Persistence Property and Asymptotic Description for DGH Equation with Strong Dissipation

where the constants α2 and γ/c 0 are squares of length scales and the constant c 0 > 0 is the critical shallow water wave speed for undisturbed water at rest at spatial infinity. Since this equation is derived by Dullin, Gottwald, and Holm, in what follows, we call this new integrable shallow water equation (1) DGH equation. If α = 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 γ = 0 and α = 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 u(x, t) represents the free surface above a flat bottom. The properties about the well-posedness, blow-up, global existence, and propagation speed have already been studied in recent works [4–13], and the generalized version of a family of dispersive equations related to Camassa-Holm equation was discussed in [14]. It is very interesting that (1) preserves the bi-Hamiltonian structure and has the following two conserved quantities:


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 wave speed for undisturbed water at rest at spatial infinity.Since this equation is derived by Dullin, Gottwald, and Holm, 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 well-posedness, blow-up, global existence, and propagation speed have already been studied in recent works [4][5][6][7][8][9][10][11][12][13], and the generalized version of a family of dispersive equations related to Camassa-Holm equation was discussed in [14].
It is very interesting that (1) preserves the bi-Hamiltonian structure and has the following two conserved quantities: Recently, in [15], local well-posedness of strong solutions to (1) was established by applying Kato's theory [16], and some sufficient conditions were found to guarantee finite time blow-up phenomenon.Moreover, Zhou [17] found the best constants for two convolution problems on the unit circle via variational method and applied the best constants on (1) to give some blow-up criteria.Later, Zhou and Guo improved the results and got some new criteria for wave breaking [18].
In general, it is quite difficult to avoid energy dissipation mechanism in the real world.Ghidaglia [19] studied the long time behavior of solutions to the weakly dissipative KdV equation as a finite dimensional dynamic system.Moreover, some results on blow-up criteria and the global existence condition for the weakly dissipative Camassa-Holm equation are presented in [20], and very related work can be found in Discrete Dynamics in Nature and Society [21,22].In this work, we are interested in the following model, which can be viewed as the DGH equation with dissipation where  ∈ R,  > 0, (1 −  2  2  ) is the weakly dissipative term and  is a positive dissipation parameter.Set  = (1 −  2  2  ) 1/2 , then the operator  −2 can be expressed by for all  ∈  2 (R) with () = (1/2) −||/ .With this in hand, we can rewrite (3) as a quasilinear equation of hyperbolic type It is the dissipative term that causes the previous conserved quantities () and () to be no longer conserved for (3), and this model could also be regarded as a model of a type of a certain rate-dependent continuum material called a compressible second grade fluid [23].Our consideration is based on this fact.Furthermore, we will show how the dissipation term affects the behavior of solutions in our forthcoming paper.As a whole, the current dissipation model is of great importance mathematically and physically, and it is worthy of being considered.In what follows, we assume that  0 + / 2 = 0 and  > 0 just for simplicity.Since (, ) is bounded by its  1 -norm, 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 as follows: where The rest of this paper is organized as follows.In Section 2, we list the local well-posedness theorem for (6) with initial datum  0 ∈   ,  > 3/2 and collect some auxiliary results.In Section 3, we establish the condition for global existence in view of the initial potential.Persistence properties of the strong solutions are explored in Section 4. Finally, in Section 5, we give a detailed description of the corresponding solution with compactly supported initial data.

Preliminaries
In this section, we make some preparations for our consideration.Firstly, the local well-posedness of the Cauchy problem of (6) with initial data  0 ∈   with  > 3/2 can be obtained by applying Kato's theorem [16].More precisely, we have the following local well-posedness result.
The maximal value of  in Theorem 1 is called the lifespan of the solution in general.If  < ∞, that is lim sup  →  ‖(⋅, )‖   = ∞, we say that the solution blows up in finite time, otherwise, the solution exists globally in time.Next, we show that the solution blows up if and only if its first-order derivative blows up.
Lemma 2. Given that  0 ∈   ,  > 3/2, the solution  = (⋅,  0 ) of (3) blows up in finite time  < +∞ if and only if Proof.We first assume that  0 ∈   for some  ∈ N,  ≥ 4. Equation ( 6) can be written into the following form in terms Multiplying ( 10) by  = (1 −  2  2  ), and integrating by parts, we have Differentiating (10) with respect to the spatial variable , then multiplying by   = (1 −  2  2  )  , and integrating by parts again, we obtain Summarizing ( 11) and ( 12), we obtain If   is bounded from below on [0, ), for example,   ≥ −,  is a positive constant, then we get by ( 13) and Gronwall's inequality the following: where  = max {5, (3−2/ 2 )}.Therefore, the  3 -norm of the solution to (10) 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.
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 (10) by  and integrating by parts on R, we get Note that Then, we have Thus, we easily get and, therefore, By integration from 0 to , we get Hence, the lemma is proved.
We also need to introduce the standard particle trajectory method for later use.Consider now the following initial value problem as follows: where  ∈  1 ([0, ),  −1 ) is the solution to (6) with initial data  0 ∈   , ( > 3/2) and  > 0 is the maximal time of existence.By direct computation, we have Then, which means that (, ⋅) : R → R is a diffeomorphism of the line for every  ∈ [0, ).Consequently, the  ∞ -norm of any function V(, ⋅) is preserved under the family of the diffeomorphism (, ⋅), that is, Similarly, Moreover, one can verify the following important identity for the strong solution in its lifespan: We get that where (, ) is defined by (, ) = (1− 2  2  )(, ), for  ≥ 0 in its lifespan.

Global Existence
It is shown that it is the sign of initial potential not the size of it that can guarantee the global existence of strong solutions.
for some point  0 ∈ R.Then, the solution (, ) to (6) exists globally in time.
In fact, if  0 is regarded as ±∞, we prove this corollary immediately from Theorem 4.

Persistence Properties
In this section, we will investigate the following property for the strong solutions to (6) in  ∞ -space which behave algebraically at infinity as their initial profiles do.The main idea comes from the recent work of Himonas and his collaborators [7].Theorem 6. Assume that for some  > 0 and  > 3/2,  ∈ ([0, ];   ) is a strong solution of the initial value problem associated to (6), and that  0 () = (, 0) satisfies for some  ∈ (0, 1) and  ≥ 1.Then, uniformly in the time interval [0, ].
Proof.The proof is organized as follows.Firstly, we will estimate ‖(, )‖  ∞ and ‖  (, )‖  ∞ .Then, we apply the weight function to obtain the desired result.In the following proof, we denote some constants by ; they may be different from instance to instance, changing even within the same line.Multiplying ( 6) by  2−1 with  ∈ Z + , then integrating both sides with respect to  variable, we can get The first term of the above identity is and the estimates of the second term is In view of Hölder's inequality, we can obtain the following estimate for the third term in (39) Then, differentiating (6) with respect to variable  produces the following equation: Again, multiplying (49) by  2−1

𝑥
with  ∈ Z + , integrating the result in  variable, and considering the second term and the third term in the above identity with integration by parts, one gets so, we have Similarly, the following inequality holds and therefore as before, we obtain Taking the limits in (53), we obtain Next, we will introduce the weight function to get our desired result.This function   () with  ∈ Z + is independent of  as the following: From ( 6) and (49), we get the following two equations: We need some tricks to deal with the following term as in [18]: where we have used the fact 0 ≤    () ≤   (), a.e. ∈ R. Similar technique is used for the term ∫ R (  ) 2−1  2−1      .Hence, as in the weightless case, we get the following inequality in view of ( 48) and ( 54 which completes the proof of Theorem 6.

Asymptotic Description
The following result is to give a detailed description on the corresponding strong solution (, ) in its lifespan with  0 () being compactly supported.