J. Differential Equations 248 (2010) 1376–1400
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Journal of Differential Equations www.elsevier.com/locate/jde
Renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and L 1 data ✩ Chao Zhang ∗ , Shulin Zhou LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, PR China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 30 March 2009 Revised 28 October 2009 Available online 3 December 2009 MSC: primary 35D05 secondary 35D10, 46E35
In this paper we prove the existence and uniqueness of both renormalized solutions and entropy solutions for nonlinear parabolic equations with variable exponents and L 1 data. And moreover, we obtain the equivalence of renormalized solutions and entropy solutions. © 2009 Elsevier Inc. All rights reserved.
Keywords: Variable exponents Renormalized solutions Entropy solutions Existence Uniqueness
1. Introduction Suppose that Ω is a bounded open domain of R N with Lipschitz boundary ∂Ω , T is a positive number. In this paper we study the following nonlinear parabolic problem
⎧ ∂u ⎪ ⎪ − div |∇ u | p (x)−2 ∇ u = f ⎨ ∂t u=0 ⎪ ⎪ ⎩ u (x, 0) = u 0 (x)
in Q ≡ Ω × (0, T ), on Γ ≡ ∂Ω × (0, T ),
(1.1)
on Ω,
¯ → (1, +∞) is a continuous function, f ∈ L 1 ( Q ) and u 0 ∈ L 1 (Ω). where the variable exponent p : Ω ✩
*
This work was supported in part by the NBRPC under Grant 2006CB705700 and the NSFC under Grant 10990013. Corresponding author. E-mail addresses:
[email protected] (C. Zhang),
[email protected] (S. Zhou).
0022-0396/$ – see front matter doi:10.1016/j.jde.2009.11.024
© 2009 Elsevier Inc.
All rights reserved.
C. Zhang, S. Zhou / J. Differential Equations 248 (2010) 1376–1400
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The study of differential equations and variational problems with nonstandard growth conditions arouses much interest with the development of elastic mechanics, electro-rheological fluid dynamics and image processing, etc. We refer the readers to [31,32,36,15] and references therein. p (x)-growth conditions can be regarded as a very important class of nonstandard ( p , q)-growth conditions. There are already numerous results for such kind of problems (see [1–3,19,20,18,5]). The functional spaces to deal with these problems are the generalized Lebesgue spaces L p (x) (Ω) and the generalized Lebesgue– Sobolev spaces W k, p (x) (Ω). Under our assumptions, it is reasonable to work with entropy solutions or renormalized solutions, which need less regularity than the usual weak solutions. The notion of renormalized solutions was first introduced by DiPerna and Lions [17] for the study of Boltzmann equation. It was then adapted to the study of some nonlinear elliptic or parabolic problems and evolution problems in fluid mechanics. We refer to [14,16,8,10,9,26] for details. At the same time the notion of entropy solutions has been proposed by Bénilan et al. in [7] for the nonlinear elliptic problems. This framework was extended to related problems with constant p in [13,30,11,4,28]. Recently, Sanchón and Urbano in [33] studied a Dirichlet problem of p (x)-Laplace equation and obtained the existence and uniqueness of entropy solutions for L 1 data, as well as integrability results for the solution and its gradient. The proofs rely crucially on a priori estimates in Marcinkiewicz spaces with variable exponents. Besides, Bendahmane and Wittbold in [6] proved the existence and uniqueness of renormalized solutions to nonlinear elliptic equations with variable exponents and L 1 data. The aim of this paper is to extend the results in [33,6] to the case of parabolic equations. As far as we know, there are no papers concerned with the nonlinear parabolic equations involving variable exponents and L 1 data. Inspired by [29] and [30], we develop a refined method. The advantage of our method is that we cannot only obtain the existence and uniqueness of renormalized solutions for problem (1.1), but also find that the renormalized solution is equivalent to the entropy solution for problem (1.1). We first employ the difference and variation methods to prove the existence and uniqueness of weak solutions for the approximate problem of (1.1) under appropriate assumptions. Then we construct an approximate solution sequence and establish some a priori estimates. Next, we draw a subsequence to obtain a limit function, and prove this function is a renormalized solution. Based on the strong convergence of the truncations of approximate solutions, we obtain that the renormalized solution of problem (1.1) is also an entropy solution, which leads to an equality in the entropy formulation. By choosing suitable test functions, we prove the uniqueness of renormalized solutions and entropy solutions, and thus the equivalence of renormalized solutions and entropy solutions. For the convenience of the readers, we recall some definitions and basic properties of the generalized Lebesgue spaces L p (x) (Ω) and generalized Lebesgue–Sobolev spaces W k, p (x) (Ω). ¯ = {h ∈ C (Ω) ¯ : minx∈Ω¯ h(x) > 1}. For any h ∈ C + (Ω) ¯ we define Set C + (Ω)
h+ = sup h(x) and h− = inf h(x). x∈Ω
x∈Ω
¯ , we introduce the variable exponent Lebesgue space L p (·) (Ω) to consist of all For any p ∈ C + (Ω) measurable functions such that
u (x) p (x) dx < ∞,
Ω
endowed with the Luxemburg norm
u (x) p (x) |u | p (·) = inf λ > 0: dx 1 , λ
Ω
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C. Zhang, S. Zhou / J. Differential Equations 248 (2010) 1376–1400
which is a separable and reflexive Banach space. The dual space of L p (x) (Ω) is L p (x) (Ω), where 1/ p (x) + 1/ p (x) = 1. If p (x) is a constant function, then the variable exponent Lebesgue space coincides with the classical Lebesgue space. The variable exponent Lebesgue spaces is a special case of Orlicz–Musielak spaces treated by Musielak in [27]. For any positive integer k, denote
W k, p (x) (Ω) = u ∈ L p (x) (Ω): D α u ∈ L p (x) (Ω), |α | k , where the norm is defined as
D α u
u W k, p(x) =
|α |k
p (x)
.
W k, p (x) (Ω) is called generalized Lebesgue–Sobolev space, which is a special generalized Orlicz– Sobolev space. An interesting feature of a generalized Lebesgue–Sobolev space is that smooth functions are not dense in it without additional assumptions on the exponent p (x). This was observed by Zhikov [35] in connection with Lavrentiev phenomenon. However, when the exponent p (x) is logHölder continuous, i.e., there is a constant C such that
p (x) − p ( y ) for every x, y ∈ Ω with |x − y |
1 , 2
C
(1.2)
− log |x − y |
then smooth functions are dense in variable exponent Sobolev 1, p (·)
(Ω), spaces and there is no confusion in defining the Sobolev space with zero boundary values, W 0 as the completion of C 0∞ (Ω) with respect to the norm u W 1, p(·) (see [21]). ¯ satisfies the log-Hölder continuity condiThroughout this paper we assume that p (x) ∈ C + (Ω) tion (1.2). Let T k denote the truncation function at height k 0:
T k (r ) = min k, max{r , −k} =
⎧ ⎨k ⎩
r
if r k, if |r | < k,
−k if r −k,
and its primitive Θk : R → R+ by
r2
r Θk (r ) =
T k (s) ds = 0
if |r | k,
2
k|r | −
2
k 2
if |r | k.
It is obvious that Θk (r ) 0 and Θk (r ) k|r |. We denote 1, p (·)
T0
1, p (·) ( Q ) = u: Ω¯ × (0, T ] → R is measurable T k (u ) ∈ L p − 0, T ; W 0 (Ω)
N with ∇ T k (u ) ∈ L p (·) ( Q ) , for every k > 0 . 1, p (·)
Next we define the very weak gradient of a measurable function u ∈ T0 from Lemma 2.1 of [7] due to the fact that
1, p (·) W0 (Ω)
⊂
1, p W 0 − (Ω).
( Q ). The proof follows
C. Zhang, S. Zhou / J. Differential Equations 248 (2010) 1376–1400
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1, p (·)
Proposition 1.1. For every measurable function u ∈ T0 ( Q ), there exists a unique measurable function v : Q → R N , which we call the very weak gradient of u and denote v = ∇ u, such that
∇ T k (u ) = v χ{|u | 0,
where χ E denotes the characteristic function of a measurable set E. Moreover, if u belongs to L 1 (0, T ; 1, 1 W 0 (Ω)), then v coincides with the weak gradient of u. The notion of the very weak gradient allows us to give the following definitions of renormalized solutions and entropy solutions for problem (1.1). 1, p (·)
Definition 1.1. A function u ∈ T0 ( Q ) ∩ C ([0, T ]; L 1 (Ω)) is a renormalized solution to problem (1.1) if the following conditions are satisfied:
(i) limn→∞ {(x,t )∈ Q : n|u (x,t )|n+1} |∇ u | p (x) dx dt = 0; (ii) for every function ϕ ∈ C 1 ( Q¯ ) with ϕ (·, T ) = 0 and S in W 2,∞ (R) which is piecewise C 1 satisfying that S has a compact support,
−
T S (u 0 )ϕ (x, 0) dx −
S (u )
∂ϕ dx dt ∂t
0 Ω
Ω
T +
S (u )|∇ u | p (x)−2 ∇ u · ∇ ϕ + S (u )|∇ u | p (x) ϕ dx dt =
0 Ω
T
f S (u )ϕ dx dt
(1.3)
0 Ω
holds. 1, p (·)
Definition 1.2. A function u ∈ T0
( Q ) ∩ C ([0, T ]; L 1 (Ω)) is an entropy solution to problem (1.1) if
Θk (u − φ)( T ) dx − Ω
Θk u 0 − φ(0) dx +
Ω
|∇ u | p (x)−2 ∇ u · ∇ T k (u − φ) dx dt =
+ Q
T
φt , T k (u − φ) dt
0
f T k (u − φ) dx dt ,
(1.4)
Q
for all k > 0 and φ ∈ C 1 ( Q¯ ) with φ|Γ = 0. Now we state our main results. Theorem 1.1. Assume that condition (1.2) holds. Then there exists a unique renormalized solution for problem (1.1). Theorem 1.2. Assume that condition (1.2) holds. Then the renormalized solution u in Theorem 1.1 is also an entropy solution for problem (1.1). And the entropy solution is unique. Remark 1.1. The renormalized solution for problem (1.1) is equivalent to the entropy solution for problem (1.1).
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The rest of this paper is organized as follows. In Section 2, we state some basic results that will be used later. We will prove the main results in Section 3. In the following sections C will represent a generic constant that may change from line to line even if in the same inequality. 2. Preliminaries In this section, we first state some elementary results for the generalized Lebesgue spaces L p (x) (Ω) and the generalized Lebesgue–Sobolev spaces W k, p (x) (Ω). The basic properties of these spaces can be found from [23], and many of these properties were independently established in [20]. Lemma 2.1. (See [20,23].)
(1) The space L p (·) (Ω) is a separable, uniform convex Banach space, and its conjugate space is L p (·) (Ω) where 1/ p (x) + 1/ p (x) = 1. For any u ∈ L p (·) (Ω) and v ∈ L p (·) (Ω), we have
uv dx 1 + 1 |u | p (x) | v | p (x) 2|u | p (x) | v | p (x) ; p− ( p − ) Ω
¯ , p 1 (x) p 2 (x) for any x ∈ Ω , then there exists the continuous embedding L p 2 (x) (Ω) → (2) If p 1 , p 2 ∈ C + (Ω) L p 1 (x) (Ω), whose norm does not exceed |Ω| + 1. Lemma 2.2. (See [20].) If we denote
|u | p (x) dx,
ρ (u ) =
∀u ∈ L p (x) (Ω),
Ω
then
p
p
p
p
+ − + min |u | p − (x) , |u | p (x) ρ (u ) max |u | p (x) , |u | p (x) .
Lemma 2.3. (See [20].) W k, p (x) (Ω) is a separable and reflexive Banach space.
¯ satisfy the log-Hölder continuity condition (1.2). Then, for u ∈ Lemma 2.4. (See [22,23].) Let p ∈ C + (Ω) 1, p (·) W0 (Ω), the p (·)-Poincaré inequality
|u | p (x) C |∇ u | p (x) holds, where the positive constant C depends on p and Ω .
Lemma 2.5. Assume that u 0 ∈ L 2 (Ω) and f ∈ L ( p − ) (0, T ; L p (x) (Ω)). Then the following problem
⎧ ∂u ⎪ ⎪ − div |∇ u | p (x)−2 ∇ u = f ⎨ ∂t u=0 ⎪ ⎪ ⎩ u (x, 0) = u 0 1, p (·)
admits a unique weak solution u ∈ L p − (0, T ; W 0 that for any ϕ ∈ C 1 ( Q¯ ) with ϕ (·, T ) = 0,
in Q , on Γ, on Ω,
(Ω)) ∩ C ([0, T ]; L 2 (Ω)) with ∇ u ∈ ( L p (·) ( Q ))N such
C. Zhang, S. Zhou / J. Differential Equations 248 (2010) 1376–1400
−
T u 0 (x)ϕ (x, 0) dx +
−u ϕt + |∇ u | p (x)−2 ∇ u · ∇ ϕ dx dt =
0 Ω
Ω
1381
T f ϕ dx dt 0 Ω
holds. Proof. By employing the difference and variation methods (see [34]), we give a sketched proof. Let n be a positive integer. Denote h = T /n. We first consider the following time-discrete problem
u −u k k −1 h uk |∂Ω = 0,
− div |∇ uk | p (x)−2 ∇ uk = [ f ]h (k − 1)h , k = 1, 2, . . . , n ,
where [ f ]h denotes the Steklov average of f defined by
[ f ]h (x, t ) =
1
t +h f (x, τ ) dτ .
h t
It is easy to see that [ f ]h (·) ∈ L p (·) (Ω). For k = 1, we introduce the variational problem
min J (u ) u ∈ W , where
1, p (x)
W = u ∈ W0
(Ω) ∩ L 2 (Ω)
and functional J is
J (u ) =
1
u 2 dx +
2h Ω
1 p (x)
|∇ u | p (x) dx −
1
u 0 u dx −
h
Ω
Ω
[ f ]h (0)u dx. Ω
We will establish that J (u ) has a minimizer u 1 (x) in W . By Lemmas 2.1, 2.4, Young’s inequality and Lemma 2.2, we have
[ f ]h (0)u dx 2 [ f ]h (0) |u | p (x) p (x) Ω
C [ f ]h (0) p (x) |∇ u | p (x) ( p ) p ε |∇ u | p −(x) + C (ε ) [ f ]h (0) p (−x) ε Ω
ε
Ω
β p − ( p ) |∇ u | p (x) dx + C (ε ) [ f ]h (0) p (−x) ( p ) |∇ u | p (x) dx + 1 + C (ε ) [ f ]h (0) p (−x) ,
(2.1)
1382
where
C. Zhang, S. Zhou / J. Differential Equations 248 (2010) 1376–1400
ε is a small positive number and β=
Choosing
1 p−
if |∇ u | p (·) 1,
1 p+
if |∇ u | p (·) 1.
ε sufficiently small and using Young’s inequality, we obtain J (u )
1 2p +
|∇ u | p (x) dx + Ω
1
u 2 dx − C
4h Ω
( p − )
u 20 dx + [ f ]h (0) p (x) + 1 ,
Ω
and thus J (u ) is lower bounded and coercive on W . On the other hand, J (u ) is weakly lower semicontinuous on W . Therefore, there exists a function u 1 ∈ W such that
J (u 1 ) = inf J (u ). u∈W
Thus the function u 1 is a weak solution of the corresponding Euler–Lagrange equation of J (u ), which is (2.1) in the case k = 1. And it is unique. Following the same procedures, we find weak solutions uk of (2.1) for k = 2, . . . , n. It follows that, for every ϕ ∈ W ,
u k − u k −1 h
Ω
|∇ uk | p (x)−2 ∇ uk · ∇ ϕ dx =
ϕ dx + Ω
[ f ]h (k − 1)h ϕ dx.
Ω
For every h = T /n, we define the approximate solutions
⎧ u 0 (x), ⎪ ⎪ ⎪ ⎪ ⎪ u 1 (x), ⎪ ⎪ ⎪ ⎨..., u h (x, t ) = ⎪ u j (x), ⎪ ⎪ ⎪ ⎪ ⎪ ..., ⎪ ⎪ ⎩ un (x), Taking
t = 0, 0 < t h,
..., ( j − 1)h < t jh, ..., (n − 1)h < t nh = T .
ϕ = uk in (2.2), we can obtain an a priori estimate
T u h2 (x, t ) dx +
∇ uh (x, t ) p (x) dx dt
0 Ω
Ω
T u 20 dx +
( p )
| f | p (−x) dt ,
C 0
Ω
which implies from Lemma 2.2 that
T
min 0
p |∇ uh | p +(x) ,
p |∇ uh | p −(x) dt
T |∇ uh | p (x) dx dt C
0 Ω
and
uh L ∞ (0,T ; L 2 (Ω)) + |∇ uh | p (x), Q + uh L p− (0,T ;W 1, p(x) (Ω)) C . 0
(2.2)
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Thus we may choose a subsequence (we also denote it by the original sequence for simplicity) such that
uh u ,
weakly-∗ in L ∞ 0, T ; L 2 (Ω) ,
uh u ,
weakly in L p − 0, T ; W 0
1, p (x)
N |∇ uh | p (x)−2 ∇ uh ξ, weakly in L p (x) ( Q ) .
(Ω) ,
Following the arguments in [34] with necessary changes in detail, we use the monotonicity 1, p (x) (Ω)) ∩ method to show that ξ = |∇ u | p (x)−2 ∇ u a.e. in Q . Recalling the fact that u ∈ L p − (0, T ; W 0
L ∞ (0, T ; L 2 (Ω)) and ut ∈ L ( p − ) (0, T ; W −1, p (x) (Ω)) from the equation, we conclude that u belongs to C ([0, T ]; L 2 (Ω)). Therefore, we obtain the existence of weak solutions. For uniqueness, suppose there exist two weak solutions u and v of problem (1.1). Then w = u − v satisfies the following problem
⎧ ∂w ⎪ ⎪ − div |∇ u | p (x)−2 ∇ u − |∇ v | p (x)−2 ∇ v = 0 in Q , ⎨ ∂t w =0 on Γ, ⎪ ⎪ ⎩ w (x, 0) = 0 on Ω. Choosing w as a test function in the above problem, we have, for almost every t ∈ (0, T ),
1
t 2
w (t ) dx +
2 Ω
|∇ u | p (x)−2 ∇ u − |∇ v | p (x)−2 ∇ v · ∇(u − v ) dx ds = 0.
0 Ω
Since the two terms on the left-hand side are nonnegative, we have u = v a.e. in Q . This finishes the proof. 2 3. The proofs of main results Now we are ready to prove the main results. Some of the reasoning is based on the ideas developed in [29] and [30] for the constant exponent case. First we prove the existence and uniqueness of renormalized solutions for problem (1.1). Proof of Theorem 1.1. (1) Existence of renormalized solutions. We first introduce the approximate problems. Find two sequences of functions { f n } ⊂ C 0∞ ( Q ) and {u 0n } ⊂ C 0∞ (Ω) strongly converging respectively to f in L 1 ( Q ) and to u 0 in L 1 (Ω) such that
f n L1 ( Q ) f L1 ( Q ) ,
u 0n L 1 (Ω) u 0 L 1 (Ω) .
(3.1)
Then we consider the approximate problem of (1.1)
⎧ ∂ un ⎪ ⎪ − div |∇ un | p (x)−2 ∇ un = f n in Q , ⎨ ∂t u on Γ, ⎪ n=0 ⎪ ⎩ un (x, 0) = u 0n on Ω. 1, p (·)
(3.2)
By Lemma 2.5, we can find a weak solution un ∈ L p − (0, T ; W 0 (Ω)) with ∇ un ∈ ( L p (·) ( Q ))N for problem (3.2). Our aim is to prove that a subsequence of these approximate solutions {un } converges to a measurable function u, which is a renormalized solution of problem (1.1). We will divide the
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C. Zhang, S. Zhou / J. Differential Equations 248 (2010) 1376–1400
proof into several steps. Although some of the arguments are not new, we present a self-contained proof for the sake of clarity and readability. Step 1. Prove the convergence of {un } in C ([0, T ]; L 1 (Ω)) and find its subsequence which is almost everywhere convergent in Q . Let m and n be two integers, then from (3.2) we can write the weak form as
T
T
|∇ un | p (x)−2 ∇ un − |∇ um | p (x)−2 ∇ um · ∇φ dx dt
(un − um )t , φ dt +
0
0 Ω
T =
( f n − f m )φ dx dt , 0 Ω
for all φ ∈ L p − (0, T ; W 0 (Ω)) ∩ L ∞ ( Q ) with ∇φ ∈ ( L p (·) ( Q ))N . Choosing φ = T 1 (un − um )χ(0,t ) with t T and discarding the positive term, we get 1, p (·)
Θ1 (un − um )(t ) dx Ω
Θ1 (u 0n − u 0m ) dx + f n − f m L 1 ( Q ) Ω
u 0n − u 0m L 1 (Ω) + f n − f m L 1 ( Q ) := an,m . Therefore, we conclude that
|un − um |2 (t ) 2
{|un −um | 0, we define the regularization in time of the function T k (u ) given by
t
T k (u ) μ (x, t ) := μ
e μ(s−t ) T k u (x, s) ds,
−∞ 1, p (·)
extending T k (u ) by 0 for s < 0. Observe that ( T k (u ))μ ∈ L p − (0, T ; W 0 ∇( T k (u ))μ ∈ ( L p (·) ( Q ))N , it is differentiable for a.e. t ∈ (0, T ) with
(Ω)) ∩ L ∞ ( Q ) with
T k (u ) (x, t ) k 1 − e −μt < k a.e. in Q , μ ∂( T k (u ))μ = μ T k (u ) − T k (u ) μ . ∂t After computation, we can get
N ∇ T k (u ) μ → ∇ T k (u ) strongly in L p (·) ( Q ) .
set
Let us take now a sequence {ψ j } of C 0∞ (Ω) functions that strongly converge to u 0 in L 1 (Ω), and
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C. Zhang, S. Zhou / J. Differential Equations 248 (2010) 1376–1400
ημ , j ( u ) ≡ T k ( u )
μ +e
−μt
T k (ψ j ).
The definition of ημ, j , which is a smooth approximation of T k (u ), is needed to deal with a nonzero initial datum (see also [29]). Note that this function has the following properties:
⎧ ημ, j (u ) t = μ T k (u ) − ημ, j (u ) , ⎪ ⎪ ⎪ ⎪ ⎨ ημ, j (u )(0) = T k (ψ j ), ημ, j (u ) k, ⎪ ⎪ ⎪ ⎪ N ⎩ ∇ ημ, j (u ) → ∇ T k (u ) strongly in L p (·) ( Q ) , as μ → +∞.
(3.7)
Fix a positive number k. Let h > k. We choose
w n = T 2k un − T h (un ) + T k (un ) − ημ, j (u )
as a test function in (3.2). The use of w n as a test function to prove the strong convergence of truncations was first introduced in the elliptic case in [25], then adapted to parabolic equations in [29]. If we set M = 4k + h, then it is easy to see that ∇ w n = 0 where |un | > M. Therefore, we may write the weak form of (3.2) as
T
T T p (x)−2 ∂ un , w n dt + ∇ T M (un ) · ∇ w n dx dt = f n w n dx dt . ∇ T M (u n ) ∂t
0
0 Ω
0 Ω
In the following, denote w (n, μ, j , h) all quantities such that
lim
lim
h→+∞ j →+∞
lim
lim w (n, μ, j , h) = 0.
μ→+∞ n→+∞
First as far as the first term is concerned, that is
T
∂ un , w n dt . ∂t
0
Since |ημ, j (u )| k, w n can be written as
w n = T h+k un − ημ, j (u ) − T h−k un − T k (un ) . Applying Lemma 2.1 in [29], we can obtain that
T
∂ un , w n dt w (n, j , h). ∂t
0
From the above estimate, we have
T 0 Ω
∇ T M (un ) p (x)−2 ∇ T M (un ) · ∇ w n dx dt
T f n w n dx dt + w (n, j , h). 0 Ω
C. Zhang, S. Zhou / J. Differential Equations 248 (2010) 1376–1400
1387
Splitting the integral in the left-hand side on the sets where |un | k and where |un | > k and discarding some nonnegative terms, we find
T
∇ T M (un ) p (x)−2 ∇ T M (un ) · ∇ T 2k un − T h (un ) + T k (un ) − ημ, j (u ) dx dt
0 Ω
T 0 Ω
∇ T k (un ) p (x)−2 ∇ T k (un ) · ∇ T k (un ) − ημ, j (u ) dx dt
−
∇ T M (un ) p (x)−2 ∇ T M (un ) ∇ ημ, j (u ) dx dt .
{|un |>k}
It follows from the above inequality that
T
∇ T k (un ) p (x)−2 ∇ T k (un ) · ∇ T k (un ) − ημ, j (u ) dx dt
0 Ω
∇ T M (un ) p (x)−2 ∇ T M (un ) ∇ ημ, j (u ) dx dt +
{|un |>k}
T f n w n dx dt + w (n, μ, j , h). 0 Ω
Using the fact that ∇ ημ, j (u ) → ∇ T k (u ) strongly in ( L p (·) ( Q )) N as
T
μ → +∞, we conclude that
∇ T k (un ) p (x)−2 ∇ T k (un ) · ∇ T k (un ) − T k (u ) dx dt
0 Ω
∇ T M (un ) p (x)−2 ∇ T M (un ) ∇ ημ, j (u ) dx dt +
{|un |>k}
T f n w n dx dt + w (n, μ, j , h). 0 Ω
Furthermore, we have
T
∇ T k (un ) p (x)−2 ∇ T k (un ) − ∇ T k (u ) p (x)−2 ∇ T k (u ) ∇ T k (un ) − T k (u ) dx dt
0 Ω
∇ T M (un ) p (x)−2 ∇ T M (un ) · ∇ ημ, j (u ) dx dt
{|un |>k}
T +
f n T 2k un − T h (un ) + T k (un ) − ημ, j (u ) dx dt 0 Ω
T −
∇ T k (u ) p (x)−2 ∇ T k (u ) · ∇ T k (un ) − T k (u ) dx dt + w (n, μ, j , h)
0 Ω
= I 1 + I 2 + I 3 + w (n, μ, j , h).
(3.8)
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Now we show the limits of I 1 , I 2 and I 3 are zeros when n, μ and then h tend to infinity respectively.
Limit of I 1 . We observe that |∇ T M (un )| p (x)−2 ∇ T M (un ) is bounded in L p (x) ( Q ), and by the dominated convergence theorem χ{|un |>k} |∇ ημ, j (u )| converges strongly in L p (x) ( Q ) to χ{|u |>k} |∇ T k (u )|, which is zero, as n and μ tends to infinity. Thus we obtain
lim I 1 =
lim
μ→+∞ n→+∞
lim
∇ T M (un ) p (x)−2 ∇ T M (un ) ∇ T k (u ) dx dt = 0. (3.9)
lim
μ→+∞ n→+∞
{|un |>k}
Limit of I 2 . Notice that
T I2
| f n − f | T 2k un − T h (un ) + T k (un ) − ημ, j (u ) dx dt
0 Ω
T +
f T 2k un − T h (un ) + T k (un ) − ημ, j (u ) dx dt
0 Ω
T 2k
T | f n − f | dx +
0 Ω
f T 2k un − T h (un ) + T k (un ) − ημ, j (u ) dx dt .
0 Ω
Since f n is strongly compact in L 1 ( Q ), using (3.3), the definition of nated convergence theorem, we have
T lim
h→+∞
lim
lim | I 2 | lim
μ→+∞ n→+∞
h→+∞
ημ, j and the Lebesgue domi-
f T 2k u − T h (u ) dx dt = 0.
(3.10)
0 Ω
Limit of I 3 . Recalling (3.5), we have
lim I 3 = 0.
(3.11)
n→+∞
Therefore, passing to the limits in (3.8) as n, (3.10) and (3.11), we deduce that
μ, j, and then h tend to infinity, by means of (3.9),
lim E (n) = 0,
n→+∞
where
T E (n) =
∇ T k (un ) p (x)−2 ∇ T k (un ) − ∇ T k (u ) p (x)−2 ∇ T k (u ) · ∇ T k (un ) − T k (u ) dx dt .
0 Ω
We recall the following well-known inequalities: for any two real vectors a, b ∈ R N ,
a|a| p −2 − b|b| p −2 (a − b) c ( p )|a − b| p ,
if p 2
C. Zhang, S. Zhou / J. Differential Equations 248 (2010) 1376–1400
and for every
1389
ε ∈ (0, 1],
|a − b| p c ( p )ε ( p −2)/ p a|a| p −2 − b|b| p −2 (a − b) + ε |b| p , 1− p
if 1 < p < 2,
2− p
where c ( p ) = 2p −1 when p 2 and c ( p ) = 3p −1 when 1 < p < 2. Therefore, we have
∇ T k (un ) − ∇ T k (u ) p (x) dx dt 2 p + −1 ( p + − 1) E (n)
(3.12)
{(x,t )∈ Q : p (x)2}
and
∇ T k (un ) − ∇ T k (u ) p (x) dx dt
{(x,t )∈ Q : 1< p (x) 0,
N ∇ T k (un ) → ∇ T k (u ) strongly in L p (·) ( Q )
(3.14)
∇ T k (un ) p (x)−2 ∇ T k (un ) → ∇ T k (u ) p (x)−2 ∇ T k (u ) in L p (·) ( Q ) N .
(3.15)
and
Thanks to Lemma 2.2, we know that
T k (u n ) → T k (u )
1, p (·)
strongly in L p − 0, T ; W 0
(Ω) .
Step 3. Show that u is a renormalized solution. For given a, k > 0, define the function T k,a (s) = T a (s − T k (s)) as
⎧ ⎨ s − k sign(s) if k |s| < k + a, T k,a (s) = a sign(s) if |s| k + a, ⎩ 0 if |s| k. Using T k,a (un ) as a test function in (3.2), we find
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C. Zhang, S. Zhou / J. Differential Equations 248 (2010) 1376–1400
Θa (un ∓ k)( T ) dx − {|un |>k}
{|u 0n |>k}
|∇ un | p (x)−2 ∇ un · ∇ un dx dt
Θa (u 0n ∓ k) dx + {k|un |k+a}
f n T k,a (un ) dx dt , Ω
which yields that
|∇ un | p (x) dx dt a {k|un |k+a}
| f n | dx dt + {|un |>k}
|u 0n | dx .
{|u 0n |>k}
Recalling the convergence of {un } in C ([0, T ]; L 1 (Ω)), we have
lim meas (x, t ) ∈ Q : |un | > k = 0 uniformly with respect to n.
k→+∞
Therefore, passing to the limit first in n then in k, we conclude that
|∇ u | p (x) dx dt = 0.
lim
k→+∞
{(x,t )∈ Q : k|u (x,t )|k+a}
Choosing a = 1, we obtain the renormalized condition, i.e.,
|∇ u | p (x) dx dt = 0.
lim
k→+∞
{(x,t )∈ Q : k|u (x,t )|k+1}
Let S ∈ W 2,∞ (R) be such that supp S ⊂ [− M , M ] for some M > 0. For every ϕ (x, T ) = 0, S (un )ϕ is a test function in (3.2). It yields
T
∂ S (u n ) ϕ dx dt + ∂t
0 Ω
T
ϕ ∈ C ∞ ( Q¯ ) with
S (un )|∇ un | p (x)−2 ∇ un · ∇ ϕ + S (un )|∇ un | p (x) ϕ dx dt
0 Ω
T =
f n S (un )ϕ dx dt .
(3.16)
0 Ω
First we consider the first term on the left-hand side of (3.16). Since S is bounded and continuous, (3.3) implies that S (un ) converges to S (u ) a.e. in Q and weakly-∗ in L ∞ ( Q ). Then ∂ S∂(ut n ) converges
to ∂ S∂(tu ) in D ( Q ) as n → +∞, that is
T 0 Ω
∂ S (u n ) ϕ dx dt → ∂t
T
∂ S (u ) ϕ dx dt . ∂t
0 Ω
For the other terms on the left-hand side of (3.16), because of supp S ⊂ [− M , M ] we know
p (x)−2 S (un )|∇ un | p (x)−2 ∇ un = S (un ) ∇ T M (un ) ∇ T M (u n )
C. Zhang, S. Zhou / J. Differential Equations 248 (2010) 1376–1400
1391
and
p (x) S (un )|∇ un | p (x) = S (un ) ∇ T M (un ) . Using (3.3), (3.14) and (3.15), we have
p (x)−2
S (un ) ∇ T M (un )
p (x)−2 N ∇ T M (un ) → S (u ) ∇ T M (u ) ∇ T M (u ) in L p (·) ( Q )
and
p (x)
p (x)−2
p (x)
S (un ) ∇ T M (un )
p (x) → S (u ) ∇ T M (u ) in L 1 ( Q ).
Noting that
S (u ) ∇ T M (u )
S (u ) ∇ T M (u )
∇ T M (u ) = S (u )|∇ u | p (x)−2 ∇ u ,
= S (u )|∇ u | p (x) ,
we deduce
S (un )|∇ un | p (x)−2 ∇ un → S (u )|∇ u | p (x)−2 ∇ u
N
in L p (·) ( Q )
and
S (un )|∇ un | p (x) → S (u )|∇ u | p (x)
in L 1 ( Q ).
For the right-hand side of (3.16), thanks to the strong convergence of f n , it is easy to pass to the limits. Therefore, we obtain
−
T S (u 0 )ϕ (x, 0) dx −
∂ϕ S (u ) dx dt + ∂t
0 Ω
Ω
T =
T
S (u )|∇ u | p (x)−2 ∇ u · ∇ ϕ + S (u )|∇ u | p (x) ϕ dx dt
0 Ω
f S (u )ϕ dx dt .
0 Ω
This completes the proof of the existence of renormalized solutions. (2) Uniqueness of renormalized solutions. Now we prove the uniqueness of renormalized solutions for problem (1.1) by choosing an appropriate test function motivated by [9] and [6]. Let u and v be two renormalized solutions for problem (1.1). Fix a positive number k. For σ > 0, let S σ be the function defined by
⎧ S σ (r ) = r if |r | < σ , ⎪ ⎪ ⎪ ⎪ 1 1 ⎪ 2 ⎨ S (r ) = σ + ∓ r ∓ (σ + 1) if σ ±r σ + 1, σ 2 2 ⎪ ⎪ ⎪ 1 ⎪ ⎪ if ±r > σ + 1. ⎩ S σ (r ) = ± σ + 2
It is obvious that
(3.17)
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⎧ if |r | < σ , ⎪ ⎨ S σ (r ) = 1 S σ (r ) = σ + 1 − |r | if σ |r | σ + 1, ⎪ ⎩ S σ (r ) = 0 if |r | > σ + 1. It is easy to check S σ ∈ W 2,∞ (R) with supp S σ ⊂ [−σ − 1, σ + 1] and supp S σ ⊂ [σ , σ + 1] ∪ [−σ − 1, −σ ]. Therefore, we may take S = S σ in (1.3) to have
T
∂ S σ (u ) ϕ dx dt + ∂t
0 Ω
T
S σ (u )|∇ u | p (x)−2 ∇ u · ∇ ϕ + S σ (u )|∇ u | p (x) ϕ dx dt
0 Ω
T
f S σ (u )ϕ dx dt
= 0 Ω
and
T
∂ S σ (v ) ϕ dx dt + ∂t
0 Ω
T
S σ ( v )|∇ v | p (x)−2 ∇ v · ∇ ϕ + S σ ( v )|∇ v | p (x) ϕ dx dt
0 Ω
T
f S σ ( v )ϕ dx dt .
= 0 Ω
We plug obtain that
ϕ = T k ( S σ (u ) − S σ ( v )) as a test function in the above equalities and subtract them to
J 0 + J 1 + J 2 = J 3,
(3.18)
where
T J0 =
∂( S σ (u ) − S σ ( v )) , T k S σ (u ) − S σ ( v ) dt , ∂t
0
T J1 =
S σ (u )|∇ u | p (x)−2 ∇ u − S σ ( v )|∇ v | p (x)−2 ∇ v · ∇ T k S σ (u ) − S σ ( v ) dx dt ,
0 Ω
T J2 =
S σ (u )|∇ u | p (x) − S σ ( v )|∇ v | p (x) T k S σ (u ) − S σ ( v ) dx dt ,
0 Ω
T J3 =
f S σ (u ) − S σ ( v ) T k S σ (u ) − S σ ( v ) dx dt .
0 Ω
We estimate J 0 , J 1 , J 2 and J 3 one by one. Recalling the definition of Θk (r ), J 0 can be written as
C. Zhang, S. Zhou / J. Differential Equations 248 (2010) 1376–1400
J0 =
Θk S σ (u ) − S σ ( v ) ( T ) dx −
Ω
1393
Θk S σ (u ) − S σ ( v ) (0) dx.
Ω
Due to the same initial condition for u and v, and the properties of Θk , we get
Θk S σ (u ) − S σ ( v ) ( T ) dx 0.
J0 = Ω
Writing
T
∇ S σ (u ) p (x)−2 ∇ S σ (u ) − ∇ S σ ( v ) p (x)−2 ∇ S σ ( v ) · ∇ T k S σ (u ) − S σ ( v ) dx dt
J1 = 0 Ω
T
+
p (x)−2
p (x)−2 |∇ v | p (x)−2 ∇ v · ∇ T k S σ (u ) − S σ ( v ) dx dt
S σ (u ) − S σ (u ) S σ (u )
|∇ u | p (x)−2 ∇ u · ∇ T k S σ (u ) − S σ ( v ) dx dt
0 Ω
T
−
S σ ( v ) − S σ ( v ) S σ ( v )
0 Ω
:= and setting
J 11
+ J 12 + J 13 ,
σ k, we have
J 11
|∇ u | p (x)−2 ∇ u − |∇ v | p (x)−2 ∇ v · ∇(u − v ) dx dt .
(3.19)
{|u − v |k}∩{|u |,| v |k}
Recalling supp S σ ⊂ [−σ − 1, σ + 1] and supp S σ ⊂ [σ , σ + 1] ∪ [−σ − 1, −σ ], we obtain
2 J 2
|∇ u | p (x) dx dt
1
{σ |u |σ +1}
+
|∇ u |
{σ |u |σ +1}∩{| v |σ +1}∩{| S σ (u )− S σ ( v )|k}
p (x)−1
|∇ v | dx dt
|∇ u | p (x) dx dt +
2 {σ |u |σ +1}
|∇ u | p (x)−1 |∇ v | dx dt
{σ |u |σ +1}∩{σ −k| v |σ +1}
|∇ v | p (x) dx dt .
|∇ u | p (x) dx dt +
C {σ |u |σ +1}
{σ −k| v |σ +1}
And we may get the similar estimate for J 13 . Furthermore, we have
|∇ u | p (x) dx dt +
| J 2| C {σ |u |σ +1}
{σ | v |σ +1}
|∇ v | p (x) dx dt .
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C. Zhang, S. Zhou / J. Differential Equations 248 (2010) 1376–1400
From the above estimates and (i) in Definition 1.1, we obtain
lim
2 3 J + J + | J 2 | = 0. 1
σ →+∞
1
Observing
f S σ (u ) − S σ ( v ) → 0 strongly in L 1 ( Q ) as
σ → +∞ and using the Lebesgue dominated convergence theorem, we deduce that lim | J 3 | = 0.
σ →+∞
Therefore, sending
σ → +∞ in (3.18) and recalling (3.19), we have |∇ u | p (x)−2 ∇ u − |∇ v | p (x)−2 ∇ v · ∇(u − v ) dx dt = 0,
{|u | 2k ,| v | 2k }
which implies ∇ u = ∇ v a.e. on the set {|u |
k , 2
| v | 2k }. Since k is arbitrary, we conclude that 1, p (x)
∇ u = ∇ v a.e. in Q . Then, set ξn = T 1 ( T n+1 (u ) − T n+1 ( v )). We have ξn ∈ L p − (0, T ; W 0
(Ω)) and
⎧ on |u | n + 1, | v | n + 1 , ⎪ ⎨0
∇ξn = ∇ u χ{|u −T n+1 ( v )|1} on |u | n + 1, | v | > n + 1 , ⎪
⎩ −∇ v χ{|T n+1 (u )− v |1} on |u | > n + 1, | v | n + 1 , such that
|∇ξn | p (x) dx dt
{n|u |n+1}
Q
|∇ u | p (x) dx dt +
|∇ v | p (x) dx dt .
{n| v |n+1}
Thanks to Lemma 2.2 and (i) in Definition 1.1, we deduce that ξn → 0 strongly in L p − (0, T ; 1, p (x) W0 (Ω)). Since ξn → T 1 (u − v ) a.e. in Q , we conclude that T 1 (u − v ) = 0, hence u = v a.e. in Q . Therefore we obtain the uniqueness of renormalized solutions. This completes the proof of Theorem 1.1. 2 Next, we prove that the renormalized solution u is also an entropy solution of problem (1.1) and the entropy solution of problem (1.1) is unique. Proof of Theorem 1.2. (1) The renormalized solution is an entropy solution. Now we choose v n = T k (un − φ) as a test function in (3.2) for k > 0 and φ ∈ C 1 ( Q¯ ) with φ|Γ = 0. We note that, if L = k + φ L ∞ ( Q ) , then
T |∇ un | p (x)−2 ∇ un · ∇ T k (un − φ) dx dt 0 Ω
T = 0 Ω
∇ T L (un ) p (x)−2 ∇ T L (un ) · ∇ T k T L (un ) − φ dx dt
C. Zhang, S. Zhou / J. Differential Equations 248 (2010) 1376–1400
1395
and
T
(un )t , T k (un − φ) dt +
0
T
∇ T L (un ) p (x)−2 ∇ T L (un ) · ∇ T k T L (un ) − φ dx dt
0 Ω
T =
f n T k (un − φ) dx dt . 0 Ω
Since (un )t = (un − φ)t + φt , we have
T
(un )t , T k (un − φ) dt
0
=
Θk (un − φ)( T ) dx −
Ω
T Θk (un − φ)(0) dx +
φt , T k (un − φ) dt ,
0
Ω
which yields that
Θk (un − φ)( T ) dx − Ω
T Θk (un − φ)(0) dx + 0
Ω
T +
φt , T k T L (un ) − φ dt
∇ T L (un ) p (x)−2 ∇ T L (un ) · ∇ T k T L (un ) − φ dx dt
0 Ω
T =
f n T k (un − φ) dx dt .
(3.20)
0 Ω
Recalling un converges to u in C ([0, T ]; L 1 (Ω)), hence ∀t T , un (t ) → u (t ) in L 1 (Ω). Since Θk is Lipschitz continuous, we get
Θk (un − φ)( T ) dx → Ω
Θk (u − φ)( T ) dx Ω
and
Θk (un − φ)(0) dx → Ω
Θk u 0 − φ(0) dx,
Ω
as n → +∞. Using the strong convergence of f n , (3.5) and (3.15), we can pass to the limits as n tends to infinity for the other terms to conclude
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C. Zhang, S. Zhou / J. Differential Equations 248 (2010) 1376–1400
Θk (u − φ)( T ) dx − Ω
Θk u 0 − φ(0) dx +
T 0
Ω
T |∇ u | p (x)−2 ∇ u · ∇ T k (u − φ) dx =
+
φt , T k (u − φ) dt
0 Ω
f T k (u − φ) dx, Ω
for all k > 0 and φ ∈ C 1 ( Q¯ ) with φ|Γ = 0. Therefore, we finish the proof of the existence of entropy solutions. (2) Uniqueness of entropy solutions. Suppose that u and v are two entropy solutions of problem (1.1). Let {un } be a sequence constructed in (3.2), which satisfies ∇ T k (un ) strongly converges to ∇ T k (u ) in ( L p (·) ( Q )) N , for every k > 0. Choosing S σ (un ) as a test function in (1.4) for entropy solution v, we have
Θk v − S σ (un ) ( T ) dx −
Ω
Θk u 0 − S σ (u 0n ) dx +
+
(un )t , S σ (un ) T k ( v − S σ (un ) dt
0
Ω
T
T
|∇ v | p (x)−2 ∇ v · ∇ T k v − S σ (un ) dx dt
0 Ω
T =
f T k v − S σ (un ) dx dt .
(3.21)
0 Ω
In order to deal with the third term on the left-hand side of (3.21), we take S σ (un )Ψ with Ψ = T k ( v − S σ (un )) as a test function for problem (3.2) to obtain
T
T
(un )t , S σ (un )Ψ dt +
0
T
S σ (un )Ψ |∇ un |
p (x)
dx dt +
0 Ω
T =
S σ (un )|∇ un | p (x)−2 ∇ un · ∇Ψ dx dt
0 Ω
f n S σ (un )Ψ dx dt .
(3.22)
0 Ω
Thus we deduce from (3.21) and (3.22) that
Θk v − S σ (un ) ( T ) dx −
Ω
Θk u 0 − S σ (u 0n ) dx
Ω
T −
S σ (un ) T k v − S σ (un ) |∇ un | p (x) dx dt
0 Ω
T − 0 Ω
S σ (un )|∇ un | p (x)−2 ∇ un · ∇ T k v − S σ (un ) dx dt
C. Zhang, S. Zhou / J. Differential Equations 248 (2010) 1376–1400
T +
1397
|∇ v | p (x)−2 ∇ v · ∇ T k v − S σ (un ) dx dt
0 Ω
T =
f T k v − S σ (un ) dx dt −
0 Ω
T
f n S σ (un ) T k v − S σ (un ) dx dt .
0 Ω
We will pass to the limit as n → +∞ and σ → +∞ successively. Let us denote A 3 for the third term on the left-hand side of the above equality for simplicity. Recalling supp S σ ⊂ [σ , σ + 1] ∪ [−σ − 1, −σ ], we have
| A3| k
|∇ un | p (x) dx dt .
{σ |un |σ +1}
Observe that S σ (un )|∇ un | p (x)−2 ∇ un = S σ (un )|∇ T σ +1 (un )| p (x)−2 ∇ T σ +1 (un ), then we get
Θk v − S σ (un ) ( T ) dx −
Ω
Θk u 0 − S σ (u 0n ) dx
Ω
T +
p (x)−2 |∇ v | p (x)−2 ∇ v − S σ (un ) ∇ T σ +1 (un ) ∇ T σ +1 (un ) · ∇ T k v − S σ (un ) dx dt
0 Ω
T
f − f n S σ (un ) T k v − S σ (un ) dx dt + k
|∇ un | p (x) dx dt .
{σ |un |σ +1}
0 Ω
Thanks to the fact that ∇ T k (un ) → ∇ T k (u ) strongly in ( L p (·) ( Q )) N and the Lebesgue dominated convergence theorem, letting n → +∞, we obtain
Θk v − S σ (u ) ( T ) dx −
Ω
Θk u 0 − S σ (u 0 ) dx
Ω
T +
p (x)−2 |∇ v | p (x)−2 ∇ v − S σ (u ) ∇ T σ +1 (u ) ∇ T σ +1 (u ) · ∇ T k v − S σ (u ) dx dt
0 Ω
T
f 1 − S σ (u ) T k v − S σ (u ) dx dt + k
|∇ u | p (x) dx dt .
{σ |u |σ +1}
0 Ω
Let us denote A 3 for the third term on the left-hand side of (3.23). Then we can write A 3 as
A 3 =
T 0 Ω
|∇ v | p (x)−2 ∇ v − S σ (u )|∇ u | p (x)−2 ∇ u · ∇ T k v − S σ (u ) dx dt
(3.23)
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C. Zhang, S. Zhou / J. Differential Equations 248 (2010) 1376–1400
T =
p (x)−2 |∇ v | p (x)−2 ∇ v − ∇ S σ (u ) ∇ S σ (u ) · ∇ T k v − S σ (u ) dx dt
0 Ω
T +
S (u ) p (x)−2 S (u ) − S (u ) |∇ u | p (x)−2 ∇ u · ∇ T k v − S σ (u ) dx dt σ σ σ
0 Ω
= I1 + I2. Recalling the definition of S σ , we have
|∇ u | p (x) dx dt +
|I2| 2 {σ |u |σ +1}
|∇ u | p (x)−1 ∇ v dx dt
{σ |u |σ +1}∩{| v − S σ (u )|k}
|∇ u | p (x) dx dt +
2 {σ |u |σ +1}
{σ |u |σ +1}∩{σ −k| v |σ +k+1}
C
|∇ u |
p (x)
dx dt +
{σ |u |σ +1}
Now we let
|∇ u | p (x)−1 ∇ v dx dt
|∇ v |
p (x)
dx dt .
(3.24)
{σ −k| v |σ +k+1}
σ → +∞. Since
Θk v − S σ (u ) ( T ) k v ( T ) + u ( T ) ,
Θk u 0 − S σ (u 0 ) k|u 0 |,
by the Lebesgue dominated convergence theorem, we have
Θk u 0 − S σ (u 0 ) dx → 0,
Ω
Θk v − S σ (u ) ( T ) dx →
Ω
Θk ( v − u )( T ) dx. Ω
According to the fact that
|∇ u | p (x) dx dt = 0
lim
k→+∞
{(x,t )∈ Q : k|u (x,t )|k+1}
and Fatou’s lemma, we deduce from (3.23) and (3.24) that
Θk ( v − u )( T ) dx + Ω
|∇ v | p (x)−2 ∇ v − |∇ u | p (x)−2 ∇ u · ∇( v − u ) dx dt 0.
{|u | 2k ,| v | 2k }
Using the positivity of Θk , we have ∇ u = ∇ v a.e. in Q , for all k. Similar to the case of renormalized solutions, we conclude that u = v a.e. in Q . Therefore we obtain the uniqueness of entropy solutions. This completes the proof of Theorem 1.2. 2 Remark 3.1. Furthermore, we may improve the integrability of the renormalized solution or entropy 1 . Then we can prove that solution u for problem (1.1) by assuming that p − > 2 − N + 1
u Lq (0,T ;W 1,q (Ω)) C , 0
C. Zhang, S. Zhou / J. Differential Equations 248 (2010) 1376–1400
1399
with
1q
