sgnm.
Then, the solutionut,x of (1) possesses bounded from blow x-derivative, which implies the global existence of the solution ut,x in time t.Proof.
Sinceu=G∗m, with Gx=coshx−x−1/2/2sinh1/2,x∈ℝ, we get
(72)ut,x=12sinh1/2∫Scoshx−ξ−x−ξ−1/2mt,ξdξ=ex4sinh1/2∫0xe−ξ−1/2mt,ξdξ+ex4sinh1/2∫x1e−ξ+1/2mt,ξdξ+e−x4sinh1/2∫0xeξ+1/2mt,ξdξ+e−x4sinh1/2∫x1eξ−1/2mt,ξdξ.
Differentiating this representation ofu with respect to x gives
(73)uxt,x=ex4sinh1/2∫0xe−ξ−1/2mt,ξdξ+ex4sinh1/2∫x1e−ξ+1/2mt,ξdξ−e−x4sinh1/2∫0xeξ+1/2mt,ξdξ−e−x4sinh1/2∫x1eξ−1/2mt,ξdξ.
By combining (72) and (73), we have
(74)ut,x+uxt,x=ex2sinh1/2∫0xe−ξ−1/2mt,ξdξ+ex2sinh1/2∫x1e−ξ+1/2mt,ξdξ,(75)uxt,x−ut,x=−e−x2sinh1/2∫0xeξ+1/2mt,ξdξ−e−x2sinh1/2∫x1eξ−1/2mt,ξdξ.
Since thatsgnm=sgnm0, it follows from (72), (74), and (75) that if m0x≥0, then ut,x≥0 and ut,x≥−uxt,x or if m0x≤0, then ut,x≤0 and uxt,x≥ut,x.
The above facts are enough to ensure that(76)uxt,x≥−ut,xL∞,t,x∈0,T×S.
Consider thatuL∞≤1/√2uH1≤1/√2u0H1, we conclude that uxt,x≥−1/√2u0H1. The above inequality and Theorem 10 imply T=∞, which shows the solution ut,x≥0 or ut,x≤0 exists globally in time.Remark 13.
It is observed that if the conditions of Theorem12 are satisfied and if m0∈H1S, then u does not blow up in finite time by considering Theorem 9. This is similar to the previous results on the line [33].Remark 14.
From Lemma11, we can observe that the presence of function hu is the main obstacle of our investigation of existence of the global solutions. In fact, if hu=0, it follows from Lemma 11 that the second condition of Theorem 12 would hold automatically as a consequence of its first condition. Therefore, the sign invariance of m is essential to prove the global existence of solutions.Remark 15.
Similarly for the Camassa-Holm equation and other analogous equations (see [4, 5, 7, 25, 33]), (1) admits the wave-breaking phenomenon, but differently from the Camassa-Holm equation, we cannot assure the global existence of solutions through the way we followed here [30, 33].



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