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

u_{t}=−∇[D(u)∇(∆u−f(u))] + (∆u−f(u)), in Q_{T}, (1.1)
whereQT = Ω×(0, T), Ω is a bounded domain inR^{n} with aC^{3}-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,

C_{0}≤D(u)≤C_{1}|u|^{m}, if|u| ≥δ, (1.2)
for some constantsC_{0}, C_{1}, δ >0, where 0< m <∞ifn= 1,2 and _{n}^{4} < m < _{n−2}^{4}
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) =u_{0}(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−u^{2})^{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

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 ∈C^{2}(R) satisfies

k_{0}(|s|^{r+1}−1)≤F(s)≤k_{1}(|s|^{r+1}+ 1), (1.5)

|F^{0}(s)| ≤k_{2}(|s|^{r}+ 1), (1.6)

|F^{00}(s)| ≤k_{3}(|s|^{r−1}+ 1), (1.7)
for some constantsk0, k1, k2, k3>0 where 1≤r <∞ifn= 1,2 and 1≤r≤ _{n−2}^{n}
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−u^{2})^{2}
studied by Karali and Nagase [9], the logarithmic functionf(u) =−θcu+^{θ}_{2}ln^{1+u}_{1−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

u_{t}=∇(D(u)∇v)−v, in Q_{T},
v=−∆u+f(u), in Q_{T},

u(x,0) =u0(x), in Ω, u=v= 0, on∂Ω.

(1.9)

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

Z

Ω

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

Ω

vu_{t}dx=
Z

Ω

v[∇(D(u)∇v)−v]dx=− Z

Ω

D(u)|∇v|^{2}+v^{2}
dx≤0.

Notation. Define the usual Lebesgue norms and theL^{2}-inner-product
kukp=kuk_{L}p(Ω) and (u, v) = (u, v)_{L}2(Ω).

The duality pairing between the spaceH^{2}(Ω) and its dual (H^{2}(Ω))^{0}will be denoted
using the form h·,·i. For simplicity, 2^{∗} := _{n−2}^{2n} . 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 mobilityD_{ε}(u) defined for anεsatisfying 0< ε < δ by

Dε(u) :=

(|u|^{m}, if|u|> ε,

ε^{m}, if|u| ≤ε. (2.1)
So we consider the problem

ut=∇(Dε(u)∇v)−v, in QT,
v=−∆u+f(u), in Q_{T},

u(x,0) =u0(x), in Ω, u=v= 0, on∂Ω.

(2.2)

Theorem 2.1. Supposeu_{0}∈H^{1}(Ω), under assumptions(1.2)and (1.5)–(1.7), for
any T >0, there exists a pair of functions(u_{ε}, v_{ε})such that

(1) uε∈L^{∞}(0, T;H_{0}^{1}(Ω))∩C([0, T];L^{p}(Ω))∩L^{2}(0, T;H^{3}(Ω)), where1≤p <

∞if n= 1,2 and2≤p < _{n−2}^{2n} if n≥3,
(2) ∂tuε∈L^{2}(0, T; (H^{2}(Ω))^{0}),

(3) uε(x,0) =u0(x)for allx∈Ω,
(4) v_{ε}∈L^{2}(0, T;H_{0}^{1}(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+F^{00}(uε)∇uε

· ∇φ dx dt

(2.3)

for all test functions φ ∈ L^{2}(0, T;H^{2}(Ω)∩H_{0}^{1}(Ω)). In addition, u_{ε} satisfies the
energy inequality

E(uε) + Z t

0

Z

Ω

Dε(uε(x, τ))|∇vε(x, τ)|^{2}+|vε(x, τ)|^{2}

dx dτ ≤E(u0), (2.4)

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 L^{2}(Ω) with Dirichlet boundary
condition, i.e.,

−∆φJ =λ_{J}φ_{J}, in Ω,

φJ= 0, on∂Ω. (2.5)

The eigenfunctions{φJ}^{∞}_{j=1}form an orthogonal basis forL^{2}(Ω), H^{1}(Ω) andH^{2}(Ω).

Hence, for initial data u_{0} ∈ H^{1}(Ω), we can find sequences of scalars (u^{0}_{N,j}; j =
1,2, . . . , N)^{∞}_{N}_{=1} such that

lim

N→∞

N

X

j=1

u^{0}_{N,j}φJ=u0, inH^{1}(Ω). (2.6)
LetVN denote the linear span of (φ1, . . . , φN) andPN be the orthogonal projection
fromL^{2}(Ω) toVN, that is

PNφ:=

N

X

j=1

Z

Ω

φφJdx φJ.

Letu^{N}(x, t) =PN

j=1c^{N}_{J}(t)φJ(x),v^{N}(x, t) =PN

j=1d^{N}_{J}(t)φJ(x) be the approximate
solution of (2.2) inVN; that is,u^{N}, v^{N} satisfy the g system of equations

Z

Ω

∂_{t}u^{N}φ_{J}dx=−
Z

Ω

D_{ε}(u^{N})∇v^{N}· ∇φ_{J}dx−
Z

Ω

v^{N}φ_{J}dx, (2.7)
Z

Ω

v^{N}φJdx=
Z

Ω

∇u^{N}· ∇φJ+f(u^{N})φJdx, (2.8)
u^{N}(x,0) =

N

X

j=1

u^{0}_{N,j}φJ(x), (2.9)

forj= 1, . . . , N andu^{0}_{N,j} =R

Ωu0φJdx.

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

∂_{t}c^{N}_{J}(t) =−

N

X

k=1

d^{N}_{k}(t)
Z

Ω

D_{ε}

N

X

i=1

c^{N}_{i} (t)φ_{i}(x)

∇φk∇φJdx−d^{N}_{J}(t), (2.10)

d^{N}_{J}(t) =λ_{J}c^{N}_{J}(t) +
Z

Ω

f

N

X

i=1

c^{N}_{i} (t)φ_{i}(x)

φ_{J}dx, (2.11)

c^{N}_{J}(0) =u^{0}_{N,j} = (u0, φJ), (2.12)
which has to hold forj= 1, . . . , N.

Define X(t) = c^{N}_{1} (t), . . . , c^{N}_{N}(t)

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

fJ(t,X(t)) =−

N

X

k=1

Z

Ω

Dε

X^{N}

i=1

c^{N}_{i} (t)φi(x)

∇φk∇φJdx

×

λ_{k}c^{N}_{k}(t) +
Z

Ω

f

N

X

i=1

c^{N}_{i} (t)φ_{i}(x)
φ_{k}dx

−λJc^{N}_{J}(t)−
Z

Ω

fX^{N}

k=1

c^{N}_{k} (t)φk(x)
φJdx

forj= 1, . . . , N. Then problem (2.10)-(2.12) is equivalent to the problem
X^{0}(t) =F(t,X(t)), X(0) = (u^{0}_{N,1}, . . . , u^{0}_{N,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)∈
C^{1}[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 onu^{N}.
Lemma 2.2. For any T >0, we have

ku^{N}k_{L}∞(0,T;H^{1}_{0}(Ω))≤C, for allN,
k∂tu^{N}k_{L}2(0,T;(H^{2}(Ω))^{0})≤C, for allN,
whereC independent of N.

Proof. For any fixedN ∈N^{+}, we multiply (2.7) byd^{N}_{J}(t) and sum overj= 1, . . . , N
to obtain

Z

Ω

∂tu^{N}v^{N}dx=−
Z

Ω

Dε(u^{N})|∇v^{N}|^{2}dx−
Z

Ω

|v^{N}|^{2}dx. (2.13)
Multiply (2.8) by∂tc^{N}_{J}(t) and sum over j= 1, . . . , N to obtain

Z

Ω

v^{N}∂tu^{N}dx=
Z

Ω

∇u^{N}∂t∇u^{N} +f(u^{N})∂tu^{N}
dx,

= d dt

Z

Ω

1

2|∇u^{N}|^{2}+F(u^{N})
dx.

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

dt Z

Ω

1

2|∇u^{N}|^{2}+F(u^{N})
dx=−

Z

Ω

D_{ε}(u^{N})|∇v^{N}|^{2}dx−
Z

Ω

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

Z

Ω

1

2|∇u^{N}(x, t)|^{2}+F(u^{N}(x, t))

dx

+ Z t

0

Z

Ω

D_{ε}(u^{N}(x, τ))|∇v^{N}(x, τ)|^{2}+|v^{N}(x, τ)|^{2}
dx dτ

= Z

Ω

1

2|∇u^{N}(x,0)|^{2}+F(u^{N}(x,0))

dx

≤1

2k∇u^{N}(x,0)k^{2}_{2}+k_{1}ku^{N}(x,0)k^{r+1}_{r+1}+k_{1}|Ω|.

≤1

2k∇u_{0}k^{2}_{2}+k_{1}Cku_{0}k^{r+1}_{H}_{1}_{(Ω)}+k_{1}|Ω| ≤C.

The last inequality follows fromu_{0}∈H^{1}(Ω). This implies
Z

Ω

1

2|∇u^{N}(x, t)|^{2}+k0|u^{N}|^{r+1}
dx

+ Z t

0

Z

Ω

Dε(u^{N}(x, τ))|∇v^{N}(x, τ)|^{2}+|v^{N}(x, τ)|^{2}

dx dτ ≤C.

(2.15)

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

ku^{N}k_{H}1(Ω)≤C, fort >0.

This estimate implies that the coefficients{c^{N}_{J} :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

u^{N} ∈L^{∞}(0, T;H_{0}^{1}(Ω)), ku^{N}k_{L}∞(0,T;H^{1}_{0}(Ω))≤C, for allN. (2.16)
Inequality (2.15) implies

k q

Dε(u^{N})∇v^{N}k_{L}2(QT)≤C, for allN, (2.17)
kv^{N}k_{L}2(Q_{T})≤C, for allN. (2.18)
By the Sobolev embedding theorem, the growth condition (1.2) and (2.1), for|u|>

ε, we obtain Z

Ω

|D_{ε}(u^{N})|^{n/2}dx≤(C_{1}+ 1)
Z

Ω

|u^{N}|^{m·}^{n}^{2} dx,≤Cku^{N}k^{mn/2}_{H}_{1}_{(Ω)}≤C.

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

kDε(u^{N})k_{L}∞(0,T;L^{n/2}(Ω))≤C, for allN. (2.19)
For any φ ∈ L^{2}(0, T;H^{2}(Ω)), we obtain PNφ = PN

j=1aJ(t)φJ, where aJ(t) = R

ΩφφJdx. 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

Ω

∂_{t}u^{N}φ dx dt

=

Z T

0

Z

Ω

∂tu^{N}PNφ dx dt

=

Z T

0

Z

Ω

Dε(u^{N})∇v^{N}∇PNφ+v^{N}PNφ
dx dt

≤ Z T

0

k q

D_{ε}(u^{N})knk
q

D_{ε}(u^{N})∇v^{N}k2k∇PNφk2^{∗}dt+
Z T

0

kv^{N}k2kPNφk2dt

≤C Z T

0

kq

Dε(u^{N})∇v^{N}k2kφk_{H}2+kv^{N}k2kφk_{H}2dt

≤C k q

Dε(u^{N})∇v^{N}k_{L}2(QT)+kv^{N}k_{L}2(QT)

kφk_{L}2(0,T;H^{2}(Ω))

≤Ckφk_{L}2(0,T;H^{2}(Ω)).
Hence,

k∂_{t}u^{N}k_{L}2(0,T;(H^{2}(Ω))^{0})≤C for allN. (2.20)

The proof is complete.

Lemma 2.3. Supposeu_{0} ∈H^{1}(Ω), under assumptions (1.2) and (1.5)-(1.7), for
any T >0, there exists a pair of functions(u_{ε}, v_{ε})such that

(1) uε∈L^{∞}(0, T;H_{0}^{1}(Ω))∩C([0, T];L^{p}(Ω)), where 1≤p <∞if n= 1,2 and
2≤p < _{n−2}^{2n} if n≥3,

(2) ∂_{t}u_{ε}∈L^{2}(0, T; (H^{2}(Ω))^{0}),
(3) u_{ε}(x,0) =u_{0}(x)for allx∈Ω,

(4) vε∈L^{2}(0, T;H_{0}^{1}(Ω)),
which satisfies

Z T

0

h∂tuε, φidt=− Z T

0

Z

Ω

Dε(uε)∇vε· ∇φ dx dt− Z T

0

Z

Ω

vεφ dx dt.

Proof. Since the embeddingH_{0}^{1}(Ω),→L^{p}(Ω) is compact for 1≤p <∞ifn= 1,2
and 1 ≤ p < _{n−2}^{2n} if n ≥3, L^{p}(Ω) ,→ (H^{2}(Ω))^{0} is continuous for p≥1 if n ≤3,
p >1 ifn= 4 and p≥ _{n+4}^{2n} if n≥5. Using the Aubin-Lions lemma (Lions [17]),
we can find a subsequence which we still denote byu^{N} anduε∈L^{∞}(0, T;H_{0}^{1}(Ω)),
such that asN → ∞

u^{N} * u_{ε}, weak-* inL^{∞}(0, T;H_{0}^{1}(Ω)), (2.21)
u^{N} →uε, strongly inC([0, T];L^{p}(Ω)), (2.22)
u^{N} →u_{ε}, strongly inL^{2}(0, T;L^{p}(Ω)) and almost everywehre inQ_{T}, (2.23)

∂tu^{N} * ∂tuε, weakly inL^{2}(0, T; (H^{2}(Ω))^{0}), (2.24)
where 2≤p <2^{∗}ifn≥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

Ω

∂tu^{N}aJ(t)φJdx dt

=− Z T

0

Z

Ω

Dε(u^{N})∇v^{N} ·aJ(t)∇φJdx dt−
Z T

0

Z

Ω

v^{N}aJ(t)φJdx dt.

(2.25)

To pass to the limit in (2.25), we need the convergence ofv^{N} andD_{ε}(u^{N})∇v^{N}. By
(2.17) andDε(u^{N})≥ε^{m}, then

k∇v^{N}k_{L}2(Q_{T})≤Cε^{−}^{m}^{2} <∞, for anyε >0. (2.26)
This implies that {∇v^{N}} is a bounded sequence in L^{2}(QT), thus there exists a
subsequence, not relabeled, andζε∈L^{2}(QT) such that

∇v^{N} * ζε, weakly inL^{2}(QT). (2.27)
By (2.26) and Poincar´e’s inequality, we have

kv^{N}k_{L}2(0,T;H_{0}^{1}(Ω))≤Cε^{−}^{m}^{2} <∞, for anyε >0.

Hence we can find a subsequence of v^{N}, not relabeled, and vε ∈ L^{2}(0, T;H_{0}^{1}(Ω))
such that

v^{N} * vε, weakly inL^{2}(0, T;H_{0}^{1}(Ω)). (2.28)
For anyg∈L^{2}(0, T;H_{0}^{1}(Ω)), by (2.26) and (2.27) we have

Nlim→∞

Z T

0

Z

Ω

∇v^{N}g dx dt=
Z T

0

Z

Ω

ζεg dx dt

= lim

N→∞

Z T

0

Z

Ω

v^{N}∇g dx dt=
Z T

0

Z

Ω

∇vεg dx dt.

Henceζε=∇vεalmost all in QT and

∇v^{N} *∇vε, weakly inL^{2}(Q_{T}). (2.29)

By (2.18), we can extract a further sequence ofv^{N}, not relabeled, andηε∈L^{2}(QT)
such that

v^{N} * ηε, weakly inL^{2}(QT). (2.30)
By (2.28) and (2.30) for anyg∈L^{2}(QT)⊂L^{2}(0, T;H^{−1}(Ω)), we have

N→∞lim Z T

0

Z

Ω

v^{N}g dx dt=
Z T

0

Z

Ω

v_{ε}g dx dt=
Z T

0

Z

Ω

η_{ε}g dx dt.

This impliesηε=vεalmost allQT and

v^{N} * v_{ε}, weakly inL^{2}(Q_{T}). (2.31)
Consequently we have the bound

Z

Q_{T}

|vε|^{2}dx dt≤C. (2.32)

For anyt∈[0, T], byDε(u^{N})≤C(1 +|u^{N}|^{m}), we have
Dε(u^{N})n/2

≤C(1 +|u^{N}|^{m})^{n/2}≤(C(1 +|u^{N}|))^{mn/2},

where 2≤ ^{mn}_{2} <2^{∗}. By (2.22), C(1 +|u^{N}|)→C(1 +|uθ|) inL^{mn/2}(Ω). SinceD_{ε}
is continuous and (2.23), we obtain

Dε(u^{N})→Dε(uε), a.e. in Ω.

The generalized Lebesgue convergence theorem [1] gives
D_{ε}(u^{N})→D_{ε}(u_{ε}), inL^{n/2}(Ω).

This implies

kDε(u^{N})−D_{ε}(u_{ε})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(u^{N})−Dε(uε)kn/2→0, asN → ∞.

This implies

Dε(u^{N})→Dε(uε), strongly inC(0, T;L^{n/2}(Ω)). (2.33)
Byp

Dε(u^{N})≤C(1 +|u^{N}|^{m}^{2}), (2.22), (2.23) and the generalized Lebesgue conver-
gence theorem, similarly, we have

q

D_{ε}(u^{N})→p

D_{ε}(u_{ε}), strongly inC(0, T;L^{n}(Ω)). (2.34)
For anyϕ∈L^{2}(0, T;L^{2}^{∗}(Ω)), by H¨older’s inequality we have

Z Z

Q_{T}

qD_{ε}(u^{N})∇v^{N}ϕ−p

D_{ε}(u_{ε})∇vεϕ
dx dt

=

Z Z

Q_{T}

[ q

D_{ε}(u^{N})−p

D_{ε}(u_{ε})]∇v^{N}ϕ+p

D_{ε}(u_{ε})[∇v^{N}ϕ− ∇vεϕ]

dx dt

≤ Z T

0

k q

Dε(u^{N})−p

Dε(uε)knk∇v^{N}k2kϕk2^{∗}dt
+

Z Z

Q_{T}

pDε(uε)ϕ[∇v^{N} − ∇vε]dx dt

≤ sup

t∈[0,T]

kq

Dε(u^{N})−p

Dε(uε)knk∇v^{N}k_{L}2(QT)kϕk_{L}2(0,T;L^{2}^{∗}(Ω))

+

Z Z

Q_{T}

pDε(uε)ϕ[∇v^{N} − ∇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|^{2}dx dt≤
Z T

0

Z

Ω

Dε(uε)^{n/2}

dxn/2Z

Ω

|ϕ|^{n−2}^{2n} dx^{n−2}_{n}
dt

≤ sup

t∈[0,T]

kp

D_{ε}(u_{ε})k^{2}_{n}
Z T

0

kϕk^{2}_{L}2∗(Ω)dt

≤Ckϕk^{2}_{L}2(0,T;L^{2}^{∗}(Ω)).
This implies

pDε(uε)ϕ∈L^{2}(QT). (2.35)
ThusII →0 asN → ∞by (2.29). Hence

q

D_{ε}(u^{N})∇v^{N} *p

D_{ε}(u_{ε})∇vε, weakly inL^{2}(0, T;L^{n+2}^{2n} (Ω)). (2.36)
Next we consider the convergence ofDε(u^{N})∇v^{N}. By (2.17), (2.36) andL^{2}(QT)⊂
L^{2}(0, T;L^{n+2}^{2n} (Ω)), we can extract a further sequence, not relabeled, such that

q

D_{ε}(u^{N})∇v^{N} *p

D_{ε}(u_{ε})∇vε, weakly inL^{2}(Q_{T}). (2.37)
By H¨older’s inequality and (2.17), we have

Z Z

Q_{T}

q

D_{ε}(u^{N})∇v^{N} ·p

D_{ε}(u_{ε})∇vεdx dt

≤ k q

D_{ε}(u^{N})∇v^{N}kL^{2}(Q_{T})kp

D_{ε}(u_{ε})∇vεkL^{2}(Q_{T})

≤Ckp

D_{ε}(u_{ε})∇v_{ε}k_{L}2(Q_{T}),

(2.38)

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

kp

D_{ε}(u_{ε})∇v_{ε}k_{L}2(Q_{T})≤C. (2.39)
For anyϕ∈L^{2}(0, T;L^{2}^{∗}(Ω)), by H¨older’s inequality we obtain

Z Z

Q_{T}

D_{ε}(u^{N})∇v^{N}ϕ−D_{ε}(u_{ε})∇vεϕ
dx dt

≤

Z Z

Q_{T}

[ q

D_{ε}(u^{N})−p
D_{ε}(u_{ε})]

q

D_{ε}(u^{N})∇v^{N}ϕ dx dt

+

Z Z

Q_{T}

pD_{ε}(u_{ε})[

q

D_{ε}(u^{N})∇v^{N}ϕ−p

D_{ε}(u_{ε})∇vεϕ]dx dt

≤ Z T

0

k q

Dε(u^{N})−p

Dε(uε)knk q

Dε(u^{N})∇v^{N}k2kϕk2^{∗}dt
+

Z Z

Q_{T}

pDε(uε)ϕ[

q

Dε(u^{N})∇v^{N} −p

Dε(uε)∇vε]dx dt

≤ sup

t∈[0,T]

kq

Dε(u^{N})−p

Dε(uε)knkq

Dε(u^{N})∇v^{N}k_{L}2(QT)kϕk_{L}2(0,T;L^{2}^{∗}(Ω))

+

Z Z

Q_{T}

pDε(uε)ϕ[

q

Dε(u^{N})∇v^{N} −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(u^{N})∇v^{N} * Dε(uε)∇vε, weakly inL^{2}(0, T;L^{n+2}^{2n} (Ω)). (2.40)
For any φ ∈ L^{2}(0, T;H^{2}(Ω)∩H_{0}^{1}(Ω)), we obtain Pnφ = Pn

j=1aJ(t)φJ, where aJ(t) =R

ΩφφJdx, then Pnφconverges strongly toφinL^{2}(0, T;H^{2}∩H_{0}^{1}(Ω)) and
aJ(t)∈L^{2}(0, T). For φJ ∈H^{2}(Ω), by Sobolev embedding theorem, we obtain

k∇φJk2^{∗}≤Ck∇φJkH^{1}(Ω)≤C.

ThusaJ(t)∇φJ ∈L^{2}(0, T;L^{2}^{∗}) and

a_{J}(t)φ_{J}∈L^{2}(0, T;H^{2}∩H_{0}^{1}(Ω))⊂L^{2}(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∂_{t}u_{ε}, a_{J}(t)φ_{J}idt

=− Z T

0

Z

Ω

Dε(uε)∇vε·aJ(t)∇φJdx dt− Z T

0

Z

Ω

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

Ω

Dε(uε)∇vε· ∇Pnφ dx dt− Z T

0

Z

Ω

vεPnφ dx dt.

(2.42)

SincePnφconverges strongly toφin L^{2}(0, T;H^{2}(Ω)), thus asn→ ∞,
Z T

0

k∇Pnφ− ∇φk^{2}_{2}∗dt≤
Z T

0

k∇Pnφ− ∇φk^{2}_{H}1dt

≤ Z T

0

kPnφ−φk^{2}_{H}2dt→0.

This implies that ∇Pnφ converges strongly to ∇φ in L^{2}(0, T;L^{2}^{∗}(Ω)). Thus we
obtain

Pnφ * φ, weakly inL^{2}(0, T;H^{2}(Ω)∩H_{0}^{1}(Ω)), (2.43)

∇Pnφ *∇φ, weakly inL^{2}(0, T;L^{2}^{∗}(Ω)). (2.44)
ByL^{2}(0, T;H_{0}^{1}(Ω))⊂L^{2}(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

Ω

Dε(uε)∇vε· ∇φ dx dt− Z T

0

Z

Ω

vεφ dx dt. (2.45)

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

u^{N}(x,0)→u_{0}(x) in L^{2}(Ω).

By (2.22),u_{ε}(x,0) =u_{0}(x) inL^{2}(Ω). The proof is complete.

Proof of Theorem 2.1. We need only to check that uε ∈ L^{2}(0, T;H^{3}(Ω)), vε =

−∆uε+f(uε) and∇vε=−∇∆uε+F^{00}(uε)∇uε. First we consider the convergence
of∇u^{N} andf(u^{N}). By (2.21), we have

Z T

0

k∇u^{N}k^{2}_{2}dt≤C.

Hence we can find a subsequence ofu^{N}, not relabeled, andυ∈L^{2}(QT), such that

∇u^{N} * υ weakly inL^{2}(Q_{T}). (2.46)
For anyφ∈L^{2}(0, T;H_{0}^{1}(Ω)), by integration by parts we have

lim

N→∞

Z T

0

Z

Ω

∇u^{N}φ dx dt= lim

N→∞

Z T

0

Z

Ω

u^{N}∇φ dx dt.

By (2.21), (2.46) and∇φ∈L^{2}(QT)⊂L^{1}(0, T;H^{−1}(Ω)) we have
Z T

0

Z

Ω

υφ dx dt= Z T

0

Z

Ω

uε∇φ dx dt= Z T

0

Z

Ω

∇uεφ dx dt.

Henceυ=∇uεalmost all in Ω×[0, T] and

∇u^{N} *∇uε weakly inL^{2}(QT). (2.47)
By|F^{0}(u^{N})| ≤C(1 +|u^{N}|^{r}), (2.22), (2.23) and the general dominated convergence
theorem, similarly, we have

F^{0}(u^{N})→F^{0}(u_{ε}) strongly in C(0, T;L^{q}(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(u^{N})k^{2}_{L}2(Ω)=

Z

Ω

(F^{0}(u^{N}))^{2}dx

≤C Z

Ω

(|u^{N}|^{r}+ 1)^{2}dx

≤2C Z

Ω

|u^{N}|^{2r}dx+ 2C|Ω|

≤Cku^{N}k^{2r}_{H}1(Ω)+C.

Thus there exists aw∈L^{∞}(0, T;L^{2}(Ω)) such that

F^{0}(u^{N})* w weakly-* inL^{∞}(0, T;L^{2}(Ω)).

This implies

lim

N→∞

Z T

0

Z

Ω

F^{0}(u^{N})g dx dt=
Z T

0

Z

Ω

wg dx dt, (2.49)
for anyg∈L^{1}(0, T;L^{2}(Ω)).

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

Z Z

Q_{T}

F^{0}(uε)−w
g dx dt

≤ Z Z

QT

|F^{0}(uε)−F^{0}(u^{N})||g|dx dt+

Z Z

QT

[F^{0}(u^{N})−w]g dx dt

≤ Z T

0

kF^{0}(u_{ε})−F^{0}(u^{N})k2kgk2dt+

Z Z

Q_{T}

[F^{0}(u^{N})−w]g dx dt
≤0,
for anyg∈L^{1}(0, T;L^{2}(Ω)). Hence F^{0}(uε) =wa.e. inQT and

F^{0}(u^{N})* F^{0}(uε) weak-* inL^{∞}(0, T;L^{2}(Ω)). (2.50)
Multiplying (2.8) byaJ(t) and integrating (2.8) overt∈[0, T], we obtain

Z T

0

Z

Ω

v^{N}a_{J}(t)φ_{J}dx dt

= Z T

0

Z

Ω

∇u^{N} ·aJ(t)∇φJ+F^{0}(u^{N})aJ(t)φJ

dx dt.

(2.51)

For any φ ∈ L^{2}(0, T;H_{0}^{1}(Ω)), we obtain Pnφ = Pn

j=1a_{J}(t)φ_{J}, where a_{J}(t) ∈
L^{2}(0, T). Thusa_{J}(t)φ_{J} ∈ L^{2}(0, T;H_{0}^{1}(Ω)) and a_{J}(t)∇φJ ∈ L^{2}(Q_{T}). 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

Ω

vεaJ(t)φJdx dt= Z T

0

Z

Ω

(∇uεaJ(t)∇φJ+F^{0}(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

Ω

vεPnφ dx dt= Z T

0

Z

Ω

(∇uε· ∇Pnφ+F^{0}(uε)Pnφ)dx dt. (2.53)
SincePnφconverges strongly toφin L^{2}(0, T;H_{0}^{1}(Ω)), we have asn→ ∞

Z T

0

k∇Pnφ− ∇φk^{2}_{2}dt≤
Z T

0

kPnφ−φk^{2}_{H}1
0

dt→0.

This implies that∇Pnφconverges strongly to∇φin L^{2}(QT). Thus we obtain
Pnφ * φ weakly inL^{2}(0, T;H_{0}^{1}(Ω)) (2.54)

∇Pnφ *∇φ weakly inL^{2}(Q_{T}). (2.55)
By L^{2}(0, T;H_{0}^{1}(Ω)) ⊂L^{2}(0, T;H^{−1}(Ω)) andL^{∞}(0, T;L^{2}(Ω))⊂L^{2}(0, T;H^{−1}(Ω)),
we take the limit asn→ ∞on both sides (2.53), and we obtain

Z Z

Q_{T}

v_{ε}φ dx dt=
Z Z

Q_{T}

(∇u_{ε}· ∇φ+F^{0}(u_{ε})φ)dx dt.

SinceF^{0}(uε)∈L^{∞}(0, T;L^{2}(Ω)) andvε∈L^{2}(0, T;H_{0}^{1}(Ω)), it follows from regularity
theory [6] thatuε∈L^{2}(0, T;H^{2}(Ω)). Hence

vε=−∆uε+F^{0}(uε) almost everywhere inQT. (2.56)
Next we showF^{0}(u_{ε})∈L^{2}(0, T;H^{1}(Ω)). By H¨older’s inequality, the Sobolev em-
bedding theorem and (1.7), we have

Z T

0

Z

Ω

|∇F^{0}(u_{ε})|^{2}dx dt=
Z T

0

Z

Ω

|F^{00}(u_{ε})|^{2}|∇uε|^{2}dx dt

≤ Z T

0

Z

Ω

|F^{00}(uε)|^{2×}^{n}^{2} dx2/n Z

Ω

|∇uε|^{2×}^{n−2}^{n} dx^{n−2}_{n}
dt

≤C Z T

0

Z

Ω

(1 +|uε|^{r−1})^{n}dx2/n

k∇uεk^{2}2n
n−2

dt

≤C Z T

0

1 + Z

Ω

|uε|^{(r−1)n}dx^{2/n}

k∇uεk^{2}_{H}1(Ω)dt

≤C Z T

0

1 +ku_{ε}k

4 n−2

2n n−2

ku_{ε}k^{2}_{H}2(Ω)dt

≤C 1 +kuεk

4 n−2

L^{∞}(0,T;H^{1}(Ω))

Z T

0

kuεk^{2}_{H}2(Ω)dt

≤C 1 +kuεk

4 n−2

L^{∞}(0,T;H^{1}(Ω))

kuεk^{2}_{L}2(0,T;H^{2}(Ω))≤C.

Thus ∇F^{0}(uε) ∈ L^{2}(QT) and F^{0}(uε) ∈ L^{2}(0, T;H^{1}(Ω)). Combined with vε ∈
L^{2}(0, T;H_{0}^{1}(Ω)), by (2.56) and regularity theory we haveuε∈L^{2}(0, T;H^{3}(Ω)) and

∇vε=−∇∆uε+F^{00}(uε)∇uε, almost everywhere inQT. (2.57)
By (2.45), (2.56) and (2.57), we obtain

Z T

0

h∂tu_{ε}, φidt+
Z T

0

Z

Ω

(−∆uε+F^{0}(u_{ε}))φ dx dt

=− Z T

0

Z

Ω

Dε(uε)(−∇∆uε+F^{00}(uε)∇uε)· ∇φ dx dt,

(2.58)

for allφ∈L^{2}(0, T;H^{2}(Ω)∩H_{0}^{1}(Ω)).

Last we show that a weak solutionu_{ε} to (2.2) satisfies energy inequality (2.4).

Replacingt byτ in (2.14) and integrating overτ ∈[0, T], we have
E(u^{N}(x, t)) +

Z t

0

Z

Ω

Dε(u^{N}(x, τ))|∇v^{N}(x, τ)|^{2}dx dτ
+

Z t

0

Z

Ω

|v^{N}(x, τ)|^{2}dx dτ =E(u^{N}(x,0)).

(2.59)

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

Z

Ω

F(u^{N}(t))−F(uε(t))
dx

≤ Z

Ω

|F^{0}(ξ)||u^{N}(t)−uε(t)|dx

≤ Z

Ω

C(|u^{N}(t)|^{r}+|uε(t)|^{r}+ 1)|u^{N}(t)−uε(t)|dx,

(2.60)

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

Z

Ω

|u^{N}(t)|^{r}|u^{N}(t)−uε(t)|dx≤ ku^{N}(t)−uε(t)k2ku^{N}(t)k^{r}_{2r}. (2.61)
Since the Sobolev embedding theorem says that H_{0}^{1}(Ω) ,→L^{p}(Ω) for 1 ≤p ≤2^{∗}
and the embedding is compact if 1 ≤ p < 2^{∗}, by (2.21), then for a subsequence,
not relabeled, we haveu^{N} →uεstrongly inL^{∞}(0, T;L^{2}(Ω)) andu^{N} is bounded in
L^{∞}(0, T;L^{2r}(Ω)). Hence, it follows from (2.61) that

Z

Ω

|u^{N}(t)|^{r}|u^{N}(t)−uε(t)|dx→0, (2.62)

asN → ∞, for almost allt∈[0, T].

Similarly, we can prove that Z

Ω

(|u_{ε}(t)|^{r}+ 1)|u^{N}(t)−u_{ε}(t)|dx→0, (2.63)
asN → ∞, for almost allt∈[0, T], by (2.60), (2.62) and (2.63), we have

lim

N→∞

Z

Ω

F(u^{N}(t))dx=
Z

Ω

F(uε(t))dx. (2.64)
Sinceu^{N}(x,0)→u_{0}(x) strongly inL^{2}(Ω), we obtain

lim

N→∞

Z

Ω

F(u^{N}(0))dx=
Z

Ω

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

Z

Ω

1

2|∇uε(x, t)|^{2}+F(uε(x, t))
dx

+ Z t

0

Z

Ω

D_{ε}(u_{ε}(x, τ))|∇v_{ε}(x, τ)|^{2}+|v_{ε}(x, τ)|^{2}
dx dτ

≤ lim

N↑∞inf Z

Ω

1

2|∇u^{N}(x, t)|^{2}+F(u^{N}(x, t))

dx

+ lim

N↑∞inf Z Z

Qt

Dε(u^{N}(x, τ))|∇v^{N}(x, τ)|^{2}+|v^{N}(x, τ)|^{2}
dx dτ

= lim

N↑∞infE(u^{N}(x,0)).

(2.66)

Sinceu^{N}(x,0)→u_{0}(x) strongly inH^{1}(Ω), by (2.65) we have

Nlim→∞E(u^{N}(x,0)) =
Z

Ω

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 u_{ε}_{i} 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∈H^{1}(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;H_{0}^{1}(Ω))∩C([0, T];L^{p}(Ω))∩L^{2}(0, T;H^{2}(Ω)), where 1 ≤p <

∞if n= 1,2 and2≤p < _{n−2}^{2n} if n≥3,
(2) ∂tu∈L^{2}(0, T; (H^{2}(Ω))^{0}),

(3) u(x,0) =u0(x) for allx∈Ω,

which satisfies (1.9)in the following weak sense:

(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 →R^{n}
satisfyingχ_{A∩P}D(u)ζ∈L^{2}(0, T;L^{n+2}^{2n} (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

Ω

(−∆u+f(u))φ dx dt

(3.1)

for all test functionsφ∈L^{2}(0, T;H^{2}(Ω)∩H_{0}^{1}(Ω)).

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

δJ forδJ>0}=TJ×SJ such that

u∈L^{2}(TJ;H^{3}(SJ)),
ζ=−∇∆u+F^{00}(u)∇u, in EJ.
In addition, usatisfies the energy inequality

E(u) + Z Z

Q_{t}∩A∩P

D(u(x, τ))|ζ(x, τ)|^{2}dx dτ

+ Z Z

Q_{t}

| −∆u+f(u)|^{2}dx dτ ≤E(u_{0}),

(3.2)

for allt >0.

Proof. We consider a sequence of positive numbersε_{i} monotonically decreasing to
0 as i→ ∞. Fixu_{0} ∈H^{1}(Ω), for any fixedε_{i}, here, for the sake of simplicity, we
writeu_{i}:=u_{ε}_{i} andD_{i}(u_{i}) :=D_{ε}_{i}(u_{ε}_{i}). By Theorem 2.1, there exists a functionu_{i}
such that

(1) ui∈L^{∞}(0, T;H_{0}^{1}(Ω))∩C([0, T];L^{p}(Ω))∩L^{2}(0, T;H^{3}(Ω)), where 1≤p <

∞ifn= 1,2 and 2≤p < _{n−2}^{2n} ifn≥3,
(2) ∂tui∈L^{2}(0, T; (H^{2}(Ω))^{0}),

Z T

0

h∂tu_{i}, φidt=−
Z T

0

Z

Ω

D_{i}(u_{i})∇vi· ∇φ dx dt−
Z T

0

Z

Ω

v_{i}φ dx dt (3.3)
for all test functionsφ∈L^{2}(0, T;H^{2}(Ω)∩H_{0}^{1}(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

kuik_{L}∞(0,T;H_{0}^{1}(Ω))≤C, (3.5)
k∂_{t}u_{i}k_{L}2(0,T;(H^{2}(Ω))^{0})≤C, (3.6)
kp

Di(ui)∇vik_{L}2(QT)≤C, (3.7)

kv_{i}k_{L}2(Q_{T})≤C. (3.8)