On the Cauchy Problem for a Class of Weakly Dissipative One-Dimensional Shallow Water Equations
We investigate a more general family of one-dimensional shallow water equations with a weakly dissipative term. First, we establish blow-up criteria for this family of equations. Then, global existence of the solution is also proved. Finally, we discuss the infinite propagation speed of this family of equations.
Recently, in , the following one-dimensional shallow water equations were studied: where and . A detailed description of the corresponding strong solution with the initial data was also given by them in .
When , , and , (1) reduces to the Camassa-Holm equation, which was derived physically by Camassa and Holm in  (found earlier by Fuchssteiner and Fokas  as a bi-Hamiltonian generalization of the KdV equation) by approximating directly the Hamiltonian for Euler’s equations in the shallow water region with representing the free surface above a flat bottom. The Camassa-Holm equation is completely integrable and has infinite conservation laws. Local well-posedness for the initial datum with was proved in [5, 6]. One of the remarkable features of Camassa-Holm equation is the presence of breaking waves and global solutions. Necessary and sufficient condition for wave breaking was established by Mckean  in 1998. A new and direct proof was also given in . The solitary waves of Camassa-Holm equation are peaked solitons. The orbital stability of the peakons was shown by Constantin and Strauss in  (see also ). The property of propagation speed of solutions to the Camassa-Holm equation, which was presented by Himonas and his collaborators in their work is worthy of being mentioned here .
In this paper, we consider the following weakly dissipative one-dimensional shallow water equation: where is the weakly dissipative term.
The paper is organized as follows. In Section 2, we establish the local well-posedness of the initial-value problem associated with (2) and present the precise blow-up scenario. Some blow-up results are given in Section 3. In Section 4, we establish a sufficient condition added on the initial data to guarantee global existence. We will consider the infinite propagation speed in Section 5.
2. Local Well-Posedness and Blow-Up Scenario
System (2) is equivalent to the following system: where , means doing convolution.
Theorem 1. Given , , then there exist a and a unique solution to (2) such that
To make the paper concise, we would like to omit the detailed proof, since one can find similar ones for these types of equations in .
3. Blow-Up Phenomenon
In this section, we will give some conditions to guarantee the finite time blowup. Motivated by Mckean’s deep observation for the Camassa-Holm equation , we can consider the similar particle trajectory as where is the lifespan of the solution: then is a diffeomorphism of the line. Taking derivative (5) with respect to , we obtain Therefore Hence, from (2), the following identity can be proved: In fact, direct calculation yields
Motivated by , we give the following theorem.
Theorem 2. Let , : suppose that , and there exists a such that , Then the corresponding solution to (2) with as the initial datum blows up in finite time.
Proof. Suppose that the solution exists globally. From (8) and initial condition (10), we have and
for all . Due to , we can write and as
for all .
By direct calculation, for , we have Similarly, for , we have So for any fixed , combination of (15) and (16), we obtain for all .
From the expression of in terms of , differentiating with respect to , we have where we have used (17), and the inequality . In addition, we also used the equation , which is obtained by differentiating equation (3).
For (11), we know that
Claim. is decreasing. and , for all .
Suppose that there exists a such that and on ; then or .
Now, let Firstly, differentiating , we have Secondly, by the same argument, we obtain Therefore, it follows from (21), (22), and the continuity property of ODEs that for all . This implies that can be extended to the infinity.
Moreover, using (21) and (22) again, we have the following equation for : where we use .
Now, recalling (18), we have
Putting (25) into (24), it yields Before finishing the proof, we need the following technical lemma.
Lemma 3 (see ). Suppose that is twice continuously differential satisfyingThen blows up in finite time. Moreover the blow-up time can be estimated in terms of the initial datum as
Let ; then (26) is an equation of type (27) with . The proof is complete by applying Lemma 3.
Theorem 5. Let . Suppose that and there exists a such that , Then the corresponding solution to (2) with as the initial datum blows up in finite time.
Proof. We easily obtain
Differentiating at the point with respect to , we get
Process of the proof is similar to Theorem 2. Thus to be concise, we omit the detailed proof.
When , using , (2) can be reformulated into which is the well-known Camassa-Holm equation. Meanwhile, we also find that the condition in Theorem 5 can be reformulated into which is one of the sufficient conditions to guarantee blow-up add-on initial data for the Camassa-Holm equation.
So, we show the necessary and sufficient condition for the special case and in the following theorem.
Theorem 6. When and , then the nonlinear wave equation (2) breaks if and only if some portion of the positive part of lies to the left of some portion of its negative part.
Proof. As studied in , when and , rewriting (2) yields
Recalling Mckean’s theorem in , (32) breaks if and only if some portion of the positive part of lies to the left of some portion of its negative part.
So (34) breaks if and only if some portion of the positive part of lies to the left of some portion of its negative part.
This completes the proof.
4. Global Existence
Now, let us try to find a condition for global existence. Unfortunately, When , like the Degasperis-Procesi equation , only the following easy one can be proved at present.
Theorem 8. Suppose that , and is one sign. Then the corresponding solution to (2) exists globally.
Proof. We can assume that . It is sufficient to prove that has a lower and upper bound for all . In fact, Therefore, we have This completes the proof.
5. Infinite Propagation Speed
In this section, we will give a more detailed description on the corresponding strong solution to (2) in its life span with initial data being compactly supported. The main theorem reads as follows.
Theorem 9. Let . Assume that for some and , is a strong solution of (2). If has compact support [a,c], 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 , from (8), so does has compact support in in in its lifespan. Hence the following functions are well-defined:
Thus, for , we obtain
Similarly, for , we have
Hence, as consequences of (40) and (41), we get
On the other hand,
It is easy to get
Putting the identity (44) into , we have
where we have used (42).
Therefore, in the lifespan of the solution, we get 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 , respectively.
This work is partially supported by Zhejiang Innovation Project (T200905), ZJNSF (Grant no. R6090109), and NSFC (Grant no. 10971197 and 11101376).