Research Article | Open Access
Xuefei Liu, Mingxuan Zhu, Zaihong Jiang, "On the Cauchy Problem for the b-Family Equations with a Strong Dispersive Term", Journal of Applied Mathematics, vol. 2011, Article ID 513467, 15 pages, 2011. https://doi.org/10.1155/2011/513467
On the Cauchy Problem for the b-Family Equations with a Strong Dispersive Term
In this paper, we consider b-family equations with a strong dispersive term. First, we present a criterion on blow-up. Then global existence and persistence property of the solution are also established. Finally, we discuss infinite propagation speed of this equation.
Recently, Holm and Staley  studied the exchange of stability in the dynamics of solitary wave solutions under changes in the nonlinear balance in a evolutionary partial differential equation related both to shallow water waves and to turbulence. They derived the following equations (the b-family equations): where denotes the velocity field and .
Detailed description of the corresponding strong solutions to (1.1) with being its initial data was given by Zhou . He established a sufficient condition in profile on the initial data for blow-up in finite time. The necessary and sufficient condition for blow-up is still a challenging problem for us at present. More precious, Theorem 3.1 in  means that no matter what the profile of the compactly supported initial datum is (no matter whether it is positive or negative), for any in its lifespan, the solution is positive at infinity and negative at negative infinity; it is really a very nice property for the b-family equations.
In this paper, we consider the following b-family equations with a strong dispersive term: where , , and is the strong dispersive term.
Let ; then, the operator can be expressed by its associated Green's function as . So (1.2) is equivalent to the following equation:
Theorem 1.1. Given , , then there exist a and a unique solution to (1.3) such that
To make the paper concise, we would like to omit the detailed proof.
The paper is organized as follows. In Section 2, we get a criterion on blow-up. A condition for global existence is found in Section 3. Persistence property is considered in Section 4. In Section 5, the infinite propagation speed will be established analogous to the b-family equation.
The maximum value of in Theorem 1.1 is called the lifespan of the solution, in general. If , that is, , we say the solution blows up in finite time.
The following lemma tells us that the solution blows up if and only if the first-order derivative blows up.
Lemma 2.1. Assume that , . If , then the solution of (1.2) will exist globally in time. If , then the solution blows up if and only if becomes unbounded from below in finite time. If , the solution blows up in finite time if and only if becomes unbounded from above in finite time.
Proof. By direct computation, one has
Applying on (1.2) and integration by parts, we obtain
If , then . Hence, Equation (2.4) implies the corresponding solution exists globally.
If , due to the Gronwall inequality, it is clear that, from (2.3), is bounded from below on and then the -norm of the solution is also bounded on . On the other hand, Therefore where we use (2.2). Hence, (2.6) tells us if -norm of the solution is bounded, then the -norm of the first derivative is bounded.
By the same argument, we can get the similar result for .
This completes the proof.
Motivated by Mckean's deep observation for the Camassa-Holm equation , we can do the similar particle trajectory as where is the lifespan of the solution; then, is a diffeomorphism of the line. Differentiating the first equation in (2.7) with respect to , one has Hence
Since it follows that
Then we establish sufficient condition on the initial data to guarantee blow-up for (1.2).
Theorem 2.2. Let . Suppose that and there exists an such that , Then the corresponding solution to (1.2) with as the initial datum blows up in finite time.
Proof. Suppose that the solution exists globally. Due to (2.11) and the initial condition (2.12), we have , and
for all . Since , one can write and as
for all .
For any fixed , if , then Similarly, for , we also have Combining (2.16) and (2.17) together, we get that for any fixed , for all .
Differentiating (1.3), we get Differentiating with respect to , where is the diffeomorphism defined in (2.7), where we use (2.18) and the following inequality: .
Claim 1. is decreasing and for all .
Suppose not; that is, there exists a such that on and . Now, let Firstly, differentiating , we have Secondly, by the same argument, we get Hence, it follows from (2.22), (2.23), and the continuity property of ODEs that Moreover, due to (2.22) and (2.23) again, we have the following equation for : Now, substituting (2.20) into (2.25), it yields Before completing the proof, we need the following technical lemma.
Lemma 2.3. Suppose that is twice continuously differential satisfying Then blows up in finite time. Moreover, the blow-up time can be estimated in terms of the initial datum as
Let ; then, (2.26) is an equation of type (2.27) with . The proof is complete by applying Lemma 2.3.
Remark 2.4. Mckean got the necessary and sufficient condition for the Camassa-Holm equation in . It is worth pointing out that Zhou and his collaborators  gave a new proof to Mckean's theorem. However, the necessary and sufficient condition for (1.2) is still a challenging problem for us at present.
3. Global Existence
In this section, a global existence result is proved.
Theorem 3.1. Supposing that , is one sign. Then the corresponding solution to (1.2) exists globally.
Proof. We can assume that . It is sufficient to prove has a lower and supper bound for all . In fact, so This completes the proof.
4. Persistence Property
Now, we will investigate the following property for the strong solutions to (1.2) in -space which asymptotically exponentially decay at infinity as their initial profiles. The main idea comes from a recent work of Zhou and his collaborators  for the standard Camassa-Holm equation (for slower decay rate, we refer to ).
Theorem 4.1. Assume that for some and , is a strong solution of (1.2) and that satisfies that for some , Then uniformly in the time interval .
Proof. First, we will introduce the weight function to get the desired result. This function with is independent on as follows: which implies that From (1.3), we can get Multiplying (4.5) by with and integrating the result in the -variable, we get from which we can deduce that Denoting and by Gronwall's inequality, we obtain Taking the limits in (4.8), we get Next differentiating (1.3) in the -variable produces the equation Using the weight function, we can rewrite (4.10) as Multiplying (4.11) by with and integrating the result in the -variable, it follows that For the second term on the right side of (4.12), we know Using the above estimate and the Hölder inequality, we deduce that Thanks to Gronwall's inequality, it holds that Taking the limits in (4.15), we have Combining (4.9) and (4.16) together, it follows that A simple calculation shows that there exists , depending only on , such that for any , Thus, for any appropriate function and , one sees that Similarly, we can get Thus, inserting the above estimates into (4.17), there exists a constant such that Hence, for any and any , we have Finally, taking the limit as goes to infinity in (4.22), we find that for any which completes the proof of the theorem.
5. Infinite Propagation Speed
Recently, Himonas and his collaborators established infinite propagation speed for the Camassa-Holm equation in . Later, Guo [15, 16] considered a similar problem on the weakly dissipative Camassa-Holm equation and the weakly dissipative Degasperis-Procesi equation. Recently, infinite propagation speed for a class of nonlocal dispersive -equations was established in . The purpose of this section is to give a more detailed description on the corresponding strong solution to (1.2) in its life span with initial data being compactly supported. The main theorem is as follows.
Theorem 5.1. Let . Assume that for some and , is a strong solution of (1.2). If has compact support , then for , one has where and denote continuous nonvanishing functions, with and for . Furthermore, is strictly increasing function, while is strictly decreasing function.
Proof. Since has compact support in in the interval , from (2.11), so does in the interval in its lifespan. Hence the following functions are well defined:
Then for , we have
Similarly, when , we get
Hence, as consequences of (5.4) and (5.5), we have
On the other hand,
It is easy to get
Substituting identity (5.8) into , we obtain
where we use (5.6).
Therefore, in the lifespan of the solution, we have By the same argument, one can check that the following identity for is true: In order to complete the proof, it is sufficient to let and .
Remark 5.2. The main result in  is that any nontrivial classical solution of the b-family equation with dispersive term will not have compact support if its initial data has this property. But Theorem 4.1 means that no matter what the profile of the compactly supported initial datum is (no matter whether it is positive or negative), for any in its lifespan, the solution is positive at infinity and negative at negative infinity. So Theorem 4.1 is an improvement of that in .
This work is partially supported by the Zhejiang Innovation Project (T200905), ZJNSF (Grant no. R6090109), and NSFC (Grant no. 11101376).
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