On a Fractional Nonlinear Hyperbolic Equation Arising from Relative Theory

Abstract

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 𝕋^𝑛=ℝ^𝑛/ℤ^𝑛: 𝜕²𝑢𝜕𝑡²−Λ^2𝛼𝑢+|𝑢|^𝜌𝑢()𝑢=𝑓,𝑥,0=𝑢₀(𝑥),𝜕𝑢()𝜕𝑡𝑥,0=𝑢₁(𝑥), (1)(2)(3) where Λ=(−Δ)^1/2 is the square root of the Laplacian operator, 𝛼∈(0,1), 𝜌∈[0,∞) are two real parameters, and 𝑓(𝑥,𝑡) is given. Equations (1)–(3) play an important role in nuclear force and relativistic theory.

The fractional diffusion operator Λ^2𝛼 is nonlocal except when 𝛼∈ℕ, which means that Λ^2𝛼𝑢(𝑥) depends not only on 𝑢(𝑦) for 𝑦 near 𝑥, but on 𝑢(𝑦) for all 𝑦∈ℝ^𝑛. 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 [1–4] and references therein. In the past ten years, the quasi-geostrophic equation with fractional dissipation has been extensively studied; see Constantin et al. [5–9] 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 𝛼=1, (1)–(3) become the standard equations, which were intensively studied in the past century. The readers are referred to[10–12] 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 −Δ, Λ is given by (in terms of Fourier series) Λ𝑓_𝑘=|𝑘|𝑓_𝑘,𝑘∈ℤ^𝑛, (4) where 𝑓_𝑘 is the Fourier coefficients of 𝑓: 𝑓_𝑘=1(2𝜋)^𝑛_𝕋^𝑛𝑓(𝑥)𝑒^{−𝑖𝑘⋅𝑥}d𝑥,𝑘∈ℤ^𝑛. (5) More generally, Λ^2𝛼𝑓 for 𝛼∈ℝ can be defined as Λ^2𝛼𝑓_𝑘=|𝑘|^2𝛼𝑓_𝑘,𝑘∈ℤ^𝑛. (6)

We will also invoke the notion of homogeneous Sobolev space 𝐻̇^𝑠(𝕋^𝑛)(𝑠∈ℝ), which comprises all tempered distributions 𝑓 on ℝ^𝑛 such that 𝑓_{̇𝐻𝑠(𝕋𝑛)}=_{𝑘∈ℤ𝑛⧵{0}}|𝑘|^2𝑠𝑢_𝑘^21/2. (7) We also recall the meaning of by the weak (weakly^∗) convergence in 𝐿^𝑝(0,𝑇;𝐿^𝑞(𝕋^𝑛)). Since the dual of 𝐿^𝑟(𝕋^𝑛)(1≤𝑟<∞) is 𝐿^{𝑟/(𝑟−1)}(𝕋^𝑛) and the space 𝐶^∞(𝕋^𝑛) is dense in 𝐿^𝑟(𝕋^𝑛) (noticing the periodic boundary conditions), we have 𝑓_𝑘⇀𝑓in𝐿^𝑝0,𝑇;𝐿^𝑞𝕋³ (8) with 1≤𝑝,𝑞<+∞ if and only if ^𝑇_0𝕋𝑛𝑓_𝑘𝜙d𝑥d𝑡⟶^𝑇_0𝕋𝑛𝑓𝜙d𝑥d𝑡,∀𝜙∈𝐶^∞₀0,𝑇;𝐶^∞𝕋^𝑛,𝑓_𝑘⇀𝑓in𝐿^𝑝0,𝑇;𝐿^𝑞𝕋³ (9)(10) with 𝑝=+∞ or 𝑞=+∞ 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 𝑋 be a Banach space, 𝑓∈𝐿^𝑝(0,𝑇;𝑋), and 𝜕𝑓/𝜕𝑡∈𝐿^𝑝(0,𝑇;𝑋)(1≤𝑝≤∞), then 𝑓∈𝐶([0,𝑇];𝑋).

Lemma 2 (see [15]). Let 𝒟 be a bounded domain in ℝ^𝑛×ℝ₊, and {𝑔_𝑘}^∞_𝑘=1,𝑔 belong to 𝐿^𝑞(𝒟)(1<𝑞<∞) with 𝑔_{𝑘𝐿𝑞(𝒟)}≤𝐶,𝑔_𝑘⟶𝑔𝑎.𝑒.𝒟. (11) Then 𝑔_𝑘⇀𝑔 in 𝐿^𝑞(𝒟) weakly.

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

Definition 3. Let 𝑇>0, 𝑢₀∈𝐻̇^𝛼(𝕋^𝑛)∩𝐿^𝜌+2(𝕋^𝑛),𝑢₁∈𝐿²(𝕋^𝑛), and 𝑓∈𝐿²(0,𝑇;𝐿²(𝕋^𝑛)). A measurable function 𝑢(𝑥,𝑡) is said to be a weak solution on [0,𝑇] to (1)–(3) if the following conditions hold: (1)𝑢∈𝐿^∞(0,𝑇;𝐻̇^𝛼(𝕋^𝑛)∩𝐿^𝜌+2(𝕋^𝑛)) and 𝜕𝑢/𝜕𝑡∈𝐿^∞(0,𝑇;𝐿²(𝕋^𝑛)); (2) (1) holds in the sense of distributions; that is, ^𝑇_0𝕋𝑛𝜕𝑢⋅𝜕𝑡𝜕𝜙𝜕𝑡d𝑥d𝑡−^𝑇_0𝕋𝑛Λ^𝛼𝑢⋅Λ^𝛼+𝜙d𝑥d𝑡_{𝑇0𝕋𝑛}|𝑢|^𝜌=𝑢𝜙d𝑥d𝑡_𝕋𝑛𝑢₁𝜙_𝑡d𝑥+^𝑇_0𝕋𝑛𝑓𝜙d𝑥d𝑡, (12)
  for each 𝜙∈𝐶^∞(𝕋^𝑛×(0,𝑇)); (3)𝑢(𝑥,0)=𝑢₀(𝑥) a.e. in 𝕋^𝑛; (4)(𝜕𝑢/𝜕𝑡)(𝑥,0)=𝑢₁(𝑥) a.e. in 𝕋^𝑛.

To see how item (4) in Definition 3 makes sense, we rewrite (1) as 𝜕²𝑢𝜕𝑡²=𝑓+Λ^2𝛼𝑢−|𝑢|^𝜌𝑢. (13) Noticing that [16] Λ^2𝛼iscontinuousfrom𝐻̇^𝛼𝕋^𝑛to𝐻̇^−𝛼𝕋^𝑛, (14) we have Λ^2𝛼𝑢∈𝐿^∞0,𝑇;𝐻̇^−𝛼𝕋^𝑛. (15) Also, due to the fact that |𝑢|^𝜌𝑢_{𝐿(𝜌+2)/(𝜌+1)(𝕋𝑛)}=‖𝑢‖_𝐿^𝜌+1_{𝜌+2(𝕋𝑛)}, (16) we deduce from (13) that 𝜕²𝑢𝜕𝑡²∈𝐿²0,𝑇;𝐿²𝕋^𝑛+𝐿^∞0,𝑇;𝐻̇^−𝛼𝕋^𝑛+𝐿^{(𝜌+2)/(𝜌+1)}𝕋^𝑛. (17) Thus 𝜕²𝑢𝜕𝑡²∈𝐿²0,𝑇;𝐻̇^−𝛼𝕋^𝑛+𝐿^{(𝜌+2)/(𝜌+1)}𝕋^𝑛. (18) Hence, by item (1) of Definition 3, (18), and Lemma 1, we gather that 𝜕𝑢[];𝐻̇𝜕𝑡∈𝐶0,𝑇^−𝛼𝕋^𝑛+𝐿^{(𝜌+2)/(𝜌+1)}𝕋^𝑛. (19) 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 𝑇>0, 𝑢₀∈𝐻̇^𝛼𝕋^𝑛𝐿^𝜌+2𝕋^𝑛𝑢₁,∈𝐿²𝕋^𝑛,𝑓∈𝐿²0,𝑇;𝐿²𝕋^𝑛; (20)(21)(22) then there exists at least one weak solution 𝑢 on [0,𝑇] to (1)–(3), taking 𝑢₀,𝑢₁ as initial data.

Proof. We use Galerkin method to establish the existence of such a solution.
Step   1 (construction of approximate solution). Let {𝑤_𝑖}^∞_𝑖=1 be a dense and total basis in 𝐻̇^𝛼(𝕋^𝑛)∩𝐿^𝜌+2(𝕋^𝑛), and consider the approximate solution which has the form 𝑢_𝑚(𝑡)=^𝑚_𝑖=1𝑐_𝑖𝑚(𝑡)𝑤_𝑖, (23) where 𝑐_𝑖𝑚 satisfy the following ordinary differential system: 𝜕²𝑢_𝑚𝜕𝑡²,𝑤_𝑗−Λ^2𝛼𝑢_𝑚,𝑤_𝑗+𝑢_𝑚^𝜌𝑢_𝑚,𝑤_𝑗=𝑓,𝑤_𝑗,𝑢1≤𝑗≤𝑚,_𝑚(0)=𝑢_0𝑚=^𝑚_𝑖=1𝛼_𝑖𝑚𝑤_𝑖⟶𝑢₀in𝐻̇^𝛼𝕋^𝑛𝐿^𝜌+2𝕋^𝑛,as𝑚⟶∞,𝜕𝑢_𝑚(0)𝜕𝑡=𝑢_1𝑚=^𝑚_𝑖=1𝛽_𝑖𝑚𝑤_𝑖⟶𝑢₁in𝐿²𝕋^𝑛,as𝑚⟶∞. (24)(25)(26) Here and hereafter, we denote (𝑢,𝑣)=∫_𝕋𝑛𝑢𝑣𝑑𝑥 for 𝑢,𝑣∈𝐿²(𝕋^𝑛).
The system (24)–(26) is nonsingular because {𝑤_𝑖} 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 [0,𝑡_𝑚], for some 𝑡_𝑚>0. We will then, in the next step, establish some a priori estimates of the obtained solutions which will ensure that 𝑡_𝑚=𝑇.
Step   2 (a priori estimates). Taking the inner product of (24) with 𝑐^′_𝑖𝑚(𝑡) in 𝐿²(ℝ^𝑛), we obtain 12𝑑𝑑𝑡𝜕𝑢_𝑚(𝑡𝜕𝑡)²_{𝐿2(𝕋𝑛)}+Λ^𝛼𝑢_𝑚(𝑡),Λ^𝛼𝑢_𝑚(𝑡)+1𝑑𝜌+2𝑑𝑡_𝕋^𝑛𝑢_𝑚(𝑥,𝑡)^𝜌+2𝑑𝑥=𝑓(𝑡),𝜕𝑢_𝑚(𝑡).𝜕𝑡 (27) Integrating over [0,𝑡] and invoking Hölder inequality then yield 12𝜕𝑢_𝑚(𝑡𝜕𝑡)²_{𝐿2(𝕋𝑛)}+Λ^𝛼𝑢_𝑚(𝑡)²_{𝐿2(𝕋𝑛)}+1𝑢𝜌+2^𝑚(𝑡)_𝐿^𝜌+2_{𝜌+2(𝕋𝑛)}≤12𝜕𝑢_𝑚(0𝜕𝑡)²_{𝐿2(𝕋𝑛)}+Λ^𝛼𝑢_𝑚(0)²_{𝐿2(𝕋𝑛)}+1𝑢𝜌+2^𝑚(0)_𝐿^𝜌+2_{𝜌+2(𝕋𝑛)}+^𝑡₀𝑓(𝜏)_{𝐿2(𝕋𝑛)}𝜕𝑢_𝑚(𝜏𝜕𝜏)_{𝐿2(𝕋𝑛)}[].𝑑𝜏,∀𝑡∈0,𝑇 (28) Due to (20), (25), and (26), we have that ‖(𝜕𝑢_𝑚/𝜕𝑡)(0)‖²_{𝐿2(𝕋𝑛)},‖𝑢_𝑚(0)‖²,and‖𝑢_𝑚(0)‖_𝐿^𝜌+2_{𝜌+2(𝕋𝑛)} are uniformly bounded. Thus, (28) becomes 12𝜕𝑢_𝑚𝜕𝑡(𝑡)²_{𝐿2(𝕋𝑛)}+Λ^𝛼𝑢_𝑚(𝑡)²_{𝐿2(𝕋𝑛)}+1𝑢𝜌+2^𝑚(𝑡)_𝐿^𝜌+2_{𝜌+2(𝕋𝑛)}1≤𝐶+2^𝑡₀𝑓(𝜏)²_{𝐿2(𝕋𝑛)}+1𝑑𝜏2^𝑡₀𝜕𝑢_𝑚(𝜏𝜕𝜏)²_{𝐿2(𝕋𝑛)}𝑑𝜏,∀𝑡∈[0,𝑇]. (29) Noticing that 𝑓∈𝐿²(0,𝑇;𝐿²(𝕋^𝑛)), we have 12𝜕𝑢_𝑚𝜕𝑡(𝑡)²_{𝐿2(𝕋𝑛)}+Λ^𝛼𝑢_𝑚(𝑡)²_{𝐿2(𝕋𝑛)}+1𝑢𝜌+2^𝑚(𝑡)_𝐿^𝜌+2_{𝜌+2(𝕋𝑛)}≤12^𝑡₀𝜕𝑢_𝑚𝜕𝜏(𝜏)²_{𝐿2(𝕋𝑛)}𝑑𝜏+𝐶,∀𝑡∈[0,𝑇]. (30) Applying Gronwall's inequality then yields 𝜕𝑢_𝑚(𝑡𝜕𝑡)_{𝐿2(𝕋𝑛)}≤𝐶, (31) and hence,𝑢_𝑚(𝑡)_{𝐿𝜌+2(𝕋𝑛)}≤𝐶,Λ^𝛼𝑢_𝑚(𝑡)_{𝐿2(𝕋𝑛)}≤𝐶. (32)
Step   3 (passage to limit 𝑚→∞).
By (31) and (32), we have, up to a subsequence, still denoted by 𝑢_𝑚, that 𝑢_𝑚⇀𝑢weakly∗in𝐿^∞0,𝑇;𝐻̇^𝛼𝕋^𝑛𝐿^𝜌+2𝕋^𝑛,as𝑚⟶∞,𝜕𝑢_𝑚⇀𝜕𝑡𝜕𝑢𝜕𝑡weakly∗in𝐿^∞0,𝑇;𝐿²𝕋^𝑛,as𝑚⟶∞, (33) and also that 𝑢_𝑚 are uniformly bounded in 𝐿²(0,𝑇;𝐻̇^𝛼(𝕋^𝑛)). Thus by Lemma 1, we find that 𝑢_𝑚⟶𝑢stronglyin𝐿²0,𝑇;𝐿²𝕋^𝑛,a.e.𝕋^𝑛,as𝑚⟶∞. (34) Hence, there exists a function 𝑤(𝑥,𝑡) such that 𝑢_𝑚^𝜌𝑢_𝑚⟶𝑤(𝑥,𝑡)weakly∗in𝐿^∞×0,𝑇;𝐿^{(𝜌+2)/(𝜌+1)}𝕋^𝑛,as𝑚⟶∞. (35) By Lemma 2 and the fact that 𝑢_𝑚^𝜌𝑢_𝑚⟶|𝑢|^𝜌𝑢,a.e.in𝕋^𝑛, (36) we know then that 𝑤=|𝑢|^𝜌𝑢. (37) Fixing 𝑗, we now pass to the limit 𝑚→∞ in (24) to deduce that 𝜕²𝑢𝜕𝑡²(𝑡),𝑤_𝑗+Λ^2𝛼𝑢(𝑡),𝑤_𝑗+|𝑢(𝑡)|^𝜌𝑢(𝑡),𝑤_𝑗=𝑓(𝑡),𝑤_𝑗. (38) A simple density argument then shows that 𝜕²𝑢𝜕𝑡²(𝑡),𝑣+Λ^2𝛼𝑢(𝑡),𝑣+|𝑢(𝑡)|^𝜌𝑢(𝑡),𝑣=𝑓(𝑡),𝑣, (39) for all 𝑣∈𝐻̇^𝛼(𝕋^𝑛)∩𝐿^𝜌+2(𝕋^𝑛).
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 𝑢_𝑚(0)⇀𝑢(0) weakly in 𝐿²(𝕋^𝑛), and from (25), 𝑢_𝑚(0)→𝑢₀ in 𝐻̇^𝛼(𝕋^𝑛)∩𝐿^𝜌+2(𝕋^𝑛); thus item (3) of Definition 3 is verified.
By (24) and (39), we see that 𝜕²𝑢_𝑚𝜕𝑡²(0),𝑤_𝑗⟶𝜕²𝑢𝜕𝑡²(0),𝑤_𝑗weakly∗, (40) and by Lemma 1, we have 𝜕𝑢_𝑚(0)𝜕𝑡,𝑤_𝑗⟶𝜕𝑢(0)𝜕𝑡,𝑤_𝑗. (41) On the other hand, (26) implies that 𝜕𝑢_𝑚(0)𝜕𝑡,𝑤_𝑗⟶𝑢₁,𝑤_𝑗. (42) Hence 𝜕𝑢(0)𝜕𝑡,𝑤_𝑗=𝑢₁,𝑤_𝑗,∀𝑗∈ℕ. (43) 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 0≤𝜌≤2𝛼/(𝑛−2𝛼). 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 0≤𝜌≤2𝛼/(𝑛−2𝛼).

Proof. Let 𝑢, 𝑣 be two weak solutions for (1)–(3) given in Theorem 4 with the same datum. Then 𝑤=𝑢−𝑣 satisfies 𝜕²𝑤𝜕𝑡²+Λ^2𝛼𝑤=|𝑣|^𝜌𝑣−|𝑢|^𝜌𝑢, (44)𝑤(0)=0,𝜕𝑤(0)𝜕𝑡=0. (45) Also, we have 𝑤∈𝐿^∞0,𝑇;𝐻̇^𝛼𝐿^𝜌+2𝕋^𝑛,𝜕𝑤𝜕𝑡∈𝐿^∞0,𝑇;𝐿²𝕋^𝑛. (46)
Taking the inner product of (44) with 𝜕𝑤/𝜕𝑡 in 𝐿²(𝕋^𝑛), we obtain 12𝑑𝑑𝑡𝜕𝑤𝜕𝑡²_𝐿²_(𝕋^𝑛₎+‖𝑤‖²_{̇𝐻𝛼(𝕋}^𝑛₎=_𝕋𝑛|𝑣|^𝜌𝑣−|𝑢|^𝜌𝑢𝜕𝑤𝜕𝑡𝑑𝑥. (47)
Invoking Hölder and Sobolev inequalities, we obtain _𝕋𝑛|𝑣|^𝜌𝑣−|𝑢|^𝜌𝑢𝜕𝑤≤𝜕𝑡𝑑𝑥𝜌+1_𝕋𝑛sup|𝑢|^𝜌,|𝑣|^𝜌|𝑤|𝜕𝑤𝑢|𝜕𝑡𝑑𝑥≤𝐶|^𝜌_{𝐿𝑛/𝛼(𝕋𝑛)}+|𝑣|^𝜌_{𝐿𝑛/𝛼(𝕋𝑛)}‖𝑤‖_𝐿^𝑞_(𝕋^𝑛₎𝜕𝑤𝜕𝑡_𝐿²_(𝕋^𝑛₎≤𝐶‖𝑢‖^𝜌_{̇𝐻𝛼(𝕋𝑛)}+‖𝑣‖^𝜌_{̇𝐻𝛼(𝕋𝑛)}‖𝑤‖_𝐿^𝑞_(𝕋^𝑛₎𝜕𝑤𝜕𝑡_𝐿²_(𝕋^𝑛₎≤𝐶‖𝑢‖^𝜌_{̇𝐻𝛼(𝕋𝑛)}+‖𝑣‖^𝜌_{̇𝐻𝛼(𝕋𝑛)}‖𝑤‖_̇𝐻^𝛼_(𝕋^𝑛₎𝜕𝑤𝜕𝑡_𝐿²_(𝕋^𝑛₎, (48) where (1/𝑞)+(𝛼/𝑛)+(1/2)=1, and we use 0≤𝜌≤2𝛼/(𝑛−2𝛼).
Thus, (47) becomes 𝜕𝑤𝜕𝑡²_𝐿²_(𝕋^𝑛₎+‖𝑤‖²_{̇𝐻𝛼(𝕋}^𝑛₎≤𝐶∫^𝑡₀𝜕𝑤𝜕𝑡²_𝐿²_(𝕋^𝑛₎+‖𝑤‖²_{̇𝐻𝛼(𝕋}^𝑛₎𝑑𝜏, (49) and we get 𝑤=0, 𝑢=𝑣 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 𝑢₀∈𝐻̇^2𝛼𝕋^𝑛𝐻̇^𝛼𝕋^𝑛𝐿^𝜌+2𝕋^𝑛,𝑢₁∈𝐻̇^𝛼𝕋^𝑛𝐿²𝕋^𝑛,𝑓∈𝐿^∞0,𝑇;𝐿²𝕋^𝑛,𝜕𝑓𝜕𝑡∈𝐿²0,𝑇;𝐿²𝕋^𝑛, (50) and 0≤𝜌≤2𝛼/(𝑛−2𝛼). Then there exists a unique weak solution for (1)–(3). Furthermore, 𝑢∈𝐿^∞0,𝑇;𝐻̇^2𝛼𝕋^𝑛,𝜕𝑢𝜕𝑡∈𝐿^∞0,𝑇;𝐻̇^𝛼𝕋^𝑛,𝜕²𝑢𝜕𝑡²∈𝐿^∞0,𝑇;𝐿²𝕋^𝑛. (51)

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 𝑛0≤𝜌≤⟹𝑛𝑛−2𝛼2𝑛𝜌+1≥−𝛼+2, (52) we see by Sobolev inequality that 𝑢₀^𝜌𝑢_{0𝐿2(𝕋𝑛)}≤𝑢_0𝐿^𝜌+1_{2(𝜌+1)(𝕋𝑛)}≤𝑢_0̇𝐻^𝜌+1_{𝛼(𝕋𝑛)}. (53) By formula (13), we have 𝜕²𝑢𝜕𝑡²(0)∈𝐿²𝕋^𝑛. (54)
Step  2 (bounds for 𝜕𝑢/𝜕𝑡, 𝜕²𝑢/𝜕²𝑡). Differentiating (1) with respect to 𝑡, we find that 𝜕³𝑢𝜕𝑡³−Λ^2𝛼𝜕𝑢+|𝑢|𝜕𝑡𝜌+1^𝜌𝜕𝑢=𝜕𝑡𝜕𝑓.𝜕𝑡 (55) Taking the inner product of (55) with 𝜕²𝑢/𝜕𝑡², we obtain by integration by parts that 12𝑑𝜕𝑑𝑡²𝑢𝜕𝑡²²_{𝐿2(𝕋𝑛)}+Λ^𝛼𝜕𝑢𝜕𝑡²_𝐿²_(𝕋^𝑛₎≤𝜕𝑓,𝜕𝜕𝑡²𝑢𝜕𝑡²−𝜌+1|𝑢|^𝜌𝜕𝑢,𝜕𝜕𝑡²𝑢𝜕𝑡²≡𝐼₁+𝐼₂. (56) The first term 𝐼₁ can be easily dominated by using Hölder inequality as 𝐼₁≤𝜕𝑓𝜕𝑡_𝐿²_(𝕋^𝑛₎𝜕²𝑢𝜕𝑡²_{𝐿2(𝕋𝑛)}≤12𝜕𝑓𝜕𝑡²_𝐿²_(𝕋^𝑛₎+12𝜕²𝑢𝜕𝑡²²_{𝐿2(𝕋𝑛)}. (57) To tackle 𝐼₂, we invoke Hölder and Sobolev inequalities to deduce that 𝐼₂≤𝐶|𝑢|^𝜌_{𝐿𝑝(𝕋𝑛)}𝜕𝑢𝜕𝑡_𝐿^𝑞_(𝕋^𝑛₎𝜕²𝑢𝜕𝑡²_{𝐿2(𝕋𝑛)}≤𝐶‖𝑢‖^𝜌_{𝐿𝜌𝑝(𝕋𝑛)}𝜕𝑢𝜕𝑡_𝐿^𝑞_(𝕋^𝑛₎𝜕²𝑢𝜕𝑡²_{𝐿2(𝕋𝑛)}≤𝐶Λ^𝛼𝑢^𝜌_{𝐿2(𝕋𝑛)}Λ^𝛼𝜕𝑢𝜕𝑡_𝐿²_(𝕋^𝑛₎𝜕𝑢𝜕𝑡_𝐿^𝑞_(𝕋^𝑛₎≤𝐶Λ^𝛼𝜕𝑢𝜕𝑡²_𝐿²_(𝕋^𝑛₎+𝜕𝑢𝜕𝑡²_𝐿^𝑞_(𝕋^𝑛₎, (58) where 𝑝, 𝑞 are chosen so that 𝑛𝑛𝜌𝑝=−𝛼+2=𝑛𝑞, (59) and thus 1𝑝+1𝑞+12≤1. (60) Gathering (57) and (58) into (56), it follows that 𝑑𝜕𝑑𝑡²𝑢𝜕𝑡²²_{𝐿2(𝕋𝑛)}+Λ^𝛼𝜕𝑢𝜕𝑡²_𝐿²_(𝕋^𝑛₎≤𝜕𝑓𝜕𝑡²_𝐿²_(𝕋^𝑛₎+𝐶Λ^𝛼𝜕𝑢𝜕𝑡²_𝐿²_(𝕋^𝑛₎+𝜕𝑢𝜕𝑡²_𝐿^𝑞_(𝕋^𝑛₎. (61) Gronwall inequality then implies that 𝜕²𝑢𝜕𝑡²²_{𝐿2(𝕋𝑛)}+Λ^𝛼𝜕𝑢𝜕𝑡²_𝐿²_(𝕋^𝑛₎≤𝑒^𝐶𝑡𝜕²𝑢𝜕𝑡²(0)²_{𝐿2(𝕋𝑛)}+Λ^𝛼𝑢₁²_{𝐿2(𝕋𝑛)}+𝑒^𝐶𝑡∫^𝑡₀𝜕𝑓𝜕𝑠(𝑠)²_{𝐿2(𝕋𝑛)}𝑑𝑠<∞. (62)
Step  3 (bounds for 𝑢). Rewriting (1) as Λ^2𝛼𝜕𝑢=²𝑢𝜕𝑡²+|𝑢|^𝜌𝑢−𝑓, (63) we have 𝑢∈𝐿^∞(0,𝑇;𝐻̇^2𝛼(𝕋^𝑛)), in view of similar inequalities satisfied by |𝑢|^𝜌𝑢 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.

Acknowledgments

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).