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The stability of strong solutions to a generalized BBM equation
Journal of Inequalities and Applications volume 2014, Article number: 3 (2014)
A nonlinear generalized Benjamin-Bona-Mahony equation is investigated. Using the estimates of strong solutions derived from the equation itself, we establish the stability of the solutions under the assumption that the initial value lies in the space .
Benjamin, Bona and Mahony  established the model
where a, b and k are constants. Equation (1) is often used as an alternative to the equation which describes unidirectional propagation of weakly long dispersive waves . As a model that characterizes long waves in nonlinear dispersive media, the equation, like the equation, was formally derived to describe an approximation for surface water waves in a uniform channel. Equation (1) covers not only the surface waves of long wavelength in liquids, but also hydromagnetic waves in cold plasma, acoustic waves in anharmonic crystals, and acoustic gravity waves in compressible fluids (see [2, 3]). Nonlinear stability of nonlinear periodic solutions of the regularized Benjamin-Ono equation and the Benjamin-Bona-Mahony equation with respect to perturbations of the same wavelength is analytically studied in . Unique continuation property and control for the Benjamin-Bona-Mahony equation on a periodic domain are discussed in . The () asymptotic property of solutions for the Benjamin-Bona-Mahony-Burgers equations is studied in  under certain assumptions on the initial data. The tanh technique is employed in  to get the compact and noncompact solutions for KP- and ZK- equations.
Applying the tanh method and the sine-cosine method, Wazwaz  obtained compactons, solitons, solitary patterns and periodic solutions for the following generalized Benjamin-Bona-Mahony equation
where , and are constants, and is an integer.
The objective of this work is to investigate Eq. (2). Using the methods of the Kruzkov’s device of doubling the variables presented in Kruzkov’s paper , we obtain the stability of strong solutions. Namely, for any solutions and satisfying Eq. (2), we will derive that
where T is the maximum existence time of solutions and and c depends on and . From our knowledge, we state that the stability of strong solutions for Eq. (2) has never been acquired in the literature.
This paper is organized as follows. Section 2 gives several lemmas and Section 3 establishes the proofs of the main result.
2 Several lemmas
Let for an arbitrary . We denote the space of all infinitely differentiable functions with compact support in by . We define to be a function which is infinitely differentiable on such that , for and . For any number , we let . Then we have that is a function in and
Assume that the function is locally integrable in . We define the approximation of function as
We call a Lebesgue point of function if
At any Lebesgue point of the function , we have . Since the set of points which are not Lebesgue points of has measure zero, we get as almost everywhere.
We introduce notations connected with the concept of a characteristic cone. For any , we define . Let ℧ denote the cone . We let represent the cross section of the cone ℧ by the plane , . Let , where .
Lemma 2.1 ()
Let the function be bounded and measurable in cylinder . If for any and any number , then the function
In fact, for Eq. (2), we have the conservation law
from which we have
where c only depends on b.
We write the equivalent form of Eq. (2) in the form
where the operator for any .
Lemma 2.2 Let , and . For any , it holds that
where the constant C is independent of time t.
Proof We have
Using (5)-(6), the integral and (8)-(9), we obtain the proof of Lemma 2.2. □
Lemma 2.3 Let u be the strong solution of Eq. (2), . Then
where k is an arbitrary constant.
Proof Let be an arbitrary twice smooth function on the line . We multiply Eq. (7) by the function , where . Integrating over and transferring the derivatives with respect to t and x to the test function f, we obtain
Let be an approximation of the function and set . Letting , we complete the proof. □
In fact, the proof of (10) can also be found in .
Lemma 2.4 Assume that and are two strong solutions of Eq. (2) associated with the initial data and . Then, for any ,
where c depends on and and f.
Proof Using (9), we have
in which we have used and . Using the Fubini theorem completes the proof. □
3 Main results
Theorem 3.1 Let and be two local or global strong solutions of Eq. (2) with initial data and , respectively. Let be the maximum existence time of solutions and . For any , it holds that
where c depends on and .
Proof For an arbitrary , set . Let . We assume that outside the cylinder
where and . The function is defined in (4). Note that
Taking and and assuming outside the cylinder ⨄, from Lemma 2.3, we have
Similarly, it holds
from which we obtain
We will show that
We note that the first term in the integrand of (20) can be represented in the form
By the choice of g, we have outside the region
Considering the estimate and the expression of function , we have
where the constant c does not depend on ε. Using Lemma 2.1, we obtain as . The integral does not depend on ε. In fact, substituting , , , and noting that
By Lemmas 2.1 and 2.2, we have as . Using (26), we have
From (28) and (32), we prove that inequality (21) holds.
We define the following increasing function
and choose two numbers and , . In (21), we choose
When h is sufficiently small, we note that function outside the cone ℧ and outside the set ⨄. For , we have the relation
Applying (21), (34)-(37) and the increasing properties of , we have the inequality
From (38), we obtain
Using Lemma 2.4, we have
where c is defined in Lemma 2.4.
Letting in (40) and letting , we have
By the properties of the function for , we have
where c is independent of ε.
Using the similar proof of (42), we get
from which we obtain
Similarly, we have
Then we get
Letting , and , from (41), (42) and (47), for any , we have
from which we complete the proof of Theorem 3.1 by using the Gronwall inequality. □
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The authors are very grateful to the reviewers for their helpful and valuable comments, which have led to a meaningful improvement of the paper. This work is supported by both the Fundamental Research Funds for the Central Universities (JBK120504) and the Applied and Basic Project of Sichuan Province (2012JY0020).
The authors declare that they have no competing interests.
The article is a joint work of two authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.
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Lai, S., Wang, A. The stability of strong solutions to a generalized BBM equation. J Inequal Appl 2014, 3 (2014). https://doi.org/10.1186/1029-242X-2014-3
- generalized BBM equation
- strong solutions