Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 100, pp. 1–17.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
WEAK SOLUTIONS FOR NONLOCAL EVOLUTION
VARIATIONAL INEQUALITIES INVOLVING GRADIENT
CONSTRAINTS AND VARIABLE EXPONENT
MINGQI XIANG, YONGQIANG FU
Abstract. In this article, we study a class of nonlocal quasilinear parabolic variational inequality involving p(x)-Laplacian operator and gradient
constraint on a bounded domain. Choosing a special penalty functional according to the gradient constraint, we transform the variational inequality to a
parabolic equation. By means of Galerkin’s approximation method, we obtain
the existence of weak solutions for this equation, and then through a priori
estimates, we obtain the weak solutions of variational inequality.
1. Introduction
In this article, we are concerned with the existence of weak solutions for nonlocal
(Kirchhoff type) parabolic variational inequality involving variable exponent. More
precisely, we shall find a function u ∈ K = {w(x, t) ∈ V (QT ) ∩ L∞ (0, T ; L2 (Ω)) :
w(x, 0) = 0, |∇w(x, t)| ≤ 1 a.e. (x, t) ∈ QT } satisfying the follow inequality
Z
Z T Z
Z
∂v
(v − u) dx dt +
|∇u|p(x)−2 ∇u∇(v − u) dx dt
a t,
|∇u|p(x) dx
∂t
QT
Ω
0
Ω
Z
≥
f (v − u) dx dt,
QT
(1.1)
0
for all v ∈ V (QT ) with ∂v
∈
V
(Q
),
v(x,
0)
=
0,
|∇v(x,
t)|
≤
1
a.e.
(x,
t)
∈
QT ,
T
∂t
where V 0 (QT ) is the dual space of variable exponent Sobolev space V (QT ) (see
Definition 2.3 below).
In recent years, the research of nonlinear problems with variable exponent growth
conditions has been an interesting topic. p(·)-growth problems can be regarded as
a kind of nonstandard growth problems and these problems possess very complicated nonlinearities, for instance, the p(x)-Laplacian operator − div(|∇u|p(x)−2 ∇u)
is inhomogeneous. And these problems have many important applications in nonlinear elastic, electrorheological fluids and image restoration (see [9, 27, 30, 31, 32]).
Many results have been obtained on this kind of problems, see [1, 2, 5, 6, 11, 12, 14,
15, 16, 25]. Especially, in [6, 25], the authors studied the existence and uniqueness
2000 Mathematics Subject Classification. 35K30, 35K86, 35K59.
Key words and phrases. Nonlocal evolution variational inequality; variable exponent space;
Galerkin approximation; penalty method.
c
2013
Texas State University - San Marcos.
Submitted March 7, 2013. Published April 19, 2013.
1
2
M. XIANG, Y. FU
EJDE-2013/100
of weak solutions for anisotropy parabolic variation inequalities in the framework
of variable exponent Sobolev spaces. Motivating by their works, we study a class
variational inequalities with gradient constrain and variable exponent. To the best
of our knowledge, there are no papers dealing with parabolic equalities involving
variable growth and gradient constraints. For the fundamental theory about variable exponent Lebesgue and Sobolev spaces, we refer to [13, 21]. The basic theory
about Variational inequalities, we refer the reader to [7, 26] for the details.
The study of Kirchhoff-type problems has received considerable attention in
recent years, see [3, 4, 10, 19, 18, 20, 28, 29]. This interest arises from their contributions to the modeling of many physical and biological phenomena. We refer
the reader to [17, 24] for some interesting results and further references. In [3, 4],
the authors discussed the asymptotic stability for Kirchhoff systems with variable
exponent growth conditions
utt − M (F u(t))∆p(x) u + Q(t, x, u, ut ) + f (x, u) = 0
u(t, x) = 0
in R+
0 ×Ω
on R+
0 × ∂Ω,
a + bτ γ−1 , τ ≥ 0 with a, b ≥ 0, a + b > 0 and γ > 1, and F u(t) =
Rwhere M (τ ) =
p(x)
{|Du(x, t)|
/p(x)}dx, ∆p(x) = div(|Du|p(x)−2 Du).
Ω
On the one hand, our motivation for investigating (1.1) arises from reactiondiffusion equations that model population density or heat propagation (see [8]). The
following equation describes the density of a population (for instance of bacteria)
subject to spreading
ut = a(u)∆u + F (u)
u(x, t) = 0
in Ω × (0, T ),
on ∂Ω × (0, T ),
u(x, 0) = u0 (x)
in Ω.
The diffusion coefficient a depends on a nonlocal quantity related to the total
population in the domain Ω; that is, the diffusion of individuals is guided by the
global state of the population in the medium. From an experimentalist point view,
it certainly makes sense to introduce nonlocal quantities, since measurements are
often averages. The function F describes the reaction or growth of the population.
On the other hand, we can use problem (1.1) to describe the motion of a nonstationary fluid or gas in a nonhomogeneous and anisotropic medium and the nonlocal
term a appearing in (1.1) can describe a possible change in the global state of the
fluid or gas caused by its motion in the considered medium.
This article is organized as follows. In section 2, we will give some necessary
definitions and properties of variable exponent Lebesgue spaces and Sobolev spaces.
Moreover, we introduce the space V (QT ) and give some necessary properties, which
provides a basic framework to solve our problem. In section 3, using the penalty
method, we consider class of parametrized parabolic equations, and obtain weak
solutions by Galerkin’s approximation. In section 4, we give the proof of main
theorem to this paper.
2. Preliminaries
In this section, we first recall some important properties of variable exponent
Lebesgue spaces and Sobolev spaces, see [12, 13, 21] for details.
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WEAK SOLUTIONS
3
2.1. Variable exponent Lebesgue space and Sobolev space. Let Ω ⊂ RN be
a domain. A measurable function p : Ω → [1, ∞) is called a variable exponent and
we define p− = ess inf x∈Ω p(x) and p+ = ess supx∈Ω p(x). If p+ is finite, then the
exponent p is said to be bounded. The variable exponent Lebesgue space is
Z
Lp(x) (Ω) = {u : Ω → R is a measurable function; ρp(x) (u) =
|u(x)|p(x) dx < ∞}
Ω
with the Luxemburg norm
kukLp(x) (Ω) = inf{λ > 0 : ρp(x) (λ−1 u) ≤ 1},
then Lp(x) (Ω) is a Banach space, and when p is bounded, we have the following
relations
−
+
−
+
min{kukpLp(x) (Ω) , kukpLp(x) (Ω) } ≤ ρp(x) (u) ≤ max{kukpLp(x) (Ω) , kukpLp(x) (Ω) }.
That is, if p is bounded, then norm convergence is equivalent to convergence with
respect to the modular ρp(x) . For bounded exponent the dual space (Lp(x) (Ω))0
0
can be identified with Lp (x) (Ω), where the conjugate exponent p0 is defined by
p
. If 1 < p− ≤ p+ < ∞, then the variable exponent Lebesgue space
p0 = p−1
Lp(x) (Ω) is separable and reflexive.
In the variable exponent Lebesgue space, Hölder’s inequality is still valid. For
0
all u ∈ Lp(x) (Ω), v ∈ Lp (x) (Ω) with p(x) ∈ (1, ∞) the following inequality holds
Z
1
1 |uv|dx ≤ − + 0 − kukLp(x) (Ω) kvkLp0 (x) (Ω) ≤ 2kukLp(x) (Ω) kvkLp0 (x) (Ω) .
p
(p )
Ω
Definition 2.1 ([11, 12]). We say a bounded exponent p : Ω → R is globally
log-Hölder continuous if p satisfies the following two conditions:
(1) there is a constant c1 > 0 such that
c1
|p(y) − p(z)| ≤
log(e + |y − z|−1 )
for all points y, z ∈ Ω;
(2) there exist constants c2 > 0 and p∞ ∈ R such that
c2
|p(y) − p∞ | ≤
log(e + |y|−1 )
for all y ∈ Ω.
The Variable exponent Sobolev space W 1,p(x) (Ω) is defined as
W 1,p(x) (Ω) = {u ∈ Lp(x) (Ω) : |∇u| ∈ Lp(x) (Ω)}
and equipped with the norm
kukW 1,p(x) (Ω) = kukLp(x) (Ω) + k∇ukLp(x) (Ω) ,
1,p(x)
then the space W 1,p(x) (Ω) is a Banach space. The space W0
(Ω) is defined as
the closure of C0∞ (Ω) with the norm of k · kW 1,p(x) (Ω) . If 1 < p− ≤ p+ < ∞, then
the space W 1,p(x) (Ω) is separable and reflexive.
Theorem 2.2 ([11, 12]). Let Ω ⊂ RN be a bounded domain and assume that
p : RN → (1, ∞) is a bounded globally log-Hölder continuous exponent such that
1,p(x)
p− > 1, then for every u ∈ W0
(Ω) we have
kukLp(x) (Ω) ≤ c diam(Ω)k∇ukLp(x) (Ω) ,
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M. XIANG, Y. FU
EJDE-2013/100
where the constant c only depends on the dimension N and the log-Hölder constant
of p.
2.2. Variable exponent Sobolev space V (QT ).
Definition 2.3. Let Ω ⊂ RN be a bounded domain with smooth boundary. Denote
QT = Ω × (0, T ), 0 < T < ∞. Suppose that p(x) is a bounded globally log-Hölder
continuous function on Ω with p− > 1, we set
1,p(x)
V (QT ) = {u ∈ L2 (QT ) : |∇u| ∈ Lp(x) (QT ), u(·, t) ∈ W0
(Ω)
a.e. t ∈ (0, T )},
with the norm
kuk = kukL2 (QT ) + k∇ukLp(x) (QT ) .
Remark 2.4. Following the standard proof for Sobolev spaces, we can prove that
V (QT ) is a Banach space, and it’s easy to check that V (QT ) can be continuously
−
embedded into the space Lr (0, T ; W01,p (Ω) ∩ L2 (Ω)), where r = min{p− , 2}. It is
worth to mention the paper [6] where the space V (QT ) is defined in a similar way.
By the same method in [11], we have the following theorem.
Theorem 2.5 ([11]). The space C0∞ (QT ) is dense in V (QT ).
Since C0∞ (QT ) ⊂ C ∞ (0, T ; C0∞ (Ω)), we have the following result.
Lemma 2.6. The space C ∞ (0, T ; C0∞ (Ω)) is dense in V (QT ).
Let V 0 (QT ) denote the dual space of V (QT ).
Theorem 2.7 ([6, 11]). A function g ∈ V 0 (QT ) if and only if there exist ḡ ∈ L2 (QT )
0
and Ḡ ∈ (Lp (x) (QT ))N such that
Z
Z
Z
gϕ dx dt =
ḡϕ dx dt +
Ḡ∇ϕ dx dt.
(2.1)
QT
QT
QT
Remark 2.8. It follows from the proof of Theorem 2.7 that V (QT ) is reflexive and
0
V 0 (QT ) ,→ Ls (0, T ; W −1,(p
+ 0
)
(Ω) + L2 (Ω)),
where s = max{p+ , 2}.
Similar to that in [11], we give the following definition.
0
Definition 2.9. We define the space W (QT ) = {u ∈ V (QT ) : ∂u
∂t ∈ V (QT )} with
the norm
∂u kukW (QT ) = kukV (QT ) + V 0 (Q ) ,
T
∂t
∂u
where ∂t is the weak derivative of u with respect to time variable t defined by
Z
Z
∂u
∂ϕ
ϕ dx dt = −
u
dx dt, for all ϕ ∈ C0∞ (QT ).
∂t
∂t
QT
QT
Lemma 2.10 ([11]). The space W (QT ) is a Banach space.
By the method in [11], we have the following result.
Theorem 2.11. The space C ∞ (0, T ; C0∞ (Ω)) is dense in W (QT ).
The following theorem can be proved similarly to that in [11], thus we omit its
proof.
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WEAK SOLUTIONS
5
Theorem 2.12 ([1, 11]). W (QT ) can be embedded continuously in C(0, T ; L2 (Ω)).
Furthermore, for all u, v ∈ W (QT ) and s, t ∈ [0, T ] the following rule for integration
by parts is valid
Z tZ
Z
Z
Z tZ
∂u
∂v
v dx dτ =
dx dτ.
u(x, t)v(x, t)dx −
u(x, s)v(x, s)dx −
u
∂t
s
Ω
Ω
Ω
s
Ω ∂t
The following theorem gives a relation between almost everywhere convergence
and weak convergence.
Theorem 2.13 ([9]). Let p(x) : QT → R be a bounded globally log-Hölder continp(x)
uous function, with p− > 1. If {un }∞
(QT ) and un → u a.e.
n=1 is bounded in L
(x, t) ∈ QT as n → ∞, then there exists a subsequence of {un } still denoted by {un }
such that un → u weakly in Lp(x) (QT ) as n → ∞.
We will give a compact embedding for V (QT ) in the following.
Theorem 2.14 ([23]). Let B0 ⊂ B ⊂ B1 be three Banach spaces, where B0 , B1
are reflexive, and the embedding B0 ⊂ B is compact. Denote W = {v : v ∈
p1
Lp0 (0, T ; B0 ), ∂v
∂t ∈ L (0, T ; B1 )}, where T is a fixed positive number, 1 < pi < ∞,
i = 0, 1, then W can be compactly embedded into Lp0 (0, T ; B).
Theorem 2.15. Let F be a bound subset in V (QT ) and { ∂u
∂t : u ∈ F } be bounded
0
r
2
in V (QT ), then F is relatively compact in L (0, T ; L (Ω)).
Proof. Since p− >
2N
N +2
−
(N ≥ 2), the embedding W01,p (Ω) ,→ L2 (Ω) is compact.
−
By Remarks 2.4 and 2.8, the embeddings V (QT ) ,→ Lr (0, T ; W01,p (Ω) ∩ L2 (Ω))
and
0
V 0 (QT ) ,→ Ls (0, T ; W −1,(p
+ 0
)
0
(Ω) + L2 (Ω)) ,→ Ls (0, T ; W −1,λ (Ω))
are continuous, where λ = min{2, (p+ )0 }. As the embedding L2 (Ω) ,→ W −1,λ (Ω)
is continuous, by Theorem 2.14, F is relatively compact in Lr (0, T ; L2 (Ω)).
3. Existence of solutions for parabolic equations
In this section, for ε ∈ (0, 1), we consider the following nonlocal parabolic equation with Diriclet boundary-value conditions:
Z
∂u
− a t,
|∇u|p(x) dx div(|∇u|p(x)−2 ∇u)
∂t
Ω
1
− div (|∇u|p(x)−2 − 1)+ ∇u = f (x, t), (x, t) ∈ Ω × (0, T ),
(3.1)
ε
u(x, t) = 0, (x, t) ∈ ∂Ω × (0, T ),
u(x, 0) = 0,
x ∈ Ω,
where (|∇u|p(x)−2 − 1)+ = max{|∇u|p(x)−2 − 1, 0}. We assume that
(H1) a(t, s) : [0, ∞) × [0, ∞) → (0, ∞) is a continuous function and there exists
two positive constants a0 and a1 such that
a0 ≤ a(t, s) ≤ a1
for each (t, s) ∈ [0, ∞) × [0, ∞).
(H2) p(x) : Ω → (1, ∞) is a global log-Hölder continuous function. Denote
p− = inf x∈Ω p(x), p+ = supx∈Ω p(x). And there holds
2 < p− ≤ p(x) ≤ p+ < ∞ for each x ∈ Ω.
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M. XIANG, Y. FU
EJDE-2013/100
(H3) f ∈ V 0 (QT ).
0
ε
Definition 3.1. A function uε ∈ V (QT ) ∩ C(0, T ; L2 (Ω)) with ∂u
∂t ∈ V (QT ) is
called a weak solution of (3.1), if
Z
Z T Z
Z
∂uε
a t,
|∇uε |p(x) dx
ϕ dx dt +
|∇uε |p(x)−2 ∇uε ∇ϕ dx dt
0
Ω
QT ∂t
Ω
Z
Z
1
+
(|∇uε |p(x)−2 − 1)+ ∇uε ∇ϕ dx dt =
f ϕ dx dt,
QT ε
QT
for all ϕ ∈ V (QT ).
Since f ∈ V 0 (QT ), there exists a sequence fn ∈ C0∞ (QT ) such that limn→∞ fn =
∞
f in V 0 (QT ). Similar to that in [14, 15], we choose a sequence {wj }∞
j=1 ⊂ C0 (Ω)
C 1 (Ω̄)
and {wj }∞
such that C0∞ (Ω) ⊂ ∪∞
j=1 is a standard orthogonal basis in
n=1 Vn
2
L (Ω), where Vn = span{w1 , w2 , . . . , wn }.
Theorem 3.2. Let assumptions (H1)–(H3) hold and let ε ∈ (0, 1) be fixed. Then
there exists a weak solution for equation (3.1).
Proof. (i) Galerkin approximation. For each n ∈ N, we want to find the approximate solutions to problem (3.1) in the form
un (x, t) =
n
X
(ηn (t))j wj (x).
j=1
First we define a vector-valued function Pn (t, υ) : [0, 1] × Rn → Rn as
n
n
n
Z X
Z X
X
(Pn (t, ν))i = a t,
|
νj ∇ωj |p(x) dx
|
νj ∇wj |p(x)−2
νj ∇wj ∇wi dx
Z
+
Ω
Ω j=1
n
X
Ω j=1
p(x)−2
+
1 νj ∇wj −1
ε j=1
j=1
n
X
νj ∇wj ∇wi dx,
j=1
where ν = (ν1 , · · ·, νn ). Since a and p are continuous, from the definition of Pn (t, ν),
Pn (t, ν) is continuous with respect to t and ν.
We consider the following ordinary differential systems
η 0 (t) + Pn (t, η(t)) = Fn ,
η(0) = 0,
R
where (Fn )i = Ω fn wi dx.
Multiplying (3.2) by η(t), we arrive at the equality
η 0 (t)η(t) + Pn (t, η(t)) η(t) = Fn η(t).
Since
n
n
Z X
Z X
p(x)
Pn (t, η)η = a t,
|
ηj ∇ωj |
dx
|
ηj ∇wj |p(x) dx
Z
+
Ω
Ω j=1
n
X
Ω j=1
n
X
+
1
(|
νj ∇wj |p(x)−2 − 1) |
νj ∇wj |2 dx ≥ 0,
ε j=1
j=1
(3.2)
EJDE-2013/100
WEAK SOLUTIONS
7
by Young’s inequality, there holds
1 ∂|η(t)|2
1
1
≤ |Fn ||η(t)| ≤ |Fn |2 + |η(t)|2 .
2 ∂t
2
2
Then integrating with respect to t from 0 to t, we obtain
Z t
|η(t)|2 ≤ Cn +
|η(s)|2 ds.
0
By Gronwall’s inequality, we obtain that |η(t)| ≤ Cn (T ). We denote
Ln =
max
(t,η)∈[0,T ]×B(η(0),Cn (T ))
|Fn − Pn (t, η)|,
τn = min{T,
Cn (T )
},
Ln
where B(η(0), Cn (T )) is the ball of radius Cn (T ) with the center at the point η(0)
in Rn . By Peano’s Theorem we know that (3.2) admits a C 1 solution in [0, τn ]. Let
η(τn ) be a initial value, then we can repeat the above process and get a C 1 solution
in [tn , 2τn ]. Without lost of generality, we assume that T = [ τTn ]τn + ( τTn )τn , 0 <
( τTn ) < 1, where [ τTn ] is the integer part of τTn , ( τTn ) is the decimal part of τTn . We
can divide [0, T ] into [(i − 1)τn , iτn ], i = 1, . . . , L and [Lτn , T ] where L = [ τTn ], then
there exist C 1 solution ηni (t) in [(i − 1)τn , iτn ], i = 1, . . . , L and ηnL+1 (t) in [Lτn , T ].
Therefore, we obtain a solution ηn (t) ∈ C 1 [0, T ] defined by


ηn1 (t),
if t ∈ [0, τn ],



2


if t ∈ (τn , 2τn ],
ηn (t),
ηn (t) = . . .


ηnL (t),
if t ∈ ((L − 1)τn , Lτn ],



η L+1 (t), if t ∈ (Lτ , T ].
n
n
Pn
Thus, we obtain the approximate solutions sequence un =
j=1 (ηn (t))j wj (x).
From (3.2), for 1 ≤ i ≤ n, we have
Z
Z
Z
∂un
p(x)
wi dx + a t,
|∇un |
dx
|∇un |p(x)−2 ∇un ∇wi dx
Ω ∂t
Ω
Ω
Z
1
+
(|∇un |p(x)−2 − 1)+ ∇un ∇wi dx
(3.3)
ε
Ω
Z
=
fn wi dx.
Ω
Multiplying by (ηn (t))i , summing up i from 1 to n, and integrating with respect to
t from 0 to τ , where τ ∈ (0, T ], we obtain
Z τ Z
Z τZ
Z
∂un
un dx dt +
a t,
|∇un |p(x) dx
|∇un |p(x) dx dt
0
Ω ∂t
0
Ω
Ω
Z τZ
1
(3.4)
+
(|∇un |p(x)−2 − 1)+ |∇un |2 dx dt
0
Ω ε
Z τZ
=
fn un dx dt.
0
Ω
Remark 3.3. The approximate solutions un depends on ε; For convenience, we
omit the ε. For all ϕ ∈ C 1 (0, T ; Vk ), k ≤ n, there holds
Z τZ
Z τ Z
Z
∂un
p(x)
ϕ dx dt +
a t,
|∇un |
∇un dx
|∇un |p(x)−2 ∇un ∇ϕ dx dt
0
Ω ∂t
0
Ω
Ω
8
M. XIANG, Y. FU
Z
τ
Z
+
Z0 τ ZΩ
=
EJDE-2013/100
1
(|∇un |p(x)−2 − 1)+ ∇un ∇ϕ dx dt
ε
fn ϕ dx dt.
0
Ω
(ii) A priori estimates. By (3.4), assumption (H1) and integration by parts,
we arrive at the inequality
Z
Z τZ
1
2
2
|un (x, τ )| − |un (x, 0)| dx + a0
|∇un |p(x) dx dt ≤ kfn kV 0 (Qτ ) kun kV (Qτ ) ,
2 Ω
0
Ω
where Qτ = Ω × (0, τ ), τ ∈ (0, T ]. Since un (x, 0) = 0 and fn → f in V 0 (QT ),
kfn kV 0 (QT ) ≤ C, where C independent of τ and n. Thus, we obtain
Z
Z τZ
u2n (x, τ )dx+a0
|∇un |p(x) dx dt ≤ C(kun kL2 (Qτ ) +k∇un kLp(x) (Qτ ) ). (3.5)
Ω
0
Ω
Without lost generality, we assume that k∇un kLp(x) (Qτ ) ≥ 1. Then
Z
p−
k∇un kL
≤
|∇un |p(x) dx dt.
p(x) (Q )
τ
Qτ
By (3.5) and Young’s inequality, there holds
Z Z
Z
a0 τ
|∇un |p(x) dx dt ≤ C(kun kL2 (Qτ ) + 1).
u2n (x, τ )dx +
2 0 Ω
Ω
By Gronwall’s inequality, we obtain kun kL∞ (0,T ;L2 (Ω)) ≤ C. Therefore,
kun kL∞ (0,T ;L2 (Ω)) + kun kV (QT ) ≤ C.
(3.6)
Combining assumption (H1), with (3.6), we have
Z Z
p0 (x)
dx dt ≤ C.
|∇un |p(x) dx |∇un |p(x)−2 ∇un a t,
QT
Ω
Z
0
|(|∇un |p(x)−2 − 1)+ ∇un |p (x) dx dt ≤ C(ε),
QT
where C(ε) is a constant independent of n on ε and C(ε) → ∞ as ε → ∞. Thus
we obtain
Z
a(t,
|∇un |p(x) dx)|∇un |p(x)−2 ∇un Lp0 (x) (Q ) ≤ C,
T
(3.7)
Ω
p(x)−2
+
(|∇un |
− 1) ∇un Lp0 (x) (Q ) ≤ C(ε).
T
By Lemma 2.6, for all ϕ ∈ V (QT ), there exists a sequence ϕn ∈C 1 (0, T ; Vn ) such
that ϕn → ϕ strongly in V (QT ). By Remark 3.3, we have
Z
∂un
ϕn dx dt
∂t
QT
Z
Z
=−
a t,
|∇un |p(x) dx |∇un |p(x)−2 ∇un ∇ϕn dx dt
Q
Ω
Z
Z T
1
p(x)−2
+
−
(|∇un |
− 1) ∇un ϕn dx dt +
fn ϕn dx dt
ε
QT
QT
Z
∇ϕn p(x)
≤ C a t,
|∇un |p(x) dx |∇un |p(x)−2 ∇un p0 (x,t)
L
(Q )
Ω
L
(QT )
T
EJDE-2013/100
WEAK SOLUTIONS
9
1
+ k(|∇un |p(x)−2 − 1)+ ∇un kLp0 (x,t) QT k∇ϕn k + kfn kV 0 (QT ) kϕn kV (QT )
ε
≤ C(ε)kϕn kV (QT ) ,
where C(ε) is a constant independent of n on ε. We immediately get that
∂un
kV 0 (QT ) ≤ C(ε).
(3.8)
∂t
(iii) Passage to the limit. From (3.6)-(3.8), we obtain a subsequence of {un }
(still denoted by {un }) such that
k
un * uε
weakly* in L∞ (0, T ; L2 (Ω)),
un * uε
Z
a t,
weakly in V (QT ),
|∇un |p(x) dx |∇un |p(x)−2 ∇un * ξ
0
weakly in Lp (x) (QT )
N
,
Ω
(|∇un |p(x)−2 − 1)+ ∇un * η
0
weakly in (Lp (x) (QT ))N
∂uε
∂un
*
weakly in V 0 (QT ).
∂t
∂t
R
Since Ω u2n (x, T )dx ≤ C, there exist a subsequence of {un (x, T )} (still denoted
by {un (x, T )}) and a function ũ in L2 (Ω) such that un (x, T ) → ũ weakly in L2 (Ω).
Then for any ϕ(x) ∈ C0∞ (Ω) and η(t) ∈ C 1 [0, T ], there holds
Z TZ
∂un
ϕη dx dt
0
Ω ∂t
Z
Z
Z TZ
=
un (x, T )ϕη(T )dx −
un (x, 0)ϕη(0)dx −
un ϕη 0 (t) dx dt.
ω
Ω
0
Ω
Letting n → ∞, by integration by parts, we obtain
Z
Z
(ũ − uε (x, T ))η(T )ϕdx +
uε (x, 0)η(0)ϕdx = 0.
Ω
Ω
Choosing η(T ) = 1, η(0) = 0 or η(T ) = 0, η(0) = 1, by the density of C0∞ (Ω) in
L2 (Ω), we have ũ = uε (x, T ) and uε (x, 0) = 0 for almost every x ∈ Ω. That is
un (x, T ) → uε (x, T ) weakly in L2 (Ω), as n → ∞, thus
Z
Z
2
uε (x, T )dx ≤ lim inf
u2n (x, T )dx.
(3.9)
Ω
n→∞
Ω
1
In view of Remark 3.3, for all ϕ ∈ C (0, T ; Vk ) where k ≤ n, letting n → ∞
there holds
Z
Z
∂uε
1
ϕ + ξ∇ϕ + η∇ϕ dx dt =
f ϕ dx dt,
(3.10)
ε
QT ∂t
QT
1
1
since C 1 0, T ; ∪∞
k=1 Vk is dense in C (0, T ; C (Ω)), the above equality is valid for
1
∞
all ϕ ∈ C (0, T ; C0 (Ω)). Moreover, for all ϕ ∈ V (QT ), the above equality is valid.
Thus, we can take ϕ = uε . By integration by parts, we have
Z
Z
Z
1
1
2
|uε (x, T )| dx +
ξ∇uε + η∇uε dx dt =
f uε dx dt.
(3.11)
2 Ω
ε
QT
QT
We denote
Z
Z
Yn =
a t,
|∇un |p(x) dx |∇un |p(x)−2 ∇un − |∇uε |p(x)−2 ∇uε
QT
Ω
10
M. XIANG, Y. FU
EJDE-2013/100
× ∇un − ∇uε dx dt
Z
1
+
(|∇un |p(x)−2 − 1)+ ∇un − (|∇uε |p(x)−2 −)+ ∇uε ∇un − ∇uε ) dx dt.
ε QT
By (3.4), we obtain
Z
Z
1
0 ≤ Yn =
fn un −
|un (x, T )|2 − |un (x, 0)|2 dx
2
Q
Ω
Z T Z
−
a t,
|∇un |p(x) dx |∇un |p(x)−2 ∇un ∇uε dx dt
Q
Ω
Z T Z
−
a t,
|∇un |p(x) dx |∇uε |p(x)−2 ∇uε (∇un − ∇uε ) dx dt
(3.12)
QT
Ω
Z
1
(|∇un |p(x)−2 − 1)+ ∇un ∇uε dx dt
−
ε QT
Z
1
(|∇uε |p(x)−2 − 1)+ ∇uε (∇un − ∇uε ) dx dt
−
ε QT
R
By assumption (H1), the sequence {a t, Ω |∇un |p(x) dx }∞
n=1 is equi-integrable and
uniformly bounded in L1 (0, T ). Therefore, there exist a subsequence of {un } (still
denoted by {un }) and ā(t) such that
Z
a t,
|∇un |p(x) dx → ā(t) a.e. t ∈ [0, T ].
Ω
As
Z
p0 (x)
|∇un |p(x) dx |∇uε |p(x)−2 ∇uε ≤ C|∇uε |p(x) ∈ L1 (QT ),
a t,
Ω
by the Lebesgue dominated convergence theorem, we obtain
Z h Z
p0 (x)
i
dx dt → 0.
|∇un |p(x) dx − ā(t) |∇uε |p(x)−2 ∇uε a t,
QT
Ω
That is,
Z
a t,
|∇un |p(x) dx |∇uε |p(x)−2 ∇uε → ā(t)|∇uε |p(x)−2 ∇uε
N
0
in Lp (x) (QT ) .
Ω
(3.13)
Thus, from (3.9), (3.11)-(3.13), we obtain
Z
Z
1
|u(x, T )|2 dx
0 ≤ lim sup Yn ≤
f u dx dt −
2 Ω
n→∞
QT
Z
Z
1
η∇uε dx dt = 0;
−
ξ∇uε dx dt −
ε QT
QT
therefore limn→∞ Yn = 0. Furthermore, by assumption (H1), there holds
Z
|∇un |p(x)−2 ∇un − |∇uε |p(x)−2 ∇uε ∇un − ∇uε dx dt → 0.
QT
Since p(x, t) ≥ p− > 2, as n → ∞, there holds
Z
|∇un − ∇uε |p(x) dx dt
QT
Z
≤C
(|∇un |p(x)−2 ∇un − |∇uε |p(x)−2 ∇uε )(∇un − ∇uε ) dx dt → 0.
QT
(3.14)
EJDE-2013/100
WEAK SOLUTIONS
11
Therefore, from (3.14), we obtain ∇un → ∇uε in (Lp(x) (QT ))N . Thus, there
exists a subsequence of {un } still denoted by {un } such that
Z
|∇un − ∇uε |p(x) dx → 0 a.e. t ∈ [0, T ].
(3.15)
Ω
Since
Z
|∇un |p(x) − |∇uε |p(x) dx
Ω
Z
p(x)−1 |∇un | − |∇uε |dx
≤
p(x)|∇un | + θ(|∇un | − |∇uε |)
Ω
≤ C |∇un |p(x)−1 + |∇uε |p(x)−1 Lp0 (x) (Ω) k∇un − ∇uε kLp(x) (Ω) ,
where 0 ≤ θ ≤ 1, by (3.15), we have
Z
Z
p(x)
|∇un |
→
|∇uε |p(x) dx
Ω
a.e. t ∈ [0, T ].
Ω
Thus, by the continuity of a, we obtain that
Z
ā(t) = a t,
|∇uε |p(x) dx
a.e. t ∈ [0, T ]
Ω
Since ∇un → ∇uε in (Lp(x) (QT ))N , there exists a subsequence of {un } (still
labeled by {un }) such that ∇un → ∇uε for a.e. (x, t) ∈ QT , then
Z
a t,
|∇un |p(x) dx |∇un |p(x,t−2) ∇un
Ω
Z
→ a t,
|∇uε |p(x) dx |∇uε |p(x)−2 ∇uε a.e. (x, t) ∈ QT .
Ω
R
By Theorem 2.13, we obtain ξ = a t, Ω |∇uε |p(x) dx |∇uε |p(x)−2 ∇uε . Similarly,
η = (|∇uε |p(x,t)−2 − 1)+ ∇uε .
It follows from (3.10) that
Z
Z T Z
Z
∂uε
ϕ dx dt +
a t,
|∇un |p(x) dx
|∇u|p(x)−2 ∇uε ∇ϕ
QT ∂t
0
Ω
Ω
Z
1
+ (|∇uε |p(x,t)−2 − 1)+ ∇uε ∇ϕ dx dt =
f ϕ dx dt,
ε
QT
0
for all ϕ ∈ V (QT ). Since u ∈ V (QT ) and ∂u
∂t ∈ V (QT ), by Theorem 2.12, up to a
set of measure zero, we have u ∈ C(0, T ; L2 (Ω)).
4. Existence of solutions for the variational inequality
In this section, we prove our main theorem.
Theorem 4.1. Under assumptions (H1)–(H3) there exists a function u(x, t) ∈ K
such that
Z
Z T Z
Z
∂v
(v − u) dx dt +
a t,
|∇u|p(x) dx
|∇u|p(x)−2 ∇u∇(v − u) dx dt
0
Ω
Ω
QT ∂t
Z
≥
f (v − u) dx dt
QT
for all v ∈ V (QT ) with
∂v
∂t
∈ V 0 (QT ), v(x, 0) = 0, |∇v| ≤ 1 a.e. (x, t) ∈ QT .
12
M. XIANG, Y. FU
EJDE-2013/100
Proof. We will prove this theorem in three steps.
(Step 1) A priori estimates. In Definition 3.1, we take ϕ = uε χ(0,τ ) as a test
function, where χ(0,τ ) is defined as the characteristic function of (0, τ ), τ ∈ (0, T ],
then
Z
Z
Z
∂uε
uε dx dt +
a t,
|∇uε |p(x) dx |∇uε |p(x)
Qτ ∂t
Qτ
Ω
1
+ (|∇uε |p(x)−2 − 1)+ |∇uε |2 dx dt
ε
Z
=
f (x, t)uε dx dt,
Qτ
where Qτ = Ω × (0, τ ). Similar to Section 3, we have
Z
Z
2
|∇uε |p(x) dx dt ≤ C,
|uε (x, τ )| dx +
for all τ ∈ [0, T ].
Qτ
Ω
Therefore,
Z
1
(|∇uε |p(x)−2 − 1)+ |∇uε |2 dx dt + uε L∞ (0,T ;L2 (Ω)) + uε V (Q ) ≤ C. (4.1)
T
ε QT
Since
Z
p0 (x)
a t,
|∇uε |p(x,t) dx |∇uε |p(x)−2 ∇uε dx dt
Ω
QT
Z
|∇uε |p(x) dx dt
≤C
Z
QT
p−
p+
≤ C max{∇uε Lp(x) (Q ) , ∇uε Lp(x) (Q ) } ≤ C,
T
T
there holds
Z
|∇uε |p(x) dx |∇uε |p(x)−2 ∇uε a t,
Lp0 (x) (QT )
Ω
≤ C.
(Step 2) Passage to the limit. By (4.1)-(??), there exists a subsequence of
{uε }ε>0 , still denoted by {uε }ε>0 , such that
∗
uε * u weakly * in L∞ (0, T ; L2 (Ω)),
uε * u weakly in V (QT ),
Z
0
a t,
|∇uε |p(x) dx |∇uε |p(x)−2 ∇uε * A weakly in (Lp (x) (QT ))N .
(4.2)
Ω
For all ϕ ∈ V (QT ), there holds
Z
[(|∇uε |p(x)−2 − 1)+ ∇uε − (|∇ϕ|p(x)−2 − 1)+ ∇ϕ](∇uε − ∇ϕ) dx dt ≥ 0.
QT
Since
Z
p(x)−2
|(|∇uε |
+
− 1) ∇uε |
QT
p0 (x)
Z
dx dt ≤
(|∇uε |p(x)−2 − 1)+ |∇uε |2 dx dt,
QT
R
0
by (4.1), we obtain that QT |(|∇uε |p(x)−2 − 1)+ ∇uε |p (x) dx dt → 0 as ε → 0; that
is, k(|∇uε |p(x)−2 −1)+ ∇uε kLp0 (x) (QT ) → 0. From ∇uε * u weakly in (Lp(x) (QT ))N ,
we have
Z
(|∇ϕ|p(x)−2 − 1)+ ∇ϕ(∇u − ∇ϕ) dx dt ≤ 0
QT
EJDE-2013/100
WEAK SOLUTIONS
13
We take ϕ = u + λw, where 0 < λ < 1 and w ∈ V (QT ), then
Z
(|∇(u + λw)|p(x)−2 − 1)+ ∇(u + λw)∇w dx dt ≤ 0.
QT
Since |(|∇(u + λw)|p(x)−2 − 1)+ ∇(u + λw)∇w| ≤ C(|∇u|p(x) + |∇w|p(x) ) ∈ L1 (QT )
and (|∇(u + λw)|p(x)−2 − 1)+ ∇(u + λw)∇w → (|∇u|p(x)−2 − 1)+ ∇u∇w as λ → 0,
by the Lebesgue Dominated Convergence Theorem and the arbitrariness of w, we
obtain
Z
(|∇u|p(x)−2 − 1)+ |∇u|2 dx dt = 0
QT
Thus, |∇u| ≤ 1 a.e. (x, t) ∈ QT .
0
Taking ϕ = v − uε as a test function in (3.1), where v ∈ V (QT ), ∂v
∂t ∈ V (QT ),
v(x, 0) = 0 and |∇v| ≤ 1 a.e. (x, t) ∈ QT , then
Z
Z
∂v
(v − uε ) + a t,
|∇uε |p(x) dx |∇uε |p(x)−2 ∇uε ∇(v − uε )
QT ∂t
Ω
− f (x, t)(v − uε ) dx dt
Z
Z
∂uε
(v − uε ) + a t,
|∇uε |p(x) dx |∇uε |p(x)−2 ∇uε ∇(v − uε )
=
Ω
QT ∂t
Z
∂(v − uε )
− f (x, t)(v − uε ) dx dt +
(v − uε ) dx dt
∂t
QT
Z
1
=
(|∇v|p(x)−2 − 1)+ ∇v − (|∇uε |p(x)−2 − 1)+ ∇uε (∇v − ∇uε ) dx dt
ε QT
Z
∂(v − uε )
+
(v − uε ) dx dt ≥ 0,
∂t
QT
and further
Z
a t,
|∇uε |p(x) dx |∇uε |p(x) dx dt
Ω
QT
Z
Z
≤
a t,
|∇uε |p(x) dx |∇uε |p(x)−2 ∇uε ∇u dx dt
QT
Ω
Z
Z
∂v
+
(v − uε ) dx dt −
f (x, t)(v − uε ) dx dt.
QT ∂t
QT
Z
(4.3)
For k > 0, we denote
u(k)
(k)
and uµ (x, t) = µ
(k)
Rt
0


k, u < −k,
= u, |u| ≤ k,


k, u > k,
eµ(s−t) u(k) (x, s)ds. It’s easy to check that
(k)
∂u(k)
µ
∂t
= µ(u(k) −
uµ ). From that in [6], we obtain uµ → u(k) strongly in L2 (QT ) and weakly in
V (QT ) as µ → ∞. Denote Ak = {(x, t) ∈ QT : |u| ≤ k}, then u(k) = u in Ak and
(k)
(k)
(k)
sgn(u(k) − uµ ) = sgn(u − uµ ) in QT \ Ak (because |uµ | ≤ k). Thus,
Z
(k)
∂uµ
(u(k)
µ − u) dx dt
∂t
QT
14
M. XIANG, Y. FU
EJDE-2013/100
Z
(k)
(u(k) − u(k)
µ )(uµ − u) dx dt
Z
Z
2
(k)
= −µ
(u − u(k)
)
dx
dt
−
µ
(u(k) − u(k)
µ
µ )(u − uµ ) dx dt ≤ 0.
=µ
QT
QT \Ak
Ak
By a diagonal rule, we obtain a sequence denoted by vk such
R that kvk → u strongly in
L2 (QT ) and weakly in V (QT ) as k → ∞, and lim supk→∞ QT ∂v
∂t (vk −u) dx dt ≤ 0.
Taking v = vk in (4.3), we obtain
Z
Z
lim sup
a t,
|∇uε |p(x) dx |∇uε |p(x) dx dt
ε→0
QT
Ω
Z
Z
Z
∂vk
(vk − u) dx dt −
f (x, t)(vk − u) dx dt.
≤
A∇u dx dt +
QT
QT ∂t
QT
Letting k → ∞, we have
Z
Z
lim sup
a t,
|∇uε |p(x) dx |∇uε |p(x) dx dt
ε→0
QT
Ω
Z
Z
Z
≤
A∇u dx dt = lim
a t,
|∇uε |p(x) dx |∇uε |p(x)−2 ∇uε ∇u dx dt;
ε→0
QT
QT
Ω
that is,
Z
Z
a t,
|∇uε |p(x) dx |∇uε |p(x)−2 ∇uε ∇(uε − u) dx dt ≤ 0.
lim sup
ε→0
QT
(4.4)
Ω
n R
o
As the sequence a t, Ω |∇uε |p(x) dx
is uniformly bounded and equi-integrable
ε
in L1 (QT ), there exist
a Rsubsequence of{uε } (for convenience still relabeled by {uε }
∗
) and a such that a t, Ω |∇uε |p(x) dx → a∗ for almost every t ∈ [0, T ]. Since
Z
p0 (x)
|∇uε |p(x) dx − a∗ |∇u|p(x)−2 ∇u
≤ C|∇u|p(x) ∈ L1 (QT ),
a t,
Ω
by the Lebesgue dominated convergence theorem, we obtain
Z
0
a t,
|∇uε |p(x) dx |∇u|p(x)−2 ∇u → a∗ |∇u|p(x)−2 ∇u strongly in Lp (x) (QT ).
Ω
Since
Z
Z
a t,
|∇uε |p(x) dx (|∇uε |p(x)−2 ∇uε − |∇u|p(x)−2 ∇u)(∇uε − ∇u)
Q
Ω
Z T Z
=
a t,
|∇uε |p(x) dx |∇uε |p(x)−2 ∇uε (∇uε − ∇u)
QT
Ω
Z
− a t,
|∇uε |p(x) dx |∇u|p(x)−2 ∇u(∇uε − ∇u) dx dt,
0≤
Ω
we have
Z
Z
a t,
|∇uε |p(x) dx |∇uε |p(x)−2 ∇uε ∇(uε − u) dx dt ≥ 0.
lim inf
ε→0
QT
(4.5)
Ω
From (4.4)–(4.5) and ∇uε * ∇u weakly in (Lp(x) (QT ))N , there holds
Z
Z
lim
a t,
|∇uε |p(x) dx (|∇uε |p(x)−2 ∇uε − |∇u|p(x)−2 ∇u)∇(uε − u) dx dt = 0.
ε→0
QT
Ω
EJDE-2013/100
WEAK SOLUTIONS
15
Similar to Section 3, we have ∇uε → ∇u strongly in (Lp(x) (QT ))N as ε → 0.
Thus there exists a subsequence
of {uε }, still
R
R labeled by {uε } such that ∇uε →
∇u a.e. (x, t) ∈ QT and Ω |∇uε |p(x) dx → Ω |∇u|p(x) dx a.e. t ∈ [0, T ]. Thus, we
obtain that
Z
A = a t,
|∇u|p(x) dx |∇u|p(x)−2 ∇u.
Ω
(Sep 3) Existence of weak solutions. By Fatou’s Lemma,
Z
Z
lim inf
a t,
|∇uε |p(x) dx |∇uε |p(x) dx dt
ε→0
QT
Ω
Z
Z
≥
a t,
|∇u|p(x) dx |∇u|p(x) dx dt.
QT
Ω
∈ V 0 (QT ), v(x, 0) = 0, |∇v| ≤ 1 a.e. (x, t) ∈ QT , we
For all v ∈ V (QT ) with
take ϕ = v − uε as a test function in (3.1), then
Z
Z
∂v
(v − uε ) + a t,
|∇uε |p(x) dx |∇uε |p(x)−2 ∇uε ∇(v − uε )
Ω
QT ∂t
∂v
∂t
− f (x, t)(v − uε ) dx dt
Z
1
=
(|∇v|p(x)−2 − 1)+ ∇v − (|∇uε |p(x)−2 − 1)+ ∇uε (∇v − ∇uε ) dx dt
ε QT
Z
∂(v − uε )
+
(v − uε ) dx dt ≥ 0,
∂t
QT
and furthermore,
Z
lim inf
ε→0
QT
Z
∂v
(v − uε ) + a t,
|∇uε |p(x) dx |∇uε |p(x)−2 ∇uε ∇v
∂t
Ω
− f (x, t)(v − uε ) dx dt
Z
Z
≥
a t,
|∇u|p(x) dx |∇u|p(x) dx dt.
QT
Ω
Since
Z
Z
a t,
|∇uε |p(x) dx |∇uε |p(x)−2 ∇uε * a t,
|∇u|p(x) dx |∇u|p(x)−2 ∇u
Ω
Ω
p0 (x)
N
weakly in (L
(QT )) , and uε * u weakly in V (QT ), there holds
Z
Z T Z
Z
∂v
(v − u) dx dt +
a t,
|∇u|p(x,t) dx
|∇u|p(x)−2 ∇u∇(v − u) dx dt
QT ∂t
0
Ω
Ω
Z
≥
f (x, t)(v − u) dx dt.
QT
Thus we have proved our main theorem.
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Mingqi Xiang
Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China
E-mail address: xiangmingqi hit@163.com
Yongqiang Fu
Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China
E-mail address: fuyqhagd@yahoo.cn