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Infinite propagation speed for the two component b-family system
Journal of Inequalities and Applications volume 2013, Article number: 382 (2013)
In this paper, we propose the infinite propagation speed for the two component b-family system. No matter what the profile of the compactly supported initial datum is, for any in its lifespan, the solution is positive at infinity and negative at negative infinity.
MSC:37L05, 35Q58, 26A12.
In this essay, we consider the following model, named two-component b-family system
where . As far as we known, it seems that system (1.1) appears initially by Guha in . There are two cases about this system (i) , and ; (ii) , and with . In , they applied Kato’s theory  to establish the local well-posedness for the Cauchy problem of (1.1). It is proved that there exists a unique solution for any with . The precise blow-up scenarios and some blow-up criteria were also established in .
We only consider the case ,
Obviously, under the constraint of , system (1.2) reduces to the b-family equations
which were derived physically by Holm and Staley in . Detailed description of the corresponding strong solutions to (1.3) with , being its initial data, was given by Zhou in . 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 precisely, 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. For and , (1.3) is the famous Camassa-Holm equation  and Degasperis-Procesi equation , respectively. Many papers [8–14] are devoted to their studies.
Another related system is the two component Camassa-Holm system. Recently, Constantin and Ivanov in  gave a demonstration about its derivation in view of the fluid shallow water theory from the hydrodynamic point of view. This generalization, similarly to the Camassa-Holm equation, possessed the peakon, multi-kink solutions and the bi-Hamiltonian structure  and is integrable. Well-posedness and wave breaking mechanism were discussed in , and the existence of global solutions was analyzed in . The infinite propagation speed was studied by Henry in .
In the following section, we will show our main results and give the detailed proof.
2 Main results
Motivated by Mckean’s deep observation for the Camassa-Holm equation , we can do the similar particle trajectory as
where T is the life span of the solution, then q is a diffeomorphism of the line. Differentiating the first equation in (2.1) with respect to x, one has
Our first result will show that m and ρ have compact support if their initial data have this property. m and ρ are the solutions of system (1.2).
Theorem 2.1 Assume that has compact support, contained in the interval , and that is also compactly supported, with support contained in . If is the maximal existence time of the unique classical solutions to the system (1.2) with the given initial data and , then for any , and have compact support.
Proof By the particle trajectory defined in (2.1), we find that
Since and is compactly supported, it follows from this relation that is compactly supported for all times , with support contained in the interval . Setting
it follows that is compactly supported, with its support contained in the interval , for all . □
In order to prove Theorem 2.3, we will use the following result.
Lemma 2.2 
Let be such that has compact support. Then u has compact support if and only if
Theorem 2.3 Let , , assume that the function has compact support. Let be the maximal existence time of the unique solution with initial data . If at every , has compact support, then u and ρ are identically zero.
Proof Using (1) and differentiating the left hand side of (2.2) with respect to t we get
where all boundary terms after integration by parts vanish as both and, by assumption, have compact support for all .
The expression under the integral on the right hand side of this relation must be identically zero by (2.2). This implies that all of the terms in the bracket must be identically zero, and in particular . This completes the proof. □
The theorem above means that if is a function with compact support, then the classical solution of system (1.2) must instantly lose the compactness of its support. The following theorem will give a detailed description about the profile of the solution .
Theorem 2.4 Let and , u is a nontrivial solution of (1.2). If has compact support , and is also initially compactly supported, on the interval , then for , we have
where and denote continuous nonvanishing functions with and for . Furthermore, is a strictly decreasing function, while is increasing function.
Proof Since and are compactly supported. By Theorem 2.1, m is compactly supported with its support contained in the interval . Hence the following functions are well-defined:
Then for , we have
Similarly, when , we get
Hence, as consequences of (2.3) and (2.4), we have
On the other hand,
It is easy to get
Substituting the identity (2.7) into , we obtain
where we use (2.5) and (2.6).
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 2.1 Theorem 2.3 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 negative at infinity and positive at negative infinity.
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This work is partially supported by ZJNSF (Grant No. LQ13A010008) and NSFC (Grant No. 11226176). Thanks are also given to the anonymous referees for careful reading and suggestions.
The authors declare that they have no competing interests.
ZW proposed the problems and finished the whole manuscript. YJ proved Theorems 2.1 and 2.3. LJ proved Theorem 2.4. All authors read and approved the final manuscript.
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Wei, Z., Jin, Y. & Jin, L. Infinite propagation speed for the two component b-family system. J Inequal Appl 2013, 382 (2013). https://doi.org/10.1186/1029-242X-2013-382
- two component b-family system
- infinite propagation speed