Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 103, pp. 1–14. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu CURVATURE BLOW-UP FOR THE PERIODIC CH-MCH-NOVIKOV EQUATION MIN ZHU, YING WANG, LEI CHEN Abstract. We study the CH-mCH-Novikov equation with cubic nonlinear- ity, which is derived by an asymptotic method from the classical shallow water theory. This model can be related to three different important shallow water equations: CH equation, mCH equation and Novikov equation. We show the curvature blow-up of the CH-mCH-Novikov equation by the method of char- acteristics and conserved quantities to the Riccati-type differential inequality. 1. Introduction We consider the periodic equation with cubic nonlinearity which is an asymp- totic model from the classical shallow water theory, called the CH-mCH-Novikov equation mt + k1(2uxm+ umx) + k2((u2 − u2x)m)x + k3(u2mx + 3uuxm) = 0, t > 0, x ∈ S, u(0, x) = u0(x), x ∈ S, (1.1) where m = u− uxx, and ki (i = 1, 2, 3) are constants. We know that there are two important dimensionless parameters in water-wave theory: amplitude parameter ε = a/h0 and shallowness parameter µ = h20/λ 2, where h0 is the mean depth of water, a and λ are the typical amplitude and wave- length of the waves, respectively. When we say the shallow-water (or long-wave), it means there is a presumption of small depth (compared with wavelength), i.e. µ� 1. Whereas there are at least two cases for the amplitude parameter ε = a/h0: Boussinesq scaling (weakly nonlinear regime): µ� 1, ε = O(µ); and the Camassa- Holm (CH) scaling (moderately nonlinear regime): µ� 1, ε = O( √ µ). The following equation is derived for the scaled surface elevation by using µ� 1 and ε = O(µ2/5) [3]: mt + ux − µ 4 uxxx + ε 2 (2uxm+ umx) + c1ε 2 4 ((u2 − βµu2x)m)x − c2ε 2 4 (u2mx + 3uuxm) = 0 +O(ε5, µ2), 2010 Mathematics Subject Classification. 35B44, 35G25. Key words and phrases. Camassa-Holm equation; modified Camassa-Holm equation; asymptotic method; Novikov equation; curvature blow-up. ©2021. This work is licensed under a CC BY 4.0 license. Submitted June 19, 2021. Published December 27, 2021. 1 2 M. ZHU, Y. WANG, L. CHEN EJDE-2021/103 where m = u− βµuxx. By scaling u(t, x)→ 1 2εu( √ βµt, √ βµx), we obtain mt + ux − 3 5 uxxx + 2uxm+ umx + c2((u2 − u2x)m)x + c3(u2mx + 3uuxm) = 0. Equation (1.1) is the general case of the above equation, related to the three integrable systems: Camassa-Holm equation, modified Camassa-Holm equation and Novikov equation. When k1 = 1, k2 = 0 and k3 = 0, (1.1) reduces to mt + umx + 2mux = 0, m = u− uxx. (1.2) This model (1.2) is derived by using the CH scaling µ � 1, ε = O( √ µ), which is proposed as a model to describe the uni-directional propagation of shallow water waves over a flat bottom [2, 9]. It also models the propagation of axially symmetric waves in hyperelastic rods [7, 14]. The CH equation is completely integrable for a large class of initial data, for which it can be solved by the inverse scattering method [4]. In contrast to the KdV equation, the CH equation has three remarkable distinctive properties [23, 24]. First, although CH is completely integrable, it can describe wave breaking phenomena. The second is the existence of peakons. Indeed, the CH equation has the single peakon [2] and the multi-peakon solutions [12]. It is significant that the peakons are orbitally stable: the shape is stable under small perturbations [6, 15]. These peakons capture a feature of the waves of greatest height for the free-boundary incompressible Euler equations [20]. The last one is the variety of interesting geometric formulations of the CH equation [5, 8, 14, 16]. When k1 = 0, k2 = 1 and k3 = 0, (1.1) reduces to mt + ((u2 − u2x)m)x = 0. (1.3) The mCH equation (1.3) is derived by applying the method of tri-Hamiltonian dual- ity to the bi-Hamiltonian representation of the modified Korteweg-deVries (mKdV) equation [8, 19]. The equation is formally integrable and can be rewritten as the bi-Hamiltonian form and the Lax pair [19]. Moreover, the mCH equation exhibits new features, including wave breaking and blow up criteria that do not appear in the original CH equation [10]. On the other hand, since the mCH equation also arises from an intrinsic (arc-length preserving) invariant planar curve flow in Eu- clidean geometry [10], it can be regarded as a Euclidean-invariant counterpart to the KdV equation from the viewpoint of curve flows in Klein geometries [5, 17]. When k1 = 0, k2 = 0 and k3 = 1, (1.1) becomes to the Novikov equation [21, 22]: mt + u2mx + 3uuxm = 0. It is known that the Novikov equation is integrable with Lax pair [18]. A matrix Lax pair representation to the Novikov equation was provided by Hone and Wang [13]. With that representation it can be shown that the Novikov equation is related to a negative flow in the Sawada-Kotera hierarchy. It is also noticed that the Novikov equation admits a bi-Hamiltonian structure [13]. Hone, Lundmark and Szmigielski [11] obtained multi-peakons of the Novikov equation explicitly by using the inverse scattering approach. The motivation of this study comes from the curvature blow-up for cubic non- linear models. We know that the curvature blow-up phenomena is be found in the mCH equation [10] and the generalized modified Camassa-Holm(gmCH) equation [1]. This leads to a natural question of understanding how the interaction between these cubic nonlinearities would affect the singularity formation mechanism. In this EJDE-2021/103 CURVATURE BLOW-UP OF CH-MCH-NOVIKOV EQUATION 3 paper, the key ingredients are that we will choose initial data such that m0 does not change sign. This eliminates fast local oscillation of solutions, i.e. u± ux ≥ 0. The remainder of the paper is organized as follows. In Section 2, we present some preliminary results. In Section 3, we are devoted to the precise blow-up scenario about the CH-mCH-Novikov equation. In Section 4, the curvature blow-up data are illustrated. 2. Preliminaries To discuss the wave breaking phenomenon of the periodic CH-mCH-Novikov equation (1.1), we rewrite it as ut = −k1G ∗ (2uxm+ umx)− k2G ∗ ((u2 − u2x)m)x − k3G ∗ (u2mx + 3uuxm), t > 0, x ∈ S, u(0, x) = u0(x), x ∈ S, (2.1) where G(x) = cosh(x−[x]− 1 2 ) 2 sinh(1/2) , [x] represents the largest integer part of x, and G(x) is the fundamental solution of (1 − ∂2x)−1 on the unit circle S = R/Z, that is for any x ∈ S. Let G(x) = Λ1(x)+Λ2(x), where Λ1(x) = ex−[x]− 1 2 4 sinh( 1 2 ) and Λ2(x) = e−x+[x]+ 1 2 4 sinh( 1 2 ) . Then Gx(x) = Λ2(x)− Λ1(x). Lemma 2.1. Assume that u0 ∈ Hs(S)∩L1(S) with s > 5/2. Suppose that u is the corresponding solution to (1.1) with the initial data u0. Then H0[u0] = ∫ S (u2 + u2x) dx = ∫ S (u20 + u20,x) dx. (2.2) Proof. We write (1.1) as ut − utxx + k1(2ux(u− uxx) + u(ux − uxxx)) + k2((u2 − u2x)(u− uxx))x + k3(u2(ux − uxxx) + 3uux(u− uxx)) = 0. (2.3) Multiplying (2.3) by u and integrating by parts, we have 1 2 d dt ∫ S (u2(t, x) + u2x(t, x)) dx+ k1 ∫ S (u3x − u3x) dx − k2 ∫ S (u3ux − u2uxuxx − uu3x + u3xuxxx) dx + k3 ∫ S (u3ux − u3uxxx + 3u3ux − 3u2uxuxx) dx = 0. (2.4) Then ∫ S (u2 + u2x) dx = ∫ S (u20 + u20,x) dx. Then the proof of the lemma is complete. � The following inequality is often used for the wave-breaking phenomena of the periodic CH-mCH-Novikov equation. Lemma 2.2. [24] For every f ∈ H1(S), α ∈ R we have max x∈[0,1] f2(x) ≤ µ ∫ S (f2 + α2f2x) dx, 4 M. ZHU, Y. WANG, L. CHEN EJDE-2021/103 where µ = cosh( 1 2α ) 2α sinh( 1 2α ) . Moreover, µ is the minimum value. So in this sense µ is the optimal constant which is obtained by the associated Green function G(x) = cosh( xα − [x] α − 1 2α ) 2α sinh( 1 2α ) . When α = 1, the constant µ = e+1 2(e−1) is sharp. Lemma 2.3. [25] Let − e+1 e−1 ≤ γ ≤ e+1 e−1 , α ∈ R. Then if u ∈ H1(S) such that u(t, 1) = u(t, 0), we obtain (G± γGx) ∗ (u2 + 1 2 u2x − αu) ≥ { 1 2 (u− α 2 )2 − α2 4 , |γ| ≤ 1, 1 4 ((e+ 1)− |γ|(e− 1))(u− α 2 )2 − α2 4 , 1 ≤ |γ| ≤ e+1 e−1 , (2.5) and Λ1,2 ∗ (2u2 + u2x) ≥ 1 2 u2. 3. Precise blow-up scenario The local well-posedness theorem about the periodic CH-mCH-Novikov equation can be obtained from the standard argument of [3] with a slight modification. Theorem 3.1. Let u0 ∈ Hs(S), s > 5/2. Then there exists a time T > 0 such that the periodic problem (1.1) has a unique strong solution u ∈ C([0, T ];Hs(S)) ∩ C1([0, T ];Hs−1(S)). Now we define the following characteristics associated to (1.1) as qt(t, x) = [k1u+ k2(u2 − u2x) + k3u 2](t, q(t, x)), x ∈ S, t ∈ [0, T ∗), q(0, x) = x, x ∈ S. (3.1) Then we can easily satisfy the following proposition. Proposition 3.2. Suppose that u0 ∈ Hs(S) with s > 5/2 and T > 0 be the maximal existence time of the strong solution u to the initial value problem (3.1). Then (3.1) has a unique solution q ∈ C1([0, T )×S) such that q(t, ·) is an increasing diffeomorphism of S with qx(t, x) = exp( ∫ t 0 (k1ux + 2k2mux + 2k3uux)(s, q(s, x))ds), (3.2) for (t, x) ∈ [0, T )× S. Moreover, for all (t, x) ∈ [0, T )× S there holds m(t, q(t, x)) = m0(x) exp(− ∫ t 0 (2k1ux + 2k2mux + 3k3uux)(s, q(s, x)) dx), (3.3) where m0(x) = m(0, x). Proof. From (1.1) it suffices to derive mt + (k1u+ k2(u2 − u2x) + k3u 2)mx = −(2k1ux + 2k2uxm+ 3k3uux)m. Then, using the characteristics (3.1), we obtain (3.3). � EJDE-2021/103 CURVATURE BLOW-UP OF CH-MCH-NOVIKOV EQUATION 5 Remark 3.3. Suppose u0 ∈ Hs(S) with s > 5/2. Let T > 0 be the maximal existence time of the strong solution u to the corresponding initial value problem (1.1). If m0(x) > 0 for all x ∈ S, then m(t, x) > 0 for all (t, x) ∈ [0, T ) × S. Moreover, we have u± ux ≥ 0. Similar to the other CH-type equation, (1.1) can be reformulated into a nonlocal transport form (2.1). We can get the following criterion lemma 3.4. The proof follows a similar idea as in [3], and hence we omit it. Lemma 3.4. Let u0 ∈ Hs(S), s > 5/2 and u be the solution of (1.1). Assume that T ∗ > 0 is the maximum time of existence. Then T ∗ <∞⇒ ∫ T∗ 0 ‖k1ux(τ) + k2mux(τ) + 2k3uux(τ)‖L∞dτ =∞. (3.4) Remark 3.5. The blow-up criterion (3.4) implies that the lifespan T ∗ does not depend on the regularity index s of the initial data u0. Moreover, we prove the following accurate wave-breaking criteria. Lemma 3.6. Suppose that u0 ∈ Hs(S), s > 5/2. The corresponding solution u to the periodic problem (1.1) blows up in finite time T ∗ > 0 if and only if lim t→T∗ inf x∈S {k1ux(t, x) + k2m(t, x)ux(t, x) + 2k3u(t, x)ux(t, x)} = −∞. (3.5) Proof. In view of Remark 3.5, it suffices to consider the case s = 3. Suppose that if k1ux(t, x) + k2m(t, x)ux(t, x) + 2k3u(t, x)ux(t, x) is bounded from below on [0, T ∗)× S, and there exists a constant M > 0 such that k1ux(t, x) + k2m(t, x)ux(t, x) + 2k3u(t, x)ux(t, x) ≥ −M, [0, T ∗)× S. (3.6) Multiplying (1.1) by m and integrating over S, and then integration by parts, we have 1 2 d dt ∫ S m2 dx+ 3k1 2 ∫ S m2ux dx+ ∫ S (k2uxm+ 2k3uux)m2 dx = 0. (3.7) The initial condition implies that m0 ∈ Hs−2 ⊂ Lq for any 2 ≤ q ≤ ∞. Similarly we have 1 2 d dt ∫ S m2 x dx+ k1 ∫ S (2uxm+ umx)xmx dx+ k2 ∫ S ((u2 − u2x)m)xxmx dx + k3 ∫ S (u2mx + 3uuxm)xmx dx = 0. (3.8) Integrating by parts yields k1 ∫ S (2uxm+ umx)xmx dx = −k1 ∫ S uxm 2 dx+ 5k1 2 ∫ S uxm 2 x dx, (3.9) k2 ∫ S ((u2 − u2x)m)xxmx dx = ∫ S (5k2uxm)m2 x dx− ∫ S ( 2 3 k2uxm)m2 dx, (3.10) k3 ∫ S (u2mx + 3uuxm)xmx dx = ∫ S (4k3uux)m2 x dx− ∫ S (6k3uux)m2 dx+ ∫ S 8k3umxm 2 dx. (3.11) 6 M. ZHU, Y. WANG, L. CHEN EJDE-2021/103 Plugging (3.9)-(3.11) into (3.8), we have 1 2 d dt ∫ S m2 x dx− k1 ∫ S uxm 2 dx+ 5k1 2 ∫ S uxm 2 x dx + ∫ S (5k2uxm+ 4k3uux)m2 x dx− ∫ S ( 2 3 k2uxm+ 6k3uux)m2 dx + ∫ S 8k3umxm 2 dx = 0. (3.12) So from this, (3.7), and (3.12), we have 1 2 d dt ∫ S (m2 +m2 x) dx = − ∫ S (k1ux + k2uxm+ 2k3uux)m2 dx− ∫ S ( 5 2 k1ux + 5k2uxm+ 4k3uux)m2 x dx − ∫ S ( 2 3 k2uxm+ 6k3uux)m2 dx+ ∫ S 8k3umxm 2 dx = − ∫ S (k1ux + k2uxm+ 2k3uux)m2 dx− ∫ S (5k1ux + 5k2uxm+ 10k3uux)m2 x dx + 5 2 k1 ∫ S uxm 2 x dx+ 6k3 ∫ S uuxm 2 x dx− ∫ S ( 2 3 k2uxm+ 6k3uux)m2 dx + ∫ S 8k3umxm 2 dx. Moreover, 1 2 d dt ∫ S (m2 +m2 x) dx ≤ 5 ∫ S M(m2 +m2 x) dx+ 5 2 k1 ∫ S uxm 2 x dx+ 6k3 ∫ S uuxm 2 x dx − ∫ S ( 2 3 k2uxm+ 6k3uux)m2 dx+ ∫ S 8k3umxm 2 dx ≤ 5 ∫ S M(m2 +m2 x) dx+ 5 2 |k1|‖ux‖L∞‖m‖2H1 + 6|k3|‖u‖L∞‖ux‖L∞‖m‖2H1 + 2|k2|+ 8|k3| 3 ‖u‖2H1‖m‖2H1 . Note that from Lemmas 2.1, 2.2 and 2.3, we have 1 2 d dt ‖m‖2H1 ≤ (5M + 5 2 |k1| √ µH0 + 6|k3|µH0 + 2|k2|+ 8|k3| 3 H0)‖m‖2H1 Solving the inequality, it follows that ‖m(t)‖2H1 ≤ e2(5M+ 5 2 |k1| √ µH0+6|k3|µH0+ 2|k2|+8|k3| 3 H0)t‖m0‖2H1 , for t ∈ [0, T ∗). Then Theorem 3.1 ensures that the solution does not blow-up in finite time. On the other hand, if lim t→T∗ {inf x∈S (k1ux(t, x) + k2m(t, x)ux(t, x) + 2k3u(t, x)ux(t, x))} = −∞, then either ux or m blows up in finite time. � EJDE-2021/103 CURVATURE BLOW-UP OF CH-MCH-NOVIKOV EQUATION 7 Now, we present the dynamics of a few important quantities along the charac- teristics q(t, x0). Where ′ denotes the derivative ∂t + (k1u+ k2(u2 − u2x) + k3u 2)∂x along the characteristics. Lemma 3.7. Let u0 ∈ Hs(S) with s > 5/2 and û′(t) = u′(t, q(t, x0)), ûx ′ (t) = u′x(t, q(t, x0)), m̂′(t) = m′(t, q(t, x0)), and M̂ ′(t) = (mux)(t, q(t, x0)). Then û′(t), ûx ′ (t), m̂′(t), M̂ ′(t) satisfy the following integro-differential equations û′(t) = −2 3 k2ûx 3 + ( k2 3 + k3 2 )[Λ1 ∗ (u− ux)3 − Λ2 ∗ (u+ ux)3] − k1[Λ2 ∗ (u2 + 1 2 u2x)− Λ1 ∗ (u2 + 1 2 u2x)], ûx ′ (t) = k1(û2 − 1 2 ûx 2 ) + k2( 1 3 û3 − ûûx2) + k3û 2 (û2 − ûx2) − ( k2 3 + k3 2 )[Λ1 ∗ (u− ux)3 + Λ2 ∗ (u+ ux)3] − k1[Λ1 ∗ (u2 + 1 2 u2x) + Λ2 ∗ (u2 + 1 2 u2x)], m̂′(t) = −(2k1ûx + 2k2ûxm̂+ 3k3ûûx)m̂, M̂ ′(t) = −2k2M̂ 2 + k1m̂(û2 − 5 2 ûx 2 ) + m̂û 6 [(2k2 + 3k3)û2 − (6k2 + 21k3)ûx 2 ]− ( k2 3 + k3 2 )m̂[Λ1 ∗ (u− ux)3 + Λ2 ∗ (u+ ux)3] − k1m̂[Λ1 ∗ (u2 + 1 2 u2x) + Λ2 ∗ (u2 + 1 2 u2x)]. Proof. In view of (1.1), we can obtain m̂t + (k1û+ k2(û2 − ûx2) + k3û 2)m̂x = −(2k1ûx + 2k2ûxm̂+ 3k3ûûx)m̂, (3.13) which is the equation about m̂′(t). By (3.13), we obtain ût = −k1G ∗ (2uxm+ umx)− k2G ∗ [(u2− u2x)m]x− k3G ∗ (u2m+ 3uuxm). (3.14) By a direct calculation, we have G ∗ (2uxm+ umx) = uux + Λ1 ∗ (u2 + 1 2 u2x)− Λ2 ∗ (u2 + 1 2 u2x), G ∗ [(u2 − u2x)m]x = (u2 − u2x)ux + 2 3 u3x − 1 3 [Λ1 ∗ (u− ux)3 − Λ2 ∗ (u+ ux)3], G ∗ (u2mx + 3uuxm) = u2ux − 1 2 [Λ1 ∗ (u− ux)3 − Λ2 ∗ (u+ ux)3]. Therefore, plugging the above three equations into (3.14) leads to û′(t). Differentiating (3.14) with respect to x, we obtain ûxt = −k1G∗(2uxm+umx)x−k2G∗[(u2−u2x)m]xx−k3G∗(u2m+3uuxm)x. (3.15) By the same method and calculating the following three items G ∗ (2uxm+ umx)x = uuxx + u2 − 1 2 u2x + [Λ1 ∗ (u2 + 1 2 u2x) + Λ2 ∗ (u2 + 1 2 u2x)], (3.16) G ∗ [(u2 − u2x)m]xx = (u2 − u2x)uxx + ( 1 3 u3 − uu2x)− 1 3 [Λ1 ∗ (u− ux)3 + Λ2 ∗ (u+ ux)3], (3.17) 8 M. ZHU, Y. WANG, L. CHEN EJDE-2021/103 G ∗ (u2mx + 3uuxm)x = u2uxx + u 2 (u2 − u2x)− 1 2 [Λ1 ∗ (u− ux)3 + Λ2 ∗ (u+ ux)3], (3.18) Note that ûx ′ (t) can be obtained from (3.15)-(3.18). Moreover, M̂ ′(t) = (m̂ûx)′(t) = m̂′(t)ûx(t) + m̂(t)ûx ′ (t) = −2k2M̂ 2 + k1m̂(û2 − 5 2 ûx 2 ) + m̂û 6 [(2k2 + 3k3)û2 − (6k2 + 21k3)ûx 2 ] − ( k2 3 + k3 2 )m̂[Λ1 ∗ (u− ux)3 + Λ2 ∗ (u+ ux)3] − k1m̂[Λ1 ∗ (u2 + 1 2 u2x) + Λ2 ∗ (u2 + 1 2 u2x)]. (3.19) Thus, the proof is complete. � 4. Curvature blow-up From the blow-up criterion Lemma 3.6, we know that the conservation lawH0[u0] indicates two possible scenarios for the formation of singularity, namely, the wave- breaking (ux → ∞) or curvature blow-up (uxx → ∞). Now we prove that the wave-breaking phenomena of the CH-mCH-Novikov equation (1.1) is the curvature blow-up (uxx →∞). Theorem 4.1. Suppose that k1 < 0, k2 < 0, and (1) k21 ≤ − 16 21 k2, k21 + 2 3 k2 ≤ k3 ≤ − 2 21 k2, or (2) k21 > − 16 21 k2, max{k21 + 2 3 k2,− 2 5 k2} < k3 < k21 − 2 5 k2, where m0 ∈ Hs(S) for s > 1/2 and m0 > 0. Assume that there exists some point x0 ∈ S such that m0(x0) > 0 and u0,x(x0) ≥ (2k2 + 3k3 − 3k21 4k2 )1/2 u0(x0). (4.1) Then the solution u(t, x) blows up in finite time T ∗ ≤ − 1 2k2m0(x0)u0,x(x0) . Proof. From (3.19), we have the equation M̂ ′(t) = −2k2M̂ 2 + k1m̂(û2 − 5 2 ûx 2 ) + m̂û 6 [(2k2 + 3k3)û2 − (6k2 + 21k3)ûx 2 ] − ( k2 3 + k3 2 )m̂[Λ1 ∗ (u− ux)3 + Λ2 ∗ (u+ ux)3] − k1m̂[Λ1 ∗ (u2 + 1 2 u2x) + Λ2 ∗ (u2 + 1 2 u2x)]. (4.2) EJDE-2021/103 CURVATURE BLOW-UP OF CH-MCH-NOVIKOV EQUATION 9 According to (3.3), we know that m̂, û > 0 when m0(x0) > 0. To obtain a Riccati- type inequality from (4.2), we assume that k1 < 0, k2 < 0, k2 3 + k3 2 < 0 ⇔ k3 < − 2 3 k2. (4.3) Therefore, M̂ ′(t) ≥ −2k2M̂ 2 + k1m̂(û2 − 5 2 ûx 2 ) + m̂û 6 [(2k2 + 3k3)û2 − (6k2 + 21k3)ûx 2 ]− 1 2 k1m̂û 2 = −2k2M̂ 2 + k1 2 m̂(û2 − 5ûx 2 ) + m̂û 6 [(2k2 + 3k3)û2 − (6k2 + 21k3)ûx 2 ]. (4.4) Suppose that 1− 5 ûx 2 û2 ≤ 0⇒ ûx 2 û2 ≥ 1 5 , (2k2 + 3k3)− (6k2 + 21k3) ûx 2 û2 ≥ 0⇒ ûx 2 û2 ≥ 2k2 + 3k3 6k2 + 21k3 , (4.5) where 6k2 + 21k3 ≤ 0⇔ k3 ≤ − 2 7k2. Comparing with the values of 1 5 and 2k2+3k3 6k2+21k3 , we discuss it in two cases. (1) When 1 5 ≤ 2k2 + 3k3 6k2 + 21k3 ⇔ 2 3 k2 ≤ k3 ≤ − 2 7 k2, (4.6) we have ûx 2 û2 ≥ 2k2 + 3k3 6k2 + 21k3 . (4.7) In particular, a finite-time blow-up of M̂ is realized if the ration |ux u | stays reason- able big along the characteristics. If k1 < 0, using Lemma 2.3 and ux → +∞ we have ( ûx û )′ = 1 û2 [k1û(û2 − 1 2 ûx 2 )− k1(û+ ûx)Λ1 ∗ (u2 + 1 2 u2x) − k1(û− ûx)Λ2 ∗ (u2 + 1 2 u2x)] + û2 − ûx2 û2 [( k2 3 + k3 2 )û2 − 2k2 3 ûx 2 ] − 2k2 + 3k3 6û2 [(û+ ûx)Λ1 ∗ (u− ux)3 + (û− ûx)Λ2 ∗ (u+ ux)3] = 1 û2 [k1û(û2 − 1 2 ûx 2 )− k1ûG ∗ (u2 + 1 2 u2x) + k1ûxGx ∗ (u2 + 1 2 u2x)] + û2 − ûx2 û2 [( k2 3 + k3 2 )û2 − 2k2 3 ûx 2 ] − 2k2 + 3k3 6û2 [(û+ ûx)Λ1 ∗ (u− ux)3 + (û− ûx)Λ2 ∗ (u+ ux)3]. 10 M. ZHU, Y. WANG, L. CHEN EJDE-2021/103 Moreover, ( ûx û )′ ≥ 1 û2 [k1û(û2 − 1 2 ûx 2 )− k1|ûx|(G−Gx) ∗ (u2 + 1 2 u2x)] + û2 − ûx2 û2 [( k2 3 + k3 2 )û2 − 2k2 3 ûx 2 ] ≥ û2 − ûx2 û2 [k1û+ ( k2 3 + k3 2 )û2 − 2k2 3 ûx 2 ] = û2 − ûx2 û2 [−1 2 + (−k 2 1 2 + k2 3 + k3 2 )û2 − 2k2 3 ûx 2 ], (4.8) where − 1 2 + (−k 2 1 2 + k2 3 + k3 2 )û2 − 2k2 3 ûx 2 ≥ 0⇔ ûx 2 û2 ≥ 2k2 + 3k3 − 3k21 4k2 . (4.9) We know that |ux| ≤ u from Remark 3.3. Then 0 ≤ 2k2 + 3k3 − 3k21 4k2 ≤ 1 ⇔ 2 3 k2 + k3 ≤ k21 ≤ k3 − 2 3 k2. (4.10) From (4.8) and (4.9), we have chosen the initial data so that ( ûx û ) (0) ≥ √ 2k2 + 3k3 − 3k21 4k2 . (4.11) Thus, ûx û increases initially. Moreover, ( ûx û ) (t) ≥ ( ûx û )(0) ≥ √ 2k2 + 3k3 − 3k21 4k2 . (4.12) Then we have ûx 2 û2 ≥ 2k2 + 3k3 − 3k21 4k2 ≥ 2k2 + 3k3 4k2 ≥ 2k2 + 3k3 6k2 + 21k3 , (4.13) where 2k2 + 21k3 ≤ 0 ⇔ k3 ≤ − 2 21 k2. (4.14) From (4.3), (4.6), (4.10), (4.14), we have k21 ≤ − 16 21k2, k 2 1 + 2 3k2 ≤ k3 ≤ − 2 21k2. Plugging this into (4.4) it yields that M̂ ′(t) ≥ −2k2M̂ 2, and that M̂(t) blows up in finite time with an estimate of the blow-up time T ∗ as T ∗ ≤ − 1 2k2M̂(0) = − 1 2k2m0(x0)u0,x(x0) . (2) When 1 5 > 2k2 + 3k3 6k2 + 21k3 ⇔ k3 > − 2 7 k2 or k3 < − 2 3 k2. (4.15) Then from (4.5), we have ûx 2 û2 > 1 5 . (4.16) From (4.9), we have û2x û2 ≥ 2k2 + 3k3 − 3k21 4k2 > 1 5 ⇔ k21 > k3 + 2 5 k2, k3 > − 2 5 k2, (4.17) EJDE-2021/103 CURVATURE BLOW-UP OF CH-MCH-NOVIKOV EQUATION 11 where we know that |ux| ≤ u from Remark 3.3, so 0 ≤ 2k2 + 3k3 − 3k21 4k2 ≤ 1 ⇔ 2 3 k2 + k3 ≤ k21 ≤ k3 − 2 3 k2. So, we have k3 + 2 5 k2 < k21 ≤ k3 − 2 3 k2, k3 > 2 3 k2. (4.18) From (4.8) and (4.9), we have chosen the initial data so that ( ûx û ) (0) ≥ √ 2k2 + 3k3 − 3k21 4k2 . (4.19) Thus, ûx û increases initially. Moreover, ( ûx û ) (t) ≥ ( ûx û )(0) ≥ √ 2k2 + 3k3 − 3k21 4k2 . (4.20) From (4.3), (4.17), (4.18), we obtain max{k21 + 2 3k2,− 2 5k2} < k3 < k21 − 2 5k2. Plugging this into (4.4) it yields that M̂ ′(t) ≥ −2k2M̂ 2, and that M̂(t) blows up in finite time with an estimate of the blow-up time T ∗ as T ∗ ≤ − 1 2k2M̂(0) = − 1 2k2m0(x0)u0,x(x0) . � Remark 4.2. (1) From Lemma 3.6, we obtain that the true blow-up quantity is k2mux. In Theorem 4.1, when k2 < 0, we seek data that lead to mux → +∞. So that we consider the case when k2 > 0, it leads to mux → −∞ (2) The blow-up time T ∗ is only related to the parameter k2. We know that the mCH equation plays a dominant role in blow-up phenomena when the CH equation, the mCH equation and the Novikov equation act simultaneously. Using a similar argument as above, we prove the following corollary when k2 > 0. Corollary 4.3. Suppose that k1 > 0, k2 > 0, − 2 15k2 − k3 ≤ k21 < 2 3k2 − k3 and − 2 27k2 < k3 ≤ 2 3k2. Let m0 ∈ Hs(S) for s > 1/2 and m0 > 0. Assume that there exists some point x0 ∈ S such that m0(x0) > 0 and u0,x(x0) ≤ − √ 2k2 + 3k3 + 3k21 4k2 u0(x0). (4.21) Then the solution u(t, x) blows up in finite time with an estimate of the blow-up time T ∗ as T ∗ ≤ − 1 2k2m0(x0)u0,x(x0) . Proof. Now we look for M̂ → −∞. We recall the equation M̂ ′(t) = −2k2M̂ 2 + k1m̂û 2(1− 5 2 ûx 2 û2 ) + m̂û3 6 [(2k2 + 3k3)− (6k2 + 21k3) ûx 2 û2 ] − ( k2 3 + k3 2 )m[Λ1 ∗ (u− ux)3 + Λ2 ∗ (u+ ux)3] − k1m ( Λ1 ∗ [u2(1 + 1 2 u2x u2 )] + Λ2 ∗ [u2(1 + 1 2 u2x u2 )] ) . We assume that k1 > 0, k2 > 0, k2 3 + k3 2 > 0 ⇔ k3 > − 2 3 k2. (4.22) 12 M. ZHU, Y. WANG, L. CHEN EJDE-2021/103 To obtain a Riccati-type inequality we have 1− 5ûx 2 2û2 ≤ 0⇔ ûx 2 û2 ≥ 2 5 , (2k2 + 3k3)− (6k2 + 21k3) ûx 2 û2 ≤ 0⇔ ûx 2 û2 ≥ 2k2 + 3k3 6k2 + 21k3 , (4.23) where 6k2 + 21k3 ≥ 0⇔ k3 ≥ − 2 7k2. Now we discuss the following two cases: (1) When 2k2+3k3 6k2+21k3 ≥ 2 5 ⇔ − 2k2 7 ≤ k3 ≤ − 2k2 27 , and 2k2+3k3 6k2+21k3 ≤ 1 ⇔ − 2k2 7 ≤ k3 ≤ − 2k2 9 . Then − 2k2 7 ≤ k3 ≤ − 2k2 9 . (4.24) We have ûx 2 û2 ≥ 2k2 + 3k3 6k2 + 21k3 . (4.25) In particular, a finite-time blow-up of M̂ can be realized if the ration |ux u | stays reasonable big along the characteristics. We have ( ûx û )′ = 1 û2 [k1û(û2 − 1 2 ûx 2 )− k1(û+ ûx)Λ1 ∗ (u2 + 1 2 u2x) − k1(û− ûx)Λ2 ∗ (u2 + 1 2 u2x)] + û2 − ûx2 û2 [( k2 3 + k3 2 )û2 − 2k2 3 ûx 2 ] − 2k2 + 3k3 6û2 [(û+ ûx)Λ1 ∗ (u− ux)3 + (û− ûx)Λ2 ∗ (u+ ux)3] ≤ 1 û2 [k1û(û2 − 1 2 ûx 2 )− 1 4 k1û 2(û+ ûx)− 1 4 k1û 2(û− ûx)] + û2 − ûx2 û2 [( k2 3 + k3 2 )û2 − 2k2 3 ûx 2 ]. So we have ( ûx û )′ = 1 û2 [ k1 2 û(û2 − ûx2)] + û2 − ûx2 û2 [( k2 3 + k3 2 )û2 − 2k2 3 ûx 2 ] ≤ û2 − ûx2 û2 [ 1 8 + k21 2 û2 + ( k2 3 + k3 2 )û2 − 2k2 3 ûx 2 ] = û2 − ûx2 û2 [ 1 8 + ( k21 2 + k2 3 + k3 2 )û2 − 2k2 3 ûx 2 ], (4.26) where 1 8 + ( k21 2 + k2 3 + k3 2 )û2 − 2k2 3 ûx 2 < 0 ⇔ 2k2 3 ûx 2 û2 ≥ k21 2 + k2 3 + k3 2 + 1 8û2 ⇔ 2k2 3 ûx 2 û2 ≥ k21 2 + k2 3 + k3 2 + 1 8µE0 ⇔ ûx 2 û2 ≥ 3 16µE0k2 + 2k2 + 3k3 4k2 + 3k21 4k2 ⇔ ûx 2 û2 ≥ 2k2 + 3k3 + 3k21 4k2 . EJDE-2021/103 CURVATURE BLOW-UP OF CH-MCH-NOVIKOV EQUATION 13 We know that |ux| ≤ u, so 2k2 + 3k3 + 3k21 4k2 ≤ 1⇔ k21 ≤ 2 3 k2 − k3 and 2 3 k2 − k3 > 0⇔ k3 < 2 3 k2. (4.27) We have chosen the initial data so that ( ûx û )(0) ≥ √ 2k2 + 3k3 + 3k21 4k2 . (4.28) We need ûx 2 û2 > 2k2+3k3+3k21 4k2 ≥ 2k2+3k3 4k2 ≥ 2k2+3k3 6k2+21k3 ≥ 2 5 . Therefore, 2k2 + 21k3 ≥ 0⇔ k3 ≥ − 2 21 k2. (4.29) Clearly, (4.29) contradicts (4.24), so Case 1 can not happen. (2) When 2k2+3k3 6k2+21k3 < 2 5 ⇔ k3 < − 2k2 7 or k3 > − 2k2 27 and 6k2 + 21k3 > 0⇔ k3 > − 2 7k2, we have ûx 2 û2 ≥ 2 5 . With the same method, we have chosen the initial data so that ( ûx û )(0) ≥ √ 2k2 + 3k3 + 3k21 4k2 . (4.30) We need ûx 2 û2 > 2k2+3k3+3k21 4k2 . Then ûx û decreases initially. So, we have ûx 2 û2 > 2k2 + 3k3 + 3k21 4k2 ≥ 2 5 ⇔ k21 ≥ − 2 15 k2 − k3. (4.31) From (4.22), (4.26), (4.30) and (4.31), we have − 2 15k2−k3 ≤ k 2 1 < 2 3k2−k3,− 2 27k2 < k3 < 2 3k2. Then we obtain the desired Riccati inequality for M̂ M̂ ′(t) ≤ −2k2M̂ 2, which implies that M̂(t)→ −∞ as t→ T ∗, where T ∗ ≤ − 1 2k2m0(x0)u0,x(x0) . � Acknowledgments. Min Zhu was supported by the NSF of Jiangsu Province un- der BK20201382. Ying Wang was supported by the NSF of China under 11701068. References [1] R. M. Chen, F. Guo, Y. Liu, C. Z. Qu; Analysis on the blow-up of solutions to a class of integrable peakon equations, J. Funct. Anal., 270 (2016), 2343–2374. [2] R. Camassa, D. Holm; An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661–1664. [3] R. M. Chen, T. Q. Hu, Y. Liu; The shallow-water models with cubic nonlinearity, preprint. [4] A. Constantin, H. P. McKean; A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949–982. [5] K. S. Chou, C. Z. Qu; Integrable equations arising from motions of plane curves I, Physica D, 162 (2002), 9–33. [6] A. Constantin, W. A. Strauss; Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603–610. [7] H. Dai; Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech., 127 (1998), 193–207. [8] B. Fuchssteiner; Some tricks from the symmetry-toolbox for nonlinear equations: generaliza- tions of the Camassa-Holm equation, Physica D, 95 (1996), 229–243. 14 M. ZHU, Y. WANG, L. CHEN EJDE-2021/103 [9] B. Fuchssteiner, A. Fokas; Symplectic structures, their Bäcklund transformations and hered- itary symmetries, Physica D, 4 (1981/1982), 47–66. [10] G. L. Gui, Y. Liu , P. Olver, C. Z. Qu; Wave-breaking and peakons for a modified Camassa- Holm equation, Comm. Math. Phy., 319 (2013), 731–759. [11] A. N. Hone, H. Lundmark, J. Szmigielski; Explicit multipeakon solutions of Novikov’s cubi- cally nonlinear Camassa-Holm type equation, Dyn. Partial Differ. Equ., 6 (2009), 253–289. [12] H. Holden, X. Raynaud; A convergent numerical scheme for the Camassa-Holm equation based on multipeakons, Disc. Cont. Dyn. Syst. A., 14 (2006), 505–523. [13] A. N. Hone, J. Wang; Integrable peakon equations with cubic nonlinearity, J. Phys. A, 41 (2008), 372002. [14] S. Kouranbaeva; The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J. Math. Phys., 40 (1999), 857–868. [15] J. Lenells; Traveling wave solutions of the Degasperis-Procesi equation, J. Math. Anal. Appl., 306 (2005), 72-82. [16] G. Miso lek; A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203–208. [17] P. I. Naumkin, I. Sanchez-Suarez; KdV type asymptotics for solutions to higher-order non- linear Schrodinger equations, Electron. J. Differential Equations, 2020 (2020), no. 77, 1–34. [18] V. Novikov; Generalizations of the Camassa-Holm equation, J. Phys. A, 42 (2009), 342002. [19] P. Olver, P. Rosenau; Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900–1906. [20] J. F. Toland; Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1–48. [21] A. A. Himonas, C. Holliman; The Cauchy problem for the Novikov equation, Nonlinearity, 25 (2012), 449–479. [22] A. N. Hone, H. Lundmark, J. Szmigielski; Explicit multipeakon solutions of Novikov’s cubi- cally nonlinear integrable Camassa-Holm type equation, Dyn. Partial Differ. Equ., 6 (2009), 253–289. [23] Z. Yin; On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math., 47 (2003), 649–666. [24] F. Tiglay; The periodic Cauchy problem for Novikov’s equation, Int. Math. Res. Not. IMRN., 20 (2011), 4633–4648. [25] M. Zhu, Y. Wang; Wave-breaking phenomena for a weakly dissipative shallow water equation, Z. Angew. Math. Phys., 71 (2020), no. 96, 1–20. Min Zhu (corresponding author) Department of Mathematics, Nanjing Forestry University, Nanjing, 210037, China Email address: zhumin@njfu.edu.cn Ying Wang Department of Mathematics, University of Electronic Science and Technology of China, Chengdu 611731, China Email address: nadine 1979@163.com Lei Chen Department of Mathematics, Nanjing Forestry University, Nanjing, 210037, China Email address: chenlei@njfu.edu.cn 1. Introduction 2. Preliminaries 3. Precise blow-up scenario 4. Curvature blow-up Acknowledgments References