The Local Stability of Solutions for a Nonlinear Equation

Abstract

The approach of Kruzkov’s device of doubling the variables is applied to establish the local stability of strong solutions for a nonlinear partial differential equation in the space by assuming that the initial value only lies in the space .

1. Introduction and Main Results

Coclite and Karlsen [1] studied the well-posedness for the nonlinear equation where function satisfies or where is a positive constant. The existence of entropy weak solutions and several other dynamic properties for (1) are investigated in [1].

Consider the following partial differential equation: where is a positive constant and . If and , (4) becomes the Degasperis-Procesi equation (see [2–12]). If , we note that does not satisfy conditions (2) and (3). Recently, Wu [13] established the local well-posedness of strong solutions for (4) in the Sobolev space provided that and the initial value lies in the space with . The objective of this work is to study (4) in the case . The local stability of strong solutions for this case is established in the space . We think that this stability result is a new conclusion for (4).

When , the Cauchy problem of (4) takes the form which is equivalent to where the operator for any or with a parameter satisfying .

Using the approach of Kruzkov’s device of doubling the variables in [14], we obtain the following result.

Theorem 1. Let and be two strong solutions of problem (5) or (6) with initial data and . Then, for an arbitrary , holds, where is a constant depending on , , , , and .

This paper is organized as follows. Several lemmas are given in Section 2 and the proof of Theorem 1 is completed in Section 3.

2. Several Lemmas

Lemma 1 (see [13]). Assume that initial value . Then the solution of problem (5) with satisfies where and . Moreover, there exist two constants and depending only on such that In fact, if , we know that .

Lemma 2 (see [13]). Assume that . Then where is a constant independent of .

Lemma 3. Let and . Consider that holds, where is a constant independent of .

Proof. Since we complete the proof by using Lemma 2 and .

Let for an arbitrary . is the space of all infinitely differentiable functions with compact support in . We define as 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 where is a positive constant. Assume that the function is locally integrable in . We define the approximation of function as

We know that a point is defined as a Lebesgue point of function if For the Lebesgue point of the function , we get . Since the measure for the set of points which are not the Lebesgue points of is zero, we have as almost everywhere.

For any , let . Let represent the cone , . We let represent the cross section of the cone by the plane , . Let , where .

Lemma 4 (see [14]). Let the function be bounded and measurable in cylinder . If and , then the function
satisfies .

Lemma 5 (see [14]). If the function is bounded, then the function satisfies the Lipschitz condition in and , respectively.

Lemma 6. Let be the strong solution of problem (6) and . Then where is an arbitrary constant.

Proof. Let be an arbitrary twice smooth function on the line . We multiply the first equation of problem (6) by the function , where . Integrating over and transferring the derivatives with respect to and to the test function , for any constant , we obtain in which we have used .
Integration by parts yields Let be an approximation of the function and set . Using the properties of the , (18), and (19) and sending , we have which completes the proof.

We note that the proof of (17) can also be found in [14].

Lemma 7. Assume that and are two strong solutions of problem (6) associated with the initial data and . For any , holds, where depends on , , , , , and .

Proof. We obtain in which we have used Lemma 2. Using (23) and the Fubini theorem completes the proof.

3. Proof of Theorem 1

Here we state that the techniques used in this paper to establish the local stability of solutions for problem (6) come from the methods of Kruzkov’s device of doubling the variables presented in Kruzkov’s paper [14].

Proof of Theorem 1. For an arbitrary , set . Let . We assume that outside the cylinder We define where and . The function is defined in (13). Note that
Taking and and assuming that outside the cylinder , from Lemma 6, we have Similarly, it has from which we obtain
We claim that We note that the first two terms in the integrand of (29) can be represented in the form From Lemma 5, we know that satisfies the Lipschitz condition in and , respectively. By the choice of , we have outside the region as follows: Considering the estimate and the expression of function , we have
where the constant does not depend on . Using Lemma 4, we obtain as . The integral does not depend on . In fact, substituting , , , and noting that we have Hence Since we obtain
By Lemmas 3 and 4, we have as . Using (34), we have From (36) and (38)–(40), we prove that inequality (30) holds.
Let We define and choose the two numbers and , . In (30), we choose where When is sufficiently small, we note that function outside the cone and outside the set . For , we have
Applying (30) and (42)–(45) and suitably choosing large , we have the inequality where .
From (46), we obtain Using Lemma 7, we have where is a constant as described in (21).
Letting in (48) and sending , we have By the properties of the function for , we have where is independent of .
Set Using the similar proof of (50), we get from which we obtain Similarly, we have Then, we get
Letting and , from (49), (50), and (55), for any , we have where depends on , , , , and . Using the Gronwall inequality and (56) completes the proof of Theorem 1.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

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).