Blow-up phenomena and global existence to a weakly dissipative shallow water equation

We first establish the local well-posedness for a weakly dissipative shallow water equation which includes both the weakly dissipative Camassa-Holm equation and the weakly dissipative Degasperis-Procesi equation as its special cases. Then two blow-up results are derived for certain initial profiles. Finally, We study the long time behavior of the solutions.


Introduction
Degasperis and Procesi [1] studied the following family of third-order dispersive nonlinear equations (1.1) where  , c 0 , c 1 , c 2 , c 3 and  are real constants. They found that there are only three equations that satisfy the asymptotic integrability condition, namely, the Korteweg-de Vries(KdV) equation, the Camassa-Holm(CH) equation and the Degasperis-Procesi(DP) equation.
For   c 0  c 2  c 3  0 , Eq. 1.1 becomes the well-known KdV equation which describes the unidirectional propagation of waves at the free surface of shallow water under the influence of gravity [2,3]. The Cauchy problem of Eq.  1.2  has been studied widely. Bourgain [4] proved that the Cauchy problem associated with the equation is globally well-posed in H s , H s (s  0) . Kenig [3] and Tao [5] showed that it is globally well-posed for u  L 2 S The local well-posedness of Eq. 1.2was pushed down to by Kenig et al. [6]. Whitham [7] found that the equation does not accommodate wave breaking.
which is a model describing the unidirectional propagation of shallow water waves over a flat bottom [8]. The CH equation has a bi-Hamiltonian structure [9] and is completely integrable [10,11]. It admits, in addition to smooth waves, a multitude of traveling wave solutions with singularities:peakons, cuspons, stumpons and composite waves [8,12]. Its solitary waves are stable solitons [13,14], retaining their shape and form after interactions [15]. The Cauchy problem of Eq. (1.3) has been studied extensively. Constantin [16] and Rodriguez-Blanco [17]studied the locally well-posed for initial data More interestingly, it has strong solutions that are global in time [18,19] as well as solutions that blow up in finite time [18,20,21]. On the other hand, Bressan [22] and Xin [23] showed that the Eq. (1.3) has global weak solutions with initial data (1.4) which can be used as a model for nonlinear shallow water dynamics and its asymptotic accuracy is the same as Eq. (1.3) [24]. Degasperis et al. [24] also showed that Eq. (1.4) has a bi-Hamiltonian structure with an infinite sequence of conserved quantities and admits exact peakon solutions which are analogous to Eq. (1.3) peakons [8,13,14]. Dullin etal. [25] showed that Eq. (1.4) can be obtained from the shallow water elevation equation by an appropriate Kodama transformation. The numerical stability of solitons and peakons, the multi-soliton solutions and their peakon limits, together with an inverse scattering method to compute N-peakon solutions to DP equation have been investigated respectively in [26]- [28]. After Eq. (1.4) appeared, it has attracted many researchers to discover its dynamics (see [29]- [36]). Yin [29,30] [31,33], but also blow-up solutions [33]- [35]. Apart from these, Coclite and Karlsen [36] proved that it has global entropy weak solutions in For and under suitable mathematical transforms and several restrictions on its coefficients [37], Eq. (1.1) becomes (1.5) where a  0andb  0are arbitrary positive constants. Obviously, Eq. (1.5) is a generalization of both the CH and DP equations. We can see that in models (1.3) and(1.4) , the coefficient of x uu is equal to the coefficient of plus the coefficient of uu xxx , that is 3=2+1, 4=3+1. Recently, Lai and Wu [38,39] obtained the existence of the strong solution and the global existence of its weak solutions. In general, the energy dissipation mechanisms are difficult to avoid in a real world, many authors modified those models with dissipation. Ott and Sudan [40] investigated the KdV equation with the presence of dissipation and their effect on solution of the KdV equation. The long time behavior of solutions to the weakly dissipative KdV equation was studied by Ghidaglia [41]. Recently, Wu and Yin investigated the weakly dissipative CH equation (1.6) on the line [42] and on the circle [43]. They also studied the weakly dissipative DP equation (1.7) on the line in [44,45] and on the circle [46], where  0 is a constant. In [47], Yan, Li and Zhang considered the global existence and blow-up for the weakly dissipative Novikov equation The Bulletin of Society for Mathematical Services and Standards Vol. 12 (1.8) Motivated by all the above works, in the paper, we'll consider the Cauchy problem of the following weakly dissipative shallow water equation

Local well-posedness and blow-up scenario
In this section, we prove the local well-posedness and the precise blow-up scenario of Eq. (1.9) . We now apply Kato's theory to establish the local well-posedness. For convenience, we reformulate problem (1.8) as follows: Where are arbitrary constants.
(2.2) or in the equivalent form:

BSMaSS Volume 12
The local well-posedness of the Cauchy problem of Eq. (2.2) with initial data can be obtained by applying the Kato's theorem [48]. More precisely, we have the following local well-posedness result.
with constant . By applying the Kato's theorem [48] we can finish the proof of this theorem. One can see the similar proof in [17,38] for details. Now, we present the precise blow-up scenario for sufficiently regular solutions to Eq. (1.9) .

Blow-up
In this section, we will establish two blow-up results for strong solutions to Eq. . (1.9) , from which we can see the effect of weak dissipation to the blow-up phenomena of the solution Consider now the following initial value equation Following the same argument in the proof of (3.6) in [42], we also have (4.3) From the Agmon's inequality, we obtain It follows from4.5and Theorem 2.2 thatT , i.e., the solution ut, xexists globally in time.
By means of Gronwall's inequality, we have (4.7)

Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.