Wave Breaking Phenomenon for DGH Equation with Strong Dissipation

and Applied Analysis 3 The maximal value of T in Theorem 1 is usually called the lifespan of the solution. If T < ∞, that is, lim sup t→T ‖u(⋅, t)‖ H s = ∞, we say the solution blows up in finite time. Next, we show that the corresponding solution blows up if and only if its first-order derivative blows up in finite time. Theorem 2. Given u 0 ∈ H , s > 3/2, the solution u = u(⋅, u 0 ) of (12) blows up in finite time T < +∞ if and only if lim inf t→T {inf x∈R [u x (x, t)]} = −∞. (14) Proof. We first assume that u 0 ∈ H s for some s ∈ N, s ≥ 4. Equation (12) can be written into the following form in terms of y = (1 − ∂2 x )u: y t + y x u + 2yu x + c 0 y x + λy = 0. (15) Multiplying (15) by y = (1−∂2 x )u and integrating by parts, we have


Introduction
Dullin et al. [1] derived a new equation describing the unidirectional propagation of surface waves in a shallow water regime: where the constants  2 and / 0 are squares of length scales and the constant  0 > 0 is the critical shallow water speed for undisturbed water at rest at spatial infinity.Since this equation is derived by Dullin et al., in what follows, we call this new integrable shallow water equation (1) DGH equation.
If  = 0, (1) becomes the well-known KdV equation, whose solutions are global as long as the initial data is square integrable.This is proved by Bourgain [2].If  = 0 and  = 1, (1) reduces to the Camassa-Holm equation, which was derived physically by Camassa and Holm in [3] by approximating directly the Hamiltonian for Euler's equations in the shallow water regime, where (, ) represents the free surface above a flat bottom.The properties about the wellposedness, blow-up, global existence, and propagation speed for the Camassa-Holm equation have already been studied in recent papers [4][5][6][7][8][9][10].
It is very interesting that (1) still preserves the bi-Hamiltonian structure and has the following two conserved quantities: Recently, in [11], local well-posedness of strong solutions to (1) was established by applying Kato's theory [12] and some sufficient conditions on the initial data were found to guarantee finite time blow-up phenomenon.Moreover, Zhou [13] found the best constants for two convolution problems on the unit circle via variational method and applied the best constants on (1) to give sufficient conditions on the initial data.Later, Zhou and Guo improved the results and got some new criteria on blow-up and then discussed the persistence properties of the strong solutions and infinite propagation speed in the recent work [14].
In general, it is quite difficult to avoid energy dissipation mechanism in the real world.Ghidaglia [15] studied the long time behavior of solutions to the weakly dissipative KdV equation as a finite dimensional dynamical system.Moreover, the weakly dissipative Camassa-Holm equation was investigated on the blow-up criteria and global existence in recent works [16,17], and very related work can be found in [18].In this work, we are interested in the following model, which can be viewed as the DGH equation with dissipation: where  ∈ R,  > 0,  > 0, and (1 −  2  2  ) is the dissipative term which will be demonstrated when we introduce this model in the subsequent section.Indeed, (3) can be written into the following form in terms of  = (1 −  2  2  ): Set  = (1 −  2  2  ) 1/2 ; then the operator  −2 can be expressed by for all  ∈  2 (R) with () = (1/2) where  ∈ R,  > 0. We find that () and () are no longer conserved for (3); this observation would make our research interesting.Note that the present dissipation model is of great importance mathematically and physically; it could be regarded as a model of a type of a certain ratedependent continuum material called a compressible secondgrade fluid [19].We also would like to mention another dispersive DGH model.This can be achieved by replacing )  , and some results have been obtained by Tian's group [20].The investigation of (3) in the periodic framework is attributed to [21].The motivation here is to show the influence of this dissipation on the behavior of solutions, which can be illustrated by the following blow-up criteria.Precisely, we will examine the wave breaking phenomenon for the Cauchy problem and learn how parameter ¡  plays a role in blow-up mechanism; moreover, some comparisons will be made with the wellknown DGH equation or the Camassa-Holm equation.
In what follows, we assume that  0 + / 2 = 0 and  > 0 just for simplicity.Since  is bounded by its  1norm, a general case with  0 + / 2 ̸ = 0 does not change our results essentially, but it would lead to unnecessary technical complications.So the above equation is reduced to a simpler form: where For convenience of discussion, we do scaling as follows: Therefore, (7) becomes where For concision of notations, we remove the tilde in (10) if there is no ambiguity.Then we obtain for some positive finite .The rest of this paper is organized as follows.In Section 2, we list the local well-posedness theorem for (12) with initial datum  0 ∈   ,  > 3/2, and show that the lifespan of the corresponding solution is finite if and only if its firstorder derivative blows up.In Section 3, we give some new criteria to show the generation of singularities to (12); in particular, the blow-up condition with the suitable integral form of initial momentum is involved.For simplicity, we drop R in our notations of function spaces if there is no ambiguity.Additionally, ‖ ⋅ ‖  1 denotes the norm of  1 (R) in this paper.

Preliminaries
In this section, we make some preparations for our consideration; some results here are standard, but we still give their proofs for convenience of readers.Firstly, the local wellposedness of the Cauchy problem of (12) with initial data  0 ∈   with  > 3/2 can be obtained by applying Kato's theorem [12].More precisely, we have the following local wellposedness result.
Abstract and Applied Analysis 3 The maximal value of  in Theorem 1 is usually called the lifespan of the solution.If  < ∞, that is, lim sup  →  ‖(⋅, )‖   = ∞, we say the solution blows up in finite time.Next, we show that the corresponding solution blows up if and only if its first-order derivative blows up in finite time.
Theorem 2. Given  0 ∈   ,  > 3/2, the solution  = (⋅,  0 ) of (12) blows up in finite time  < +∞ if and only if Proof.We first assume that  0 ∈   for some  ∈ N,  ≥ 4. Equation ( 12) can be written into the following form in terms of  = (1 −  2  ): Multiplying ( 15) by  = (1 −  2  ) and integrating by parts, we have Differentiating (15) with respect to , multiplying the resulting equation by   = (1 −  2  )  , and integrating by parts again, we obtain Summarizing the above two equations, we obtain If   is bounded from below on [0, ), for example,   ≥ −, where  is a positive constant, then we get by ( 18) and Gronwall's inequality Therefore the  3 -norm of the solution to (12) does not blow up in finite time.Furthermore, similar argument shows that the   -norm with  ≥ 4 does not blow up either in finite time.
Consequently, this theorem can be proved by Theorem 1 and simple density argument for all  > 3/2.On the other hand, if (14) holds, by Sobolev embedding theorem, we easily know the corresponding solution blows up in finite time.
Next, we prove that the energy ‖‖ 2  1 decays as time goes on.
Lemma 3. Let  0 ∈  1 ; then as long as the solution (, ) given by Theorem 1 exists, for any  ∈ [0, ), one has where the norm is defined as Proof.Multiplying both sides of ( 15) by  and integrating by parts on R, we get Note that Then, we have Thus, we easily have and therefore By integration from 0 to , we get Hence, (20) is proved.

Wave Breaking Phenomenon
In this section, we will establish some new criteria to guarantee the formation of singularities for the corresponding solutions to (12).The first one is as follows.
Theorem 5. Assume that  0 ∈   ,  > 3/2, satisfies the following condition: for some  0 ∈ R; then the corresponding solution to (12) blows up in finite time.
Proof.In [13], Zhou has found the best constant in R which is different from the periodic case.The inequality is as follows: Moreover, 1/2 is the best constant obtained by  =  −|−| for some ,  ∈ R.
Next, we have blow-up criterion with the following form.
Theorem 6. Assume that  0 ∈   ,  > 3/2, satisfies the following condition: and then the corresponding strong solution to (12) blows up in finite time.
Proof.From ( 12), differentiating both sides of it with respect to variable , we obtain Then we can get Next multiplying  2  on both sides of (41) and integrating by parts with respect to , one obtains In view of (41), we obtain the following inequality: By Cauchy-Schwartz inequality, we obtain and it follows that Therefore, putting it into (43), we get Let () = ∫ R where The remaining part is very similar to the above theorem.We can deduce that there exists a time  such that lim On the other hand, which shows that lim ↑ inf ∈R   (, ) = −∞.This completes the proof of Theorem 6.
Remark 7. We note that if  = 0 in the above theorems, then the blow-up conditions are nothing but the ones established by Constantin and Zhou et al. for the Camassa-Holm equation or the DGH equation.We presented them here to show that the dissipation structure of the equation not only caused energy decay but also affected the wave breaking behavior although these arguments seem to be standard.
Motivated by McKean's deep observation for the Camassa-Holm equation [23], we can do a similar particle trajectory as where (, ) is the corresponding strong solution to (12).Then for any fixed  in its lifespan, (, ) is a diffeomorphism of the line with where (, ) is defined by (, ) = (1− 2  2  )(, ), for  ≥ 0 in its lifespan.
From the expression of (, ) in terms of (, ), for all  ∈ [0, ),  ∈ R, we can rewrite (, ) and   (, ) as follows: from which we get that The following criterion shows that wave breaking occurs when the suitable integral form of initial momentum satisfies certain condition for all positive finite .This is motivated by the work in [13].We do not have direct restrictions on initial velocity slope.Compared to the result in [13], the right-hand constants can be viewed as the up and down translation of the condition for DGH equation, so this observation itself is nontrivial.Obviously, if  = 0, then our result is valid for the Camassa-Holm equation.Namely, we have the following.
Proof.We obtain by (41) that where we use inequality (32).