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Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 329, pp. 1–22.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

EXISTENCE OF SOLUTIONS TO THE

CAHN-HILLIARD/ALLEN-CAHN EQUATION WITH DEGENERATE MOBILITY

XIAOLI ZHANG, CHANGCHUN LIU

Abstract. This article we study the Cahn-Hilliard/Allen-Cahn equation with degenerate mobility. Under suitable assumptions on the degenerate mobility and the double well potential, we prove existence of weak solutions, which can be obtained by considering the limits of Cahn-Hilliard/Allen-Cahn equations with non-degenerate mobility.

1. Introduction

In this article, we consider a scalar Cahn-Hilliard/Allen-Cahn equation with degenerate mobility

ut=−∇[D(u)∇(∆u−f(u))] + (∆u−f(u)), in QT, (1.1) whereQT = Ω×(0, T), Ω is a bounded domain inRn with aC3-boundary∂Ω and f(u) is the derivative of a double-well potentialF(u) with wells ±1. The mobility D(u)∈C(R; [0,∞)) is in the form

D(u) =|u|m, if|u|< δ,

C0≤D(u)≤C1|u|m, if|u| ≥δ, (1.2) for some constantsC0, C1, δ >0, where 0< m <∞ifn= 1,2 and n4 < m < n−24 ifn≥3.

Equation (1.1) is supplemented by the boundary conditions

u|∂Ω= ∆u|∂Ω= 0, t >0, (1.3) and the initial condition

u(x,0) =u0(x). (1.4)

Equation (1.1) was introduced as a simplification of multiple microscopic mecha- nisms model [8] in cluster interface evolution. Equation (1.1) with constant mobility has been intensively studied. Karali and Nagase [9] investigated existence of weak solution to (1.1) withD(u)≡D and a quartic bistable potentialF(u) = (1−u2)2. Karali and Nagase [9] only provided existence of the solution for the deterministic case. Then Antonopoulou, Karali and Millet [2] studied the stochastic case. The

2010Mathematics Subject Classification. 35G25, 35K55, 35K65.

Key words and phrases. Cahn-Hilliard/Allen-Cahn equation; existence; Galerkin method;

degenerate mobility.

c

2016 Texas State University.

Submitted September 2, 2016. Published December 24, 2016.

1

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main result of this paper is the existence of a global solution, under a specific sub- linear growth condition for the diffusion coefficient. Path regularity in time and in space is also studied. In addition, Karali and Ricciardi [7] constructed special sequences of solutions to a fourth order nonlinear parabolic equation of the Cahn- Hilliard/Allen-Cahn equation, converging to the second order Allen-Cahn equation.

They studied the equivalence of the fourth order equation with a system of two sec- ond order elliptic equations. Karali and Katsoulakis [8] focus on a mean field par- tial differential equation, which contains qualitatively microscopic information on particle-particle interactions and multiple particle dynamics, and rigorously derive the macroscopic cluster evolution laws and transport structure. They show that the motion by mean curvature is given byV =µσκ, whereκis the mean curvature, σis the surface tension andµis an effective mobility that depends on the presence of the multiple mechanisms and speeds up the cluster evolution. This is in contrast with the Allen-Cahn equation where the velocity equals the mean curvature. Tang, Liu and Zhao [18] proved the existence of global attractor. Liu and Tang [15] ob- tained the existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions.

During the past few years, many authors have paid much attention to the Cahn- Hilliard equation with degenerate mobility. An existence result for the Cahn- Hilliard equation with a degenerate mobility in a one-dimensional situation has been established by Yin [19]. Elliott and Garcke [5] considered the Cahn-Hilliard equation with non-constant mobility for arbitrary space dimensions. Based on Galerkin approximation, they proved the global existence of weak solutions. Dai and Du [4] improved the results of the paper [5]. Liu [12] proved the existence of weak solutions for the convective Cahn-Hilliard equation with degenerate mobility.

The relevant equations or inequalities have also been studied in [10, 11, 13, 14].

Motivated by the above works, we prove the existence of weak solution to (1.1)- (1.4) under a more general range of the double-well potentialF. In particular, we assume that fors∈R,F ∈C2(R) satisfies

k0(|s|r+1−1)≤F(s)≤k1(|s|r+1+ 1), (1.5)

|F0(s)| ≤k2(|s|r+ 1), (1.6)

|F00(s)| ≤k3(|s|r−1+ 1), (1.7) for some constantsk0, k1, k2, k3>0 where 1≤r <∞ifn= 1,2 and 1≤r≤ n−2n ifn≥3. What’s more, we need the assumption on the boundary off(u),

f(u)|∂Ω= 0, t >0. (1.8) We can give examples satisfying the condition (1.8), such as F(u) = (1−u2)2 studied by Karali and Nagase [9], the logarithmic functionf(u) =−θcu+θ2ln1+u1−u, u∈(−1,1), 0< θ < θc [3].

Concerning the Allen-Cahn structure, we rewrite (1.1), (1.3), (1.4) and (1.8) to the form

ut=∇(D(u)∇v)−v, in QT, v=−∆u+f(u), in QT,

u(x,0) =u0(x), in Ω, u=v= 0, on∂Ω.

(1.9)

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We consider the free energy functionalE(u) defined in [9] given by E(u) :=

Z

1

2|∇u|2+F(u)

dx. (1.10)

For a pair of solution (u, v) of (1.9) it holds that d

dtE(u) = Z

vutdx= Z

v[∇(D(u)∇v)−v]dx=− Z

D(u)|∇v|2+v2 dx≤0.

Notation. Define the usual Lebesgue norms and theL2-inner-product kukp=kukLp(Ω) and (u, v) = (u, v)L2(Ω).

The duality pairing between the spaceH2(Ω) and its dual (H2(Ω))0will be denoted using the form h·,·i. For simplicity, 2 := n−22n . χB denotes the characteristic function ofB.

This paper is organized as follows. In Section 2, we use a Galerkin method to give a existence of weak solution for a positive mobility. Section 3 uses a sequence of non-degenerate solutions to approximate the degenerate case (1.9).

2. Existence for positive mobility

In this section, we study the Cahn-Hilliard/Allen-Cahn equation with a non- degenerate mobilityDε(u) defined for anεsatisfying 0< ε < δ by

Dε(u) :=

(|u|m, if|u|> ε,

εm, if|u| ≤ε. (2.1) So we consider the problem

ut=∇(Dε(u)∇v)−v, in QT, v=−∆u+f(u), in QT,

u(x,0) =u0(x), in Ω, u=v= 0, on∂Ω.

(2.2)

Theorem 2.1. Supposeu0∈H1(Ω), under assumptions(1.2)and (1.5)–(1.7), for any T >0, there exists a pair of functions(uε, vε)such that

(1) uε∈L(0, T;H01(Ω))∩C([0, T];Lp(Ω))∩L2(0, T;H3(Ω)), where1≤p <

∞if n= 1,2 and2≤p < n−22n if n≥3, (2) ∂tuε∈L2(0, T; (H2(Ω))0),

(3) uε(x,0) =u0(x)for allx∈Ω, (4) vε∈L2(0, T;H01(Ω)),

which satisfies equation (2.2)in the following weak sense Z T

0

h∂tuε, φidt+ Z Z

QT

−∆uε+f(uε) φ dx dt

=− Z Z

QT

Dε(uε) − ∇∆uε+F00(uε)∇uε

· ∇φ dx dt

(2.3)

for all test functions φ ∈ L2(0, T;H2(Ω)∩H01(Ω)). In addition, uε satisfies the energy inequality

E(uε) + Z t

0

Z

Dε(uε(x, τ))|∇vε(x, τ)|2+|vε(x, τ)|2

dx dτ ≤E(u0), (2.4)

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for allt >0.

To prove the above theorem, we apply a Galerkin approximation. Let{φJ}j∈N be the eigenfunctions of the Laplace operator on L2(Ω) with Dirichlet boundary condition, i.e.,

−∆φJJφJ, in Ω,

φJ= 0, on∂Ω. (2.5)

The eigenfunctions{φJ}j=1form an orthogonal basis forL2(Ω), H1(Ω) andH2(Ω).

Hence, for initial data u0 ∈ H1(Ω), we can find sequences of scalars (u0N,j; j = 1,2, . . . , N)N=1 such that

lim

N→∞

N

X

j=1

u0N,jφJ=u0, inH1(Ω). (2.6) LetVN denote the linear span of (φ1, . . . , φN) andPN be the orthogonal projection fromL2(Ω) toVN, that is

PNφ:=

N

X

j=1

Z

φφJdx φJ.

LetuN(x, t) =PN

j=1cNJ(t)φJ(x),vN(x, t) =PN

j=1dNJ(t)φJ(x) be the approximate solution of (2.2) inVN; that is,uN, vN satisfy the g system of equations

Z

tuNφJdx=− Z

Dε(uN)∇vN· ∇φJdx− Z

vNφJdx, (2.7) Z

vNφJdx= Z

∇uN· ∇φJ+f(uNJdx, (2.8) uN(x,0) =

N

X

j=1

u0N,jφJ(x), (2.9)

forj= 1, . . . , N andu0N,j =R

u0φJdx.

This gives an initial value problem for a system of ordinary differential equations for (c1, . . . , cN)

tcNJ(t) =−

N

X

k=1

dNk(t) Z

Dε

N

X

i=1

cNi (t)φi(x)

∇φk∇φJdx−dNJ(t), (2.10)

dNJ(t) =λJcNJ(t) + Z

f

N

X

i=1

cNi (t)φi(x)

φJdx, (2.11)

cNJ(0) =u0N,j = (u0, φJ), (2.12) which has to hold forj= 1, . . . , N.

Define X(t) = cN1 (t), . . . , cNN(t)

, F(t,X(t)) = f1(t,X(t)), . . . , fN(t,X(t)) , where

fJ(t,X(t)) =−

N

X

k=1

Z

Dε

XN

i=1

cNi (t)φi(x)

∇φk∇φJdx

×

λkcNk(t) + Z

f

N

X

i=1

cNi (t)φi(x) φkdx

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−λJcNJ(t)− Z

fXN

k=1

cNk (t)φk(x) φJdx

forj= 1, . . . , N. Then problem (2.10)-(2.12) is equivalent to the problem X0(t) =F(t,X(t)), X(0) = (u0N,1, . . . , u0N,N).

Since the right hand side of the above equation is continuous, it follows from the Cauchy-Peano Theorem [16] that the problem (2.10)-(2.12) has a solution X(t)∈ C1[0, TN], for someTN >0, i. e., the system (2.7)-(2.9) has a local solution.

To prove the existence of solutions, we need some a priori estimates onuN. Lemma 2.2. For any T >0, we have

kuNkL(0,T;H10(Ω))≤C, for allN, k∂tuNkL2(0,T;(H2(Ω))0)≤C, for allN, whereC independent of N.

Proof. For any fixedN ∈N+, we multiply (2.7) bydNJ(t) and sum overj= 1, . . . , N to obtain

Z

tuNvNdx=− Z

Dε(uN)|∇vN|2dx− Z

|vN|2dx. (2.13) Multiply (2.8) by∂tcNJ(t) and sum over j= 1, . . . , N to obtain

Z

vNtuNdx= Z

∇uNt∇uN +f(uN)∂tuN dx,

= d dt

Z

1

2|∇uN|2+F(uN) dx.

By (2.13) and the above identity, we have d

dt Z

1

2|∇uN|2+F(uN) dx=−

Z

Dε(uN)|∇vN|2dx− Z

|vN|2dx. (2.14) Replacingt byτ in (2.14) and integrating overτ∈[0, t], by (1.5) and the Sobolev embedding theorem we obtain

Z

1

2|∇uN(x, t)|2+F(uN(x, t))

dx

+ Z t

0

Z

Dε(uN(x, τ))|∇vN(x, τ)|2+|vN(x, τ)|2 dx dτ

= Z

1

2|∇uN(x,0)|2+F(uN(x,0))

dx

≤1

2k∇uN(x,0)k22+k1kuN(x,0)kr+1r+1+k1|Ω|.

≤1

2k∇u0k22+k1Cku0kr+1H1(Ω)+k1|Ω| ≤C.

The last inequality follows fromu0∈H1(Ω). This implies Z

1

2|∇uN(x, t)|2+k0|uN|r+1 dx

+ Z t

0

Z

Dε(uN(x, τ))|∇vN(x, τ)|2+|vN(x, τ)|2

dx dτ ≤C.

(2.15)

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By (2.15) and Poincar´e’s inequality we have

kuNkH1(Ω)≤C, fort >0.

This estimate implies that the coefficients{cNJ :j= 1, . . . , N}are bounded in time and therefore a global solution to the system (2.7)-(2.9) exists. In addition, for any T >0, we have

uN ∈L(0, T;H01(Ω)), kuNkL(0,T;H10(Ω))≤C, for allN. (2.16) Inequality (2.15) implies

k q

Dε(uN)∇vNkL2(QT)≤C, for allN, (2.17) kvNkL2(QT)≤C, for allN. (2.18) By the Sobolev embedding theorem, the growth condition (1.2) and (2.1), for|u|>

ε, we obtain Z

|Dε(uN)|n/2dx≤(C1+ 1) Z

|uN|n2 dx,≤CkuNkmn/2H1(Ω)≤C.

If|u| ≤ε, obviously we obtain the above estimate. This implies

kDε(uN)kL(0,T;Ln/2(Ω))≤C, for allN. (2.19) For any φ ∈ L2(0, T;H2(Ω)), we obtain PNφ = PN

j=1aJ(t)φJ, where aJ(t) = R

φφJdx. Multiplying (2.7) by aJ(t), summing overj = 1,2, . . . , N, by H¨older’s inequality, (2.17)-(2.19) and the Sobolev embedding theorem, we have

Z T

0

Z

tuNφ dx dt

=

Z T

0

Z

tuNPNφ dx dt

=

Z T

0

Z

Dε(uN)∇vN∇PNφ+vNPNφ dx dt

≤ Z T

0

k q

Dε(uN)knk q

Dε(uN)∇vNk2k∇PNφk2dt+ Z T

0

kvNk2kPNφk2dt

≤C Z T

0

kq

Dε(uN)∇vNk2kφkH2+kvNk2kφkH2dt

≤C k q

Dε(uN)∇vNkL2(QT)+kvNkL2(QT)

kφkL2(0,T;H2(Ω))

≤CkφkL2(0,T;H2(Ω)). Hence,

k∂tuNkL2(0,T;(H2(Ω))0)≤C for allN. (2.20)

The proof is complete.

Lemma 2.3. Supposeu0 ∈H1(Ω), under assumptions (1.2) and (1.5)-(1.7), for any T >0, there exists a pair of functions(uε, vε)such that

(1) uε∈L(0, T;H01(Ω))∩C([0, T];Lp(Ω)), where 1≤p <∞if n= 1,2 and 2≤p < n−22n if n≥3,

(2) ∂tuε∈L2(0, T; (H2(Ω))0), (3) uε(x,0) =u0(x)for allx∈Ω,

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(4) vε∈L2(0, T;H01(Ω)), which satisfies

Z T

0

h∂tuε, φidt=− Z T

0

Z

Dε(uε)∇vε· ∇φ dx dt− Z T

0

Z

vεφ dx dt.

Proof. Since the embeddingH01(Ω),→Lp(Ω) is compact for 1≤p <∞ifn= 1,2 and 1 ≤ p < n−22n if n ≥3, Lp(Ω) ,→ (H2(Ω))0 is continuous for p≥1 if n ≤3, p >1 ifn= 4 and p≥ n+42n if n≥5. Using the Aubin-Lions lemma (Lions [17]), we can find a subsequence which we still denote byuN anduε∈L(0, T;H01(Ω)), such that asN → ∞

uN * uε, weak-* inL(0, T;H01(Ω)), (2.21) uN →uε, strongly inC([0, T];Lp(Ω)), (2.22) uN →uε, strongly inL2(0, T;Lp(Ω)) and almost everywehre inQT, (2.23)

tuN * ∂tuε, weakly inL2(0, T; (H2(Ω))0), (2.24) where 2≤p <2ifn≥3 and 1≤p <∞ifn= 1,2.

By multiplying (2.7) byaJ(t) and integrating (2.7) overt∈[0, T], we obtain Z T

0

Z

tuNaJ(t)φJdx dt

=− Z T

0

Z

Dε(uN)∇vN ·aJ(t)∇φJdx dt− Z T

0

Z

vNaJ(t)φJdx dt.

(2.25)

To pass to the limit in (2.25), we need the convergence ofvN andDε(uN)∇vN. By (2.17) andDε(uN)≥εm, then

k∇vNkL2(QT)≤Cεm2 <∞, for anyε >0. (2.26) This implies that {∇vN} is a bounded sequence in L2(QT), thus there exists a subsequence, not relabeled, andζε∈L2(QT) such that

∇vN * ζε, weakly inL2(QT). (2.27) By (2.26) and Poincar´e’s inequality, we have

kvNkL2(0,T;H01(Ω))≤Cεm2 <∞, for anyε >0.

Hence we can find a subsequence of vN, not relabeled, and vε ∈ L2(0, T;H01(Ω)) such that

vN * vε, weakly inL2(0, T;H01(Ω)). (2.28) For anyg∈L2(0, T;H01(Ω)), by (2.26) and (2.27) we have

Nlim→∞

Z T

0

Z

∇vNg dx dt= Z T

0

Z

ζεg dx dt

= lim

N→∞

Z T

0

Z

vN∇g dx dt= Z T

0

Z

∇vεg dx dt.

Henceζε=∇vεalmost all in QT and

∇vN *∇vε, weakly inL2(QT). (2.29)

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By (2.18), we can extract a further sequence ofvN, not relabeled, andηε∈L2(QT) such that

vN * ηε, weakly inL2(QT). (2.30) By (2.28) and (2.30) for anyg∈L2(QT)⊂L2(0, T;H−1(Ω)), we have

N→∞lim Z T

0

Z

vNg dx dt= Z T

0

Z

vεg dx dt= Z T

0

Z

ηεg dx dt.

This impliesηε=vεalmost allQT and

vN * vε, weakly inL2(QT). (2.31) Consequently we have the bound

Z

QT

|vε|2dx dt≤C. (2.32)

For anyt∈[0, T], byDε(uN)≤C(1 +|uN|m), we have Dε(uN)n/2

≤C(1 +|uN|m)n/2≤(C(1 +|uN|))mn/2,

where 2≤ mn2 <2. By (2.22), C(1 +|uN|)→C(1 +|uθ|) inLmn/2(Ω). SinceDε is continuous and (2.23), we obtain

Dε(uN)→Dε(uε), a.e. in Ω.

The generalized Lebesgue convergence theorem [1] gives Dε(uN)→Dε(uε), inLn/2(Ω).

This implies

kDε(uN)−Dε(uε)kn/2→0, asN → ∞.

The above estimate holds for any t ∈ [0, T], and we can take supremum on both sides of the above estimate to obtain

sup

t∈[0,T]

kDε(uN)−Dε(uε)kn/2→0, asN → ∞.

This implies

Dε(uN)→Dε(uε), strongly inC(0, T;Ln/2(Ω)). (2.33) Byp

Dε(uN)≤C(1 +|uN|m2), (2.22), (2.23) and the generalized Lebesgue conver- gence theorem, similarly, we have

q

Dε(uN)→p

Dε(uε), strongly inC(0, T;Ln(Ω)). (2.34) For anyϕ∈L2(0, T;L2(Ω)), by H¨older’s inequality we have

Z Z

QT

qDε(uN)∇vNϕ−p

Dε(uε)∇vεϕ dx dt

=

Z Z

QT

[ q

Dε(uN)−p

Dε(uε)]∇vNϕ+p

Dε(uε)[∇vNϕ− ∇vεϕ]

dx dt

≤ Z T

0

k q

Dε(uN)−p

Dε(uε)knk∇vNk2kϕk2dt +

Z Z

QT

pDε(uε)ϕ[∇vN − ∇vε]dx dt

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≤ sup

t∈[0,T]

kq

Dε(uN)−p

Dε(uε)knk∇vNkL2(QT)kϕkL2(0,T;L2(Ω))

+

Z Z

QT

pDε(uε)ϕ[∇vN − ∇vε]dx dt

≡I+II.

By (2.29) and (2.34),I→0 asN → ∞. By H¨older’s inequality and (2.34) we have Z Z

QT

|p

Dε(uε)ϕ|2dx dt≤ Z T

0

Z

Dε(uε)n/2

dxn/2Z

|ϕ|n−22n dxn−2n dt

≤ sup

t∈[0,T]

kp

Dε(uε)k2n Z T

0

kϕk2L2(Ω)dt

≤Ckϕk2L2(0,T;L2(Ω)). This implies

pDε(uε)ϕ∈L2(QT). (2.35) ThusII →0 asN → ∞by (2.29). Hence

q

Dε(uN)∇vN *p

Dε(uε)∇vε, weakly inL2(0, T;Ln+22n (Ω)). (2.36) Next we consider the convergence ofDε(uN)∇vN. By (2.17), (2.36) andL2(QT)⊂ L2(0, T;Ln+22n (Ω)), we can extract a further sequence, not relabeled, such that

q

Dε(uN)∇vN *p

Dε(uε)∇vε, weakly inL2(QT). (2.37) By H¨older’s inequality and (2.17), we have

Z Z

QT

q

Dε(uN)∇vN ·p

Dε(uε)∇vεdx dt

≤ k q

Dε(uN)∇vNkL2(QT)kp

Dε(uε)∇vεkL2(QT)

≤Ckp

Dε(uε)∇vεkL2(QT),

(2.38)

whereCis independent ofε. Taking the limit of (2.38) on both sides, by (2.37) we have

kp

Dε(uε)∇vεkL2(QT)≤C. (2.39) For anyϕ∈L2(0, T;L2(Ω)), by H¨older’s inequality we obtain

Z Z

QT

Dε(uN)∇vNϕ−Dε(uε)∇vεϕ dx dt

Z Z

QT

[ q

Dε(uN)−p Dε(uε)]

q

Dε(uN)∇vNϕ dx dt

+

Z Z

QT

pDε(uε)[

q

Dε(uN)∇vNϕ−p

Dε(uε)∇vεϕ]dx dt

≤ Z T

0

k q

Dε(uN)−p

Dε(uε)knk q

Dε(uN)∇vNk2kϕk2dt +

Z Z

QT

pDε(uε)ϕ[

q

Dε(uN)∇vN −p

Dε(uε)∇vε]dx dt

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≤ sup

t∈[0,T]

kq

Dε(uN)−p

Dε(uε)knkq

Dε(uN)∇vNkL2(QT)kϕkL2(0,T;L2(Ω))

+

Z Z

QT

pDε(uε)ϕ[

q

Dε(uN)∇vN −p

Dε(uε)∇vε]dx dt

=I+II.

By (2.34) and (2.37),I→0 asN → ∞. By (2.35) and (2.37), we haveII →0 as N → ∞. Thus

Dε(uN)∇vN * Dε(uε)∇vε, weakly inL2(0, T;Ln+22n (Ω)). (2.40) For any φ ∈ L2(0, T;H2(Ω)∩H01(Ω)), we obtain Pnφ = Pn

j=1aJ(t)φJ, where aJ(t) =R

φφJdx, then Pnφconverges strongly toφinL2(0, T;H2∩H01(Ω)) and aJ(t)∈L2(0, T). For φJ ∈H2(Ω), by Sobolev embedding theorem, we obtain

k∇φJk2≤Ck∇φJkH1(Ω)≤C.

ThusaJ(t)∇φJ ∈L2(0, T;L2) and

aJ(t)φJ∈L2(0, T;H2∩H01(Ω))⊂L2(0, T;H−1(Ω)).

Taking the limit asN → ∞ on both sides of (2.25), by (2.24), (2.40) and (2.28), we have

Z T

0

h∂tuε, aJ(t)φJidt

=− Z T

0

Z

Dε(uε)∇vε·aJ(t)∇φJdx dt− Z T

0

Z

vεaJ(t)φJdx dt,

(2.41)

for allj∈N.

Then we sum overj= 1,2, . . . , non both sides (2.41) to get Z T

0

h∂tuε,Pnφidt

=− Z T

0

Z

Dε(uε)∇vε· ∇Pnφ dx dt− Z T

0

Z

vεPnφ dx dt.

(2.42)

SincePnφconverges strongly toφin L2(0, T;H2(Ω)), thus asn→ ∞, Z T

0

k∇Pnφ− ∇φk22dt≤ Z T

0

k∇Pnφ− ∇φk2H1dt

≤ Z T

0

kPnφ−φk2H2dt→0.

This implies that ∇Pnφ converges strongly to ∇φ in L2(0, T;L2(Ω)). Thus we obtain

Pnφ * φ, weakly inL2(0, T;H2(Ω)∩H01(Ω)), (2.43)

∇Pnφ *∇φ, weakly inL2(0, T;L2(Ω)). (2.44) ByL2(0, T;H01(Ω))⊂L2(0, T;H−1(Ω)), we take the limit asn→ ∞on both sides (2.42), then obtain

Z T

0

h∂tuε, φidt=− Z T

0

Z

Dε(uε)∇vε· ∇φ dx dt− Z T

0

Z

vεφ dx dt. (2.45)

(11)

As for the initial value, by (2.9) asN→ ∞,

uN(x,0)→u0(x) in L2(Ω).

By (2.22),uε(x,0) =u0(x) inL2(Ω). The proof is complete.

Proof of Theorem 2.1. We need only to check that uε ∈ L2(0, T;H3(Ω)), vε =

−∆uε+f(uε) and∇vε=−∇∆uε+F00(uε)∇uε. First we consider the convergence of∇uN andf(uN). By (2.21), we have

Z T

0

k∇uNk22dt≤C.

Hence we can find a subsequence ofuN, not relabeled, andυ∈L2(QT), such that

∇uN * υ weakly inL2(QT). (2.46) For anyφ∈L2(0, T;H01(Ω)), by integration by parts we have

lim

N→∞

Z T

0

Z

∇uNφ dx dt= lim

N→∞

Z T

0

Z

uN∇φ dx dt.

By (2.21), (2.46) and∇φ∈L2(QT)⊂L1(0, T;H−1(Ω)) we have Z T

0

Z

υφ dx dt= Z T

0

Z

uε∇φ dx dt= Z T

0

Z

∇uεφ dx dt.

Henceυ=∇uεalmost all in Ω×[0, T] and

∇uN *∇uε weakly inL2(QT). (2.47) By|F0(uN)| ≤C(1 +|uN|r), (2.22), (2.23) and the general dominated convergence theorem, similarly, we have

F0(uN)→F0(uε) strongly in C(0, T;Lq(Ω)), (2.48) for 1≤q <∞ifn= 1,2 and 2≤q < r(n−2)2n ifn≥3.

By the growth condition (1.6) and the Sobolev embedding theorem, we obtain kf(uN)k2L2(Ω)=

Z

(F0(uN))2dx

≤C Z

(|uN|r+ 1)2dx

≤2C Z

|uN|2rdx+ 2C|Ω|

≤CkuNk2rH1(Ω)+C.

Thus there exists aw∈L(0, T;L2(Ω)) such that

F0(uN)* w weakly-* inL(0, T;L2(Ω)).

This implies

lim

N→∞

Z T

0

Z

F0(uN)g dx dt= Z T

0

Z

wg dx dt, (2.49) for anyg∈L1(0, T;L2(Ω)).

By H¨older’s inequality, (2.48) and (2.49), we have asN→ ∞

Z Z

QT

F0(uε)−w g dx dt

(12)

≤ Z Z

QT

|F0(uε)−F0(uN)||g|dx dt+

Z Z

QT

[F0(uN)−w]g dx dt

≤ Z T

0

kF0(uε)−F0(uN)k2kgk2dt+

Z Z

QT

[F0(uN)−w]g dx dt ≤0, for anyg∈L1(0, T;L2(Ω)). Hence F0(uε) =wa.e. inQT and

F0(uN)* F0(uε) weak-* inL(0, T;L2(Ω)). (2.50) Multiplying (2.8) byaJ(t) and integrating (2.8) overt∈[0, T], we obtain

Z T

0

Z

vNaJ(t)φJdx dt

= Z T

0

Z

∇uN ·aJ(t)∇φJ+F0(uN)aJ(t)φJ

dx dt.

(2.51)

For any φ ∈ L2(0, T;H01(Ω)), we obtain Pnφ = Pn

j=1aJ(t)φJ, where aJ(t) ∈ L2(0, T). ThusaJ(t)φJ ∈ L2(0, T;H01(Ω)) and aJ(t)∇φJ ∈ L2(QT). By (2.28), (2.47) and (2.50), we take the limit asN → ∞on both sides of (2.51) to get

Z T

0

Z

vεaJ(t)φJdx dt= Z T

0

Z

(∇uεaJ(t)∇φJ+F0(uε)aJ(t)φJ)dx dt, (2.52) for allj∈N.

Then we sum overj= 1, . . . , non both sides (2.52), and obtain Z T

0

Z

vεPnφ dx dt= Z T

0

Z

(∇uε· ∇Pnφ+F0(uε)Pnφ)dx dt. (2.53) SincePnφconverges strongly toφin L2(0, T;H01(Ω)), we have asn→ ∞

Z T

0

k∇Pnφ− ∇φk22dt≤ Z T

0

kPnφ−φk2H1 0

dt→0.

This implies that∇Pnφconverges strongly to∇φin L2(QT). Thus we obtain Pnφ * φ weakly inL2(0, T;H01(Ω)) (2.54)

∇Pnφ *∇φ weakly inL2(QT). (2.55) By L2(0, T;H01(Ω)) ⊂L2(0, T;H−1(Ω)) andL(0, T;L2(Ω))⊂L2(0, T;H−1(Ω)), we take the limit asn→ ∞on both sides (2.53), and we obtain

Z Z

QT

vεφ dx dt= Z Z

QT

(∇uε· ∇φ+F0(uε)φ)dx dt.

SinceF0(uε)∈L(0, T;L2(Ω)) andvε∈L2(0, T;H01(Ω)), it follows from regularity theory [6] thatuε∈L2(0, T;H2(Ω)). Hence

vε=−∆uε+F0(uε) almost everywhere inQT. (2.56) Next we showF0(uε)∈L2(0, T;H1(Ω)). By H¨older’s inequality, the Sobolev em- bedding theorem and (1.7), we have

Z T

0

Z

|∇F0(uε)|2dx dt= Z T

0

Z

|F00(uε)|2|∇uε|2dx dt

≤ Z T

0

Z

|F00(uε)|n2 dx2/n Z

|∇uε|n−2n dxn−2n dt

(13)

≤C Z T

0

Z

(1 +|uε|r−1)ndx2/n

k∇uεk22n n−2

dt

≤C Z T

0

1 + Z

|uε|(r−1)ndx2/n

k∇uεk2H1(Ω)dt

≤C Z T

0

1 +kuεk

4 n−2

2n n−2

kuεk2H2(Ω)dt

≤C 1 +kuεk

4 n−2

L(0,T;H1(Ω))

Z T

0

kuεk2H2(Ω)dt

≤C 1 +kuεk

4 n−2

L(0,T;H1(Ω))

kuεk2L2(0,T;H2(Ω))≤C.

Thus ∇F0(uε) ∈ L2(QT) and F0(uε) ∈ L2(0, T;H1(Ω)). Combined with vε ∈ L2(0, T;H01(Ω)), by (2.56) and regularity theory we haveuε∈L2(0, T;H3(Ω)) and

∇vε=−∇∆uε+F00(uε)∇uε, almost everywhere inQT. (2.57) By (2.45), (2.56) and (2.57), we obtain

Z T

0

h∂tuε, φidt+ Z T

0

Z

(−∆uε+F0(uε))φ dx dt

=− Z T

0

Z

Dε(uε)(−∇∆uε+F00(uε)∇uε)· ∇φ dx dt,

(2.58)

for allφ∈L2(0, T;H2(Ω)∩H01(Ω)).

Last we show that a weak solutionuε to (2.2) satisfies energy inequality (2.4).

Replacingt byτ in (2.14) and integrating overτ ∈[0, T], we have E(uN(x, t)) +

Z t

0

Z

Dε(uN(x, τ))|∇vN(x, τ)|2dx dτ +

Z t

0

Z

|vN(x, τ)|2dx dτ =E(uN(x,0)).

(2.59)

Next, we pass to the limit in (2.59). First, by mean value theorem and (1.6) we have

Z

F(uN(t))−F(uε(t)) dx

≤ Z

|F0(ξ)||uN(t)−uε(t)|dx

≤ Z

C(|uN(t)|r+|uε(t)|r+ 1)|uN(t)−uε(t)|dx,

(2.60)

for 1≤r <∞ifn= 1,2 and 1≤r≤ n−2n ifn≥3,ξ=λuN(t) + (1−λ)uε(t) for someλ∈(0,1). By H¨older’s inequality, we have

Z

|uN(t)|r|uN(t)−uε(t)|dx≤ kuN(t)−uε(t)k2kuN(t)kr2r. (2.61) Since the Sobolev embedding theorem says that H01(Ω) ,→Lp(Ω) for 1 ≤p ≤2 and the embedding is compact if 1 ≤ p < 2, by (2.21), then for a subsequence, not relabeled, we haveuN →uεstrongly inL(0, T;L2(Ω)) anduN is bounded in L(0, T;L2r(Ω)). Hence, it follows from (2.61) that

Z

|uN(t)|r|uN(t)−uε(t)|dx→0, (2.62)

(14)

asN → ∞, for almost allt∈[0, T].

Similarly, we can prove that Z

(|uε(t)|r+ 1)|uN(t)−uε(t)|dx→0, (2.63) asN → ∞, for almost allt∈[0, T], by (2.60), (2.62) and (2.63), we have

lim

N→∞

Z

F(uN(t))dx= Z

F(uε(t))dx. (2.64) SinceuN(x,0)→u0(x) strongly inL2(Ω), we obtain

lim

N→∞

Z

F(uN(0))dx= Z

F(u0(x))dx. (2.65) By (2.47), (2.64), (2.37), (2.29), (2.59) and the weak lower semicontinuity of the Lp norms [3]. Then

Z

1

2|∇uε(x, t)|2+F(uε(x, t)) dx

+ Z t

0

Z

Dε(uε(x, τ))|∇vε(x, τ)|2+|vε(x, τ)|2 dx dτ

≤ lim

N↑∞inf Z

1

2|∇uN(x, t)|2+F(uN(x, t))

dx

+ lim

N↑∞inf Z Z

Qt

Dε(uN(x, τ))|∇vN(x, τ)|2+|vN(x, τ)|2 dx dτ

= lim

N↑∞infE(uN(x,0)).

(2.66)

SinceuN(x,0)→u0(x) strongly inH1(Ω), by (2.65) we have

Nlim→∞E(uN(x,0)) = Z

1

2|∇u0(x)|2+F(u0(x))

dx. (2.67)

Combining (2.66) with (2.67) gives the energy inequality (2.4). The proof is com-

plete.

3. Degenerate mobility

This section is devoted to the existence of weak solutions to the equations (1.9).

Here we consider the limit of approximate solutions uεi defined in section 2. The limiting value u does exist and solves the degenerate Allen-Cahn/Cahn-Hilliard equation in the weak sense.

Theorem 3.1. Supposeu0∈H1(Ω), under assumptions (1.2)and (1.5)-(1.7), for any T >0, problem (1.9)has a weak solutionu:QT →Rsatisfying

(1) u∈ L(0, T;H01(Ω))∩C([0, T];Lp(Ω))∩L2(0, T;H2(Ω)), where 1 ≤p <

∞if n= 1,2 and2≤p < n−22n if n≥3, (2) ∂tu∈L2(0, T; (H2(Ω))0),

(3) u(x,0) =u0(x) for allx∈Ω,

which satisfies (1.9)in the following weak sense:

(15)

(1) DefineP as the set where D(u)is non-degenerate, that is P :={(x, t)∈QT :|u| 6= 0}.

There exists a set A⊂QT with|QT \A|= 0 and a functionζ:QT →Rn satisfyingχA∩PD(u)ζ∈L2(0, T;Ln+22n (Ω)), such that

Z T

0

h∂tu, φidt

=− Z T

0

Z

A∩P

D(u)ζ· ∇φ dx dt− Z T

0

Z

(−∆u+f(u))φ dx dt

(3.1)

for all test functionsφ∈L2(0, T;H2(Ω)∩H01(Ω)).

(2) For eachj ∈N, there exists EJ :={(x, t)∈QT;ui →uuniformly, |u|>

δJ forδJ>0}=TJ×SJ such that

u∈L2(TJ;H3(SJ)), ζ=−∇∆u+F00(u)∇u, in EJ. In addition, usatisfies the energy inequality

E(u) + Z Z

Qt∩A∩P

D(u(x, τ))|ζ(x, τ)|2dx dτ

+ Z Z

Qt

| −∆u+f(u)|2dx dτ ≤E(u0),

(3.2)

for allt >0.

Proof. We consider a sequence of positive numbersεi monotonically decreasing to 0 as i→ ∞. Fixu0 ∈H1(Ω), for any fixedεi, here, for the sake of simplicity, we writeui:=uεi andDi(ui) :=Dεi(uεi). By Theorem 2.1, there exists a functionui such that

(1) ui∈L(0, T;H01(Ω))∩C([0, T];Lp(Ω))∩L2(0, T;H3(Ω)), where 1≤p <

∞ifn= 1,2 and 2≤p < n−22n ifn≥3, (2) ∂tui∈L2(0, T; (H2(Ω))0),

Z T

0

h∂tui, φidt=− Z T

0

Z

Di(ui)∇vi· ∇φ dx dt− Z T

0

Z

viφ dx dt (3.3) for all test functionsφ∈L2(0, T;H2(Ω)∩H01(Ω)), where

vi =−∆ui+f(ui),almost everywhere inQT. (3.4) By the arguments in the proof of Theorem 2.1, the bounds on the right hand side of (2.16), (2.20), (2.39) and (2.32) depend only on the growth conditions of the mobility and potential, so there exists a constant C > 0 independent of εi such that

kuikL(0,T;H01(Ω))≤C, (3.5) k∂tuikL2(0,T;(H2(Ω))0)≤C, (3.6) kp

Di(ui)∇vikL2(QT)≤C, (3.7)

kvikL2(QT)≤C. (3.8)

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