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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 619068, 5 pages
http://dx.doi.org/10.1155/2013/619068
Research Article

The Local Strong Solutions and Global Weak Solutions for a Nonlinear Equation

Department of Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, China

Received 31 March 2013; Accepted 26 April 2013

Academic Editor: Shaoyong Lai

Copyright © 2013 Meng Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The existence and uniqueness of local strong solutions for a nonlinear equation are investigated in the Sobolev space provided that the initial value lies in with . Meanwhile, we prove the existence of global weak solutions in for the equation.

1. Introduction

Coclite and Karlsen [1] investigated the well posedness in classes of discontinuous functions for the generalized Degasperis-Procesi equation: which is subject to the condition or where is a positive constant. The existence and stability of entropy weak solutions belonging to the class are established for (1) in paper [1].

In this work, we study the following model: where is a positive constant and . If and , (4) reduces to the classical Degasperis-Procesi model (see [213]). Here, we notice that assumptions (2) and (3) do not include the case . In this paper, we will study the case , and is an arbitrary positive constant.

In fact, the Cauchy problem of (4) in the case is equivalent to the following system: Using the operator to multiply the first equation of the problem (5), we obtain

It is shown in this work that there exists a unique local strong solution in the Sobolev space by assuming that the initial value belongs to with . In addition, we prove the existence of global weak solutions in for the system (6).

This paper is organized as follows. Section 2 investigates the existence and uniqueness of local strong solutions. The result about global weak solution is given in Section 3.

2. Local Existence

In this section, we will use the Kato theorem in [14] for abstract differential equation to establish the existence of local strong solution for the problem (6). Let us consider the following problem: Let and be Hilbert spaces such that is continuously and densely embedded in , and let be a topological isomorphism. Let be the space of all bounded linear operators from to . In the case of , we denote this space by . We illustrate the following conditions in which , , , and are constants depending only on .(i) for with and (i.e., is quasi-m-accretive), uniformly on bounded sets in .(ii), where is bounded, uniformly on bounded sets in . Moreover, (iii) extends to a map from into , is bounded on bounded sets in , and satisfies

Kato Theorem (see [14]). Assume that conditions (i), (ii), and (iii) hold. If , there is a maximal depending only on and a unique solution to the problem (7) such that Moreover, the map is a continuous map from to the following space:

In order to apply the Kato theorem to establish the local well posedness for the problem (6), we let , , , , , and . We know that is an isomorphism of onto . Now, we cite the following Lemmas.

Lemma 1. The operator with , belongs to .

Lemma 2. Assume that with and . Then, for all . Moreover,

Lemma 3. For , and , it holds that for and

The above three Lemmas can be found in Ni and Zhou [15].

Lemma 4. Let with and . Then, is bounded on bounded sets in and satisfies

Proof. For , we know that . Consequently, we have

Using the Kato Theorem, Lemmas 14, we immediately obtain the local well-posedness theorem.

Theorem 5. Assume that with . Then, there exists a such that the system (5) or the problem (6) has a unique solution satisfying

3. Weak Solutions

In this section, our aim is to establish the existence of global weak solutions for the system (6). Firstly, we prove that the solution of the problem (5) is bounded in the space and .

Lemma 6. The solution of the problem (5) with satisfies where , and . Moreover, there exist two constants and depending only on such that

Proof. Setting and and using the first equation of the problem (5), we obtain and Using the Parseval identity and (21), we obtain (19) and (20).

From Theorem 5, we know that for any with , there exists a maximal and a unique strong solution to the problem (6) such that Firstly, we study the following differential equation:

Lemma 7. Let , , and let be the maximal existence time of the solution to the problem (6). Then, the problem (23) has a unique solution . Moreover, the map is an increasing diffeomorphism of with for .

Proof. Using Theorem 5, we obtain and . Therefore, we know that functions and are bounded, Lipschitz in space, and in time. Using the existence and uniqueness theorem for ordinary differential equations derives that the problem (23) has a unique solution .
Differentiating (23) with respect to gives rise to the following: from which we obtain For every , using the Sobolev imbedding theorem yields that
It is inferred that there exists a constant such that for . It completes the proof.

Lemma 8. Assume that . Let be the maximal existence time of the solution to the problem (6). Then, it has where is a constant independent of .

Proof. Let , we have for all and . Using a simple density argument presented in [7], it suffices to consider to prove this lemma. Let be the maximal existence time of the solution to the problem (6) with the initial value such that . From (6), we have Since from (29), we have Using Lemma 6 and (30) derives that where is a positive constant independent of . Using (31) results in the following: Therefore, Using the Sobolev embedding theorem to ensure the uniform boundedness of for with , from Lemma 7, for every , we get a constant such that We deduce from (34) that the function is strictly increasing on with as long as . It follows from (33) that Using the Gronwall inequality and (35) derives that (27) holds.

For a real number with , suppose that the function is in , and let be the convolution of the function and such that the Fourier transform of satisfies , , and for any . Then, we have . It follows from Theorem 5 that for each satisfying , the Cauchy problem, has a unique solution . Using Lemmas 6 and 8, for every , we obtain Sending , we know that inequalities (37) are still valid. This means that for , (37) hold.

Now, we state the concepts of weak solutions.

Definition 9 (weak solution). We call a function a weak solution of the Cauchy problem (5) provided that(i);(ii) in , that is, for all there holds the following identity:

Theorem 10. Let . Then, there exists a weak solution to the problem (5).

Proof. Consider the problem (36). For an arbitrary , choosing a subsequence , from (37), we know that is bounded in and is uniformly bounded in . Therefore, we obtain that is bounded in . Therefore, there exist subsequences and , still denoted by and , are weakly convergent to in . Noticing (38) completes the proof.

Acknowledgment

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

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