UNIQUENESS AND COMPARISON THEOREMS FOR ... - UZH

COMMUNICATIONS ON PURE AND APPLIED ANALYSIS Volume 12, Number 4, July 2013

doi:10.3934/cpaa.2013.12.1527 pp. 1527–1546

UNIQUENESS AND COMPARISON THEOREMS FOR SOLUTIONS OF DOUBLY NONLINEAR PARABOLIC EQUATIONS WITH NONSTANDARD GROWTH CONDITIONS

Stanislav Antontsev CMAF, University of Lisbon, Portugal

Michel Chipot University of Zurich, Switzerland

Sergey Shmarev University of Oviedo, Spain

(To the memory of Professor I. V. Skrypnik) Abstract. The paper addresses the Dirichlet problem for the doubly nonlinear parabolic equation with nonstandard growth conditions: ut = div a(x, t, u)|u|α(x,t) |∇u|p(x,t)−2 ∇u + f (x, t) with given variable exponents α(x, t) and p(x, t). We establish conditions on the data which guarantee the comparison principle and uniqueness of bounded weak solutions in suitable function spaces of Orlicz-Sobolev type.

1. Introduction. We study the Dirichlet problem for the doubly nonlinear parabolic equation  ut = div a(z, u)|u|α(z) |∇u|p(z)−2 ∇u + f (z)     for z = (x, t) ∈ Q = Ω × (0, T ], (1)  u(x, 0) = u0 (x) in Ω,    u = 0 on Γ = ∂Ω × [0, T ]. Equation (1) is formally parabolic, but may degenerate or become singular at the points where u and/or |∇u| vanish or become infinite. Let us introduce the functions Z u u|u|γ(z) α(z) , v(z) = |s|γ(z) ds = , γ(z) = p(z) − 1 γ(z) + 1 (2) 0 1

−γ

u(z) = Φ0 (z, v) = (1 + γ) 1+γ |v| 1+γ v, 2000 Mathematics Subject Classification. Primary: 35K55, 35K65, 35K67. Key words and phrases. Nonlinear parabolic equations, double nonlinearity, nonstandard growth conditions. The first author was partially supported by FCT, Financiamento Base 2008-ISFL-1-209 and by the research project PTDC/MAT/110613/2009, FCT (Portugal). The second author acknowledges the support of the Swiss National Science Foundation under the grant #200021-129807/1. The third author acknowledges the support of the research projects MTM2007-65088 and MTM201018427 (Spain).

1527

1528

STANISLAV ANTONTSEV, MICHEL CHIPOT AND SERGEY SHMAREV

and rewrite problem (1) in the form:  p(z)−2  ∂ Φ (z, v) = div b(z, v)|∇v + B(v)| (∇v + B(v) +f t 0    v = 0 on Γ,  u0 |u0 |γ(x,0)   in Ω,  v(x, 0) ≡ v0 (x) = 1 + γ(x, 0)

in Q, (3)

with Z b(z, v) ≡ a(z, Φ0 (z, v)),

B(v) = −∇γ ·

u

|s|γ(z) ln |s| ds.

0

Problem (3) will be the subject of the further study. Equations of the types (1) and (3) with constant exponents α and p arise in the mathematical modelling of various physical processes such as flows of incompressible turbulent fluids or gases in pipes, processes of filtration in porous media, glaciology - see [5, 6, 16, 17, 22, 33] and further references therein. The questions of existence and uniqueness of solutions to equations like (1) and (3) with constant exponents of nonlinearity α and p were studied by many authors - see [6, 14, 15, 16, 24, 28, 29] for equations of the type (1) and [17, 21] for the equations of the type (3) with the prescribed function B ≡ B(x, t) independent of the solution v. Existence, uniqueness, and qualitative properties of solutions for parabolic equations with variable nonlinearity corresponding to the special cases α(x, t) = 0, or p(x, t) = 2 were studied in [1, 2, 3, 4, 8, 9, 10], see also [7] for a review of results concerning elliptic equations with variable nonlinearity, and [12] for elliptic equations with triple variable nonlinearity. The Cauchy problem for doubly nonlinear parabolic equations with constant exponents of nonlinearity is studied in [30, 31, 32]. In the present work we prove comparison principle and uniqueness of weak solutions for the Dirichlet problem (3) in which the exponents α and p are allowed to be variable. The paper is organized as follows. In Section 2 we prove several auxiliary assertions and collect some known facts from the theory of Orlicz–Sobolev spaces. The precise assumptions on the data and main results are given in Section 3. Besides, in this section we recall the known existence theorem for problem (3) published in [11]. In Section 4 we derive formulas of integration by parts for the elements of the main function spaces used throughout the paper. In Sections 5, 6 we give the proofs of the main comparison theorems. The comparison principle and uniqueness are proved for the solutions subject to some additional restrictions, but under weaker assumptions on the data, and are independent of the proof of the existence theorem. To be precise, the comparison principle and uniqueness are true for the weak solutions with ∂t Φ0 (z, v) ∈ L1 (Q). In order to ensure that this class of solutions is nonempty, in the final Section 7 we give a sketch of the proof of the existence theorem from [11], formulated in Section 3, and show that the already constructed solution belongs to the class of uniqueness, provided that the data of the problem satisfy some additional conditions. 2. The function spaces. 1,p(·)

2.1. Spaces Lp(·) (Ω) and W0 (Ω). The definitions of the function spaces used throughout the paper and a brief description of their properties follow [18, 19, 23, 25]. Further references can be found in the review papers [20, 26]. Let Ω ⊂ Rn be

DOUBLY NONLINEAR PARABOLIC EQUATIONS

1529

a bounded domain, ∂Ω be Lipschitz-continuous, and let p(x) be log-continuous in Ω: ∀ x, y ∈ Ω such that |x − y| < 21 1 = C < ∞. (4) |p(x) − p(y)| ≤ ω(|x − y|) with limτ →0+ ω(τ ) ln τ By Lp(·) (Ω) we denote the space of measurable functions f (x) on Ω such that Z Ap(·) (f ) = |f (x)|p(x) dx < ∞. Ω

The set Lp(·) (Ω) equipped with the norm kf kp(·),Ω ≡ kf kLp(·) (Ω) = inf λ > 0 : Ap(·) (f /λ) ≤ 1 1, p(·)

becomes a Banach space. The Banach space W0 (Ω) with p(x) ∈ [p− , p+ ] ⊂ (1, ∞) is defined by  n o 1, p(·)  (Ω) = f ∈ Lp(·) (Ω) : |∇ f |p(x) ∈ L1 (Ω), u = 0 on ∂Ω ,  W0 X (5) kDi ukp(·),Ω + kukp(·),Ω .   kukW01,p(·) (Ω) = i

Throughout the paper we use the following properties of the functions from the 1,p(·) spaces W0 (Ω): 1, p(·) • if condition (4) is fulfilled, then C0∞ (Ω) is dense in W0 (Ω), and the space 1, p(·) W0 (Ω) can be defined as the closure of C0∞ (Ω) with respect to the norm (5) – see [27, 34, 35, 36]; • if p(x) ∈ C 0 (Ω), the the space W 1,p(·) (Ω) is separable and reflexive; • if 1 < q(x) ≤ sup q(x) < inf p∗ (x) with Ω

Ω

  p(x)n p∗ (x) = n − p(x)  ∞

if p(x) < n,

(6)

if p(x) > n,

1,p(·)

then the embedding W0 (Ω) ,→ Lq(·) (Ω) is continuous and compact; • it follows directly from the definition that p− p+ p− p+ min kf kp(·) , kf kp(·) ≤ Ap(·) (f ) ≤ max kf kp(·) , kf kp(·) ;

(7)

0

p • for all f ∈ Lp(·) (Ω), g ∈ Lp (·) (Ω) with p(x) ∈ (1, ∞), p0 = p−1 H¨older’s inequality holds, Z 1 1 |f g| dx ≤ + 0 − kf kp(·) kgkp0 (·) ≤ 2 kf kp(·) kgkp0 (·) . (8) p− (p ) Ω

2.2. Parabolic spaces Lp(·,·) (Q) and W(Q). Let p(z), z = (x, t) ∈ Q, satisfy condition (4) in the cylinder Q. For every fixed t ∈ [0, T ] we introduce the Banach space n o Vt (Ω) = u(x) : u(x) ∈ L2 (Ω) ∩ W01,1 (Ω), |∇u(x)|p(x,t) ∈ L1 (Ω) , kukVt (Ω) = kuk2,Ω + k∇ukp(·,t),Ω ,

1530


and denote by Vt0 (Ω) its dual. By W(Q) we denote the Banach space n o ( W(Q) = u : [0, T ] 7→ Vt (Ω)| u ∈ L2 (Q), |∇u|p(z) ∈ L1 (Q) ,

(9)

kukW(Q) = k∇ukp(·),Q + kuk2,Q . W0 (Q) is the dual of W(Q) (the space of linear functionals over W(Q)):  2 p0 (·)   w = (w0 , w1 , . . . , wn ), wZ0 ∈ L (Q), wi ∈ L !(Q), X w ∈ W0 (Q) ⇐⇒  w0 φ + wi Di φ dz.  ∀φ ∈ W(Q) hhw, φii = QT

i

The norm in W0 (Q) is defined by kvkW0 (Q) = sup hhv, φii| φ ∈ W(Q), kφkW(Q) ≤ 1 . Set

n o + V+ (Ω) = u(x)| u ∈ L2 (Ω) ∩ W01,1 (Ω), |∇u| ∈ Lp (Ω) .

Since V+ (Ω) is separable, it is a span of a countable set of linearly independent functions {ψk (x)} ⊂ V+ (Ω). We will need two elementary inequalities. Proposition 1 ([16]). For every p ≥ 2, |a| ≥ |b| ≥ 0 ||a|p−2 a − |b|p−2 b| ≤ C(p)|a − b|(|a| + |b|)p−2 . This proposition is an immediate byproduct of the easily verified relation 1 − tp−1 ≤ C(p)(1 − t)(1 + t)p−2

∀ p ≥ 2,

t ∈ [0, 1].

Proposition 2 ([16]). For 2 − p < β < 1 and |a| ≥ |b| ≥ 0 ||a|p−2 a − |b|p−2 b| ≤ C(p)|a − b|1−β (|a| + |b|)p−2+β . The assertion follows from the inequality 1 − tp−1 ≤ C(p)(1 − t)1−β (1 + t)p−2+β ,

t ∈ [0, 1]

with the same p and β. 3. Assumptions and results. The existence result is established for the problem ( ∂t Φ0 (z, v) = div b(z, v)|∇v + B(v)|p(z)−2 (∇v + B(v) + f in Q, (10) v(x, 0) in Ω, v = 0 on Γ with b, Φ0 , B defined in (2), (3). Problem (10) is formally equivalent to problem (1). Throughout the paper we assume that the coefficient a(z, r) and the exponents on nonlinearity p(z), α(z) satisfy the following conditions: • a(z, r) is a Carath´eodory function such that there exist constants a± such that ∀ z ∈ Q, r ∈ R

a− ≤ a(z, r) ≤ a+ < ∞,

(11) ±

• α(z), p(z) are measurable and bounded in Q, there exist constants α , p± such that − 1 < α− ≤ α(z) ≤ α+ < ∞, • the exponent γ(z) =

α(z) p(z)−1

1 < p− ≤ p(z) ≤ p+ < ∞,

α− + p− > 1,

(12)

satisfies

|∇γ(z)|p(z) ∈ L1 (Q),

∂t γ(z) ∈ L2 (Q).

The solution of problem (10) is understood in the following sense.

(13)


1531

Definition 3.1. A function v(z) is called weak solution of problem (10) if 1. v ∈ W(Q) ∩ L∞ (Q), ∂t Φ0 (z, v) ∈ W0 (Q), 2. for every φ ∈ W(Q) Z φ ∂t Φ0 (z, v) + b(z, v)|∇v + B(v)|p−2 (∇v + B(v)) · ∇φ − f φ dz = 0,

(14)

Q

3. ∀ φ(x) ∈ C0∞ (Ω) Z Z Φ0 (z, v(z)) φ(x) dx → Φ0 ((x, 0), v0 (x)) φ(x) dx Ω

as t → 0.

Ω

The main existence result is given in the following theorem. Theorem 3.2 ([11]). Let conditions (11), (12), (13), (4) be fulfilled. Then for every f ∈ L1 (0, T ; L∞ (Ω)), u0 , v0 ∈ L∞ (Ω) problem (10) has at least one weak solution v(z) in the sense of Definition 3.1. The uniqueness result is proved for the solutions satisfying the additional restriction, not included into Definition 3.1: it is required that ∂t Φ0 (z, v(z)) ∈ L1 (Q). In Section 7 we review the proof of Theorem 3.2 given in [11] and show that the class of uniqueness is nonempty, provided that the problem data possess some additional regularity. Another restriction is that either a(z, v) ≡ 1, or α(z) ≡ 0. In the latter case Φ0 (z, v) ≡ v and the equation transforms into the evolutional p(z)-Laplacian equation. Theorem 3.3. Let us assume that the data of problem (10) satisfy the conditions a(z, u) ≡ 1,

Φ0 (z, s) ∈ C 1 (Q × R).

Let conditions (12), (13) be fulfilled. Then for every weak solutions v1 , v2 , such that ∂t Φ0 (z, vi ) ∈ L1 (Q), and t ∈ (0, T ) kΦ0 (z, v1 (z)) − Φ0 (z, v2 (z))kL1 (Ω) ≤ kΦ0 (x, 0, v01 ) − Φ0 (x, 0, v02 )kL1 (Ω) + kf1 − f2 kL1 (Q) . Theorem 3.4. Let v1 , v2 be two weak solutions of problem (10) with α(z) ≡ 0. Let the coefficient a(z, s) be H¨ older-continuous with respect to s, |a(z, s) − a(z, r)| ≤ C |s − r|β ,

C = const,

β ∈ [1/2, 1].

If conditions conditions (11), (12) are fulfilled and ∂t ui ∈ L1 (Q), then for every t ∈ (0, T ) kv1 (x, t) − v2 (x, t)kL1 (Ω) ≤ kv01 − v02 kL1 (Ω) + kf1 − f2 kL1 (Q) . The uniqueness is proved in a narrower class of functions than the existence, but since the proofs of Theorems 3.3, 3.4 are practically independent on the proof of Theorem 3.2, the conditions on the exponents α(z), p(z) are less restrictive. For the sake of completeness of presentation, in the end of the paper we present the conditions on the data of problem (10) which guarantee that the corresponding solution satisfy the conditions of the comparison and uniqueness theorems.

1532


4. Formulas of integration by parts. Let ρ be the Friedrich’s mollifying kernel ( Z 1 κ exp − 1−|s| if |s| < 1, 2 ρ(s) = κ = const : ρ(z) dz = 1. Rn+1 0 if |s| > 1, Given a function v ∈ L1 (QT ), we extend it to the whole Rn+1 by a function with compact support (keeping the same notation for the continued function) and then define Z s 1 , h > 0. vh (z) = v(s)ρh (z − s) ds with ρh (s) = n+1 ρ h h Rn+1 Lemma 4.1. If u ∈ W(QT ) with the exponent p(z) satisfying (4) in Q, then kuh kW(Q) ≤ C 1 + kukW(Q) and kuh − ukW(Q) → 0 as h → 0. Lemma 4.1 is an immediate byproduct of [36, Theorem 2.1]. Lemma 4.2 ([10]). Let in the conditions of Proposition 4.1 ut ∈ W0 (Q). Then (uh )t ∈ W0 (Q), and for every ψ ∈ W(Q) hh(uh )t , ψii → hhut , ψii as h → 0. Lemma 4.3 (Integration by parts). Let v, w ∈ W(Q) and vt , wt ∈ W0 (Q) with the exponent p(z) satisfying (4) in Q. Then Z t2 Z Z t2 Z Z t=t2 . ∀ a.e. t1 , t2 ∈ (0, T ] v wt dz + vt w dz = v w dx t1

Ω

t1

Ω

t=t1

Ω

Proof. Let t1 < t2 . Take   0 for t ≤ t1 ,      k(t − t ) for t1 ≤ t ≤ t1 + k1 , 1  χk (t) = 1 for t1 + k1 ≤ t ≤ t2 − k1 ,    k(t2 − t) for t2 − k1 ≤ t ≤ t2 ,    0 for t ≥ t2 . For every k ∈ N and h > 0 Z Z Z 0= (vh wh χk )t dz ≡ (vh wh )t χk dz − k Q

θ

1 θ− k

Q

Z Ω

(15)

θ=t2 vh wh dz

1 θ=t1 + k

.

The last two integrals on the right-hand side exist because vh , wh ∈ L2 (Q). Letting h → 0, we obtain the equality Z Z t2 Z Z t1 + k1 Z lim (vh (wh )t + (vh )t wh ) χk (t) dz = k v w dz − k v w dz. h→0

1 t2 − k

Q

Ω

t1

Ω

According to Lemmas 4.1, 4.2 vh → v in W(Q), (wh )t = (wt )h → wt weakly in W0 (Q) as h → 0, and kvkW , k(wh )t kW0 are uniformly bounded. It follows that Z Z lim vh (wh )t χk (t) dz = lim (vh − v)(wh )t χk (t) dz h→0 Q h→0 Q Z Z Z + lim v ((wh )t − wt )χk (t) dz + v wt χk (t) dz = v wt χk (t) dz. h→0

Q

Q

Q

In the same way we check that Z Z lim (vh )t wh χk (t) dz = v wt χk (t) dz. h→0

Q

Q


1533

By the Lebesgue differentiation theorem Z θ Z Z ∀ a.e. θ > 0 lim k v w dx dt = v w dx, k→0

1 θ− k

Ω

Ω

whence for almost every t1 , t2 ∈ [0, T ] Z t2 Z Z (v wt + vt w) dz = lim (v wt + vt w)χk (t) dz t1

k→∞

Ω

Q

Z

θ

Z

= lim k k→∞

1 θ− k

Ω

Z t=t2 t=t2 . = v w dx v w dx θ=t1

Ω

θ=t1

Corollary 1. Let u ∈ W(Q) and ut ∈ W0 (Q) with the exponent p(z) satisfying (4). Then Z t2 Z t=t2 1 . ∀ a.e. t1 , t2 ∈ (0, T ] u ut dz = kuk22,Ω 2 t=t1 Ω t1 Lemma 4.4. Let u ∈ W(Q) ∩ L∞ (Q), ut ∈ W0 (Q), and let the exponent p(z) satisfy (4). Introduce the function Z u γ(z) v= ( + |s|) ds, > 0, 0

with the exponent γ(z) ≥ γ − > −1 such that γt ∈ L2 (Q) and |∇γ(z)|p(z) ∈ L1 (Q). For a.e. t1 , t2 ∈ [0, T ] Z Z t2 Z Z t=t2 Z t2 Z u v t=t2 uv v ut v dz = + dx γt dz + dx γ + 2 γ + 2 γ + 2 t=t t=t1 1 Ω Ω Ω t1 t1 Ω Z t2 Z Z u γt (16) + ( + |s|)γ ln ( + |s|) ds dz γ +2 0 t1 Ω Z t2 Z γt − v dz ≡ µ (u, v). 2 t1 Ω (γ + 2) Proof. Let uh ∈ C ∞ (Q) be the mollification of u ∈ W(Q) and Z uh sign uh γ(z) vh = ( + |s|) ds ≡ ( + |uh |)γ+1 − γ+1 . γ+1 0 Since u and uh are bounded by a constant 1 + K0 , and γ(z) ≥ γ − > −1, it follows from Propositions 1, 2 that n o − |vh − v| ≤ C max |vh − v|, |vh − v|1+min{0, γ } , C ≡ C(, p± , α± , K0 ). The inclusion u ∈ L∞ (Q) entails the convergence kvh − vkLr (Q) → 0 as h → 0 for every r > 1. Explicitly calculating the primitive, in the same way we check that for every r > 1

Z uh

γ(z)

( + |s|) ln ( + |s|) ds → 0 as h → 0.

u

Lr (Q)

1534


k (t) with the function χk introduced in (15). Following the proof of Let ψk (z) = χγ+2 Lemma 4.3, we find: Z θ Z θ=t2 Z t2 Z uh vh χk (t)(uh )t vh dz = k dt dx 1 γ + 2 θ=t1 t1 Ω θ− k Ω Z t2 Z uh v h − γt χk (t) dz t1 Ω γ+2 (17) Z uh Z t2 Z γt χk (t) ( + |s|)γ ln ( + |s|) ds dz − γ+2 0 t1 Ω Z θ Z t2 Z Z θ=t2 γt vh . dz + k dx + χk (t) vh dt 2 1 (γ + 2) θ=t1 θ− k t1 Ω Ω γ+2

Since u ∈ W(Q) ∩ L∞ (Q) and γ − > −1, v ∈ W(Q) for every > 0. Indeed: since kukL∞ (Q) ≤ M , we have the estimates

Z

|u|

kvkL∞ (Q) ≤ M1 (γ ± , M ), ( + s)γ | ln ( + s)| ds ≤ M2 (γ ± , M ),

0

∞ L

(Q)

which provide the inequality γ(z)

|∇v| ≤ ( + |u|)

|u|

Z

( + s)γ(z) | ln ( + s)| ds a.e. in Q

|∇u| + |∇γ| 0

and the inclusion |∇v(z)|p(z) ∈ L1 (Q). By Lemma 4.1 kvh kW(Q) ≤ C (1 + kvkW(Q) )

and kvh − vkW(Q) → 0

as h → 0.

We may now pass to the limit as h → 0 in every term of (17), following the proof of Lemma 4.3: Z θ Z t2 Z Z θ=t2 Z t2 Z uv uv k = χk (t)ut v dz − dt dx γt χk (t) dz 1 γ + 2 γ +2 θ=t 1 θ− k t1 Ω t1 Ω Ω Z t2 Z Z u γt − χk (t) ( + |s|)γ ln ( + |s|) ds dz γ+2 0 Ω t1 Z θ Z Z t2 Z θ=t2 γt v . dz + k dt dx + χk (t) v 2 1 (γ + 2) γ + 2 θ=t1 θ− k t1 Ω Ω Letting k → ∞ and applying the Lebesgue differentiation theorem, we arrive at (16). γ

|u| Remark 1. Let = 0, u ∈ W(Q), ut ∈ W0 (Q), and let v = uγ+1 ∈ W(Q). Under the foregoing conditions on the exponents p(z) and γ(z) the following formula of integration by parts holds: ∀ a.e. t1 , t2 ∈ [0, T ] Z t2 Z Z t=t2 Z t2 Z u v uv ut v dz = dx + γt dz ≡ µ(u, v). γ + 2 t=t1 t1 Ω Ω t1 Ω γ+2

Let us introduce the function space V(Q) ≡ v(z) : v ∈ W(Q) ∩ L∞ (Q), ∂t Φ0 (z, v) ∈ L1 (Q) ∩ W0 (Q) . with Φ0 defined in (2) and define the functions s Tδ (s) = √ , δ > 0, 2 δ + s2


1535

φk,δ,θ (z) = χk,θ (t) Tδ (v(z))

(18)

and with   0      k t χk,θ (t) = 1    k (θ − t)    0

for for for for for

t ≤ 0, 0 ≤ t ≤ k1 , 1 1 k ≤ t ≤ θ − k, 1 θ − k ≤ t ≤ θ, t ≥ θ,

1 < θ ≤ T. k

It is easy to see that Tδ0 (s) =

Tδ (s) → sign s as δ → 0,

δ2

−1 ≤ sTδ0 (s) ≤ 1 for s ∈ R.

> 0,

3

(δ 2 + s2 ) 2

Lemma 4.5. Let vi ∈ V(Q), v = v1 − v2 and w = w1 − w2 ≡ Φ0 (z, v1 ) − Φ0 (z, v2 ). For a.e. θ ∈ (0, T ) there exists the limit Z Z t=θ φk,δ,θ ∂t w dz = |w| dx . lim lim δ→0 k→∞

Q

t=0

Ω

Proof. From now on, we will denote Qτ = Q ∩ {t < τ },

τ ∈ [0, T ).

∞

Since w ∈ L (Q) and φ = φk,δ,θ are uniformly bounded, it follows from the dominated convergence theorem that Z Z χk,θ (t) Tδ (v) ∂t w dz → Tδ (v) ∂t w dz as k → ∞, Qθ = Q ∩ {t < θ}, Q

Qθ

and, because sign v = sign w, Z Z φk,δ,θ ∂t w dz = Tδ (v) ∂t w dz lim k→∞ Q Q Z Z → sign v ∂t w dz ≡ Qθ

sign w ∂t w dz = J

as δ → 0.

Qθ

On the other hand, repeating the same arguments with the test-function φk,δ,θ ≡ χk,θ (t) Tδ (w), we find that Z J = lim lim Tδ (w) χk,θ (t) ∂t w dz. δ→0 k→∞

Q

The straightforward computation shows that Z Z Z Tδ (w) χk,θ (t) ∂t w dz = χk,θ (t) ∂t Q

Q θ

Z Z

=k

dt θ−1/k

Z

Ω 1/k

−k w

Tδ (s) ds = 0

p

w

Tδ (s) ds dx

0

Z Z dt

0

Z

Tδ (s) ds dz

0

Z

where

w

δ 2 + w2 − δ →

Ω

√

w

Tδ (s) ds dx,

0

w2 = |w| as δ → 0.

1536


Letting k → ∞, δ → 0 and applying the Lebesgue differentiation theorem, we find that for a.e. θ ∈ (0, T ) Z p Z Z Z w t=θ t=θ = Tδ (s) ds dx Tδ (w) χk,θ (t) ∂t w dz = δ 2 + w2 dx t=0 t=0 Ω 0 Q Ω Z t=θ →J = |w| dx . t=0

Ω

5. Proof of Theorem 3.3. Let vi ∈ V(Q) be two bounded weak solutions of problem (3) with the data (fi , v0i ), i = 1, 2. Introduce the functions w = Φ0 (z, v1 ) − Φ0 (z, v2 ),

v = v1 − v2 ,

p(z)−2

F (s) = |B(v) + ∇v|

(B(v) + ∇v) .

By (14) for every test-function φ ∈ W(Q) Z Z φ ∂t w + (F (v1 ) − F (v2 )) · ∇φ dz = (f1 − f2 ) φ dz. Q

(19)

Q

Taking for the test-function φk,δ,θ defined in (18) and applying Lemma 4.5 we have that for a.e. θ ∈ (0, T ) there exists the limit of the first term on the left-hand side of (19): Z Z t=θ φk,δ,θ ∂t w dz → |w| dx as k → ∞, δ → 0. (20) Q

Ω

t=0

On the other hand, the rest of the terms in (19) are continuous functions of θ because of the property of absolute continuity of the integral. It follows that (20) is true for all t ∈ [0, T ]. The second term on the left-hand side of (19) with φ(z) = χk,θ (t) Tδ (v(z)) is represented in the form Z Z I2 = (F (v1 ) − F (v2 )) · ∇φ dz = χk,θ (F (v1 ) − F (v2 )) ∇Tδ (v) dz Q Q Z (21) = χk,θ Tδ0 (v) (F (v1 ) − F (v2 )) ∇v dz. Q

Let us denote ζi = ∇vi + B(vi ),

i = 1, 2,

so that ∇vi = ζi − B(vi ),

F (vi ) = |ζi |p(z)−2 ζi ,

0

ζi = |F (vi )|p (z)−2 F (vi )

(recall that B(s) is defined in (3)). Passing to the limit as k → ∞, for every fixed δ and θ we obtain the equality Z lim I2 = Tδ0 (v) F (v1 ) − F (v2 ) · ∇v dz k→∞ Q Z θ = Tδ0 (v) F (v1 ) − F (v2 ) (ζ1 − ζ2 ) dz Qθ Z − Tδ0 (v) F (v1 ) − F (v2 ) (B(v1 ) − B(v2 )) dz Qθ

≡ J1 (δ) − J2 (δ).


1537

Making use of the well-known inequality ∀ ξ, η ∈ Rn |ξ|p−2 ξ − |η|p−2 η (ξ − η) ≥

 −p p  2 |ξ − η|  (p − 1)

if 2 ≤ p < ∞, |ξ − η|2

(|ξ|p + |η|p )

2−p p

if 1 < p < 2,

we may write Z

Tδ0 (v) |ζ1 |p(z)−2 ζ1 − |ζ2 |p(z)−2 ζ2 (ζ1 − ζ2 ) dz Qθ Z − 0 p(z) Tδ0 (v) |F (v1 ) − F (v2 )| ≥ 2−(p ) dz

J1 (δ) =

Qθ ∩{p(z)≥2}

Z

+ (p− − 1)

Tδ0 (v) |F (v1 ) − F (v2 )|

2

Qθ ∩{p(z)∈(1,2)}

× |F (v1 )|p(z) + |F (v2 )|p(z)

p(z)−2 p(z)

dz.

Next, Z

Tδ0 (v) |F (v1 ) − F (v2 )| |B(v1 ) − B(v2 )| dz

J2 (δ) ≤ Qθ ∩{z: p(z)≥2}

Z

. . . ≡ J (1) (δ) + J (2) (δ).

+ Qθ ∩{z: 1