We obtain the existence of a weak solution to a fractional nonlinear hyperbolic equation arising from relative theory by the Galerkin method. Its uniqueness is also discussed. Furthermore, we show the regularity of the obtained solution. In our proof, we use harmonic analysis techniques and compactness arguments.

1. Introduction

This paper is concerned with the following fractional partial differential equations in : where is the square root of the Laplacian operator, , are two real parameters, and is given. Equations (1)–(3) play an important role in nuclear force and relativistic theory.

The fractional diffusion operator is nonlocal except when , which means that depends not only on for near , but on for all . Fractional differential equations, arising from mathematical physics such as viscoelasticity, electrochemistry, control theory, porous media, and electromagnetism, now attract the interests of many mathematicians; see [14] and references therein. In the past ten years, the quasi-geostrophic equation with fractional dissipation has been extensively studied; see Constantin et al. [59] and references therein. The relativistic equation shares some similar difficulties with the quasi-geostrophic equation. However, the equations studied in this paper are more complicated in that the fractional diffusion operator and the nonlinear term in (1)–(3) bring new difficulties in passing to the limits of the approximate solutions, and hence, new devices must be introduced to overcome these obstacles.

When , (1)–(3) become the standard equations, which were intensively studied in the past century. The readers are referred to[1012] for more details.

Interestingly enough, the parabolic version of (1)–(3) with convection corresponds to the Navier-Stokes equations with damping; see [13, 14].

We now collect the notations in this paper. The square root of the negative Laplacian , is given by (in terms of Fourier series) where is the Fourier coefficients of : More generally, for can be defined as

We will also invoke the notion of homogeneous Sobolev space , which comprises all tempered distributions on such that We also recall the meaning of by the weak () convergence in . Since the dual of is and the space is dense in (noticing the periodic boundary conditions), we have with if and only if with or if and only if (9) holds.

We now close this introduction by outlining the rest of this paper. In Section 2, we prove the existence of a weak solution to (1)–(3); see Theorem 4. The uniqueness of such weak solutions is discussed in Section 3; see Theorem 5. Finally, a regularity result is obtained in Section 4; see Theorem 6.

2. Existence of a Weak Solution

First, let us recall the following two fundamental lemmas.

Lemma 1 (see [15]). Let be a Banach space, , and , then .

Lemma 2 (see [15]). Let be a bounded domain in , and belong to with Then in weakly.

Let us now give the weak formulation of (1)–(3).

Definition 3. Let , ,, and . A measurable function is said to be a weak solution on to (1)–(3) if the following conditions hold: (1) and ; (2) (1) holds in the sense of distributions; that is,
  for each ; (3) a.e. in ; (4) a.e. in .

To see how item (4) in Definition 3 makes sense, we rewrite (1) as Noticing that [16] we have Also, due to the fact that we deduce from (13) that Thus Hence, by item (1) of Definition 3, (18), and Lemma 1, we gather that From this, we see that item (4) of Definition 3 makes sense, as claimed.

Now, we state our existence results in the following theorem.

Theorem 4. Let , then there exists at least one weak solution on to (1)–(3), taking as initial data.

Proof. We use Galerkin method to establish the existence of such a solution.
Step   1 (construction of approximate solution). Let be a dense and total basis in , and consider the approximate solution which has the form where satisfy the following ordinary differential system: Here and hereafter, we denote for .
The system (24)–(26) is nonsingular because are linear independent. Thus, we may apply standard theory of ordinary differential equations to obtain the existence of a local solution to (24)–(26) on , for some . We will then, in the next step, establish some a priori estimates of the obtained solutions which will ensure that .
Step   2 (a priori estimates). Taking the inner product of (24) with in , we obtain Integrating over and invoking Hölder inequality then yield Due to (20), (25), and (26), we have that , are uniformly bounded. Thus, (28) becomes Noticing that , we have Applying Gronwall's inequality then yields and hence,
Step   3 (passage to limit ).
By (31) and (32), we have, up to a subsequence, still denoted by , that and also that are uniformly bounded in . Thus by Lemma 1, we find that Hence, there exists a function such that By Lemma 2 and the fact that we know then that Fixing , we now pass to the limit in (24) to deduce that A simple density argument then shows that for all .
Up to now, we have proved items (1) and (2) in Definition 3. Let us turn our attention to items (3) and (4) in Definition 3.
By (33) and Lemma 1, we know weakly in , and from (25), in ; thus item (3) of Definition 3 is verified.
By (24) and (39), we see that and by Lemma 1, we have On the other hand, (26) implies that Hence This verifies item (4) in Definition 3.

3. Uniqueness of Weak Solutions

In this section, we will discuss the uniqueness of weak solutions of (1)–(3). We only obtain partial results in case . More precisely, we have the following theorem.

Theorem 5. Assuming as in Theorem 4, then there exists an unique weak solution of (1)–(3), in case .

Proof. Let , be two weak solutions for (1)–(3) given in Theorem 4 with the same datum. Then satisfies Also, we have
Taking the inner product of (44) with in , we obtain
Invoking Hölder and Sobolev inequalities, we obtain where , and we use ).
Thus, (47) becomes and we get , as desired.

4. Regularity of the Weak Solution

Now we discuss the regularity of solutions for (1)–(3). If the initial value and force are more regular, then so is the solution.

Theorem 6. Let and . Then there exists a unique weak solution for (1)–(3). Furthermore,

Proof. We just establish the a priori bounds, since the verification follows directly from passing to the limit for Galerkin approximate solutions.
Step  1 (bounds for initial data). Observing that we see by Sobolev inequality that By formula (13), we have
Step  2 (bounds for , ). Differentiating (1) with respect to , we find that Taking the inner product of (55) with , we obtain by integration by parts that The first term can be easily dominated by using Hölder inequality as To tackle , we invoke Hölder and Sobolev inequalities to deduce that where , are chosen so that and thus Gathering (57) and (58) into (56), it follows that Gronwall inequality then implies that
Step  3 (bounds for ). Rewriting (1) as we have , in view of similar inequalities satisfied by as (53).
This completes the proof of Theorem 6.

Conflict of Interests

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


Zujin Zhang was partially supported by the Youth Natural Science Foundation of Jiangxi Province (20132BAB211007), the Science Foundation of Jiangxi Provincial Department of Education (GJJ13658, GJJ13659), and the National Natural Science Foundation of China (11326138, 11361004). Xiaofeng Wang was partially supported by Guangzhou Higher Education Science and Technology Projection (2012A018).