On the Study of Global Solutions for a Nonlinear Equation

Abstract

The well-posedness of global strong solutions for a nonlinear partial differential equation including the Novikov equation is established provided that its initial value satisfies a sign condition and with . If the initial value and the mean function of satisfies the sign condition, it is proved that there exists at least one global weak solution to the equation in the space in the sense of distribution and .

1. Introduction

Recently, Wu [1] obtained the existence of local solutions in the space with for the following nonlinear equation: where is a natural number, , and is a constant. Letting and , (1) becomes the Camassa-Holm equation [2]. If , and , (1) reduces to the Novikov equation [3].

A lot of works have been carried out to study various dynamic properties for the Camassa-Holm and the Novikov equations. Xin and Zhang [4] proved that there exists a global weak solution for the Camassa-Holm equation in the space without the assumption of sign conditions on the initial value. Coclite et al. [5] investigated the global weak solutions for a generalized hyperelastic rod wave equation or a generalized Camassa-Holm equation. It is shown in Constantin and Escher [6] that the blowup occurs in the form of breaking waves; namely, the solution remains bounded but its slope becomes unbounded in finite time. After wave breaking, the solution can be continued uniquely either as a global conservative weak solution [7] or a global dissipative solution [8–10]. The periodic and the nonperiodic Cauchy problems for the Novikov equation were discussed by Grayshan [11] in the Sobolev space. Using the Galerkin-type approximation method, Himonas and Holliman [12] established the well-posedness for the Novikov model in the Sobolev space with on both the line and the circle. The scattering theory was employed in Hone et al. [13] to find nonsmooth explicit soliton solutions with multiple peaks for the Novikov equation. Wu and Zhong [14] proved the existence of local strong and weak solutions for a generalized Novikov equation.

The objective of this work is to study (1) with . Namely, we investigate the problem where , and are described in (1). Assuming that the initial value satisfies a sign condition and , we will show that there exists a unique global strong solution in the Sobolev space . If the initial value and the mean function of satisfies the sign condition, it is shown that there exists at least one global weak solution to the equation in the space in the sense of distribution and .

The structure of this paper is as follows. The main results are given in Section 2. Several lemmas are given in Section 3. Section 4 establishes the proof of the main results.

2. Main Results

We define and let with . For the convolution , we know that for any with . Notation (or equivalently ) means that the mean function of is nonnegative; namely, (or equivalently ) for an arbitrary sufficiently small . For and nonnegative number , we let denote the Frechet space of all continuous -valued functions on and write .

We state the result of global strong solutions for problem (2).

Theorem 1. Let , and for all or for all . Then problem (2) has a unique strong solution satisfying

Definition 2. A function is called a global weak solution to problem (2) if for every and all , it holds that with .

Now we give the main result of global weak solution for problem (2).

Theorem 3. Let , (or equivalently ). Then problem (2) has a unique global weak solution in the sense of distribution and .

3. Several Lemmas

Lemma 4 (see [1]). Let with . Then the Cauchy problem (2) has a unique local solution where depends on .

Using the first equation of system (2) derives which yields the conservation law

Lemma 5 (see [1]). Let and the function is a solution of problem (2) and the initial data . Then the following inequalities hold:
For , there is a constant such that
For , there is a constant such that

Consider the differential equation where is the solution of problem (2) and is the maximal existence time of the solution.

Lemma 6. Let , and let be the maximal existence time of the solution to problem (2). Then system (12) has a unique solution . Moreover, the map is an increasing diffeomorphism of with for .

Proof. From Lemma 4, we know that there exists a unique solution
The Sobolev imbedding theorem derives . This means that two functions and are bounded, Lipschitz in space and in time. Using the existence and uniqueness theorem of ordinary differential equations, we derive that problem (12) has a unique solution .
Differentiating (12) with respect to gives rise to from which we obtain For every , applying the Sobolev imbedding theorem results in
Therefore, we know that there exists a constant such that for . The proof is completed.

Lemma 7. Let with , and let be the maximal existence time of the problem (2); it holds that where and .

Proof. We have from which we have Using completes the proof.

Lemma 8. If , or , then the solution of problem (2) satisfies

Proof. We only need to prove this lemma for the case since the proof of the other case is similar. It follows from Lemmas 6 and 7 that . Letting , we have which derives On the other hand, we have The inequalities (22) and (23) derive that inequality (20) is valid.

Lemma 9. For , and , it holds that where is a constant independent of .

The proof of this lemma can be found in [15, 16].

From Lemma 4, it derives that the Cauchy problem has a unique solution depending on the parameter . We write to represent the solution of problem (25). Using Lemma 4 derives that since .

Lemma 10. Provided that , and (or equivalently ), then there exists a constant independent of and such that the solution of problem (25) satisfies

Proof. Using Lemmas 5 and 9, if with , we have where is independent of and .
From Lemma 8, we have which completes the proof.

4. Proof of Main Results

Proof of Theorem 1. Since and taking in inequality (10), we have from which we obtain Applying Lemma 8 yields from which we complete the proof of Theorem 1.

Provided that , for problem (25), applying Lemmas 5, 8, and 10, and the Gronwall’s inequality, we obtain the inequalities where , and is a constant independent of and . Using the Aubin compactness theorem, we know that that there is a subsequence of such that and their temporal derivatives converge weakly to a function and its derivative in the space and , respectively, where is an arbitrary fixed positive number. In addition, for any real number , converges strongly to the function in the space for and converges strongly to in the space for .

Proof of Theorem 3. For an arbitrary fixed , using Lemma 10, we know that is bounded in the space . Therefore, we derive that the sequences , , , and converge weakly to , , , and in for any , separately. Applying the identity , we conclude that satisfies the equation where . We know that is a separable Banach space and is a bounded sequence in the dual space of . Thus, there exists a subsequence of , still denoted by , weakly star convergent to a function in . Since weakly converges to in , it derives that almost everywhere. Therefore, we obtain . Since is an arbitrary number, we complete the proof of existence of global weak solutions to problem (2).

Conflict of Interests

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

Acknowledgment

This work is supported by the Fundamental Research Funds for the Central Universities (JBK120504).