Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 46, pp. 1–21.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE OF SOLUTIONS TO A PARABOLIC p(x)-LAPLACE
EQUATION WITH CONVECTION TERM VIA L∞ ESTIMATES
ZHONGQING LI, BAISHENG YAN, WENJIE GAO
Abstract. This article is devoted to the study of the existence of weak solutions to an initial and boundary value problem for a parabolic p(x)-Laplace
equation with convection term. Using the De Giorgi iteration technique, the
authors establish the critical a priori L∞ -estimates and thus prove the existence of weak solutions.
1. Introduction
In this article, we consider the initial and boundary value problem for parabolic
p(x)-Laplace equation
→
−
∂u
− div |∇u|p(x)−2 ∇u = B(x, t)|∇u|p(x) − div F (x, t), (x, t) ∈ QT ,
∂t
(1.1)
u(x, t) = 0, (x, t) ∈ ΓT ,
u(x, 0) = u0 (x) ∈ L∞ (Ω),
x ∈ Ω.
Here, Ω ⊂ RN is a bounded domain with smooth boundary ∂Ω, QT = Ω × (0, T ),
→
−
ΓT = ∂Ω × (0, T ), T > 0 is finite, and p(x), B(x, t), F (x, t) are given quantities
satisfying conditions to be specified later.
Recently, partial differential equations involving variable exponents, such as the
p(x)-Laplace equation in (1.1), have been extensively investigated, owing to their
physical importance and powerful application. The mathematical model of Problem
(1.1) originates from heat and mass transfer in nonhomogeneous media and nonNewtonian fluids with thermo-convective effects [2]. Equations of this type also
appear in the study of digital image recovery [4] and electrorheological fluids [16].
It describes the evolution diffusion and filtration process. In particular, the models
like (1.1) with variable exponent provide a good mathematical interpretation for
the mechanical properties of certain viscous electrorheological fluids characterized
by their abilities to undergo significant changes when an electric field is applied.
We focus on mathematical analysis concerning the existence of solutions to Problem (1.1). Similar problems with constant exponents or L1 data have been studied
by many authors; see, e.g., [3, 5, 13, 14, 15, 18, 21, 24]. To study our problem,
2000 Mathematics Subject Classification. 35K65, 35K55, 46E35.
Key words and phrases. Parabolic p(x)-Laplace equation; convection term;
De Giorgi iteration; L∞ estimates.
c
2015
Texas State University - San Marcos.
Submitted November 28, 2014. Published February 17, 2015.
1
2
Z. LI, B. YAN, W. GAO
EJDE-2015/46
we encounter several difficulties arising from the variable exponents. To deal with
(1.1), one must face the typical difficulty of how to define the solution space to (1.1).
When p(x) = p is a constant, it is well known that Lp (0, T ; W01,p (Ω)) can be taken as
the solution space. However, in the nonconstant case and p− = inf p(x) > 1, if the
−
1,p(x)
1,p(x)
solution space is defined to be Lp(x) (0, T ; W0
(Ω)), or Lp (0, T ; W0
(Ω)),
etc., then it leads to an unfavorable fact that the p(x)-Laplace operator is not
bounded and not continuous from this space into its dual. To conquer this difficulty, we adopt the appropriate solution space V as defined below, which helps
us to define a weak solution to (1.1). However, other difficulties arise from it at
the same time. On one hand, one must verify the chain rule in the variable exponent space, as given in Lemma 2.2 with its proof in the Appendix, even if this
is an obvious fact in the case when p is a constant [5, 13]. On the other hand,
we will get the existence result for Problem (1.1) through a limit process in which
Simon’s compactness theorem [17] plays a crucial role. Nevertheless, the solution
space V prevents from directly employing the theorem. We take into account the
properties associated with V and surmount this difficulty. There are other differences between the variable exponent case and the constant exponent case. Some
important properties and inequalities are no longer valid. For example, the variable
exponent spaces are not translation invariant, Young’s inequality with convolution
1,p(x)
kf ∗ gkp(·) ≤ Ckf kp(·) kgk1 holds if and only if p is constant, and for u ∈ W0
(Ω),
R
R
p(x)
p(x)
|u|
dx
≤
C
|∇u|
dx
is
not
valid
for
the
variable
exponent
p,
etc.;
we
refer
Ω
Ω
to monograph [7] for details and more references.
To define an appropriate solution space for Problem (1.1), we make the following
hypotheses on the quantities appearing in (1.1).
(H1) p ∈ C(Ω), and p+ := maxΩ p(x), p− := minΩ p(x) satisfy 1 < p− ≤ p+ <
+∞; furthermore, there exists a positive constant C such that the following
log-Hölder continuous condition holds:
|p(x) − p(y)| ≤
−C
log |x − y|
for every x, y ∈ Ω satisfying |x − y| ≤
1
.
2
(1.2)
(H2) B ∈ L∞ (QT ) satisfies 0 ≤ B(x, t) ≤ b, where b > 0 is a constant, and
−
→
−
→
− −0
F is a vector field satisfying | F |(p ) ∈ Lr (QT ), where (p− )0 = p−p −1 and
N
−
→
−
→
−
0
− 0
0
. Hence, F ∈ Lp (x) (QT ) as | F | ∈ L(p ) (QT ) ,→ Lp (x) (QT );
r > Np+p
−
see the relevant definitions below.
We remark that, when p is a constant, it is well known that W01,p (Ω) (the closure of C0∞ (Ω) in W 1,p (Ω)) is identical to H01,p (Ω) := {f ∈ Lp (Ω) : |∇f | ∈
Lp (Ω) with f |∂Ω = 0}. However, when p is a function, there exists an interesting Lavrentiev phenomenon [22], which shows that the above two space are not
equivalent. The log-Hölder continuous condition (1.2) above guarantees an important fact that C0∞ (Ω) is dense in W 1,p(x) (Ω) [23]. Under this condition, one can
1,p(x)
define variable Sobolev spaces with homogeneous boundary values, W0
(Ω), as
1,p(x)
∞
(Ω); moreover, the condition makes p(x)-Poincaré’s
the closure of C0 (Ω) in W
inequality hold [1, 10, 21].
We introduce the function space
−
1,p(x)
V = {v ∈ Lp (0, T ; W0
(Ω)) : |∇v| ∈ Lp(x) (QT )},
EJDE-2015/46
EXISTENCE OF SOLUTIONS
3
endowed with the norm kukV = |∇u|Lp(x) (QT ) , or the equivalent norm kukV =
|u|Lp− (0,T ;W 1,p(x) (Ω)) + |∇u|Lp(x) (QT ) ; the equivalence follows from p(x)-Poincaré’s
0
inequality. Then V is a separable and reflexive Banach space (see [3, 21]).
We now give the definition of weak solutions to Problem (1.1).
Definition 1.1. We say that u ∈ V ∩ L∞ (QT ) is a weak solution to (1.1), provided
−
that ut ∈ V ∗ + L1 (QT ), u(x, 0) = u0 (x) in Lp (Ω), and
Z TZ
Z T
|∇u|p(x)−2 ∇u · ∇φ dx dt
hut , φidt +
0
Ω
0
(1.3)
Z TZ
Z TZ
→
−
p(x)
=
B|∇u|
φ dx dt +
∇φ · F dx dt
0
Ω
0
Ω
holds for every φ(x, t) ∈ V ∩ L∞ (QT ). Here, with ut = α(1) + α(2) ∈ V ∗ + L1 (QT ),
it is understood that
Z T
Z TZ
(1)
∗
∗
1
∞
hut , φidt := hut , φiV +L (QT ),V ∩L (QT ) = hα , φiV ,V +
α(2) φ dx dt.
0
0
Ω
When p(x) = p is a constant, sup-/sub-solution method is powerful and direct to
the existence results (see [13]). Nevertheless, it is not suitable to our problem because, due to the complicated nonlinearities of p(x)-Laplace, it may be quite difficult
to construct a supsolution u and a subsolution u in V which simultaneously satisfy
u ≤ u. Roughly speaking, in Equation (1.1), the growth power of |∇u|p(x)−2 ∇u at
the left-hand side of (1.1) is less than that of the convection term |∇u|p(x) at the
right-hand side, which leads us not to directly utilizing pseudo-monotone operator
method [12]. Instead, to obtain the existence of weak solutions to Problem (1.1), we
will employ the L∞ estimate method and get the solution through a limit process
to the approximate equations. We carry out the De Giorgi iteration, different from
the classical constant exponent case (see [24, 5, 14] and the excellent and elegant
argument therein), in the setting of variable exponent. We first give a general form
of [5, Theorem 5.1] or [24, Lemma 1], as stated in (2.5), by which we obtain the
L∞ regularity under the classification when p− ≥ 2 and when 1 < p− < 2, other
than the classification appeared in [24]. It should be remarked that, we employ the
infimum of p(x), which facilitates this iteration, however, on the other side of the
coin, it makes the iteration process more technical and complexity. By the way,
our result in Theorem 2.3 shows an interesting phenomenon: the uniformly L∞
bound of u can depend on p− other than p(x) itself as in the constant exponent
case [24]. In the limit process, the properties of solution space V and its related
variable exponent space will be frequently used, which is one of the features in the
equation with variable exponent.
The plan of this paper is as follows. In section 2, we apply the De Giorgi iteration
to Problem (1.1) to obtain a uniform bound for the bounded weak solution u ∈ V ;
this a priori L∞ -assumption is crucial for such a uniform bound, as in [5, 14]. In
section 3, we construct an approximation equation to Problem (1.1). Based on
the uniform bound of un , we obtain the strong convergence of un in the solution
space V , by virtue of which we establish the existence of solutions. Section 4 is an
Appendix in which we give some brief proofs to some lemmas in the paper.
To conclude this section, we recall some preliminary results on the Lebesgue and
Sobolev spaces with variable exponents; for more details, see [9, 10] or monograph
[7, 16]. Let p be a continuous function defined in Ω, p(x) > 1, for any x ∈ Ω.
4
Z. LI, B. YAN, W. GAO
EJDE-2015/46
1. The space
Lp(x) (Ω) := u : u is measurable in Ω and
Z
|u(x)|p(x) dx < ∞ .
Ω
This space is equipped with the Luxemburg’s norm
Z
u(x) p(x)
|u|Lp(x) (Ω) := inf λ > 0 :
|
|
dx ≤ 1 .
λ
Ω
The space Lp(x) (Ω), | · |Lp(x) (Ω) is a separable, uniformly convex Banach space.
2. The space
n
o
W 1,p(x) (Ω) := u ∈ Lp(x) (Ω) : |∇u| ∈ Lp(x) (Ω) ,
endowed with the norm
|u|W 1,p(x) (Ω) := |∇u|Lp(x) (Ω) + |u|Lp(x) (Ω) .
1,p(x)
We denote by W0
(Ω) the closure of C0∞ (Ω) in W 1,p(x) (Ω). In fact, the norm
1,p(x)
|∇u|Lp(x) (Ω) and |u|W 1,p(x) (Ω) are equivalent norms in W0
(Ω). W 1,p(x) (Ω) and
1,p(x)
W0
(Ω) are separable and reflexive Banach spaces.
3. Frequently used relationships for the estimates.
Z
−
+
−
+
min |u|pLp(x) (Ω) , |u|pLp(x) (Ω) ≤
|u(x)|p(x) dx ≤ max |u|pLp(x) (Ω) , |u|pLp(x) (Ω) .
Ω
Consequently,
Z
|uk − u|Lp(x) (Ω) → 0 ⇐⇒
|uk − u|p(x) dx → 0.
Ω
0
4. p(x)-Hölder’s inequality: For any u ∈ Lp(x) (Ω) and v ∈ Lp (x) (Ω), with
1
1
p(x) + p0 (x) = 1, we have
Z
1
1 uv dx ≤ − + 0 − |u|Lp(x) (Ω) |v|Lp0 (x) (Ω) ≤ 2|u|Lp(x) (Ω) |v|Lp0 (x) (Ω) .
p
(p )
Ω
5. Embedding relationships: If p1 and p2 are in C(Ω), and 1 ≤ p1 (x) ≤ p2 (x),
for any x ∈ Ω, then there exists a positive constant Cp1 (x),p2 (x) such that
|u|Lp1 (x) (Ω) ≤ Cp1 (x),p2 (x) |u|Lp2 (x) (Ω) .
i.e. the embedding Lp2 (x) (Ω) ,→ Lp1 (x) (Ω) is continuous. If q ∈ C(Ω) and 1 ≤
1,p(x)
q(x) < p∗ (x), for any x ∈ Ω, then the embedding W0
(Ω) ,→ Lq(x) (Ω) is continuous and compact, where
(
N p(x)
, p(x) < N,
p∗ (x) := N −p(x)
+∞,
p(x) ≥ N.
6. p(x)-Poincaré’s inequality: Under the condition (1.2), there exists a positive
constant Cp such that
|u|Lp(x) (Ω) ≤ Cp |∇u|Lp(x) (Ω) ,
1,p(x)
for all u ∈ W0
(Ω).
EJDE-2015/46
EXISTENCE OF SOLUTIONS
5
2. A priori bounds
First of all, we give some technical lemmas frequently used in the process of De
Giorgi iteration. In particular, (2.5) can be seen as a general form of [5, Theorem
5.1] or [24, Lemma 1]. Their proofs will be given in the Appendix for the convenience
of the readers.
Lemma 2.1. Assume that a, b, λ are positive constants, with λ ≥
(
eλs − 1,
s ≥ 0,
ϕ(s) =
−λs
−e
+ 1, s ≤ 0.
1
2
+ ab . Define
(2.1)
Then the following properties hold:
(1) For all s ∈ R,
a λ|s|
e .
(2.2)
2
(2) There exist constants d ≥ 0 and M > 1 such that, for all s ≥ d,
s p−
s p−
ϕ0 (s) ≤ λM ϕ −
, ϕ(s) ≤ M ϕ −
.
(2.3)
p
p
Rs
(3) Let Φ(s) = 0 ϕ(σ)dσ. If p− ≥ 2, then there exists a positive constant c∗
such that
s p−
Φ(s) ≥ c∗ ϕ −
, ∀s ≥ 0;
(2.4)
p
if 1 < p− < 2, then there exist d ≥ 0 and c∗ = c∗ (p− , d) such that
s p−
, ∀s ≥ d,
Φ(s) ≥ c∗ ϕ −
p
(2.5)
s 2
Φ(s) ≥ c∗ ϕ − , ∀0 ≤ s ≤ d.
p
|ϕ(s)| ≥ λ|s|,
aϕ0 (s) − b|ϕ(s)| ≥
Lemma 2.2. Assume that function π : R → RR is piecewise C 1 with π(0) = 0
s
and π 0 = 0 outside a compact set. Let Π(s) = 0 π(σ)dσ. If u ∈ V with ut ∈
V ∗ + L1 (QT ), then
Z T
Z
Z
hut , π(u)idt = hut , π(u)iV ∗ +L1 (QT ),V ∩L∞ (QT ) =
Π(u(T ))dx −
Π(u(0))dx.
0
Ω
Ω
(2.6)
Using the lemmas above, we begin the De Giorgi iteration to get the a priori L∞
estimate.
Theorem 2.3. Let u ∈ L∞ (QT ) ∩ V be a weak solution to Problem (1.1). Then
kukL∞ (QT ) ≤ ku0 kL∞ (Ω) + C,
→
− −0
where C is a constant depending on p− , N, T, r, b, Ω, ||| F |(p ) kLr (QT ) , but independent of u.
Proof. Let k be a real number such that k > ku0 kL∞ (Ω) and let ϕ be the function
defined in (2.1) with constant λ ≥ 21 + 2b, where b > 0 is the constant in Hypothesis
(H2). (We shall use (2.2) with a = 1 and a = 1/2 below.) Define


u − k, if u > k,
Gk (u) = u + k, if u < −k,


0,
if |u| ≤ k.
6
Z. LI, B. YAN, W. GAO
EJDE-2015/46
Note that u ∈ L∞ (QT ) ∩ V ; so does ϕ(Gk (u)). Then, for each τ ∈ [0, T ], one may
choose v = ϕ(Gk (u))χ[0,τ ] as a test function in (1.3) (where χA is the characteristic
function on the set A). Noting that ∇v = χ[0,τ ] χ{|u| > k}ϕ0 (Gk (u))∇u, we have
Z τ
Z τZ
hut , ϕ(Gk (u))idt +
|∇u|p(x) ϕ0 (Gk (u))χ{|u| > k} dx dt
0
0
Ω
Z τZ
Z τZ
→
−
p(x)
=
B|∇u|
ϕ(Gk (u)) dx dt +
χ{|u| > k}ϕ0 (Gk (u))∇u · F dx dt.
0
Ω
0
Ω
(2.7)
Denote Ak (t) = {x ∈ Ω : |u(x, t)| > k}. In what follows, we write ϕ = ϕ(Gk (u))
and ϕ0 = ϕ0 (Gk (u)) for simplicity. Thanks to the choice of k, one has
Z τ
Z
Z
hut , ϕ(Gk (u))idt =
Φ(Gk (u))(τ )dx −
Φ(Gk (u0 ))dx
0
Ω
ZΩ
Z
=
Φ(Gk (u))(τ )dx −
Φ(Gk (u0 ))dx
(2.8)
A (τ )
Ak (0)
Z k
=
Φ(Gk (u))(τ )dx.
Ak (τ )
From Young’s inequality with , it follows that
Z τZ
→
−
ϕ0 ∇u · F dx dt
0
Ak (t)
τ Z
Z
−
|∇u|p ϕ0 dx dt + C()
≤
0
τ
Z
Ak (t)
→
− −0
| F |(p ) ϕ0 dx dt.
Z
0
(2.9)
Ak (t)
Substituting (2.8) and (2.9) in (2.7) yields
Z
Z τZ
Φ(Gk (u))(τ )dx +
|∇u|p(x) (ϕ0 − B|ϕ|) dx dt
Ak (τ )
0
τ
Z
Z
|∇u|p ϕ0 dx dt + C()
≤
0
Ak (t)
−
Ak (t)
0
τ
Z
0
0
ϕ − b|ϕ| ≥ 21 eλ|Gk (u)| >
and choosing = 12 , we
Z τZ
Note that ϕ − B|ϕ| ≥
−
|∇u|p(x) ≥ |∇u|p − 1
Z
Φ(Gk (u))(τ )dx +
Ak (τ )
≤C
0
Z
τ
Ak (t)
Z
≤
0
0 by (2.2) (with a = 1). By utilizing
get from (2.10) that
Ak (t)
→
− −0
| F |(p ) ϕ0 dx dt +
Z
Z
τ
−
1 0
ϕ − B|ϕ| dx dt
2
Z
0
(2.10)
Ak (t)
|∇u|p
0
τ
Z
→
− −0
| F |(p ) ϕ0 dx dt.
Z
(ϕ0 − B|ϕ|) dx dt
(2.11)
Ak (t)
→
− −0
C| F |(p ) + 1 ϕ0 dx dt.
Ak (t)
1
1 0
1 0
1 λ|Gk (u)|
Using (2.2)
> 0. Denoting
witha = 2 , we have 2 ϕ −B|ϕ| ≥ 2 ϕ −b|ϕ| ≥ 4 e
wk = ϕ
Z τZ
0
|Gk (u)|
p−
, we proceed to estimate (2.11),
Z Z
1
|G (u)|
−
−
1 τ
λ k
ϕ0 − B|ϕ| dx dt ≥
|e p− ∇u|p dx dt
|∇u|p
2
4 0 Ak (t)
Ak (t)
(2.12)
Z Z
−
1 1 p− τ
|∇wk |p dx dt.
≥
4 λ
0
Ak (t)
EJDE-2015/46
EXISTENCE OF SOLUTIONS
7
By definition, Ak (t) \ Ak+d (t) = t{x ∈ Ω : k < |u(x, t)| ≤ k + d}; hence 0 <
|Gk (u)| ≤ d and ϕ0 (Gk (u)) = λeλ|Gk (u)| ≤ λeλd on Ak (t) \ Ak+d (t). So, from (2.3),
it follows that
Z τZ
→
− −0
C| F |(p ) + 1 ϕ0 dx dt
0
Ak (t)
Z τ
Z
≤ λM
0
τ
Z
−
→
− −0
C| F |(p ) + 1 |wk |p dx dt
Ak+d (t)
Z
+
0
Ak (t)\Ak+d (t)
τ Z
Z
≤ λM
h|wk |
0
(2.13)
→
− −0
C| F |(p ) + 1 ϕ0 dx dt
p−
dx dt + λe
τ
Z
λd
Z
h dx dt,
Ak+d (t)
0
Ak (t)\Ak+d (t)
→
− −0
where h = C| F |(p ) + 1. Putting (2.11), (2.12) and (2.13) together, we deduce
Z
Z Z
−
1 1 p− τ
Φ(Gk (u))(τ )dx +
|∇wk |p dx dt
4
λ
Ak (τ )
0
Ak (t)
Z τZ
Z τZ
(2.14)
−
≤ λM
h|wk |p dx dt + λeλd
h dx dt.
0
Ak+d (t)
0
Ak (t)\Ak+d (t)
Case 1. p− ≥ 2. In this case, by (2.4), one has
Z
Z
Φ(Gk (u))(τ )dx ≥ c∗
|wk |p dx.
Ak (τ )
−
(2.15)
Ak (τ )
Substituting (2.15) in (2.14) and taking the supremum for τ ∈ [0, t1 ], with t1 ≤ T
to be determined later, we have
Z
Z Z
−
1 1 p− t1
p−
∗
|wk | dx +
c sup
|∇wk |p dx dt
4 λ
τ ∈[0,t1 ] Ak (τ )
0
Ak (t)
(2.16)
Z t1 Z
Z t1 Z
p−
λd
≤ λM
h|wk | dx dt + λe
h dx dt.
0
Ak (t)
0
Ak (t)\Ak+d (t)
By the embedding inequality (see [6, 11]), we have
N−
Z t1 Z
− N +p−
N +p
|wk |p N dx dt
0
≤γ
Ak (t)
Z
p−
|wk |
sup
τ ∈[0,t1 ]
t1
Z
(2.17)
Z
p−
|∇wk |
dx +
Ak (τ )
0
dx dt ,
Ak (t)
where γ is a constant depending on N, p− , but independent of t1 ≤ T . Hence, from
(2.16), it follows that
Z t1 Z
N−
− N +p−
N +p
Jkt1 :=
|wk |p N dx dt
0
≤C
Z
0
t1
Ak (t)
Z
Ak (t)
−
h|wk |p dx dt +
Z
0
t1
Z
Ak (t)\Ak+d (t)
h dx dt ,
8
Z. LI, B. YAN, W. GAO
EJDE-2015/46
where C is a constant independent of t1 . Consequently, by Hölder’s inequality
−
→
− −0
(thanks to the assumption | F |(p ) ∈ Lr (QT ) with r > Np+p
), we deduce
−
Jkt1 ≤ C
Z
0
−
p− N +p
N
N− Z
N +p
dx dt
t1
0
t1 Z
t1
Z
1/r Z
h dx dt
t1
r
Ak (t)
N +p−
p−
p− −
N +p
dx dt
Ak (t)
1− r1
µ(Ak (t))dt
0
−
p− N +p
N
|wk |
Ak (t)
+ CkhkLr (Qt1 )
Z
h
0
Ak (t)
Z
Z
Z
|wk |
0
+C
≤C
t1
Z
t1
N−
p− − 1
N +p
dx dt
khkLr (Qt1 ) t1 µ(Ω) N +p− r
1− r1
µ(Ak (t))dt
,
0
where µ(Ω) represents the Lebesgue measure of Ω. Choosing t1 small enough such
that
p−
CkhkLr (Qt1 ) (t1 µ(Ω)) N +p−
− r1
≤
1
2
(2.18)
and we obtain
Jkt1 ≤ CkhkLr (QT )
t1
Z
1− r1
µ(Ak (t))dt
.
(2.19)
0
For any l > k ≥ ku0 kL∞ (Ω) , using (2.2), we conclude that
t1
N−
λGk (u) p− N +p−
N +p
N
|
dx
dt
−
p
0
Ak (t)
Z Z
−
N−
p− N +p
λ p− t1
N +p
N
≥ −
|u| − k
dx dt
p
0
Ak (t)
N−
Z t1
−
λ p−
N +p
.
≥ −
(l − k)p
µ(Al (t))dt
p
0
Jkt1 ≥
Let ψk =
R t1
0
Z
Z
|
(2.20)
µ(Ak (t))dt. It follows from (2.19) and (2.20) that
C
ψl ≤
(l − k)
(1− r1 ) N +p
N
p− (N +p− )
N
ψk
−
.
(2.21)
Case 2. 1 < p− < 2. In this case, from (2.5) (it should be remarked that the
constant d in (2.3) and (2.5) could be the same if we choose d suitably large), we
have
Z
Ak (τ )
Φ(Gk (u))(τ )dx ≥ c∗
Z
Ak+d (τ )
−
|wk |p dx + c∗
Z
Ak (τ )\Ak+d (τ )
|wk |2 dx. (2.22)
EJDE-2015/46
EXISTENCE OF SOLUTIONS
9
Substituting (2.22) into (2.14) and taking the supremum for τ ∈ [0, t1 ], where
t1 ≤ T to be chosen later, we derive
Z
Z Z
−
−
1 1 p− t1
c∗ sup
|wk |p dx +
|∇wk |p dx dt
4 λ
τ ∈[0,t1 ] Ak+d (τ )
0
Ak+d (t)
Z
Z Z
−
1 1 p− t1
∗
2
+ c sup
|wk | dx +
|∇wk |p dx dt
4
λ
τ ∈[0,t1 ] Ak (τ )\Ak+d (τ )
0
Ak (t)\Ak+d (t)
Z t1 Z
Z t1 Z
−
h dx dt.
≤ λM
h|wk |p dx dt + λeλd
0
0
Ak+d (t)
Ak (t)\Ak+d (t)
(2.23)
Again, recall the following embedding estimates [6, 11]:
Z t1 Z
− N +p−
|wk |p N dx dt
Ak+d (t)
0
≤γ
p− N +p
N
−
Z
|wk |
sup
τ ∈[0,t1 ]
t1
Z
p−
Z
|∇wk |
dx +
Ak+d (τ )
0
p−
1+ pN−
dx dt
,
Ak+d (t)
(2.24)
Z
t1
Z
0
|wk |p
− N +2
N
dx dt
Ak (t)\Ak+d (t)
≤γ
+2
p− NN
Z
τ ∈[0,t1 ]
t1
Z
|wk |2 dx
sup
Z
|∇wk |
+
0
(2.25)
Ak (τ )\Ak+d (τ )
1+ pN−
dx dt
.
p−
Ak (t)\Ak+d (t)
Combining (2.24), (2.25) with (2.23), we obtain
N−
Z t1 Z
− N +p−
N +p
(1)
Jkt :=
|wk |p N dx dt
1
0
+
Ak+d (t)
t1
Z
0
t1
Z
Z
|wk |p
Z
N−
N +p
dx dt
Ak (t)\Ak+d (t)
Z
−
h|wk |p dx dt + C
≤C
0
− N +2
N
Ak+d (t)
t1 Z
+C
Z
t1
Z
0
−
h|wk |p dx dt
Ak (t)\Ak+d (t)
h dx dt := (E1) + (E2) + (E3).
0
Ak (t)
We estimate (E1) as follows.
(E1)
≤C
Z
t1
Z
0
p− N +p
N
−
|wk |
0
≤C
Z
Ak+d (t)
t1
Z
Ak+d (t)
N− Z
N +p
dx dt
0
|wk |p
− N +p−
N
t1
Z
h
N +p−
p−
Ak+d (t)
N−
p−
−1
N +p
dx dt
khkLr (Qt1 ) (t1 µ(Ω)) N +p− r .
Using Hölder’s inequality and Young’s inequality with , we have
(E2)
p− −
N +p
dx dt
10
Z. LI, B. YAN, W. GAO
≤C
Z
1
≤
2
Z
Z
0
t1
|wk |p
NN+2 Z
dx dt
t1
Z
h
0
|wk |
+2
p− NN
N +2
2
N2+2
dx dt
Ak (t)\Ak+d (t)
N−
N +p
dx dt
Ak (t)\Ak+d (t)
t1
Z
Z
− N +2
N
Ak (t)\Ak+d (t)
Z
0
+C
1
≤
2
t1
EJDE-2015/46
Z
h
0
t1 Z
0
2−
2−p
dx dt
Ak (t)\Ak+d (t)
|wk |p
− N +2
N
N−
N +p
dx dt
Ak (t)\Ak+d (t)
Z t1
2 − (1− N2r+2 )
2−p
µ(Ak (t))dt
.
N +2
2−p−
Lr (Qt1 )
+ Ckhk
N +2
2
0
For (E3), we have
(E3) ≤ CkhkLr (Qt1 )
Z
t1
1− r1
µ(Ak (t))dt
.
0
Now select t1 ∈ (0, (µ(Ω))−1 ] sufficiently small so that
p−
CkhkLr (Qt1 ) (t1 µ(Ω)) N +p−
From the above estimates, we have
Z
(1)
Jkt ≤ CkhkLr (Qt1 )
1
t1
− r1
≤
1
.
2
1− r1
µ(Ak (t))dt
0
N +2
2−p−
Lr (Qt1 )
+ Ckhk
Z
(2.26)
t1
2 − (1− N2r+2 )
2−p
µ(Ak (t))dt
.
(2.27)
0
N +p−
p− ,
2
after a straightforward computation, we have 2−p
Noticing that r >
− (1 −
N +2
1
2r ) > 1 − r . Meanwhile, the choice of t1 ensures ψk ≤ t1 µ(Ω) ≤ 1. As a result,
(2.27) becomes
Z t1
1− r1
(1)
Jkt ≤ C
µ(Ak (t))dt
.
(2.28)
1
0
For any l > k ≥ ku0 kL∞ (Ω) , using (2.2), we deduce that
Z t1 Z
N−
λGk (u) p− N +p−
N +p
(1)
N
Jkt ≥
|
dx
dt
|
−
1
p
0
Ak+d (t)
N−
Z t1 Z
λGk (u) p− N +2
N +p
N
|
dx
dt
+
|
−
p
0
Ak (t)\Ak+d (t)
Z t1 Z
−
N−
−
p− N +p
λ p
N +p
N
≥ −
|u| − k
dx dt
p
0
Ak+d (t)
Z t1 Z
N−
+2 p− NN+2
λ p− NN+p
N +p
−
+ −
|u| − k
dx dt
p
0
Ak (t)\Ak+d (t)
Z t1
N−
λ p−
N +p
p−
≥ −
(l − k)
µ (Al (t) ∩ Ak+d (t)) dt
p
0
Z t1
N−
+2
λ p− NN+p
p− N +2−
N +p
−
N
+p
+ −
µ (Al (t)\Ak+d (t)) dt
(l − k)
.
p
0
EJDE-2015/46
EXISTENCE OF SOLUTIONS
11
In fact, we have
−
N +p
N
(1)
Jkt
≥
Z t1
−
−
λ p− N +p
p− N +p
N
N
(l
−
k)
µ (Al (t) ∩ Ak+d (t)) dt
1
p−
0
(2.29)
Z t1
+2
λ p− NN+2
p− NN
µ (Al (t)\Ak+d (t)) dt.
+ −
(l − k)
p
0
Rt
Consequently, combining (2.29) and (2.28), with ψk = 0 1 µ(Ak (t))dt, we have
again
−
C
(1− r1 ) N +p
N
ψl ≤
ψ
.
(2.30)
−
−
−
k
p (N +2)
p (N +p )
N
, (l − k) N
min (l − k)
−
guarantees
Now we have proved (2.30) and (2.21). Our hypothesis r > Np+p
−
−
1 N +p
1− r
> 1. Therefore, thanks to the iteration lemma in [24], we eventually
N
obtain that ψ(ku0 kL∞ (Ω) +D) = 0, where D > 0 is a constant depending only on
→
− −0
p− , N, t1 , r, b, Ω, k| F |(p ) kLr (Qt1 ) . This proves that, for a fixed λ validating Lemma
2.1,
ku(x, t)kL∞ (Qt1 ) ≤ ku0 kL∞ (Ω) + D.
(2.31)
Finally, partition the time interval [0, T ] into finite subintervals [0, t1 ], [t1 , t2 ] · · ·
[tn−1 , T ] such that the conditions similar to those in (2.18) and (2.26) are available
for each subinterval [ti , ti+1 ]; then, using the same method, we deduce an inequality
of the form (2.31). Eventually, we conclude that ku(x, t)kL∞ (QT ) ≤ ku0 kL∞ (Ω) + C,
→
− −0
where the constant C depends only on p− , N, T, r, b, Ω, k| F |(p ) kLr (QT ) .
3. Application to the existence of solutions to (1.1)
With the L∞ -estimate obtained above, we can prove the existence of solutions
to Problem (1.1). First, we recall a lemma from [13], which plays an important role
in our estimates.
2
2
b
3.1. Let θ(s) = seηs , s ∈ R, where η ≥ 4a
2 is fixed, and let Θ(s) =
RLemma
s
θ(τ
)dτ
.
Then
θ(0)
=
0
and
0
a
Θ(s) ≥ 0, aθ0 (s) − b|θ(s)| ≥ , ∀ s ∈ R.
(3.1)
2
We are now in a position to prove the existence of solutions to (1.1) based on
the L∞ estimate.
Theorem 3.2. Under the hypotheses (H1) and (H2), there exists a solution u ∈
L∞ (QT ) ∩ V to (1.1).
Proof. Step 1: The approximation equation. We introduce the following
approximation equation of Problem (1.1).
→
−
∂un
− div |∇un |p(x)−2 ∇un = B(x, t) min{|∇un |p(x) , n} − div F (x, t),
∂t
(x, t) ∈ QT ,
(3.2)
un (x, t) = 0,
(x, t) ∈ ΓT ,
un (x, 0) = u0 (x) ∈ L∞ (Ω), x ∈ Ω.
For each fixed n ∈ N, since min |∇un |p(x) , n is bounded, the existence of a weak
solution un ∈ L∞ ∩ V to (3.2) follows from the standard methods (for instance, the
12
Z. LI, B. YAN, W. GAO
EJDE-2015/46
pseudo-monotonicity operator theory in [12, 10, 20], or the difference and variation
methods in [21]).
We write the term B(x, t) min{|∇un |p(x) , n} in (3.2) as Bn (x, t)|∇un |p(x) , with
Bn (x, t) defined by

0,
if |∇un (x, t)| = 0,
Bn (x, t) =
p(x)
min{|∇u
(x,t)|
,n}
n
B(x, t)
, if |∇un (x, t)| =
6 0.
|∇un (x,t)|p(x)
Then Bn ∈ L∞ (QT ) satisfies 0 ≤ Bn (x, t) ≤ B(x, t) ≤ b. Hence, by Theorem 2.3,
we have the uniform bound
kun (x, t)kL∞ (QT ) ≤ ku0 kL∞ (Ω) + C,
(3.3)
→
−
− 0
where C depends only on p− , N, T, r, b, Ω, k| F |(p ) kLr (QT ) and it is independent
of n. Our goal is to show that a subsequence of the approximate solution sequence
{un } converges to a measurable function u, which coincides with a weak solution
of Problem (1.1).
−
1,p(x)
Step 2: The weak convergence un * u in Lp (0, T ; W0
(Ω)). Choosing
θ(un ) as a testing function in (3.2), we have
ZZ
Z T
∂un
, θ(un )idt +
|∇un |p(x) θ0 (un ) dx dt
h
∂t
QT
0
(3.4)
ZZ
ZZ
→
−
p(x)
0
=
B min{|∇un |
, n}θ(un ) dx dt +
θ (un )∇un · F dx dt.
QT
QT
RT
n
Lemma 2.2 yields 0 h ∂u
∂t , θ(un )idt = Ω [Θ(un (T )) − Θ(u0 )] dx. Using Young’s
inequality with in the last term of the right-hand side, (3.4) becomes
Z
ZZ
Θ(un (T ))dx +
|∇un |p(x) θ0 (un ) dx dt
Ω
QT
Z
ZZ
≤
Θ(u0 )dx +
B|∇un |p(x) |θ(un )| dx dt
Ω
QT
ZZ
ZZ
1
→
− 0
p(x) 0
− p(x)−1 | F |p (x) θ0 (un ) dx dt.
|∇un |
θ (un ) dx dt +
+
QT
R
QT
Taking = 1/2, we rewrite the above inequality as
Z
ZZ 1 0
Θ(un (T ))dx +
θ (un ) − B|θ(un )| |∇un |p(x) dx dt
Ω
QT 2
− −1 Z Z
Z
p −1
→
− 0
1
≤
Θ(u0 )dx +
| F |p (x) θ0 (un ) dx dt.
2
Ω
QT
(3.5)
With the aid of (3.1) in Lemma 3.1 (with a = 12 , and 12 θ0 (un )−B|θ(un )| ≥ 12 θ0 (un )−
b|θ(un )| ≥ 14 ), we deduce that
− −1 Z Z
ZZ
Z
p −1
→
− 0
1
1
|∇un |p(x) dx dt ≤
Θ(u0 )dx +
| F |p (x) θ0 (un ) dx dt.
4
2
QT
Ω
QT
(3.6)
Since un is uniformly bounded with respect to n and u0 ∈ L∞ (Ω), it follows that
ZZ
→
−
|∇un |p(x) dx dt ≤ C | F |Lp0 (x) (QT ) , ku0 kL∞ (Ω) , sup kun kL∞ (QT ) .
(3.7)
QT
n
EJDE-2015/46
EXISTENCE OF SOLUTIONS
13
This implies that un is uniformly bounded in V . By the way, obviously, the following
inequality holds
−
|un |p p−
1,p(x)
L (0,T ;W0
(Ω))
Z T
−
|∇un |pLp(x) (Ω) dt
=
0
≤ max
n Z Z
p(x)
|∇un |
ZZ
p−
−
1− p+
p+
p
dx dt
T
,
QT
o
|∇un |p(x) dx dt ,
QT
which implies
→
−
|un |Lp− (0,T ;W 1,p(x) (Ω)) ≤ C | F |Lp0 (x) (QT ) , ku0 kL∞ (Ω) , sup kun kL∞ (QT ) , p− , p+ , T .
0
n
(3.8)
−
1,p(x)
Therefore, un is bounded in the space L∞ (QT ) ∩ Lp (0, T ; W0
(Ω)). We can
extract a subsequence of un , still denoted by un , such that un * u, weakly in
−
1,p(x)
Lp (0, T ; W0
(Ω)). Simultaneously, un * u, weakly* in L∞ (QT ).
−
Step 3: The strong convergence un → u in Lp (0, T ; Lp(x) (Ω)). From (3.2),
we deduce that
→
−
∂un
= div |∇un |p(x)−2 ∇un − F + B min{|∇un |p(x) , n} ∈ V ∗ + L1 (QT ). (3.9)
∂t
For each v ∈ V , by the definition of the norm on V and p(x)-Hölder’s inequality,
we have
→
−
sup |hdiv |∇un |p(x)−2 ∇un − F , viV ∗ ,V |
kvkV ≤1
= sup kvkV ≤1
ZZ
→
−
−|∇un |p(x)−2 ∇un · ∇v + F · ∇v dx dt
QT
→
−
≤ sup 2||∇un |p(x)−2 ∇un |Lp0 (x) (QT ) |∇v|Lp(x) (QT ) + 2| F |Lp0 (x) (QT ) |∇v|Lp(x) (QT )
kvkV ≤1
≤ 2 max
n Z Z
|∇un |
p(x)
01 + Z Z
(p )
dx dt
,
QT
01 − o
(p )
|∇un |p(x) dx dt
QT
→
−
+ 2| F |Lp0 (x) (QT ) .
It follows from (3.7) that
− div |∇un |p(x)−2 ∇un − →
F V ∗ ≤ C,
(3.10)
where C is independent of n. Thanks to the embedding relationship
− 0
L(p
)
0
(0, T ; W −1,p (x) (Ω)) ,→ V ∗
,→ L(p
+ 0
)
0
0 −
0
(0, T ; W −1,p (x) (Ω)) = L(p ) (0, T ; W −1,p (x) (Ω)),
(3.11)
n
from (3.10), (3.7) and (3.9), we conclude that ∂u
∂t is bounded in the space
0 −
0
L(p ) (0, T ; W −1,p (x) (Ω)) + L1 (QT ).
For a fixed s such that s > N2 + 1, the following embedding relationships hold
1 s > N2 , we have H0s (Ω) ,→ L∞ (Ω), and then L1 (Ω) ,→ H −s (Ω); 2 s − 1 > N2 ,
14
Z. LI, B. YAN, W. GAO
EJDE-2015/46
0
one has H0s (Ω) ,→ W 1,p(x) (Ω), consequently, W −1,p (x) (Ω) ,→ H −s (Ω). As a result,
we have
∂un
−s
k
k 1
≤ C,
(3.12)
∂t L (0,T ;H (Ω))
1,p(x)
compact
where C is independent of n. Noticing that W0
(Ω) ,→ Lp(x) (Ω) ,→ H −s (Ω)
and by (3.8), we employ Simon’s compactness theorem in [17] to obtain that un → u,
−
strongly in Lp (0, T ; Lp(x) (Ω)).
Step 4: The convergence ∇un → ∇u a.e. in QT . From the strong convergence
obtained in Step 3, one may choose a subsequence of un , still denoted by un for
simplicity, such that un → u, a.e. in QT . We now use Egoroff’s theorem (recalling
µ(QT ) < +∞) to obtain, for fixed δ > 0, there exists a measurable closed subset
Eδ ⊂ QT such that
(1) µ(QT − Eδ ) ≤ δ;
(2) un ⇒ u uniformly on Eδ . It follows that |un − um | < k, for fixed k > 0,
and sufficiently large m, n.
Let ζ be a cut-off function satisfying ζ ∈ C0∞ (QT ); ζ = 1 on Eδ ; 0 ≤ ζ ≤ 1 on QT .
Introduce the following truncation function


if |s| < k,
s,
Tk (s) = k,
if s ≥ k,


−k, if s ≤ −k.
Subtracting Equations (3.2) for different parameters n and m, we have
∂(un − um )
− div |∇un |p(x)−2 ∇un − |∇um |p(x)−2 ∇um
∂t
= B min{|∇un |p(x) , n} − min{|∇um |p(x) , m} ,
(x, t) ∈ QT ,
(un − um )(x, t) = 0,
(un − um )(x, 0) = 0,
(3.13)
(x, t) ∈ ΓT ,
x ∈ Ω.
Since Tk is Lipschitz continuous, one may take ζTk (un − um ) as a test function in
(3.13); hence we have
Z T
∂(un − um )
h
, ζTk (un − um )idt
∂t
0
ZZ
+
|∇un |p(x)−2 ∇un − |∇um |p(x)−2 ∇um · ∇un − ∇um ζTk0 (un − um ) dx dt
Q
ZZ T
+
|∇un |p(x)−2 ∇un − |∇um |p(x)−2 ∇um · ∇ζTk (un − um ) dx dt
Q
ZZ T n
o
=
B min{|∇un |p(x) , n} − min |∇um |p(x) , m ζTk (un − um ) dx dt.
QT
(3.14)
Since ζ(x, 0) = ζ(x, T ), by Lemma 2.2, we handle the first term on the left-hand
side of (3.14) as follows,
Z T
Z Z T Z un −um
∂(un − um )
h
, ζTk (un − um )idt = −
ζt
Tk (s)dsdtdx.
∂t
0
Ω 0
0
EJDE-2015/46
EXISTENCE OF SOLUTIONS
15
Noticing that Tk is an odd function, |Tk (s)| ≤ k, we get
ZZ |∇un |p(x)−2 ∇un − |∇um |p(x)−2 ∇um · (∇un − ∇um ) ζTk0 (un − um ) dx dt
QT
ZZ
≤k
|ζt ||un − um | dx dt
Q
Z ZT
+k
||∇un |p(x)−2 ∇un − |∇um |p(x)−2 ∇um ||∇ζ| dx dt
QT
ZZ
+ bk
| min{|∇un |p(x) , n} − min{|∇um |p(x) , m}|ζ dx dt ≤ kC(δ).
QT
Noting that Tk0 ≥ 0, Tk0 (s) = 1 on |s| < k and that un converges uniformly on Eδ ,
we obtain
ZZ
|∇un |p(x)−2 ∇un − |∇um |p(x)−2 ∇um · (∇un − ∇um ) dx dt
E
Z δZ
=
|∇un |p(x)−2 ∇un − |∇um |p(x)−2 ∇um · (∇un − ∇um ) Tk0 (un − um ) dx dt
E
ZZ δ
≤
|∇un |p(x)−2 ∇un − |∇um |p(x)−2 ∇um · (∇un − ∇um ) ζTk0 (un − um ) dx dt.
QT
Hence, based on the above estimates, by (3.3), (3.7) and the arbitrariness of k, we
have
ZZ lim sup
|∇un |p(x)−2 ∇un − |∇um |p(x)−2 ∇um · (∇un − ∇um ) dx dt = 0.
n,m→+∞
Eδ
(3.15)
From (3.15) and using the method
in
[19,
24]
(or
the
method
to
be
used
in
Step
RR
p(x)
5 below), we may obtain that Eδ |∇un − ∇um |
dx dt → 0 (it is equivalent to
|∇un − ∇um |Lp(x) (Eδ ) → 0), which shows that {∇un }∞
n=1 is a Cauchy sequence in
(Lp(x) (Eδ ))N . Thus, we can extract a subsequence of un , still denoted by itself, such
−
that ∇un → α, strongly in (Lp (Eδ ))N . In step 3, we know that un → u, strongly
−
−
in Lp (0, T ; Lp(x) (Ω)), it is easy to say un → u, strongly in Lp (Eδ ). It follows
from above analysis that α = ∇u, i.e. ∇un → ∇u a.e. in Eδ . The arbitrariness of
δ indicates that ∇un → ∇u a.e. in QT .
RR
Step 5: The convergence QT |∇un − ∇u|p(x) dx dt → 0. For the function θ
defined in Lemma 3.1, it follows that θ(un − um ) ∈ L∞ (QT ) ∩ V since un , um ∈
L∞ (QT ) ∩ V . Therefore, θ(un − um ) can be taken as a test function in (3.13) to
yield that
Z T
∂(un − um )
, θ(un − um )idt
h
∂t
0
ZZ +
|∇un |p(x)−2 ∇un − |∇um |p(x)−2 ∇um · (∇un − ∇um ) θ0 (un − um ) dx dt
Q
ZZ T n
o
n
o
=
B min |∇un |p(x) , n − min |∇um |p(x) , m θ(un − um ) dx dt.
QT
(3.16)
16
Z. LI, B. YAN, W. GAO
EJDE-2015/46
Use (3.1) in Lemma 3.1 to estimate the first term on the left-hand side of (3.16) to
obtain
Z T
Z
∂(un − um )
h
, θ(un − um )idt =
Θ(un − um )(T )dx ≥ 0.
∂t
0
Ω
After a simple computation, the right-hand side of (3.16) can be estimated as
follows.
ZZ
n
o
n
o
B min |∇un |p(x) , n − min |∇um |p(x) , m θ(un − um ) dx dt
QT
ZZ |∇un |p(x) + |∇um |p(x) |θ(un − um )| dx dt
≤b
Q
ZZ T =b
|∇un |p(x)−2 ∇un · ∇um + |∇um |p(x)−2 ∇um · ∇un |θ(un − um )| dx dt
Q
Z ZT +b
|∇un |p(x)−2 ∇un − |∇um |p(x)−2 ∇um
QT
· (∇un − ∇um ) |θ(un − um )| dx dt.
Consequently, (3.16) can be estimated as
ZZ |∇un |p(x)−2 ∇un − |∇um |p(x)−2 ∇um
QT
· (∇un − ∇um ) [θ0 (un − um ) − b|θ(un − um )|] dx dt
ZZ ≤b
|∇un |p(x)−2 ∇un · ∇um + |∇um |p(x)−2 ∇um · ∇un |θ(un − um )| dx dt.
QT
(3.17)
With the help of (3.1) in Lemma 3.1 (with a = 1), since ∇un → ∇u a.e. in QT
(Step 4), we may utilize Fatou’s Lemma in (3.17) as m → +∞ to obtain that
ZZ E(n) :=
|∇un |p(x)−2 ∇un − |∇u|p(x)−2 ∇u · (∇un − ∇u) dx dt
Q
Z ZT |∇un |p(x)−2 ∇un · ∇u + |∇u|p(x)−2 ∇u · ∇un |θ(un − u)| dx dt
≤ 2b
QT
≤ 4b||∇un |p(x)−2 ∇un |Lp0 (x) (QT ) |θ(un − u)∇u|Lp(x) (QT )
+ 4b||∇u|p(x)−2 ∇uθ(un − u)|Lp0 (x) (QT ) |∇un |Lp(x) (QT )
n Z Z
01 ± o
(p )
≤ C max
|∇un |p(x) dx dt
QT
× max
1± o
p
|θ(un − u)|p(x) |∇u|p(x) dx dt
n Z Z
QT
+ C max
01 ± o
0
(p )
|θ(un − u)|p (x) |∇u|p(x) dx dt
n Z Z
QT
× max
1± o
p
|∇un |p(x) dx dt
n Z Z
QT
≤ C max
1± o
p
|θ(un − u)|p(x) |∇u|p(x) dx dt
n Z Z
QT
EJDE-2015/46
EXISTENCE OF SOLUTIONS
01 ± o
0
(p )
|θ(un − u)|p (x) |∇u|p(x) dx dt
.
n Z Z
+ C max
17
QT
In view of (3.7) and (3.3), θ(un −u) is uniformly bounded. The Lebesgue dominated
convergence theorem yields
ZZ |∇un |p(x)−2 ∇un − |∇u|p(x)−2 ∇u · (∇un − ∇u) dx dt → 0. (3.18)
E(n) :=
QT
We now estimate
ZZ
|∇un − ∇u|p(x) dx dt
QT
T
Z
Z
|∇un − ∇u|p(x) dx dt
=
(3.19)
{x∈Ω;p(x)≥2}
0
Z
T
Z
|∇un − ∇u|p(x) dx dt = I (1) + I (2) .
+
{x∈Ω;1<p(x)<2}
0
Applying the following basic inequality, for any y, z ∈ RN ,
(
+
22−p |y − z|p(x) ,
if p(x) ≥ 2,
p(x)−2
p(x)−2
|y|
y − |z|
z · (y − z) ≥
|y−z|2
−
(p − 1) (|y|+|z|)2−p(x) , if 1 < p(x) < 2.
we compute the two parts in (3.19):
Z TZ
1
I (1) ≤ 2−p+
|∇un |p(x)−2 ∇un
2
0
{x∈Ω;p(x)≥2}
p(x)−2
− |∇u|
∇u · (∇un − ∇u) dx dt
≤ 2p
+
−2
E(n) → 0.
Using p(x)-Hölder’s inequality, for I (2) , by (3.7) and (3.18), we have
Z TZ
|∇un − ∇u|p(x)
I (2) =
|∇un |
p(x)
0
{x∈Ω;1<p(x)<2} (|∇u | + |∇u|) 2 (2−p(x))
n
p(x)
2 (2−p(x))
+ |∇u|
dx dt
|∇un − ∇u|p(x)
≤ 2
2
|∇un |
p(x)
p(x) (Q )
T
2 (2−p(x)) L
(|∇un | + |∇u|)
p(x)
2 (2−p(x)) + |∇u|
2
2−p(x)
L
≤ 2 max
QT
× max
(QT )
|∇un − ∇u|2
n Z Z
n Z Z
(|∇un | + |∇u|)
2−p(x)
p(x)
(|∇un | + |∇u|)
QT
≤ C max
n
o
p±
1 p2±
2
(E(n))
p− − 1
p2± o
dx dt
±
2−p
o
2
dx dt
(3.20)
18
Z. LI, B. YAN, W. GAO
× max
n Z Z
|∇un |
p(x)
EJDE-2015/46
p(x)
+ |∇u|
±
2−p
o
2
dx dt
→ 0.
(3.21)
QT
Combining (3.19), (3.20) and (3.21), we arrive at
ZZ
|∇un − ∇u|p(x) dx dt → 0,
(3.22)
QT
which implies
|∇un − ∇u|Lp(x) (QT ) → 0;
(3.23)
that is, un → u strongly in the solution space V (simultaneously, un → u, strongly
−
1,p(x)
in Lp (0, T ; W0
(Ω))).
Step 6: Passing to the limit. It follows from (3.23), the property of Nemytskii
operator ([10, 20]) and generalized Lebesgue dominated convergence theorem that
0
N
|∇un |p(x)−2 ∇un → |∇u|p(x)−2 ∇u, strongly in Lp (x) (QT )
,
min |∇un |p(x) , n → |∇u|p(x) , strongly in L1 (QT ).
For every v ∈ V ,
h− div |∇un |p(x)−2 ∇un − |∇u|p(x)−2 ∇u , viV ∗ ,V ZZ |∇un |p(x)−2 ∇un − |∇u|p(x)−2 ∇u · ∇v dx dt
=
QT
≤ 2|∇un |p(x)−2 ∇un − |∇u|p(x)−2 ∇uLp0 (x) (Q ) |∇v|Lp(x) (QT ) .
T
It follows that
k − div |∇un |p(x)−2 ∇un − |∇u|p(x)−2 ∇u kV ∗
≤ 2||∇un |p(x)−2 ∇un − |∇u|p(x)−2 ∇u|Lp0 (x) (QT ) → 0.
Therefore, for the principal term in the approximate equation (3.2), we have
− div(|∇un |p(x)−2 ∇un ) → − div(|∇u|p(x)−2 ∇u),
strongly in V ∗ .
As a consequence, one has unt → ut , strongly in V ∗ + L1 (QT ).
On the other hand, as stated in Step 3, V ∗ + L1 (QT ) ,→ L1 (0, T ; H −s (Ω))
for s sufficiently large. Therefore, from (3.8) and (3.12), we deduce (according to W 1,1 (0, T ; H −s (Ω)) ,→ C([0, T ]; H −s (Ω)) in [8]) that un → u, strongly in
C([0, T ]; H −s (Ω)), from which un (x, 0) = u0 (x) makes a perfect sense.
Finally, since un (x, 0) → u(x, 0), strongly in H −s (Ω), it follows that u(x, 0) =
u0 (x). This proves that u ∈ V ∩ L∞ (QT ) is a weak solution to Problem (1.1). 4. Appendix
Proof of Lemma 2.1. Note that
(
0
ϕ (s) =
λeλs ,
λe−λs ,
s ≥ 0,
s ≤ 0.
(1) Obviously, |ϕ(s)| = eλ|s| − 1 ≥ λ|s|. Remember that λ ≥ 12 + ab . If s ≥ 0, then
a
aλeλs − b(eλs − 1) ≥ (aλ − b)eλs ≥ eλs .
2
EJDE-2015/46
EXISTENCE OF SOLUTIONS
19
If s ≤ 0, then
aλe−λs − b(e−λs − 1) ≥ (aλ − b)e−λs ≥
(2) The inequality λeλs ≤ λM [e
λ
s
p−
a −λs
e
.
2
s
)
p−
s
−1)
−
p
exp(λ
−
− 1]p is equivalent to [ exp(λ
−
]p ≤ M ,
which, for s ≥ d, is guaranteed by
exp(λ ps− )
lim
= 1.
exp(λ ps− − 1)
h s
ip−
λ
Likewise, the inequality eλs − 1 ≤ M e p− − 1
for s ≥ d is ensured by the limit
s→+∞
lim
s→+∞
exp(λs)
= 1.
exp(λ ps− − 1)p−
(3) We prove the case 1 < p− < 2 only; the proof of the case p− ≥ 2 is entirely
similar. The desired inequalities follow easily from the following limits:
1 λs
− 1) − s
λ (e
s
λ
s→+∞ (e p− − 1)p−
lim
=
1
;
λ
1 λs
− 1) − s
λ (e
s
λ
s→+∞ (e p− − 1)2
lim
= 2λ.
Proof of Lemma 2.2. Since π ∈ C 1 with π(0) = 0 and π, π 0 are bounded, it follows
that π(u) ∈ V ∩ L∞ (QT ). The left-hand side of (2.6) exists. By Lemma 3.2 in [3]
or [15], it follows from u ∈ V with ut ∈ V ∗ + L1 (QT ) that u ∈ C([0, T ]; L1 (Ω)) and
hence Π(u) ∈ C([0, T ]; L1 (Ω)). So, the right-hand side of (2.6) does exist. For the
decomposition of the time derivative ut = α(1) + α(2) ∈ V ∗ + L1 (QT ), noting the
embedding relationship
+
1,p(x)
Lp (0, T ; W0
−
1,p(x)
(Ω)) ,→ V ,→ Lp (0, T ; W0
(Ω)),
1,p(x)
by standard mollification method in [18], there exist un ∈ C ∞ ([0, T ]; W0
(Ω)),
(2)
(1)
(2)
(1)
∞
1
∞
−1,p0 (x)
(Ω)), αn ∈ C ([0, T ]; L (Ω)) such
unt = αn + αn , αn ∈ C ([0, T ]; W
(2)
(1)
that un → u, strongly in V ; αn → α(1) , strongly in V ∗ ; αn → α(2) , strongly in
1
1
1
1
L (0, T ; L (Ω)). Because Π(un ) ∈ C ([0, T ]; L (Ω)) and π(un ) ∈ V ∩ L∞ (QT ), we
have
Z T
Π(un (T )) − Π(un (0)) =
[Π(un )]t dt
0
ZZ
(4.1)
= hαn(1) , π(un )iV ∗ ,V +
αn(2) π(un ) dx dt.
QT
1
Since un → u, strongly in C([0, T ]; L (Ω)), we have un → u, a.e. in QT . (If
necessary, by a further subsequence to be denoted by the same un .) Furthermore,
the sequence π(un ) → π(u), a.e. in QT and remains bounded; hence π(un ) → π(u),
(2)
weakly* in L∞ (QT ). Combing with αn → α(2) , strongly in L1 (QT ), one has
RR
RR
(2)
α π(un ) dx dt → QT α(2) π(u) dx dt. Moreover, from un → u, strongly in
QT n
V and the properties of π, one has π(un ) → π(u), strongly in V . Together with
(1)
(1)
αn → α(1) , strongly in V ∗ , it yields hαn , π(un )iV ∗ ,V → hα(1) , π(u)iV ∗ ,V . Finally,
1
Π(un ) → Π(u) in C([0, T ]; L (Ω)) ,→ L1 (QT ). Meanwhile Π(unR(T )) → Π(u(T ))
and Π(un (0)) → Π(u(0)), strongly in L1 (QT ). Consequently, Ω Π(un (T ))dx −
20
Z. LI, B. YAN, W. GAO
EJDE-2015/46
R
R
R
Π(un (0))dx → Ω Π(u(T ))dx − Ω Π(u(0))dx. Hence (2.6) follows from (4.1) by
passing to the limit as n → ∞.
Ω
Acknowledgments. This research was supported by the National Science Foundation of China (11271154, 11401252), by the Key Lab of Symbolic Computation
and Knowledge Engineering of Ministry of Education, and by the 985 Program
and the Tang Ao-Qing Professorship of Jilin University. The first author is also
supported by the Graduate Innovation Fund of Jilin University (2014084).
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Zhongqing Li (corresponding author)
College of Mathematics, Jilin University, Changchun 130012, China
E-mail address: zqli jlu@163.com
Baisheng Yan
College of Mathematics, Jilin University, Changchun 130012, China.
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
E-mail address: yan@math.msu.edu
Wenjie Gao
College of Mathematics, Jilin University, Changchun 130012, China
E-mail address: wjgao@jlu.edu.cn