OSCILLATION-INDUCED BLOW-UP TO THE MODIFIED CAMASSA HOLM EQUATION WITH LINEAR DISPERSION

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1 OSCILLATION-INDUCED BLOW-UP TO THE MODIFIED CAMASSA HOLM EQUATION WITH LINEAR DISPERSION ROBIN MING CHEN, YUE LIU, CHANGZHENG QU, AND SHUANGHU ZHANG Abstract. In this paper, we provide a blow-up mechanism to the modified Camassa-Holm equation with varying linear dispersion. We first consider the case when linear dispersion is absent and derive a finite-time blow-up result with an initial data having a region of mild oscillation. A key feature of the analysis is the development of the Burgers-type inequalities with focusing property on characteristics, which can be deduced from tracing the ratio between solution and its gradient. Using the continuity and monotonicity of the solutions, we then extend this blow-up criterion to the case of negative linear dispersion, and determine that the finite time blow-up can still occur if the initial momentum density is bounded below by the magnitude of the linear dispersion and the initial datum has a local mild-oscillation region. Finally, we demonstrate that in the case of non-negative linear dispersion the formation of singularities can be induced by an initial datum with a sufficiently steep profile. In contrast to the Camassa-Holm equation with linear dispersion, the effect of linear dispersion of the modified Camassa-Holm equation on the blow-up phenomena is rather delicate. Keywords: Modified Camassa-Holm equation, oscillation-induced blow-up, integrable system, breaking wave, sign-changing momentum. AMS Subject Classification 000: 5B44, 5G5. Introduction Wave motion is ubiquitous in nature and is one of the broadest subjects. Most of wave phenomena might be described by mathematical models that are based on certain hyperbolic-type, nonlinear dispersive partial differential equations. A fundamental question in the study of those models is when and how a singularity can form, or more precisely, whether a wave breaks a wave profile remains bounded while its slope becomes unbounded in finite time in certain time. Breaking waves, both whitecaps and surf, are commonly observed in the ocean, which provide a source of turbulent energy to mix the upper layers of the ocean [5]. One common ingredient in all the current wave breaking techniques is to seek an appropriate quantity depending on the information of the solution and to obtain a differential inequality for it which can generate finite-time blow-up with well chosen data. However the way of finding such a quantity is very much case-dependent [, 7]. For quasilinear equations, in many cases the solution u remains well-defined at each point, but its gradient u may become infinite in finite time. Typical examples of this situation can be found in the case of scalar conservation laws and some non-local transport type equations, where blow-ups are due to focusing of characteristics.

2 R.M. CHEN, Y. LIU, C. QU, AND S. ZHANG In this paper, we would like to investigate the blow-up mechanism of a cubic quasilinear dispersive equation, namely the modified Camassa-Holm mch equation: { m t + u u xm + γu x x = 0, t > 0, x R,. u0, x = u 0 x, where m = xu = u u xx. represents momentum density of the system, and γ R characterizes the effect of the linear dispersion. The mch equation. can be derived by applying the method of tri-hamiltonian duality to the bi-hamiltonian representation of the focusing modified Korteweg-de Vries mkdv equation [, 5, ]. To see this, consider the focusing mkdv equation u t = u xxx + u u x. In the bi-hamiltonian form it can be written as δh u t = J δu = J δh δu, where the Hamiltonian operators J and J are expressed as with the Hamiltonians J = x, and J = x + x u x u x H = u dx, and H = R Now define two new Hamiltonian operators R 4 u4 u x dx. Ĵ = x x, and Ĵ = α x + α x + α x u x u x. Applying the recursion operator R = ĴĴ to the seed equation u t = u x we deduce a hierarchy of an integrable system, namely, u t = R n u x = ĴĴ n u x. Let u = xv and choose n =. Then the above system becomes u t = ĴĴ u x = Ĵv = α [ uv v x ] + α x v x + α v xxx. The mch equation. is then obtained from the above equation by letting α = γ, α = 0, α =. Therefore the mch equation is formally integrable possessing a bi-hamiltonian structure. A Lax representation of. was later constructed [, 4], making the mch equation amenable to the method of inverse scattering. Physically, the mch equation. models the unidirectional propagation of surface waves in shallow water over a flat bottom [], where ut, x represents the free surface elevation in dimensionless variables. The mch equation can be viewed as a cubic extension of the well-known Camassa- Holm CH equation m t + um x + mu x = 0, m = u u xx,.

3 BLOW-UP OF SOLUTIONS TO THE MCH EQUATION which was proposed as a model describing the uni-directional propagation of shallow water waves [5, 0, 6] and axially symmetric waves in hyperelastic rods [9, ]. The CH equation was originally constructed by using the recursion operator method [6], and can also be derived by applying tri-hamiltonian duality to the bi-hamiltonian structure of the KdV equation []. One of the remarkable distinctive properties that the CH equation has, in contrast to the KdV equation, is the wave-breaking phenomenon: the solution u remains bounded while its slope u x becomes infinite in finite time. It has been shown that the wave-breaking can be triggered by an initial sign-changing momentum density m 0 x = xu 0 = u 0 x u 0x. On the other hand, it can be prevented if m 0 does not change sign, c.f. [7, 8, 8, 0, 6]. In deriving such a blow-up, the global information of the data such as conservation laws, integrability, antisymmetry, etc. is needed due to the non-local feature of the equation. The geometric formulation, integrability, local well-posedness, blow-up criteria and singularity formation, existence of peaked solitons peakons, and the stability of single peakons and periodic peakons to the mch equation. were studied recently in [4, 7, 8, ]. It is shown that even if the initial momentum density m 0 x does not change sign, the solutions to the Cauchy problem. can still blow up in finite time, in contrast to the CH equation. It is noticed that a new type of blow-up criteria in the sense of the so called local-in-space is recently established in [,, 4], which illustrates that the blowup condition is merely imposed on a small neighborhood of a single point in space variable, and hence local perturbation of data around that point does not prevent the singularity formation. The main idea used there is to track the dynamics of two linear combinations u ± βu x for the CH equation, β = of u and u x along the characteristics and to make use of some convolution estimates to bound the non-local terms by the local quantities. The blow-up result then follows from the continuity and monotonicity argument. The goal of the present paper is two fold. First we want to understand how the local structure of the initial profile can affect the singularity formation. Similar to the case of the CH equation, where the blow-up is indicated from the loss of the lower bound on u x, the blow-up criterion of the mch equation asserts that blow-up occurs if and only if mu x becomes unbounded from below c.f. [7]. Applying the characteristics method we find that the evolution of mu x along the trajectories of the flow map involves competition between u and u x, along with some non-local convolutions. If any kind of convolution estimates as in the case of the CH equation is available, one may expect to produce a local-in-space blow-up. Unfortunately, the cubic nonlinearity makes it hard to control the non-local convolution terms, and hence the problem becomes more subtle and it is difficult to employ the approach in [,, 4] directly. However in the case when γ 0, the lower bound of the momentum density m is preserved under the flow, c.f..5 and 4.5, which can be used to bound the non-local terms in terms of the local quantities. The resulting estimates become an interplay between u and u x, which can in fact be measured by u x /u. Physically, this quantity is related to the local oscillation of the datum. We are able

4 4 R.M. CHEN, Y. LIU, C. QU, AND S. ZHANG to show that when the momentum density is bounded below by the weak dispersive index, blow-up can be induced by local mild oscillations, independent of the size of the oscillation region. Our main results on the oscillation-induced blow-up can be formulated in the following two theorems. Theorem. Oscillation-induced blow-up: γ = 0. Suppose γ = 0. Let m 0 H s R L R for s > / and m 0 x 0, x R. Assume there exists a point x 0 R such that m 0 x 0 > 0, and u 0,x x 0 u 0 x 0..4 Then the corresponding solution ut, x blows up in finite time with an estimate of the blow-up time T as T m 0 x 0 u 0,x x 0..5 Remark.. i By the blow-up criterion in Lemma. and a symmetric argument one can also investigate the formation of singularity with moment density data m 0 0. More specifically, assume m 0 x 0, x R. If there exists x 0 R such that m 0 x 0 < 0, and u 0,x x 0 u 0 x 0, then the solution blows up in finite time and the estimate on the blow-up time T is given in the same form as.5. ii Our blow-up result may in the mean time hint on some local information about the global solutions. Suppose that u is a given global solution to. with initial datum satisfying m 0 x > 0 for all x R. Then from the above theorem we know that at any time t it follows that u x t, x > ut, x, for all x R. Hence define an auxiliary function φt, x = e x ut, x and we can compute the x-derivative to get φ x t, x = e x u x + u t, x > 0. Thus for each fixed t, x φt, x is strictly increasing, leading to the following one-sided estimate of ut, x in terms of the information of u at x = 0: ut, x > e x ut, 0, for x > 0; ut, x < e x ut, 0, for x < 0. Moreover the sign condition mt, x > 0 implies u x t, x < ut, x, and hence a similar argument applied to the function ψt, x = e x ut, x yields ut, x < e x ut, 0, for x > 0; ut, x > e x ut, 0, for x < 0. Therefore ut, x can be bounded in terms of ut, 0 as e x ut, 0 < ut, x < e x ut, 0, for x > 0, e x ut, 0 < ut, x < e x ut, 0, for x < 0.

5 BLOW-UP OF SOLUTIONS TO THE MCH EQUATION 5 Blow-up issue in the case γ < 0 is more subtle. The sign-preservation property of m is no longer available. It is interesting to note that the quantity n = m + γ/ preserves its sign along the characteristics, which makes the previous analysis of the γ = 0 case applicable. We now state the blow-up result with the negative linear dispersion index in the following. Theorem. Oscillation-induced blow-up: γ < 0. Assume γ = β < 0. Let m 0 H s R L R for s > /, m 0 x β, x R, and H [u 0 ] β, where H [u] is defined in.8. Suppose that there exists a point x 0 R such that m 0 x 0 > β, u 0,x x 0 min u 0,x x 0 4 u 0x 0 + β. { 4β, 4 H [u] + β Then the corresponding solution ut, x blows up in finite time. }, and.6 Remark.4. i It is found that the above results are different from the ones in [,, 4] for the CH equation. In the case of the CH equation, finite-time blowup is induced by local fast oscillations in the sense that u 0,x x 0 < u 0 x 0, which implies that u 0,xx 0 u 0 x 0 >. In the case of the mch equation, however, we have to include the global information m 0 0 or n 0 0 when γ < 0 to obtain estimates on the convolution terms. Such a requirement already excludes any local fast oscillation, i. e. u 0,xx 0 u 0 x 0 < if m 0 0, for example. In other words, our result asserts that finite time blow-up to the mch equation can be triggered even by mild local oscillations. ii Our choice.6 of the initial data is not optimal in the sense of providing any sort of a threshold between blow-up and global wellposedness. A more general condition on u 0,x x 0 can be derived from the discussion in Section 4. and formulated as u 0,x x 0 min{ sβ, δk} where s and δ satisfy 4.0 and 4.4, and the relation between u 0,x x 0 and u 0 x 0 + β should satisfy 4.. The numbers we use in the theorem is simply for computational convenience. iii The main difference between the two cases γ < 0 and γ = 0, which is also the main difficulty, lies in the fact that γ 0 introduces some extra nonlocal terms in the dynamic equations of the key blow-up quantities which fail to be controlled in terms of the local terms. Bounding such terms gives rise to the additional conditions on the initial data in Theorem.. Our second aim in the present paper is to seek initial datum with a sign-changing momentum density m which can generate finite time blow-up. This is discussed in the case of non-negative dispersion γ 0. At this moment we do not have a clear picture of how to produce an oscillation-induced blow-up criterion when γ > 0. But with the help of an additional conservation law, the nonlocal terms can still be bounded by some globally conserved quantities, which hints for the blow-up data with sufficiently steep profiles. If we set r = m + γ/, the mch equation.

6 6 R.M. CHEN, Y. LIU, C. QU, AND S. ZHANG can be rewritten as the form of conservation laws [9] r t + u u xr = 0..7 x The three basic conserved quantities H 0 [u] = u dx, R H [u] = mu dx = u + u x dx, R R H [u] = u 4 + u u x 4 u4 x + γu dx, R.8 are well-known and play an important role in all analysis of the solutions. Suppose the linear dispersion parameter γ 0. If we denote wt, x = rt, x γ/ = m + γ/ γ/ 0, we may have the following new conservation law [9] wt, x dx = w 0 x dx,.9 R which is crucial in controlling the solution u and its slope u x. This observation allows us to improve the blow-up results in [7, 8] to include the case of a sign-changing initial momentum density and the effect of the linear dispersion with γ 0. Theorem.5 Blow-up for a sign-changing momentum density. Let γ 0, and suppose m 0 H s R, s > and w 0 L R. Assume further that there exists an x R such that m 0 x 0 and m 0 x u 0,x x m 0 x γa 0 + A 0,.0 where A 0 = + γ/ + γ/ w 0 L +. Then the solution ut, x blows up in finite time with an estimate of the blow-up time T as T u 4γA 0 + A 0,x x u 0,xx 4γA 0 + A m 0 x The rest of the paper is organized as follows. In Section, some preliminary estimates and results are recalled and presented. In Section the evolution equations of various quantities related to finite time blow-up analysis along the characteristics are established. Last section, Section 4, is devoted to the proofs to our main blow-up results, Theorem., Theorem., and Theorem.5. Notation. For convenience, in the following, given a Banach space X, we denote its norm by X. If there is no ambiguity, we omit the domain of function spaces. R

7 BLOW-UP OF SOLUTIONS TO THE MCH EQUATION 7. Preliminaries Let us recall some basic results concerning the formation of singularities in the mch equation.. Note that the mch equation. can be rewritten as a transport equation for the momentum density, that is, m t + u u xm x = m u x γu x.. The theory of transport equations implies that the blow-up is determined by the slope of the transport velocity u u x x = mu x = u x u u xx. In fact the local well-posedness and a blow-up criterion can be formulated as follows. The details of proof can be found in [7]. Lemma.. Let u 0 H s R with s > 5. Then there exists a time T > 0 such that the initial-value problem. has a unique strong solution u C[0, T ]; H s R C [0, T ]; H s R. Moreover, the map u 0 u is continuous from a neighborhood of the initial data u 0 in H s R into C[0, T ]; H s R C [0, T ]; H s R. In addition, if the corresponding solution u has maximum time T of existence with 0 < T <, then T 0 m u x t L dt =. Using the preceding criterion, one can derive the following precise blow-up condition [7]. Lemma.. Suppose that u 0 H s R with s > 5. Then the corresponding solution u to the initial value problem. blows up in finite time T > 0 if and only if lim t T inf x R { mt, x ux t, x } =. Certain conservative properties of the momentum density m will play a key role in establishing our new blow-up criteria. First, note that an application of the method of characteristics to the transport equation. for m requires analyzing the flow governed by the effective wave speed u u x, namely the solution qt, x to the parametrized family of ordinary differential equations dqt, x = u t, qt, x u dt xt, qt, x x R, t [0, T.. q0, x = x, x R. One can easily check that Proposition.. Suppose u 0 H s R with s > 5, and let T > 0 be the maximal existence time of the strong solution u to the corresponding initial value problem.. Then. has a unique solution q C [0, T R, R such that qt, is an increasing diffeomorphism of R with t q x t, x = exp ms, qs, x u x s, qs, x ds > 0.4 for all t, x [0, T R. 0

8 8 R.M. CHEN, Y. LIU, C. QU, AND S. ZHANG Furthermore, in the case when γ = 0, the potential m = u u xx satisfies mt, qt, xq x t, x = m 0 x, t, x [0, T R,.5 which implies that the zeros and the sign of m are preserved under the flow, whereas in the general case when γ 0 one has d dt [mt, qt, xq xt, x] = γu x q x..6 We are now in a position to present the following unusual conservation law which plays a crucial role in our proof of Theorem.5. Proposition.. Assume m 0 H s R L R with s > and γ 0. Suppose u is the corresponding solution to. with the initial data u 0. Then wt L = w 0 L = m 0x + γ/ γ/ dx, 0 t < T..7 Proof. Indeed, note that w t t, x = r t t, x. We can easily deduce from.7 that from which.7 holds true. w t + u u xw + γ/ x = 0, Denote px = e x, the fundamental solution of x on R, that is, x f = p f, and define the two convolution operators p +, p as Then we have the relation p + fx = e x p fx = ex x x e y fydy e y fydy..8 p = p + + p, p x = p p Dynamics along the characteristics It is known from most of the literatures that finite-time blow-up analysis is often carried out along the characteristics. The following lemmas give explicit information of how the blow-up quantities evolve along the characteristics. Lemma.. Let u 0 H s R, s. Then ut, x and u x t, x satisfy the following integro-differential equations: u t + u u xu x = γp x u u x + [ p+ u u x p u + u x ].. u tx + u u xu xx = γu p u + u uu x p + u + uu x u x p u + uu x + u x = γu p u + u uu x [ p+ u u x + p u + u x ]..

9 BLOW-UP OF SOLUTIONS TO THE MCH EQUATION 9 Proof. From., we have x u t + u u xu x which implies = u u xm x m u x γu x + u u xu x x u u xu x = u u xm x m u x γu x + u u xu x 6u x u xx m u xm x u u xu xxx = m u x γu x 6u x u xx m u xm x, u t + u u xu x = p m u x + γu x + 6u x u xx m + u xm x = p muu x + u xm x + γu x. Taking derivative to.4 with respect to x yields u tx + u u xu xx = mu x p x muu x + u xm x + γu x = mu x p x muu x + [ u xm p u xm ] + γ u p u = γ u p u p x muu x p u xm. In view of proof in Lemma. [8], we know..4.5 p + mu x = 6 u x + p + uu x + u x, p mu x = 6 u x + p uu x u x, p + muu x = 6 u p + u 4 uu x + p + uu x + u x, p muu x = 6 u + p u + 4 uu x p uu x u x. It then follows from.9 that p muu x + p x u xm = u x p + u + uu x u x + p u + uu x + u x = u x [ p+ u u x p u + u x ], and p x muu x + p u xm = uu x u + p + u + uu x u x + p u + uu x + u x = uu x u + [ p+ u u x + p u + u x ]..6.7 Combining.4 with.6, and.5 with.7, we complete the proofs of. and.. Next we focus on the dynamics of M := mu x along the characteristics.

10 0 R.M. CHEN, Y. LIU, C. QU, AND S. ZHANG Lemma.. Assume that the initial data m 0 H s R, s. Let T > 0 be the maximal existence time of the resulting solution mt, x to the initial value problem.. Then M := mu x satisfies M t + u u xm x = M γu x + γmu p u mp x muu x mp u xm = M γu x + γmu p u.8 + m u uu x m [ p + u u x + p u + u x ]. Proof. By simple computation, we know that M t + u u xm x = m u tx + u u xu xx + mt + u u xm x ux. Now, multiplying.5 and. with m and u x respectively, then combining them together with., we finish the proof of Finite-time blow-up data In this section we will investigate the blow-up conditions for the initial data to the mch equation. in various cases corresponding to different sign of γ. We emphasize two different blow-up mechanisms. The first one stems from the idea of local-in-space blow-up [,, 4] in the study of Camassa-Holm and hyperelastic rod equations, where the blow-up criteria assert that local perturbation of data around a single point does not prevent the singularity formation. Such type of blow-up can also be viewed as induced by local fast oscillations which is independent of the size of the oscillation region. The second mechanism involves the global information of the data, which relies on controlling the non-local terms in the transport formulation of the PDE by certain conservation laws. In view of Lemma., an approach toward the the blow-up of. is to track the evolution of M, which is given in.8. Apparently, the convolution terms in.8 contain non-local information, and do not seem to be easily controlled by any local quantity. However in the case when γ = 0, one has the sign-preservation property of the momentum mt, x c.f..5. When m does not change sign, the convolution terms have a determined sign, making it possible to obtain the oscillation-induced blow-up. In fact, the oscillation mechanism can be measured by u x /u. When γ = 0 and the sign of m remains fixed, fast oscillation is automatically excluded. But it is possible to preserve mild oscillations along the characteristics, which is enough to lead to a blow-up. For the case of general γ, one loses the sign preservation of mt, x. Therefore it becomes difficult to control the nonlocal terms in equations.,. and.8 in terms of the local terms directly. When γ < 0, however, we find a transformation that absorbs this term, making it fit in a similar structure as of the γ = 0 case, and hence an oscillation-induced blow-up follows. At this moment we don t have a clear picture of how to generate the oscillation-induced blow-up criterion when γ > 0. But with the help of the additional conservation law.7, the nonlocal terms can still be bounded by some globally conserved quantities, which makes it possible to obtain a finite time blow-up for data with sufficiently steep profiles.

11 BLOW-UP OF SOLUTIONS TO THE MCH EQUATION 4.. Oscillation-induced blow-up when γ = 0. We are now in a position to prove the oscillation-induced blow-up result for γ = 0. Proof of Theorem.. As is explained before, we will trace the dynamics along the characteristics emanating from x 0. Denote ût = ut, qt, x 0, û x t = u x t, qt, x 0, mt = mt, qt, x 0, Mt = mux t, qt, x 0, 4. and let denote the derivative along the characteristics qt, x 0. Then from.,., and.8 we have the following ODE system for û, û x, and M. û t = t + [ p+ u u x p u + u x ] t, qt, x 0, ûx û x t = û ûû x t [ p+ u u x + p u + u x ] t, qt, x 0, M t = M t + m û ûû x t mt [ p + u u x + p u + u x ] t, qt, x 0. Since we know that m 0 0, x R, in particular, m 0 x 0 > 0. So from.5, we know mt, x 0, x R and mt, qt, x 0 > 0. Therefore from the fact that ut, x = p mt, x = e x y my dy, u x t, x = p x mt, x. we have ut, x 0 and ût = ut, qt, x 0 > 0. Moreover R u u x = p + m 0, u + u x = p m 0. Hence we know u x x, t ux, t. Since ût > 0, we can track the dynamics of û x /û along the characteristics. ûx t = û û x û û x û + û x p û û û + u u x t, qt, x 0 û û x û p u + u x t, qt, x 0 < û û x û û x û. 4. So if initially û x 0 û0, we see that the right-hand side of the above estimate is non-positive. Hence û x /û decreases, and thus ûx û t < ûx û 0, which implies that ût + û x t < 0. It is then inferred that ût + û x t < 0. Meanwhile it is also found that ût û x t > 0. Hence it is concluded that û t û x t < 0.

12 R.M. CHEN, Y. LIU, C. QU, AND S. ZHANG Plugging this into the ODE for M we have M t M t + m û ûû x t = M t + mûû û x t < M t, which generates blow-up in finite time with an estimate of the blow-up time T as T M0 = m 0 x 0 u 0,x x Oscillation-induced blow-up when γ < 0. As is discussed at the beginning of this section, we look for a transformation that eliminates the γu x term in equation.. Consider the following change-of-unknown vt, x = ut, x + β 4. for some constant β to be determined. Plugging 4. into. and denoting n = v v xx, it follows n t + [ v β v xn ] x βv xn + β + γv x = 0. Thus choosing β = γ/ we obtain the equation for v as n t + [ v β v xn ] x βv xn = Note that we have the following relation between the two momenta m and n mt, x = nt, x β, and v β v x = u u x. This implies that the new equation for n has the same characteristics as defined in. for the equation.. Moreover along the characteristic qt, x we have which implies that d dt nq x = βv x nq x, nq x t, qt, x = n0, x exp t 0 βv x s, qs, x ds. 4.5 Therefore the zeros and the sign of n are preserved under the flow. Let N = nv x. Then from definition of n, v we know that N = M + βv x. From.8 and. and using the relation γ = β we deduce N t + u u xn x = M t + u u xm x + u xt + u u xu xx = N + 4βNu x + γnu p u + n u uu x n [ p + u u x + p u + u x ]. 4.6 This suggests us to look for the blow-up of Nt, x. We still want to consider the blow-up induced by oscillations. Thus from the previous argument in Section 4., we would like to check the dynamics of u x /u along the characteristics. However we no longer have the sign-preservation for u now. But instead, the sign of v can be

13 BLOW-UP OF SOLUTIONS TO THE MCH EQUATION preserved, provided that n does not change sign. Therefore we can compute the dynamics of u x /v = v x /v. Recall ût, û x t as in 4.. We further denote vt = vt, qt, x 0, v x t = v x t, qt, x 0, nt = nt, qt, x 0, Nt = Nt, qt, x0, and let the notation denote the derivative with respect to the time t along the characteristics qt, x 0. Note that the evolution of v x is the same as the one for û x : v x t = γût γp ut, qt, x 0 + û ûû x t [ p+ u u x + p u + u x ] t, qt, x 0 = β vt + β p vt, qt, x 0 + v β v β v x t 4.7 [ p+ u u x + p u + u x ] t, qt, x 0. Using 4. and the following identity vx t = v ûx û û v + βû x v, we can calculate vx t = [ γû + γβû + v + û4 û û v v x + 4 ûx γ [ v + v x p + v + v v x p v] û + β ûû x v + v x p + u u x t, qt, x 0 v v ] x p u + u x t, qt, x 0 = [ 4 vx v v β v v x + v v β 6β v + 6β [ v + v x p + v + v v x p v] v + v x p + u u x t, qt, x 0 v v x p u + u x t, qt, x 0 ]. 4.8 From the assumption that n 0 x 0, n 0 x 0 > 0 we know that nt, x 0 and nt > 0. This implies that which, in terms of u and u x, reads vt, x 0, v x t, x vt, x, vt > 0, 4.9 ut, x β, ut, x ± u x t, x β, ût > β.

14 4 R.M. CHEN, Y. LIU, C. QU, AND S. ZHANG Using these estimates together with the fact that γ < 0 we can deduce that [ p+ u u x + p u + u x ] β. v + v x p + u u x v v x p u + u x β v. 4.0 Putting together and plugging û = v β we can estimate the evolution of Nt, v x t and v x / v as [ N t N [ + γ nû p u + n v β + β ] ] v 5β v x, 4. v x t [ v β + β ] v β v x β v p v, 4. [ vx 4 vx v v β v x + v v β 6β v + β v ] t v v + β v [ v + v xp + v + v v x p v]. 4. Recall from. the evolution for n n t = m t = m + γ v x = n β n v x. 4.4 It is observed from the above equation that when n > β and v x < 0, n grows exponentially. Thus by the definition N = n v x, it is conceivable that 4. can lead to a Burgers type blow-up inequality, provided that v x is non-increasing. Therefore we would like to show that the right-hand side of 4. is nonpositive, that is v β v x [ v β + β ] β v p v, 4.5 from which we see that we need an upper bound for the non-local term p v. But it seems difficult to obtain an estimates in terms of local quantities. Instead, we make use of the following global information. p v = p v v L u L + β H [u] + β. Let K = H [u] + β. In a similar way we can estimate and hence In the case when p + v, p v v L K, v + v x p + v + v v x p v K v. 4.6 v sβ, v δk, for some s >, 0 < δ <, 4.7 a sufficient condition for 4.5 to hold is v x v β + β + 6β K 6β v. 4.8 v β

15 BLOW-UP OF SOLUTIONS TO THE MCH EQUATION 5 Controlling the right-hand side of the above leads to v β + β + 6β K 6β v v β β β δ v + + s s δ v [ ] + δ + v. s s ss δ Therefore a sufficient condition for 4.8 is [ ] v x + δ + v, 4.9 s s ss δ where a constraint on s and δ must be imposed + δ + s s ss δ <. 4.0 Next we use 4. to look for a sufficient condition to persist the local oscillation. Similarly as in the case of γ = 0, one wants the right-hand side of 4. to be negative. Plugging 4.6 into 4. and solving the inequality yield v v β 4 v x v v β +, 4. 4 where = v 4 + 6β v + β v 48β K v. 4. Further plugging in 4.7 we have [ ] 48 s + 6s δ β v. 4. Solvability of 4. requires that the discriminant 0, which holds when s + 6s δ Now we choose δ =. From condition 4.4 we may choose s = 4. This way we 4 have + δ + s s ss δ = , s + 6s δ = 9. Therefore v v β 4 4 v v v β Further plugging the values of s and δ into 4.9 yields another sufficient condition for v x t 0 as In fact the above condition implies that v x t < 0. v x 4 v. 4.6 We now illustrate our strategy of proving Theorem.. We first choose an initial datum which satisfies 4.7 and 4.6, from which we know that v x 0 0. Then v x decreases within a short time interval, which makes n increase. To continue the monotonicity further in time, we need to persist 4.7 and 4.6. But the property 4.7 can be inferred from the upper bound of v x, due to the sign-preservation of

16 6 R.M. CHEN, Y. LIU, C. QU, AND S. ZHANG n. To check 4.6 at later time, we will use the monotonicity of v x / v near /4. Now let us give the details of proof of Theorem.. Proof of Theorem.. From the assumptions of Theorem. we know that n 0 x 0, n 0 x 0 > β, v 0,x x 0 min { 4β, 4 } K, and v 0,x x 0 4 v 0x 0. So from the above argument we know that v x 0 < 0, 4.7 and hence v x decreases over a certain time interval [0, t 0 ] with t 0 > 0. So v x t < v x 0 = v 0,x x 0 min { 4β, 4 } K, t [0, t 0 ], 4.8 Together with 4.9 we know which satisfies 4.7. vt v x t > max {4β, 4 K }, 4.9 Claim 4.. The ratio estimate 4.6 holds as long as the solution exists. To see this, since 4.6 is satisfied at initial time, it may propagate over a short time interval by continuity. If the claim is not true, then there exist times t > t > 0 such that v x t 4 v t for 0 t t, v x t = 4 v t, v x t < 4 v t for t < t t. From 4.9 and the first estimate in the above we know that v x t is decreasing on [0, t ]. Hence v x t < 0. Another application of 4.9 to 4. implies that the inequality in 4.5 should be a strict inequality. Therefore from the last two estimates of the above we know by a continuity argument that there is a t > t such that condition 4. is satisfied for t [t, t ], which implies that v x / v is nonincreasing over [t, t ]. Together with the information of the sign of v and v x we know that v x / v is non-decreasing over [t, t ]. Hence v x / v /4, which is a contradiction, This completes the proof of the claim. In view of 4.9 and the above claim it follows that v x t always decreases, and that vt > 4β. This way from 4.4, the fact that v x t < 4β < 0, and the initial condition on n we know that n is increasing, more precisely, n t > 8β n0 β nt, nt > n0e 8β n0 βt. This is then deduced that Nt = nt v x t < 0.

17 BLOW-UP OF SOLUTIONS TO THE MCH EQUATION 7 Comparing 4. and 4., it is deduced from 4.8 that Therefore for t large enough, say, [ v β + β ] v 5β v x 4β v x. N t < n [ n β v x + β û p u ] [ n n β v x ] β H [u] N, t t := 8β n0 β log H [u] + β. 8 n0 Hence Nt blows up to in finite-time T with T t Nt. Finally notice that N = M + β v x, and from v x v = û + β H [u] + β, we know that in this case lim Mt =, t T which completes the proof of Theorem.. Remark 4.. As is pointed out in Remark.4, the choices for s and δ are not unique. All needed are 4.0 and 4.4, and the ratio condition 4.6 should be accordingly adjusted. For instance taking s = 6, δ = /, and v 0,x x 0 v 4 0x 0 in 4.7 can also generate finite time blow-up. 4.. Finite-time blow-up when γ > 0. In this case, we first introduce the following global estimates. Lemma 4.. Assume that the initial data m 0 H s R L R, s >. Let T > 0 be the maximal existence time of the resulting solution ut, x to the initial value problem.. Then we have where Furthermore, there holds ut, x A 0, u xt, x A 0, 4.0 A 0 = + γ/ + γ/ w 0 L +. p ut, x A 0. 4.

18 8 R.M. CHEN, Y. LIU, C. QU, AND S. ZHANG Proof. We first estimate ut, x from ut, x = p mt, x = e x y mt, y dy. 4. Taking absolute value on 4., we have ut, x e x y mt, y dy R = e x y mt, y dy + e x y mt, y dy m > m mt, y dy + e x y dy. m > m 4. Next we evaluate the first term on the right-hand side of 4.. Since wt L = m t, x + γ/ m γ/ t, x dx = m t, x + γ/ + γ/ dx = R m > m > m t, x m t, x + γ/ + γ/ dx + m t, x m t, x + γ/ + γ/ dx + γ/ + γ/ m > mt, x dx, R m m t, x m t, x + γ/ + γ/ dx we get the following estimate by noticing Proposition. mt, x dx + γ/ + γ/ w 0 L. 4.4 m > Substituting 4.4 back into 4. completes the proof of the first estimate of 4.0. By similar argument, we can get the above bound of u x t, x. An application of Young s inequality infers that p ut, x p L u L = u L A Proof of Theorem.5. Again we introduce the notation ût := ut, qt, x, û x t := u x t, qt, x, mt := mt, qt, x, Mt := Mt, qt, x = mt û x t, 4.6 where x is given in.0, and let denote the derivative with respect to t along the characteristics qt, x. Then equations. and. imply that m t = mt + γ û x t. 4.7

19 BLOW-UP OF SOLUTIONS TO THE MCH EQUATION 9 Moreover, combining. and. with Lemma 4., we find û x t γa 0 + [ ] A 0 + ût û x t ût γa 0 + { } A 0 + ût max, û x t γa 0 + ût 4 A 0, û x t γa 0 + A 0 + û x t ût γa 0 + ût 4 A 0. Integrating 4.8 from 0 to t produces γa A 0 t + û x 0 û x t 4.8 γa A 0 t + û x We will divide our argument into two cases, according to the sign of m 0 x. Case : m 0 x > 0. Then we know from 4.9 and the initial conditions that for 0 < t T +, where u x t, qt, x = û x t < T + = 4u 0,xx. 4γA 0 + A 0 Next we claim that under assumption.0, it holds mt = mt, qt, x > 0, for 0 < t T Indeed, from.6,.4 and 4.45 we know that along the characteristics Thus d dt mtq xt, x = γû x tq x t, x > 0, for 0 < t T +. mtq x t, x > m0 > 0, which in turn implies that mt > 0 on [0, T + ]. Moreover it follows from.4 that t q x t, x = exp msû x s ds <. Hence we have Thus on [0, T + ], mt mt 0 mt > m0 > 0 for 0 < t T +. is well-defined and = mt m = + γ mt Integrating 4.4 from 0 to t and using 4.9, we get [ mt t γa 0 + ] 4 A 0 t + û x 0 = γa 0 + A 0 t + u 0,x x t + û x t û x t m0 m 0 x. 4.4

20 0 R.M. CHEN, Y. LIU, C. QU, AND S. ZHANG From the assumption on the initial data we know u 0,x x < 0 and m 0 x > 0. So when 4γA 0 + A 0 u 0,x x m 0 x the right-hand side of 4.4 becomes zero at T = u 4γA 0 + A 0,x x + u 0,xx 4γA 0 + A 0 < T +. 0 m 0 x It is then deduced that there exists 0 < T T such that mt + as t T. On the other hand, 4.9 also implies that for 0 < t T [ Mt = mtû x t mt γa 0 + ] 4 A 0 t + û x 0 [ mt γa 0 + ] 4 A 0 T + u 0,x x 4.44 which thus implies that = mt u 0,x x u 0,xx 4γA 0 + A 0 m 0 x inf Mx, t Mt, as t T T. x R Therefore the solution ut, x blows up at a time 0 < T T. Case : m 0 x < 0. The argument follows along the same line as in the previous case. For the sake of completeness we give the details as follows. In view of 4.9 and the initial conditions, û x t satisfies for 0 < t T, where Hence u x t, qt, x = û x t > T = 4u 0,xx. 4γA 0 + A 0 mt < m0 < 0, for 0 < t < T, which implies that is well-defined on [0, T mt ], and we further compute to get = mt mt m = + γ û mt x t û x t. Integrating the above leads to mt γa 0 + A 0 When u 0,x x t + u 0,x x t + 4γA 0 + A 0 m 0 x, m 0 x.,

21 BLOW-UP OF SOLUTIONS TO THE MCH EQUATION the right-hand side becomes zero at T = u 4γA 0 + A 0,x x u 0,xx + 4γA 0 + A 0 < T. 0 m 0 x Hence there exists some T 0, T ] such that mt as t T. Therefore [ Mt = mtû x t mt û x 0 γa 0 + ] 4 A 0 t [ mt u 0,x x γa 0 + ] 4 A 0 T which in turn implies that = mt u 0,x x + u 0,xx + 4γA 0 + A 0 m 0 x inf Mx, t Mt, as t T T. x R Therefore in this case the solution ux, t also blows up at a time 0 < T T. This completes the proof of Theorem.5. Acknowledgments. The work of Chen is partially supported by the Central Research Development Fund, University of Pittsburgh. The work of Liu is partially supported by the NSF grant DMS and the NSF-China grant-79. The work of Qu is supported in part by the NSF-China grant The work of Zhang is partially supported by the NSF-China grant-07, Doctoral Foundation of Ministry of Education of China grant We are very grateful to the referees for various helpful suggestions and comments. References [] S. Alinhac, Blowup for nonlinear hyperbolic equations, Progress in Nonlinear Differential Equations and their Applications, 7, Birkhäuser Boston, Inc., Boston, MA, 995. [] L. Brandolese, Local-in-space criteria for blowup in shallow water and dispersive rod equations, Comm. Math. Phys., 04, DOI: 0.007/s [] L. Brandolese and M. F. Cortez, Blowup issues for a class of nonlinear dispersive wave equations, J. Differential Equations, 56 04, [4] L. Brandolese and M. F. Cortez, On permanent and breaking waves in hyperelastic rods and rings, J. Funct. Anal., 66 04, [5] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 7 99, [6] A. Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier Grenoble, , 6. [7] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 8 998, 9 4. [8] A. Constantin and J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 5 998, [9] A. Constantin and W. A. Strauss, Stability of a class of solitary waves in compressible elastic rods, Phys. Lett. A, , ,

22 R.M. CHEN, Y. LIU, C. QU, AND S. ZHANG [0] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa- Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 9 009, [] H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney- Rivlin rod, Acta Mech., 7 998, [] A. S. Fokas, On a class of physically important integrable equation, Physica D, , [] A. S. Fokas, The Korteweg-de Vries equation and beyond, Acta Appl. Math., 9 995, [4] Y. Fu, G. L. Gui, Y. Liu and C. Z. Qu, On the Cauchy problem for the integrable modified Camassa-Holm equation with cubic nonlinearity, J. Differential Equations, 55 0, [5] B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation, Physica D, , 9 4. [6] B. Fuchssteiner and A. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D, 4 98/98, [7] G. L. Gui, Y. Liu, P. Olver and C. Z. Qu, Wave-breaking and peakons for a modified Camassa-Holm equation, Comm. Math. Phy., 9 0, [8] Y. Liu, P. Olver, C. Qu and S. Zhang, On the blow-up of solutions to the integrable modified Camassa Holm equation, Anal. Appl. Singap., 04, [9] Y.Matsuno, Smooth and singular multisoliton solutions of a modified Camassa-Holm equation with cubic nonlinearity and linear dispersion, J. Phys. A, 47 04, 50. [0] H. P. Mckean, Breakdown of a shallow water equation, Asian J. Math., 998, [] P. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitarywave solutions having compact support, Phys. Rev. E, 5 996, [] Z. Qiao, B. Xia, and J.B. Li, A generalized Camassa-Holm equation and its peakon solutions, arxiv: [] C. Z. Qu, X. C. Liu, Y. Liu, Stability of peakons for an integrable modified Camassa-Holm equation with cubic nonlinearity, Comm. Math. Phys., 0, [4] J. Schiff, Zero curvature formulations of dual hierarchies, J. Math. Phys., 7 996, [5] G. B. Whitham, Linear and nonlinear waves, Pure and Applied Mathemaitcs, Wiley- Interscience, New York-London-Sydney, 974. [6] Z. Xin and P. Zhang, On the uniqueness and large time behavior of the weak solutions to a shallow water equation, Comm. Partial Differential Equations, 7 000, Robin Ming Chen Department of Mathematics, University of Pittsburgh, PA 560, USA address: mingchen@pitt.edu Yue Liu Department of Mathematics, Ningbo University, Ningbo 5, China; University of Texas at Arlington, Arlington, TX 7609, USA address: yliu@uta.edu Corresponding author Changzheng Qu Department of Mathematics, Ningbo University, Ningbo 5, P. R. China address: quchangzheng@nbu.edu.cn Shuanghu Zhang Department of Mathematics, Southwest University, Chongqing 40075, P. R. China address: zhangshh@swu.edu.cn