A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion

Abstract In this paper, we investigate the Cauchy problem for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion. Furthermore, we establish the blow-up result of the positive solutions in finite time under certain conditions on the initial datum. This result complements the early one in the literature, such as [E. Novruzov, Blow-up phenomena for the weakly dissipative Dullin-Gottwald-Holm equation, J. Math. Phys. 54 (2013), no. 9, 092703, DOI 10.1063/1.4820786] and [Z.Y. Zhang, J.H. Huang, and M.B. Sun, Blow-up phenomena for the weakly dissipative Dullin-Gottwald-Holm equation revisited, J. Math. Phys. 56 (2015), no. 9, 092701, DOI 10.1063/1.4930198].


Introduction
Di erential equations and dynamical modeling have attracted some attention from many researchers as a result of their potential applications in elds of biology [3][4][5][6], physics [7][8][9], engineering [10,11], information technology and so forth [12][13][14]. Since the seminal work by Camassa and Holm [15], Camassa-Holm type equations have been intensively investigated. In this paper, we consider the Cauchy problem for the N − abc family of Camassa-Holm type equation with both dissipation and dispersion (1.1) where N ∈ Z + , N ≥ , k, λ ≥ , and k, λ are dissipation and dispersion coe cients respectively. a, b, c are positive constants and a = b + c.
( 1.2) Eq. (1.2) was rst investigated by Himonas and Holliman [16] and they proved the local well-posedness and the nonuniform dependence of its Cauchy problem in Sobolev space H s with s > . In [17], Zhou an Mu studied the persistence properties of strong solutions and the existence of its weak solutions of (1.2). Later on, Himonas and Mantzavinos [18] showed well-posedness in H s with s > . They also provided a sharpness result on the data-to-solution map and proved that it is not uniformly continuous from any bounded subset of H s into C([ , T); H s ). Eq. (1.2) was also studied by Barostichi, Himonas and Petronilho [19] and they exhibited a power series method in abstract Banach spaces equiped with analytic initial data, and established a Cauchy-Kovalevsky type theorem.
It is important to note that (1.1) is an evolution equation with (N + )-order nonlinearities and includes three famous integrable dispersive equations: the Camassa-Holm (CH) equation, the Degasperis-Procesi (DP) equation and the Novikov equation (NE).
As c = , N = , b = , a = , k = λ = , (1.1) becomes the well-known CH equation. The local wellposedness of Cauchy problem of the CH equation has extensively been investigated in [20]. It was shown that there exist global strong solutions to the CH equation [20] and nite time blow-up strong solutions to the CH equation [20,21]. The existence and uniqueness of global weak solutions to the CH equation were studied in [22].
As c = , N = , b = , a = , k = λ = , (1.1) reads the NE in [29], which is also integrable peakon model with × Lax pairs and the peakon solution u(x, t) = √ ce −|x−ct| with c > . The most di erence between the NE and the CH and DP equations is that the former one has cubic nonlinearity and the latter ones have quadratic nonlinearity. The local well-posedness, global existence and blow-up phenomena of the NE was studied in [29][30][31][32][33][34].
It is well known that it is di culty to avoid energy dissipation in a real world. Thus it is reasonable to study the model with energy dissipation in propagation of nonlinear waves, see [35][36][37][38]. Recently, Wu and Yin [39] investigated the blow-up, blow-up rate and decay of solutions to the weakly dissipative periodic CH equation (i.e (1.1) with N = , c = , b = , a = , k = ). Thereafter, they also studied the blow-up and decay of solutions to weakly dissipative non-periodic CH equation (i.e (1.1) with N = , c = , b = , a = , k = ) [40]. Hu and Yin [41] investigated the blow-up, blow-up rate of solutions to weakly dissipative periodic rod equation. Later on, Hu [42] discussed the global existence and blow-up phenomena for a weakly dissipative periodic two component CH system. Zhou, Mu and Wang [43] considered the weakly dissipative gCH equation (i.e (1.1) with c = , a = b + , k = ). Recently, Novruzov [1] studied the Cauchy problem for the weakly dissipative Dullin-Gottwald-Holm (DGH) equation (i.e (1.1) with N = , c = , b = , a = ) and establish certain conditions on the initial datum to guarantee that the corresponding positive strong solutions blow up in nite time. The same equation for arbitrary solution has been considered in [44]. Authors showed the simple conditions on the initial data that lead to the blow-up of the solutions in nite time or guarantee that the solutions exist globally. Later on, Zhang et al. [2] improved the results of [1]. In [45], Novruzov extended the obtained "blow-up" result to the DGH equation under some conditions on the initial data. This issue is extensively studied, e.g. in [46][47][48].

Preliminaries
In this section, we recall some useful results in order to achieve our aim.
Let us rst present the local well-posedness of Cauchy problem for (1.1). Thus, we can rewrite (1.1) in the equivalent form. Let y = u − uxx . Then (1.1) becomes Hence, (2.1) can be reformulated in the form as follows: The local well-posedness of Cauchy problem for (1.1) with the initial data u (x) ∈ H s , s > , can be obtained by applying the Kato's theory, see [2,49]. It is easy to see that some results hold for (1.1). So, we omit the further details and show corresponding result directly.
. Moreover, the solution depends continuously on the initial data, i.e., the mapping is continuous and the maximal time of existence T > can be chosen to be independent of index s.
The following lemma gives necessary and su cient condition for the blow-up of the solution.
Proof. Indeed, the above result follows by standard manner in [49]. Assume u ∈ H s for some s ∈ N, s ≥ . Multiplying both sides of the rst equation of (2.1) by y = u − uxx and integrating by parts with respect to x, we get that is,  [50].
Consider now the following initial value problem where u(x, t) is the corresponding strong solution to (1.1).
After simple computations and solving (2.6), we get the following lemma.

Lemma 2.3. Let u (x) ∈ H s , s > , and let T > be the maximal existence time of the solution u to (1.1). Then, we have
In particular, if N = b, we have e −λt y L = y L .
Proof. It follow from (1.1) and (2.1) that which implies Thus, setting ξ = q(t, x), we arrive at Obviously, letting N = b leads to e −λt y L = y L . This completes the proof of Lemma 2.3.
Finally, let us now give the following lemma which will be used in the sequel. Noticing that which implies the desired result in the lemma.

Main result
We are now position to state our main result. Proof. We shall give the proof by contradiction. We assume that it is not true and solutions exist globally. That is, there exists constant C such that ux ≥ −C (due to Lemma 2.2).
Observing that u = G * y with G(x) = e −|x| , x ∈ R, we have So, we have For t ∈ [ , T), q(t, ·) is the increasing di eomorphism of the line. From Lemma 2.3, we deduce that y(t, x) ≤ , x ≥ q(t, x ). Hence, we conclude that −ux(t, x) ≥ u(t, x) ≥ , for x ≥ q(t, x ). Due to ux ≥ −C, we get Di erentiating (2.2) with respect to x and noticing that Multiplying (3.3) by ( − α)e −x (−ux) α ( < α < ) and integrating over (q(t, x ), θ) (θ < ∞), we have Next, we shall estimate I j (j = , , · · ·, ) in (10). First, integrating by parts, we have By (3.2) and G L ≤ ∂x G L = , we get Note that θ can be taken to be a su cient large. Due to and by Jensen's inequality, we arrive at Thus, we conclude that (3.11) Multiplying both sides of (3.11) by J −α −α , we obtain Integrating with respect to t, we have From q(τ, x ) = (c t u N (τ, q(t, x ))dτ + k)t + x (by (2.6)) and (3.12), we have That is, there exists a sequence (tn , xn) such that −ux(tn , xn) → ∞ as t → T which contradicts with (3.2). Thus, our main result is completed.