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Persistence Property and Asymptotic Description for DGH Equation with Strong Dissipation
The present work is mainly concerned with the Dullin-Gottwald-Holm (DGH) equation with strong dissipation. We establish a sufficient condition to guarantee global-in-time solutions, then present persistence property for the Cauchy problem, and describe the asymptotic behavior of solutions for compactly supported initial data.
Dullin et al.  derived a new equation describing the unidirectional propagation of surface waves in a shallow water regime: where the constants and are squares of length scales and the constant 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 , (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 . If and , (1) reduces to the Camassa-Holm equation which was derived physically by Camassa and Holm in  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 [4–13], and the generalized version of a family of dispersive equations related to Camassa-Holm equation was discussed in .
It is very interesting that (1) preserves the bi-Hamiltonian structure and has the following two conserved quantities:
Recently, in , local well-posedness of strong solutions to (1) was established by applying Kato's theory , and some sufficient conditions were found to guarantee finite time blow-up phenomenon. Moreover, Zhou  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 .
In general, it is quite difficult to avoid energy dissipation mechanism in the real world. Ghidaglia  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 , and very related work can be found in [21, 22]. In this work, we are interested in the following model, which can be viewed as the DGH equation with dissipation where , , is the weakly dissipative term and is a positive dissipation parameter. Set , then the operator can be expressed by for all with . 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 . 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 and just for simplicity. Since is bounded by its -norm, a general case with 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 , 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.
In this section, we make some preparations for our consideration. Firstly, the local well-posedness of the Cauchy problem of (6) with initial data with can be obtained by applying Kato's theorem . More precisely, we have the following local well-posedness result.
Theorem 1. Given that , , there exist and a unique solution to (6), such that Moreover, the solution depends continuously on the initial data; that is, the mapping is continuous, and the maximal time of existence is independent of .
Proof. Set , , ,, , and . Applying Kato's theory for abstract quasilinear evolution equation of hyperbolic type, we can obtain the local well-posedness of (6) in , and .
The maximal value of in Theorem 1 is called the lifespan of the solution in general. If , that is , 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 , , the solution of (3) blows up in finite time if and only if
Proof. We first assume that for some , . Equation (6) can be written into the following form in terms of Multiplying (10) by , and integrating by parts, we have Differentiating (10) with respect to the spatial variable , then multiplying by , and integrating by parts again, we obtain Summarizing (11) and (12), we obtain If is bounded from below on , for example, , is a positive constant, then we get by (13) and Gronwall's inequality the following: where . Therefore, the -norm of the solution to (10) does not blow up in finite time. Furthermore, similar argument shows that the -norm with does not blow up either in finite time. Consequently, this theorem can be proved by Theorem 1 and simple density argument for all .
Lemma 3. Let , then as long as the solution given by Theorem 1 exists, for any , one has where the norm is defined as
Proof. Multiplying both sides of (10) by and integrating by parts on , we get Note that Then, we have Hence, Thus, we easily get and, therefore, By integration from 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 is the solution to (6) with initial data , and is the maximal time of existence. By direct computation, we have Then, which means that is a diffeomorphism of the line for every . Consequently, the -norm of any function 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 , for in its lifespan.
From the expression of in terms of , for all , , we can rewrite and as follows: from which we get that
3. 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.
Theorem 4. Assume that , , and satisfies for some point . Then, the solution to (6) exists globally in time.
Proof. From the hypothesis and (30), we obtain that , ; , . According to (31) and (32), one can get that when , it follows that that is, is bounded below. Similarly, when , so . We also get the bounded below result as above. Therefore, the theorem is proved by Lemma 2.
Corollary 5. Assume that , , and is of one sign, then the corresponding solution to (6) exists globally.
In fact, if is regarded as , we prove this corollary immediately from Theorem 4.
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 .
Theorem 6. Assume that for some and , is a strong solution of the initial value problem associated to (6), and that satisfies for some and . Then, uniformly in the time interval .
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 with , 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) For the last term putting all the inequalities above into (39) yields Using the Sobolev embedding theorem, there exists a constant such that we have by applying Gronwall's inequality For any , we know that Taking the limits in (46) (notice that and ) from (47), we get Then, differentiating (6) with respect to variable produces the following equation: Again, multiplying (49) by with , 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 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 : where we have used the fact , a.e. . Similar technique is used for the term . Hence, as in the weightless case, we get the following inequality in view of (48) and (54) as follows: On the other hand, a simple calculation shows that there exists , depending only on and such that for any , Therefore, for any appropriate function , one obtains that and similarly, . Using the same method, we can estimate the following two terms Therefore, it follows that there exists a constant such that Hence, the following inequality is obtained for any and any : Finally, taking the limit as goes to infinity in the above inequality, we can find that for any , which completes the proof of Theorem 6.
5. Asymptotic Description
The following result is to give a detailed description on the corresponding strong solution in its lifespan with being compactly supported.
Theorem 7. Assume that the initial datum with is compactly supported in , then the corresponding solution to (6) has the following property: for any , where is defined by (24) and is its lifespan. Furthermore, and denote continuous nonvanishing functions, with and for . Moreover, is a strictly increasing function, while is strictly decreasing.
Remark 8. This is an interesting phenomenon for our model; it implies that the strong solution does not have compact -support for any in its lifespan anymore, although the corresponding is compactly supported. No matter that the initial profile is (no matter it is positive or negative), for any in its lifespan, the nontrivial solution is always positive at infinity and negative at negative infinity. Moreover, we found that the dissipative coefficient does not affect this behavior.
Proof. First, since has a compact support, so does . Equation (30) tells us that is compactly supported in in its lifespan. Hence, the following functions are well defined
And by integration by parts.
Then, for , we have where (66) is used.
Similarly, when , we get Because has a compact support in in the interval for any , we get , for or . Hence, as consequences of (68) and (69), we have On the other hand,
Substituting the identity (10) into , we obtain where we used (70). Therefore, in the lifespan of the solution, we have that is an increasing function with ; thus, it follows that for ; that is, By similar argument, one can verify that the following identity for is true: In order to finish the proof, it is sufficient to let , and to let , respectively.
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Copyright © 2013 Ke-chuang Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.