Bull. Malays. Math. Sci. Soc. DOI 10.1007/s40840-017-0468-x

A Sixth-Order Phase-Field Equation with Degenerate Mobility Ning Duan1 · Yujuan Cui2 · Xiaopeng Zhao1

Received: 13 October 2016 / Revised: 26 December 2016 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017

Abstract In this paper, we study the weak solutions of a sixth-order phase-field equation with degenerate phase-dependent diffusion mobility in 2D case. The main features and difficulties of this equation are given by a highly nonlinear sixth-order elliptic term, a strong constraint imposed by the presence of the nonlinear principal part and the lack of maximum principle. Based on the Schauder-type estimates and entropy estimates, we are able to prove the global existence of classical solutions for regularized problems. After establishing some necessary uniform estimates on the approximate solutions, we prove the existence of weak solutions by using approximation and compactness tools. In the end, we study the nonnegativity of solutions for the sixth-order degenerate phase-field equation. Keywords Sixth-order phase-field equation · Degenerate mobility · Weak solutions · Nonnegativity Mathematics Subject Classification 35B65 · 35K35 · 35K55

Communicated by Yong Zhou.

B

Xiaopeng Zhao [email protected] Ning Duan [email protected] Yujuan Cui [email protected]

1

School of Science, Jiangnan University, 214122 Wuxi, China

2

P. O. 239, No. 717 Jiefang Road Qiaokou District, 430000 Wuhan, China

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1 Introduction In the last 15–20 years, essentially sixth-order nonlinear parabolic partial differential equations, as models for applications in mechanics and physics, have become more common in the literature on pure and applied PDEs. Many papers have already been published to study the sixth-order nonlinear parabolic equations (see, e.g., [7,10,11, 17]). In order to describe strongly anisotropic crystal and epitaxial growth during the growth and coarsening of thin films, Torabi et al. [18] introduced a higher-order nonlinear Willmore regularization in the Ginzburg–Landau free energy and proposed the following modification of the Ginzburg–Landau free energy MGL =

     1 2 β 1 2 γ (v) dx, |∇u|2 + F(u) + ω  2 2 3 

(1)

∇u where γ (v) is the function describing the anisotropic effect, v = |∇u| is the unit normal,  is a small parameter which is a measure of the interface transition layer thickness  and ω = f (u) − u  u = F (u) − u is called nonlinear Willmore regularization. Moreover, F(u) = 0 f (s)ds is a nonlinear function. Usually, we take it as a doublewell potential

F(s) =

1 2 (s − 1)2 , 4

f (s) = F  (s) = s(s 2 − 1).

(2)

Based on the Ginzburg–Landau free energy and mass conservation, we end up with following sixth-order anisotropic phase-field equation ⎧ ∂u 1 ⎪ ⎪ = ∇ · (M∇μ), ⎪ ⎪ ∂t  ⎪ ⎪

⎪ ⎪ ⎨ μ = 1 γ f (u) −  2 ∇ · m + β f  (u)ω −  2 ω ,  2 ⎪ ⎪ m = γ (v)∇u + |∇u|P∇v γ (v), ⎪ ⎪ ⎪ ⎪

⎪ ⎪ ⎩ ω = 1 f (u) −  2 u , 

(3)

where M is the mobility, μ is the chemical potential function, m describes the anisotropic gradient and P = I − v ⊗ v is the projection matrix in which I is the identity matrix. Furthermore, ∇v represents the gradient with respect to the components of the normal vector. Using the method of matched asymptotic expansions, Torabi et al. [18] showed the convergence of Eq. (3) to the general sharp interface model. Moreover, the authors also presented two- and three-dimensional numerical results using an adaptive, nonlinear multigrid finite-difference method. Chen and Shen [3] constructed energy stable schemes for the time discretization of Eq. (3) using a stabilization technique. In the sequel, Makki and Miranville [15] proved the existence and uniqueness of solutions for

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Eq. (3) in 1D case. Lussardi [14] proved a full τ -convergence result for an anisotropic Modica–Mortola-type energy of Eq. (3) using the framework of τ -convergence. Recently, Miranville [16] simplified the Ginzburg–Landau free energy (1) and Eq. (3). The author supposed γ (v) = 1, β = 1,  = 1 and the equation is isotropic, provided the following modification of the Ginzburg–Landau free energy   MGL =



 1 1 2 2 |∇u| + F(u) + ω dx, 2 2

(4)

and the following sixth-order phase-field equation ⎧ ∂u ⎪ ⎪ ⎨ ∂t = ∇ · (M∇μ), ∀x ∈ , t ∈ [0, ∞), μ = f (u) − u + ω f  (u) − ω, ⎪ ⎪ ⎩ ω = f (u) − u.

(5)

The author studied the asymptotic behavior, in terms of finite-dimensional attractors for the initial-boundary value problems of Eq. (5) together with the mobility M ≡ 1. It is well known that the principal part of many types of nonlinear diffusion equations which arise from Mathematics and other branches of natural science (e.g., Physics, Mechanics, Material science and Population ecology) is nonlinearity. In the last three decades, more and more authors paid their attentions to the well posedness of solutions for higher-order diffusion equations together with nonlinear principal parts. For example, based on the framework of Campanato spaces, together with energy estimates and Schauder-type estimates, Yin et. al. [9,20] and Liu [12] studied the existence and uniqueness of classical solutions for some higher-order parabolic equations with positive mobilities. Bonetti, Dreyer and Schimperna [2] studied the existence of global weak solutions for a type of viscous Cahn–Hilliard equation with positive mobilities describing the phase separation in a binary alloy. There are also many classical results on the diffusion equation with degenerate mobilities. In [6,13,19], using the uniform estimates on the approximate solutions and function approximations, the author considered the existence of weak solutions for the Cahn–Hilliard equation and sixth-order thin film equation with degenerate mobilities. Furthermore, for the study on the Cahn– Hilliard equation with degenerate mobility in arbitrary dimensional space, we refer the reader to Dai and Du [4]. It is thus clear that the study on the sixth-order phase-field equation with nonconstant mobility ⎧ ∂u 2 ⎪ ⎪ ⎨ ∂t = ∇ · (M(u)∇μ), ∀x ∈  ⊂ R , t ∈ (0, T ), μ = ω + ω f  (u) − ω, ⎪ ⎪ ⎩ ω = f (u) − u,

(6)

is more meaningful than the equation with constant mobility. In this article, we study the initial-boundary value problem for Eq. (6). On the basis of physical consideration,

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Eq. (6) is supplemented with the following boundary value conditions u |∂ = u |∂ = 2 u |∂ = 0, t > 0,

(7)

and initial value condition u(x, 0) = u 0 (x), x ∈ .

(8)

For convenience, in this article, we suppose the mobility M(u) = |u 2 |m , where m > 0 and the polynomial forms of f is given by f (χ ) =

2k+1 

ai χ i with ai ∈ R, a2k+1 > 0, k ∈ Z+ .

(9)

i=1

For Eqs. (6)–(8), since the mobility is a degenerate function, the problem does not admit classical solutions in general. So, we introduce the weak solutions in the following sense: Definition 1.1 Suppose that Q T =  × (0, T ), a function u(x, t) is said to be a weak solution of problem (6)–(8), if the following two conditions are satisfied: 1

• u ∈ C 2 (Q T ), u ∈ L ∞ (0, T ; H 2 ()), |u|m ∇2 u ∈ L 2 (P); • For ϕ ∈ C 1 (Q T ),     ∂ϕ − u(x, T )ϕ(x, T )dx + u 0 (x)ϕ(x, 0)dx + u dxdt ∂t   QT   = |u| N ∇[2 u − u −  f (u) − f (u)u + f  (u) f (u) P

+ f (u)] · ∇ϕdxdt, where P = Q T \ ({u(x, t) = 0}



{t = 0}).

The purpose of this paper is to study the existence of a weak solution for problem (6)–(8). The main difficulties are given by a highly nonlinear sixth-order elliptic term, a strong constraint imposed by the presence of the nonlinear principal part and the lack of maximum principle. Because of the degeneracy, we first consider the existence of classical solutions for the regularized problem. Our method is based on uniform Schauder-type estimates and entropy estimates for local in time solutions via the framework of Campanato spaces. Then, based on Leray–Schauder fixed-point theorem, the result on classical solutions is obtained. Moreover, based on the uniform estimates for the approximate solutions, the existence of weak solutions for the equation with degenerate mobility can be obtained. The outline of this paper is as follows. In Sect. 2, the regularized problem is studied. In Sect. 3, the existence of weak solutions is established. In the last section, the nonnegativity of weak solutions is discussed. Throughout this paper, the same letters C and Ci (i = 0, 1, 2, . . .) shall denote positive constants that may change from line to line.

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2 Regularized Problems To discuss the existence, we adopt the method of parabolic regularization, namely, the desired solution will be obtained as the limit of some subsequence of solutions of the following regularized problem ∂u ε = ∇ · [Mε (u ε )∇με ] , in Q T , ∂t με = 2 u ε − u ε −  f (u ε ) − f  (u ε )u ε ) + f  (u ε ) f (u ε ) + f (u ε )), in Q T ,

(10)

(11)

together with u ε (x, t) = u ε (x, t) = 2 u ε (x, t) = 0, x ∈ ∂, t > 0,

(12)

u ε (x, 0) = u 0 (x), x ∈ ,

(13)

and where Mε (u ε ) = (|u ε |2 + ε)m , Q T :=  × (0, T ). From the classical approach (see [5,8]), we can obtain the existence and uniqueness of local classical solution for the problem (10)–(13). So, if we want to obtain the existence and uniqueness of global classical solution, it is sufficient to make a priori estimates. Lemma 2.1 Suppose that u ε is a smooth solution for problem (10)–(13). Then, we have (14)

u ε L ∞ () + ∇u ε L q () + u ε H 2 () ≤ C, 1 ≤ q < ∞ Proof For the regularized problem (10)–(13), we have the Ginzburg–Landau free energy    1 1 (15) |∇u ε |2 + F(u ε ) + [ f (u ε ) − u ε ]2 dx, ε (t) = 2  2 s where F(s) = 0 f (s)ds. Integrating by parts, we derive that dε (t) = dt



  ∇u ε ∇u εt + f (u ε ) + ( f (u ε ) − u ε )[ f  (u ε )u εt − u εt ] dx

   = −u ε + f (u ε ) + f (u ε ) f  (u ε ) − f  (u ε )u ε −  f (u ε ) + 2 u ε 

× =−

∂u ε dx ∂t 

 Mε (u ε ) ∇2 u ε − ∇u ε − ∇ f (u ε ) − ∇( f  (u ε )u ε )

2 +∇( f  (u ε ) f (u ε )) + ∇ f (u ε ) dx ≤ 0,

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which means ε (t) ≤ ε (0). Based on the assumption (9), we have  F(u ε ) =

uε

2 p+1

f (s)ds =

0

i=1

a2 p+1 2 p+2 ai−1 i uε ≥ u − C1 , i 4( p + 1) ε

and f  (u ε ) =

2p  2p + 1 a2 p+1 u 2ε p − C2 ≥ −C2 . (i + 1)ai+1 u iε ≥ 2 i=0

Then, the above inequality means   a2 p+1 1 1 1

∇u ε 2 + |u ε |2 p+2 dx + f 2 (u ε )dx + u ε 2 2 4p + 4  2  2  a2 p+1 1 ≤ C + C2 ∇u ε 2 + C1 || ≤ C3 + u ε 2 + |u ε |2 p+2 dx. 4 8p + 8  Hence, we have



 

Moreover,



|u ε |2 dx ≤ C.

(16)

 dx + 1 ≤ C.

(17)



 

Then

|u ε |2 p+2 dx ≤ C,

≤C

u 2ε dx

|u ε |





2 p+2





u 2ε dx +



|∇u ε |2 dx ≤ C.

(18)

Note that we consider the problem in 2D case. By Sobolev’s embedding theorem, we deduce that (19)

u ε L ∞ () + ∇u ε L q () ≤ C, 1 ≤ q < ∞, Then, Lemma 2.1 is proved. 

⎧ 0 < m < 1, ⎪ ⎪ ⎪  ⎪ ⎪ ⎨ m = 1, ln |u 0 (x)|dx < ∞, Lemma 2.2 Suppose that and u ε is a smooth solu ⎪  ⎪ ⎪ ⎪ ⎪ u 2−n ⎩ m > 1, 0 dx < ∞, 

tion for problem (10)–(13),then we have  0

123

T

 

|∇u ε |2 dxdt ≤ C.

(20)

A Sixth-Order Phase-Field Equation with Degenerate Mobility

Proof We denote  gε (s) = −

A s

dr , G ε (s) = − 2 (|r | + ε)m



A s

gε (r )dr,

where A > max |u ε | for all small ε. Then, choose the smooth approximation u 0ε of the initial function u 0 such that u 0ε > u 0 . A simple calculation shows that G ε (s) = gε (s), G ε (s) ≤ G 0 (s) = lim G ε (s), ∀s ∈ R. ε→0

In fact, we have ⎧ A2−2m s A1−2m s 2−2m ⎪ ⎪ ⎪ + + , ∀m ∈ (0, 1), ⎪ ⎪ 2 − 2m 2m − 1 (2 − 2m)(1 − 2m) ⎪ ⎪ ⎨ A s G 0 (s) = ln + − 1, m = 1, ⎪ s A ⎪ ⎪ ⎪ 2−2m ⎪ s s A1−2m A2−2m ⎪ ⎪ ⎩ + − , ∀m ∈ (1, ∞). (2m − 2)(2m − 1) 2m − 1 2m − 2

(21)

For Eq. (10), multiplying both sides by gε (u ε ), integrating over Q T , we deduce that  



 

G ε (u ε (x, t))dx + |∇u ε | dxds = − |u ε |2 dxds QT QT      + f (u ε )∇u ε ∇u ε dxds − f  (u ε ) f (u ε )|∇u ε |2 dxds QT QT     2 − f (u ε )|∇u ε | dxds + G ε (u 0ε (x))dx. 2





QT

(22)

By Lemma 2.1, Young’s inequality and (23), we deduce that  



G ε (u ε (x, t))dx + |∇u ε |2 dxds QT     2  ≤− |u ε | dxds + sup | f (u ε )| (C|∇u ε |2 + ε|∇u ε |2 )dxds QT QT      2  + sup | f (u ε ) f (u ε )| |∇u ε | dxds + sup | f (u ε )| |∇u ε |2 dxds QT QT  + G ε (u 0ε (x))dx     ≤C + |∇u ε |2 dxds + G ε (u 0ε (x))dx. (23)



QT



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Setting  small enough, we derive that  

 G ε (u ε (x, t))dx +



 |∇u ε | dxds ≤ C + 2

QT



G ε (u 0ε (x))dx.

(24)

Hence,  

 |∇u ε | dxds ≤ C + 2

QT



     G ε (u 0ε (x))dx +  G 0 (u ε (x, t))dx  . 

On the other hand, based on the assumption of this lemma, we have  

G ε (u 0ε )dx ≤ C.

Thus, we complete the proof. 

⎧ 0 < m < 1, ⎪ ⎪ ⎪  ⎪ ⎪ ⎨ m = 1, ln |u 0 (x)|dx < ∞, Lemma 2.3 Suppose that and u ε is a smooth solu ⎪  ⎪ ⎪ ⎪ ⎪ u 2−n ⎩ m > 1, 0 dx < ∞, 

tion for problem (10)–(13), then we have   QT

Mε (u ε )|∇2 u ε |2 dxdt ≤ C.

Proof Multiplying both sides of Eq. (10) by 2 u ε , using (11), integrating the resulting relation with respect to x over , we derive that  

∂u ε 2  u ε dx = ∂t



 2 u ε ∇ Mε (u ε ) ∇2 u ε − ∇u ε − ∇ f (u ε )   −∇( f  (u ε )u ε ) + ∇( f  (u ε ) f (u ε )) + ∇ f (u ε ) dx. (25)

Integrating by parts, using the boundary value conditions, we obtain 1 d 2 dt + + −



 |u ε | dx +



2





 





2

Mε (u ε )∇2 u ε ∇ f (u ε )dx 

Mε (u ε )∇ u ε ∇( f (u ε )D u ε )dx − 2



Mε (u ε )|∇ u ε | dx = 2



2





Mε (u ε )∇2 u ε ∇u ε dx

Mε (u ε )∇2 u ε ∇( f  (u ε ) f (u ε ))dx

Mε (u ε )∇2 u ε ∇ f (u ε )dx = I1 + I2 + I3 + I4 + I5 ,

123

(26)

A Sixth-Order Phase-Field Equation with Degenerate Mobility

where   2 2  I1 ≤  Mε (u ε )|∇ u ε | dx + C Mε (u ε )|∇u ε |2 dx  ≤



 I2 =





 Mε (u ε )|∇2 u ε |2 dx + C





|∇u ε |2 dx.

Mε (u ε )∇2 u ε [ f  (u ε )|∇u ε |3 + 3 f  (u ε )∇u ε u ε

+ f  (u ε )∇u ε ]dx      Mε (u ε )|∇u ε |2 dx + C sup Mε (u ε )[ f  (u ε )]2  |∇u ε |6 dx ≤ 

+ 3C sup |Mε (u ε )[ f  (u ε )]2 | + C sup |( f  (u ε ))2 Mε (u ε )|  ≤



 I3 =

 



Mε (u ε )|∇2 u ε |2 dx + C 







 =  ≤ ≤















Mε (u ε )∇2 u ε f  (u ε )∇u ε dx

Mε (u ε )[ f  (u ε )∇u ε ]2 |u ε |2 dx

2



 

Mε (u ε )|∇u ε |2 dx 

Mε (u ε )|∇2 u ε |2 dx + C13  Mε (u ε )|∇2 u ε |2 dx + C  Mε (u ε )|∇2 u ε |2 dx + C







|∇u ε |2 dx.

Mε (u ε )|∇( f  (u ε ) f (u ε ))|2 dx Mε (u ε )[( f  (u ε ) f (u ε )) ]2 |∇u ε |2 dx

     2 Mε (u ε )|∇ u ε | dx + C sup Mε (u ε )[( f (u ε ) f (u ε )) ]  |∇u ε |2 dx 2





    2 Mε (u ε )|∇ u ε | dx + C sup Mε (u ε )[ f (u ε )]  |∇u ε u ε |d x

+ C sup[ f  (u ε )]2

I4 ≤ 

|∇u ε |2 dx.

Mε (u ε )[ f  (u ε )]2 |∇u ε |d x 2







+ C

≤



Mε (u ε )∇ u ε f (u ε )∇u ε u ε dx +



≤

|∇u ε |2 dx





≤



|∇u ε u ε |2 dx

Mε (u ε )|∇2 u ε |2 dx + C12 2







2



Mε (u ε )|∇2 u ε |2 dx + C14 .

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 I5 ≤   ≤  ≤

 





Mε (u ε )|∇2 u ε |2 dx + C



Mε (u ε )[ f  (u ε )∇u ε ]2 dx

    Mε (u ε )|∇2 u ε |2 dx + C sup Mε (u ε )[ f  (u ε )]2  |∇u ε |2 dx 

Mε (u ε )|∇2 u ε |2 dx + C15 .

Summing up, we deduce that d

u ε 2 + 2(1 − 5) dt

 

   Mε (u ε )|∇2 u ε |2 dx ≤ C16 1 + |∇u ε |2 dx , 

(27) where the positive constant C can be chosen sufficiently small to satisfy 1 − 5 > 0. Then, we have   QT

Mε (u ε )|∇2 u ε |2 dxdt ≤ C.

(28)



⎧ 0 < m < 1, ⎪ ⎪ ⎪  ⎪ ⎪ ⎨ m = 1, ln |u 0 (x)|dx < ∞, Lemma 2.4 Suppose that and u ε is a smooth solu ⎪  ⎪ ⎪ ⎪ ⎪ u 2−n ⎩ m > 1, 0 dx < ∞, 

tion for problem (10)–(13), then for any (x1 , t1 ), (x2 , t2 ) ∈ Q T , we have

1 1 |u ε (x1 , t1 ) − u ε (x2 , t2 )| ≤ C |x1 − x2 | 2 + |t1 − t2 | 12 .

(29)

Proof By (17)–(19), we can obtain the following inequality: 1

|u ε (x1 , t) − u ε (x2 , t)| ≤ C|x1 − x2 | 2 .

(30)

Integrating Eq. (10) over  y × (t1 , t2 ), where 0 < t1 < t2 < T , t = t2 − t1 , 1 1  y = (y1 , y1 + (t) 12 ) × (y2 , y2 + (t) 12 ), using (11), we deduce that 

 y

[u ε (z, t2 ) − u ε (z, t1 )]dz = 

− G 1 (y1 , y, s)]dyds +

t2 

t1

t2

t1



1

y2 +(t) 12 y2 1

y1 +(t) 12

123

1

[G 2 (y, y2 + (t) 12 , s)

y1

− G 1 (y, y2 , s)]dyds  t2  1 1 1 = (t) 12 G 1 (y1 + (t) 12 , y2 + θ1∗ (t) 12 , s) t1

1

[G 1 (y1 + (t) 12 , y, s)

A Sixth-Order Phase-Field Equation with Degenerate Mobility 1

− G 1 (y1 , y2 + θ1∗ (t) 12 , s)

 1 1 1 + G 2 (y1 + θ2∗ (t) 12 , y2 + (t) 12 , s) − G 2 (y1 + θ2∗ (t) 12 , y2 , s) ds, (31)

where (G 1 , G 2 ) = Mε (u ε ){∇2 u ε − ∇u ε − ∇ f (u ε ) − ∇( f  (u ε ) f (u ε )) + ∇[ f  (u ε ) f (u ε )] + ∇ f (u ε )}(x, s).

(32)

We set  1 1 1 N (y1 , y2 , s) = (t) 12 G 1 (y1 + (t) 12 , y2 + θ1∗ (t) 12 , s) 1

− G 1 (y1 , y2 + θ1∗ (t) 12 , s) 1

1

+ G 2 (y1 + θ2∗ (t) 12 , y2 + (t) 12 , s)  1 − G 2 (y1 + θ2∗ (t) 12 , y2 , s) ds,

(33)

Therefore, (31) is converted into (t)

1 6

 =



1

I =(0,1)×(0,1) t2

1

[u(y + θ (t) 12 , t2 ) − u(y + θ (t) 12 , t1 )]dθ

N (y1 , y2 , s)ds.

t1

We integrate the above equality over x , then 1

(t) 3 [u(x ∗ , t2 ) − u(x ∗ , t1 )] =



t2

t1

 x

N (y, s)dyds. 1

Here, we have used the mean value theorem, where x ∗ = y ∗ + (θ ∗ )(t) 12 . Then, using Hölder’s inequality and (19), (28), we derive that 1

|u ε (x ∗ , t2 ) − u ε (x ∗ , t1 )| ≤ C(t) 12 . 

To prove the uniqueness and existence of classical solutions for the regularized problems (10)–(13), the key estimate is the Hölder estimate for ∇u and u. Now, we give the following result which can be seen in [12]. α Proposition 2.5 (see [12]) Assume that sup | f| < +∞, a(x, t) ∈ C α, 6 ( Q¯ T ), 0 < α < 1, and there exist two constants A0 and B0 such that 0 < A1 ≤ a(x, t) ≤ A0 , 0 < B1 ≤ b(x, t) ≤ B0 for all (x, t) ∈ Q T . If u is a smooth solution for the following linear problem

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⎧ ∂u ⎪  ⎪ ⎨ ∂t − ∇(a(x, t)∇u) + ∇(b(x, t)∇u) = ∇ f , (x, t) ∈ Q T , u(x, t)|∂ = u(x, t)|∂ = 2 u(x, t)|∂ = 0, t ∈ [0, T ], ⎪ ⎪ ⎩ u(x, 0) = u 0 (x), x ∈ , then, for any δ ∈ (0, 21 ), there is a constant K depending on A0 , B0 , A1 , B1 , δ, T ,   2 2 Q T u dxdt and Q T |∇u| dxdt, such that δ

|u(x1 , t1 ) − u(x2 , t2 )| ≤ K (1 + sup | f|)(|x1 − x2 |δ + |t1 − t2 | 6 ). Then, we give the result on the existence of unique classical solution for the regularized problems. ⎧ 0 < m < 1, ⎪ ⎪ ⎪  ⎪ ⎪ ⎨ m = 1, ln |u 0 (x)|dx < ∞, and u ε is a smooth soluLemma 2.6 Suppose that  ⎪  ⎪ ⎪ ⎪ ⎪ u 2−n ⎩ m > 1, 0 dx < ∞, 

tion for problem (10)–(13), then for each fixed ε > 0 and u 0ε ∈ C 6+α (), i u 0ε |∂ = 0, (i = 0, 1, 2), α

then for some α ∈ (0, 1), there exists a unique classical solution u ε ∈ C 6+α,1+ 6 (Q T ) for problem (10)–(13). Proof Let ω = u − u 0 . Then, ω satisfies ⎧ ∂u ⎪  ⎪ ⎨ ∂t − ∇(a(x, t)∇u) + ∇(b(x, t)∇u) = ∇ F, (x, t) ∈ Q T , u(x, t)|∂ = u(x, t)|∂ = 2 u(x, t)|∂ = 0, t ∈ [0, T ], ⎪ ⎪ ⎩ u(x, 0) = u 0 (x), x ∈ , where a(x, t) = M(u), b(x, t) = M(u) and F = M(u)[−∇ f (u) − ∇( f  (u)u) + ∇( f  (u) f (u)) + ∇ f (u)]. Hence, based on Lemma 2.5, we get α

α

|u(x1 , t1 ) − u(x2 , t2 )| ≤ C(|x1 − x2 | 2 + |t1 − t2 | 12 ).

(34)

The conclusion follows immediately from the classical theory, since we can transform Eq. (10) into the form ∂u + a1 (x, t)3 u + b1 (x, t)∇2 u + a2 (x, t)2 u + b2 (x, t)∇u ∂t + a3 (x, t)u + b3 (x, t)∇u = 0,

123

(35)

A Sixth-Order Phase-Field Equation with Degenerate Mobility

where the Hölder norms on a1 (x, t) = −Mε (u ε ), b1 (x, t) = −Mε (u ε )∇u ε , a2 (x, t) = Mε (u ε )

+ 2Mε (u ε ) f  (u ε ), b2 (x, t) = Mε (u ε )∇u ε + Mε (u ε ) f  (ε )∇u ε + 6Mε (u ε ) f  (u ε )∇u ε

+ Mε (u ε ) f  (u ε )∇u ε , a3 (x, t) = 4Mε (u ε ) f  (u ε )|∇u ε |2 + 4Mε (u ε ) f  (u ε )u ε + 7Mε (u ε ) f  (u ε )|∇u ε |2 − Mε (u ε )| f  (u ε )|2 − Mε (u ε ) f (u ε ) f  (u ε ) − Mε (u ε ) f  (u ε ), b3 (x, t) = Mε (u ε ) f  (u ε )|∇u ε |2 ∇u ε + Mε (u ε ) f (4) (u ε )|∇u ε |2 ∇u ε − 3Mε (u ε ) f  (u ε ) f  (u ε )∇u ε − Mε (u ε ) f (u ε ) f  (u ε )∇u ε − Mε (u ε ) f (u ε ) f  (u ε )∇u ε − Mε (u ε )| f  (u ε )|2 ∇u ε − Mε (u ε ) f  (u ε )∇u ε − Mε (u ε ) f  (u ε )∇u ε ,

have been estimated in the above discussion. Then, the proof is completed.



3 Existence of Solutions for Degenerate Equation After the discussion of the regularized problem, we can now turn to the study of the existence of weak solutions for problem (6)–(8). ⎧ 0 < m < 1, ⎪ ⎪ ⎪  ⎪ ⎪ ⎨ m = 1, ln |u 0 (x)|dx < ∞, Theorem 3.1 Suppose that u 0 ∈ H02 () and then  ⎪  ⎪ ⎪ ⎪ ⎪ u 2−n ⎩ m > 1, 0 dx < ∞, there exists at least one weak solution u(x, t).



Proof Suppose that u ε is the approximate solution of problem (10)–(13) constructed in the previous section. Using the estimates (29), (17) and (34), we can extract a subsequence from {u ε }, denoted also by {u ε }, such that u ε (x, t) → u(x, t) uniformly in Q T , ∇u ε (x, t) → ∇u(x, t) uniformly in P, u ε (x, t) → u(x, t) uniformly in P, and the limiting function 1

1

u ∈ C 2 , 12 (Q T )



L ∞ (0, T ; H 2 ()).

(36)

Now, let δ > 0 be fixed and set Pδ = {(x, t) : |u|2m (x, t) > δ}. We choose ε0 (δ) > 0, such that δ (37) (|u ε |2 + ε)m ≥ , (x, t) ∈ Pδ , 0 < ε < ε0 (δ), 2 and |u ε |2m ≤ 2δ, (x, t) ∈ Q T \ Pδ , 0 < ε < ε0 (δ).

(38)

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Then, from (28), we derive that  



C |∇u ε |2 + |∇2 u ε |2 dxdt ≤ , δ Pδ

(39)

where the constant C is independent of ε and δ. By employing a diagonal selection, we obtain a subsequence from {u ε }, denoting also by {u ε }, such that ∇u ε → ∇u, weakly in L 2 (Pδ ). and ∇2 u ε → ∇2 u, weakly in L 2 (Pδ ). Note that   |u|

2m

Pδ

   |∇ u| dxdt ≤ 2  2

2

|u|2m ∇2 u(∇2 u     |u|2m |∇2 u|2 dxdt. −∇2 u ε )dxdt  + Pδ

Pδ

(40) This and the fact that   lim ε→0

Pδ

|u|2m ∇2 u(∇2 u − ∇2 u ε )dxdt = 0,

lim |(|u ε |2 + ε)m − |u|2m ||∇2 u ε |d xdt = 0,

ε→0

yield  

  |u| Pδ

2m

|∇ u| dxdt ≤ limε→0 2

2

Pδ

(|u ε |2 + ε)m |∇2 u ε |2 dxdt ≤ C.

To prove the integral equality in the definition of solutions, it suffices to pass the limit as ε → 0 in     ∂ϕ − u ε (x, T )ϕ(x, T )dx + u 0ε ϕ(x, 0)dx + u ε dxdt ∂t   QT     = (|u ε |2 + ε)m ∇2 u ε ∇ϕdxdt − (|u ε |2 + ε)m ∇u ε ∇ϕdxdt Q QT  T 2 m − (|u ε | + ε) ∇ f (u ε )dxdt QT   − (|u ε |2 + ε)m ∇[ f  (u ε )u ε ]∇ϕdxdt QT

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  +  

QT

+ QT

(|u ε |2 + ε)m ∇[ f  (u ε ) f (u ε )]∇ϕdxdt (|u ε |2 + ε)m ∇ f (u ε )∇ϕdxdt.

Note that the limits  lim

ε→0

 u ε (x, T )ϕ(x, T )dx = u(x, T )ϕ(x, T )dx,    lim u 0ε ϕ(x, 0)dx = u 0 (x)ϕ(x, 0)dx, ε→0 



and   lim

ε→0

QT

∂ϕ u ε dxdt = ∂t

  u QT

∂ϕ dxdt, ∂t

are obvious. It remains to show  

  lim

ε→0

 

QT

lim

ε→0

QT

lim

=

QT

 

P

 

(|u ε |2 + ε)m ∇[ f  (u ε )u ε )]∇ϕdxdt

|u|2m ∇[ f  (u)u]∇ϕdxdt,

(43)

P

lim

ε→0

|u|2m ∇2 u∇ϕdxdt, (41)   (|u ε |2 + ε)m ∇ f (u ε )∇ϕdxdt = |u|2m ∇ f (u)∇ϕdxdt, P

(42)

  ε→0

(|u ε |2 + ε)m ∇2 u ε ∇ϕdxdt =

QT

 

(|u ε |2 + ε)m ∇[ f  (u ε ) f (u ε )]∇ϕdxdt

|u|2m ∇[ f  (u) f (u)]∇ϕdxdt,

=

(44)

P

and  

  (|u ε | + ε) ∇ f (u ε )∇ϕdxdt = 2

lim

ε→0

QT

|u|2m ∇ f (u)∇ϕdxdt.

m

(45)

P

In fact, for any fixed δ > 0,  

 

lim

ε→0

(|u ε | + ε) ∇ f (u ε )∇ϕdxdt = lim 2

QT

m

ε→0

QT

(|u ε |2 + ε)m [ f  (u ε )∇u ε

+ 3 f  (u ε )∇u ε u ε + f  (u ε )|∇u ε |3 ]∇ϕdxdt,

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and        2 m  2m   (|u ε | + ε) f (u ε )∇u ε ∇ϕdxdt − |u| f (u)∇u∇ϕdxdt   Q Q  T    T   2 m  2m   ≤ (|u ε | + ε) f (u ε )∇u ε ∇ϕdxdt − |u| f (u)∇u∇ϕdxdt  P P  δ  δ   +  (|u ε |2 + ε)m f  (u ε )∇u ε ∇ϕdxdt  Q T \Pδ      2m   + |u| f (u)∇udxdt  P\P    δ       ≤  (|u ε |2 + ε)m − |u|2m  | f  (u ε )∇u ε ||∇ϕ|dxdt  P  δ      +  |u|2m  f  (u ε )(∇u ε − ∇u) + ∇u( f  (u ε ) − f  (u)) |∇ϕ|dxdt  P   δ    2 m   + (|u ε | + ε) f (u ε )∇u ε ∇ϕdxdt  Q \P   T δ    2m   + |u| f (u)∇u∇ϕdxdt  P\Pδ

  C   ≤ sup (|u ε |2 + ε)m − |u|2m  |∇ϕ| √ δ        2m       f (u ε )(∇u ε − ∇u) + ∇u( f (u ε ) − f (u)) |∇ϕ|dxdt  |u| + Pδ

√ m + C(δ + ε) 2 sup |∇ϕ| + C δ sup |∇ϕ|, ε ∈ (0, ε0 (δ)). 2

By the boundary value conditions and the embedding theorem, we have      2 m 2m    [(|u | + ε) − |u| ]| f (u )∇u u ||∇ϕ|dxdt ε ε ε ε   Pδ     ≤ sup (|u ε |2 + ε)m − |u|2m  sup | f  (u ε )| ∇u ε L 4 (Pδ ) u ε L 4 (Pδ ) ∇ϕ L 2 (Pδ )     ≤ C sup (|u ε |2 + ε)m − |u|2m  u ε H 2 (Pδ ) ∇u ε L 2 (Pδ ) ∇ϕ L 2 (Pδ )   C   ≤ sup (|u ε |2 + ε)m − |u|2m  |∇ϕ| √ , δ and      2 m 2m  3  [(|u ε | + ε) − |u| ] f (u ε )|∇u ε | ∇ϕdxdt   Pδ     ≤ sup (|u ε |2 + ε)m − |u|2m  sup | f  (u ε )| ∇u ε 3L 6 (P ) ∇ϕ L 2 (Pδ ) δ

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A Sixth-Order Phase-Field Equation with Degenerate Mobility

    ≤ C sup (|u ε |2 + ε)m − |u|2m  u ε 2H 2 (P ) ∇u ε L 2 (Pδ ) ∇ϕ L 2 (Pδ ) δ   C  2 m 2m  ≤ sup (|u ε | + ε) − |u|  |∇ϕ| √ . δ Therefore,     

(|u ε |2 + ε)m f  (u ε )∇u ε u ε ∇ϕdxdt     2m  |u| f (u)∇uu∇ϕdxdt  − Q   T    2 m 2m   ≤ [(|u ε | + ε) − |u| ]| f (u ε )∇u ε u ε ||∇ϕ|dxdt  P  δ    2m   + |u| f (u ε )∇u ε (u ε − u)∇ϕdxdt  P   δ    +  |u|2m f  (u ε )(∇u ε − ∇u)u∇ϕdxdt  P   δ    +  |u|2m ( f  (u ε ) − f  (u))∇uu|∇ϕ|dxdt  P   δ    2m   + |u| f (u)∇uu∇ϕdxdt  P\P   δ    2 m   + (|u ε | + ε) f (u ε )∇u ε u ε ∇ϕdxdt  Q T \Pδ      C  2 m 2m  ≤ sup (|u ε | + ε) − |u|  |∇ϕ| √ +  |u|2m f  (u ε )∇u ε (u ε δ Pδ − u)∇ϕdxdt|      m |u|2m f  (u ε )(∇u ε − ∇u)u∇ϕdxdt  + C(δ 2 + ε) 2 sup |∇ϕ| +  P √ δ + C δ sup |∇ϕ|      +  |u|2m ( f  (u ε ) − f  (u))∇uu|∇ϕ|dxdt  , ε ∈ (0, ε0 (δ)) QT

Pδ

and     

QT

(|u ε |2 + ε)m f  (u ε )|∇u ε |3 ∇ϕdxdt

  − QT

  |u|2m f  (u)|∇u|3 ∇ϕdxdt 

     ≤  [(|u ε |2 + ε)m − |u|2m ] f  (u ε )|∇u ε |3 ∇ϕdxdt  Pδ

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     +  |u|2m ( f  (u ε )|∇u ε |3 − f  (u ε )|∇u|3 )∇ϕdxdt  Pδ      2m   3  + |u| [ f (u ε ) − f (u)]|∇u| )∇ϕdxdt  Pδ

     2 m  3  + (|u ε | + ε) f (u ε )|∇u ε | ∇ϕdxdt  Q T \Pδ      2m  3  + |u| f (u)|∇u| ∇ϕdxdt  P\Pδ

  C   ≤ sup (|u ε |2 + ε)m − |u|2m  |∇ϕ| √ δ      |u|2m ( f  (u ε )|∇u ε |3 − f  (u ε )|∇u|3 )∇ϕdxdt  +  Pδ      +  |u|2m [ f  (u ε ) − f  (u)]|∇u|3 )∇ϕdxdt  Pδ

√ m + C(δ + ε) 2 sup |∇ϕ| + C δ sup |∇ϕ|, ε ∈ (0, ε0 (δ)). 2

Then        2 m 2m  lim (|u ε | + ε) ∇ f (u ε )∇ϕdxdt − |u| ∇ f (u)∇ϕdxdt  ε→0  QT P √ ≤ C δ sup |∇ϕ|. By the arbitrariness of δ, we can check that the limit (42) holds. Similarly, using the same method, we can obtain the limit (41), (43)–(45). Then, the proof is completed. 

4 Nonnegativity Note that in Theorem 3.1, we assume that u 0 (x) ≥ 0 when m ≥ 1. In fact, we also have the nonnegativity of solutions for the weak solution u(x, t) when m > 21 and u 0 (x) ≥ 0. Theorem 4.1 Suppose that u 0 (x) ≥ 0, • If 21 < m < 1, then the weak solution u(x, t) obtained in the above section satisfies u(x, t) ≥ 0; ⎧ ⎪ ⎪ | ln u 0 (x)|dx < ∞, if m = 1, ⎨  then the weak solution • If m ≥ 1 and ⎪ ⎪ ⎩ u 2−n dx < ∞, if m > 1, 0 

obtained in the above section satisfies u(x, t) ≥ 0. Moreover, the set {u(x, t) = 0}

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A Sixth-Order Phase-Field Equation with Degenerate Mobility

has zero measure and ⎧ ⎪ ⎪ | ln u(x, t)|dx < ∞, ∀t ∈ [0, T ], if m = 1, ⎨  ⎪ ⎪ ⎩ [u(x, t)]2−n dx < ∞, ∀t ∈ [0, T ], if m > 1.

(46)



Proof It is easy to check that G 0 (s) > 0 in (21) when m > 21 . We prove the results by reduction. If it is not true, then there exists a point (x0 , t0 ) ∈ Q T such that u(x0 , t0 ) < 0. Note that u ε → u uniformly, there exists δ > 0 and ε0 > 0 such that u ε (x, t0 ) < −δ, if |x − x0 | ≤ δ, x ∈ , ε < ε0 . But, for such x,  G ε (u ε (x, t0 )) = −



A u ε (x,t0 )

gε (s)ds ≥ −

0 −δ

gε (s)ds → +∞, as ε → 0,

which contradicts (24). Moreover, we prove that for each T ∈ (0, T0 ), the set {u(·, T ) = 0} has measure zero. If it is not true, then for some t0 ∈ (0, T0 ), the set E = {u(·, t0 ) = 0} has positive measure. Since u ε → u uniformly, for any given δ > 0, there exists ε0 > 0, such that u ε < δ, ∀x ∈ E and ε < ε0 . For any x ∈ E and for any δ > 0,  G ε (u ε (x, t0 )) ≥ −

A δ

 gε (r )dr → −

A δ

g0 (r )dr ≡ λ(δ) as ε → 0.

It is easy to see that lim λ(δ) = +∞,

δ→0

then  lim

ε→0 

G ε (u ε (x, t0 ))dx ≥ λ(δ)measE,

Letting δ → 0, we can get a contradiction to (24). At the points (x, ·) where u(x, t) > 0, G ε (u ε (x, t)) → G 0 (u(x, t)). (47) Note that the set {u(·, t) = 0} has measure zero for any t, then for any t, (47) holds for almost all x. It then follows from (24) and Fatou’s lemma, and we get  G 0 (u(x, t))dx ≤ C, 

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which yields the assertions. Then, we complete the proof.



Theorem 4.2 Suppose that u 0 (x) > 0 and m ≥ 2. Then, the weak solution obtained in the above section satisfies u(x, t) > 0. Moreover, such a solution is unique. Proof u(x, t) ≥ 0 has been proved in the above theorem. If the conclusion were false, then there would exist a point (x0 , t0 ) ∈ Q T such that u(x0 , t0 ) = 0. Based on the 1 Hölder continuity of u, we can obtain u(x, t0 ) ≤ C|x − x0 | 2 . Since m ≥ 2, we get 





u(x, t0 )

2−2m

dx ≥ C



|x − x0 |1−m dx = ∞.

Moreover, by the proof of Theorem 4.1, 

 

u(x, t0 )2−n dx ≤ lim

ε→0 

G ε (u ε )dx ≤ C,

which is a contradiction. We prove the uniqueness in the following. Suppose that v is another solution. Note that 0 < C17 ≤ u(x, t), v(x, t) ≤ C18 , (x, t) ∈ Q T . Let w(x, t) = u(x, t) − v(x, t), then w(x, t) satisfies  ∂w = ∇ · (|u|2m ∇2 u − |v|2m ∇2 v) − (|u|2m ∇u − |v|2m ∇v) ∂t − (|u|2m ∇ f (u) − |v|2m ∇ f (v)) − (|u|2m ∇( f  (u)u) − |v|2m ∇( f  (v)v))  + |u|2m ∇( f  (u) f (u) + f (u)) − |v|2m ∇( f (v) f  (v) + f (v)) . Multiplying by 2 w, we derive that 1 d 2 dt



 |w|2 dx +

(|u|2m ∇2 u − |v|2m ∇2 v)∇2 wdx

  2m = (|u| ∇u − |v|2m ∇v)∇2 wdx   + (|u|2m ∇ f (u) − |v|2m ∇ f (v))∇2 wdx  + (|u|2m f  (u)∇u − |v|2m f  (v)∇v)∇2 wdx  + (|u|2m f  (u)∇uu − |v|2m f  (v)∇vv)∇2 wdx  − |u|2m [ f  (u) f (u) + f (u)] ∇u − |v|2m [ f  (v) f (v) 

123

(48)

A Sixth-Order Phase-Field Equation with Degenerate Mobility

 + f (v)] ∇v ∇2 wdx := I6 + I7 + I8 + I9 + I10 .

(49)

Note that ||u|2m −|v|2m | ≤ C19 |w| and f (i) (u)− f (i) (v) ≤ C20 |w|, where i = 1, 2, 3. Hence,  (|u|2m ∇2 u − |v|2m ∇2 v)∇2 wdx   2m 2 2 = |u| |∇ w| dx + (|u|2m − |v|2m )∇2 v∇2 w.





(50)



Now, we estimate I6 –I10 . 

(|u|2m f  (u)∇u − |v|2m f  (v)∇v)∇2 wdx  + 3 (|u|2m f  (u)∇uu − |v|2m f  (v)∇vv)∇2 wdx   + (|u|2m f  (u)|∇u|3 − |v|2m f  (v)|∇v|3 )∇2 wdx    = |u|2m f  (u)∇w∇2 wd x + |u|2m ( f  (u)

I7 =







− f  (v))∇v∇2 wdx   2m 2m  2 + (|u| − |v| ) f (v)∇v∇ wdx + 3 |u|2m f  (u)∇uw∇2 wdx     +3 |u|2m f  (u)v∇w∇2 wdx + 3 |u|2m ( f  (u) 



− f  (v))∇vv∇2 wd x   2m 2m  2 + 3 (|u| − |v| ) f (v)∇vv∇ wdx + |u|2m f  (u)(|∇u|3 



− |∇v|3 )∇2 wdx   |u|2m ( f  (u) − f  (v))|∇v|3 ∇2 wdx + (|u|2m +  2m

− |v|





) f (v)|∇v| ∇ wdx   2m 2 2 2 ≤ C17 |∇ w| d x + C |∇w| dx + C w 2 |∇v|2 dx       +C |∇uw|2 dx + C |v∇w|2 dx + C |w∇vv|2 dx     2 2 2 2 +C |∇w| (|∇u| + ∇u∇v + |∇v| ) dx + C w 2 |∇v|6 dx      2m 2 2 2 ≤ C17 |∇ w| dx + C |∇w| dx + C w 2 |∇v|2 dx 



3

2





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 1  1  2 2 |∇u|4 dx |w|4 dx + C sup |∇w|2 · |v|2 dx x∈      + C sup |∇w|2 · |∇vv|2 dx + C sup |∇w|2 · |∇v|6 dx 

+C



x∈



x∈

 1 

2 3 (|∇u|2 + ∇u∇v + |∇v|2 )dx     2m 2 2 ≤ C17 |∇ w| dx + C |∇w|2 dx + C w 2 |∇v|2 dx       2m 2 2 2 ≤ 2C17 |∇ w| dx + C23 |∇w| dx + C24 w 2 |∇v|2 dx,      I6 = |u|2m ∇w∇2 wdx + (|u|2m − |v|2m )∇v∇2 wdx     2m 2 2 ≤ C17 |∇ w| dx + C |∇w|2 dx + C w 2 |∇v|2 dx.       2m 2 2 2 ≤ 2C17 |∇ w| dx + C21 |∇w| dx + C22 w 2 |∇v|2 dx,      I8 = |u|2m f  (u)∇w∇2 wdx + |u|2m ( f  (u) − f  (v))∇v∇2 wdx    + (|u|2m − |v|2m ) f  (v)∇v∇2 wdx     2m ≤ C17 |∇2 w|2 dx + C |∇w|2 dx + C w 2 |∇v|2 dx       2m 2 2 2 ≤ 2C17 |∇ w| dx + C25 |∇w| dx + C26 w 2 |∇v|2 dx,      2m  2 I9 = |u| f (u)∇uw∇ wdx + |u|2m f  (u)∇wv∇2 wdx    + |u|2m ( f  (u) − f  (v))∇vv∇2 wdx  + (|u|2m − |v|2m ) f  (v)∇vv∇2 wdx 

|∇w|6 dx

+C



≤

I10

2m C17

3

 1 





2

1 2

|∇ w| dx + C |∇u| d x |w| d x     + C sup |∇w|2 · |v|2 d x + C sup |∇w|2 · |∇vv|2 dx x∈ x∈     2m 2 2 2 ≤ C17 |∇ w| dx + C |∇w| dx     2m ≤ 2C17 |∇2 w|2 dx + C27 |∇w|2 dx,    = |u|2m [ f  (u) f (u) + f (u)] ∇w∇2 wdx 

123

2

2

4

2

A Sixth-Order Phase-Field Equation with Degenerate Mobility

 +





(|u|2m − |v|2m )[ f  (v) f (v) + f (v)] ∇v∇2 wdx

|u|2m {[ f  (u) f (u) + f (u)] − [ f  (v) f (v) + f (v)] }∇v∇2 wdx    2m ≤ C17 |∇2 w|2 dx + C |∇w|2 dx + C w 2 |∇v|2 dx       2m 2 2 2 2 ≤ C17 |∇ w| dx + C |∇w| dx + C sup |w| · |∇v|2 dx x∈       2m 2 2 2 ≤ C17 |∇ w| dx + C28 |∇w| dx + C29 |w|2 dx. +









On the other hand, we have  

 |∇w|2 dx ≤ 



 w 2 dx + C

 

|w|2 dx ≤ C



 |∇w|2 dx + C



|w|2 dx,

where the constant  is small enough and it satisfies 1 − C > 0. Hence, 

 

|∇w|2 dx ≤ C



|w|2 dx.

We also have  −



 

2m (|u|2m − |v|2m )∇2 v∇2 w ≤ C17

≤

2m C17

 ·



 

|∇2 w|2 dx + C



w 2 |∇2 v|2 dx

|∇2 w|2 dx + C sup |w|2 x∈

|∇2 v|2 dx   2m ≤ C17 |∇2 w|2 dx + C |w|2 dx    2 2 × |∇ v| dx. 



Summing up, we have 1 d 2 dt





|w| dx |∇2 w|2 dx      2 2 2 2 |w| dx |∇ v| dx + |∇v| dx + 1 , ≤ C30 2





2m + (1 − 12)C17





where  is small enough and it satisfies 1 − 12 > 0. Integrating (51) over  × (0, t), we deduce that

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 sup 0


 t  |w|2 dx |∇2 v|2 dx 0
|w|2 dx ≤ 2C30 sup



(51)

This shows that for small t, we must have w = 0. Using Poincaré’s inequality, we get w = 0. Then, the uniqueness of solutions is obtained. Theorem 4.2 is proved.  If we define u 0δ = u 0 (x) + δ, denote by u δ (x, t) the solution u constructed in Theorem 3.1 for the initial data u 0δ , which then satisfies all the properties asserted in Theorems 4.1–4.2. Then, u δ satisfies the estimate (36). Taking a subsequence u δ → u, we obtain the following result. Theorem 4.3 Suppose that m > 21 and u 0 (x) ≥ 0. Then the weak solution u(x, t) obtained in Sect. 3 satisfies u(x, t) ≥ 0. Acknowledgements The authors are indebted to the referees for careful reading of the paper and helpful suggestions. This work is partially supported by “Fundamental Research Funds for the Central Universities (Grant No. JUSRP116030),” “Natural Science Foundation of China for Young Scholar (Grant No. 11401258)” and “Natural Science Foundation of Jiangsu Province for Young Scholar (Grant No. BK20140130).”

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