Parabolic equations with double variable nonlinearities

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Mathematics and Computers in Simulation 81 (2011) 2018–2032

Parabolic equations with double variable nonlinearities S. Antontsev a,1 , S. Shmarev b,∗,2 b

a CMAF, University of Lisbon, Portugal Department of Mathematics, University of Oviedo, Spain

Received 15 November 2009; received in revised form 26 May 2010; accepted 7 December 2010 Available online 21 December 2010

Abstract The paper is devoted to the study of the homogeneous 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 existence of bounded weak solutions in suitable Sobolev–Orlicz spaces. © 2011 IMACS. Published by Elsevier B.V. All rights reserved. MSC: 35K57; 35K65; 35B40 Keywords: Parabolic equation; Double nonlinearity; Variable nonlinearity; Nonstandard growth conditions

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) z = (x, t) ∈ Q = Ω × (0, T ], u(x, 0) = u0 (x) in Ω, u = 0 on Γ = ∂Ω × [0, T ].

(1.1)

Eq. (1.1) is formally parabolic, but it may degenerate or become singular at the points where u = 0 or | ∇ u| = 0. Introducing the functions α(z) γ(z) = , p(z) − 1

 v(z) = 0

u

|s|

γ(z)

u|u|γ(z) ds = , γ(z) + 1

−γ

u(z) = Φ0 (z, v) =

∗

|v| 1+γ v 1

(1 + γ) 1+γ

,

(1.2)

Corresponding author. E-mail address: [email protected] (S. Shmarev). 1 The author was partially supported by FCT, Financiamento Base 2008-ISFL-1-209, Portugal, and by the research project MTM2008-06208 of the Ministerio de Ciencia e Innovacion, Spain. 2 The author acknowledges the support of the research project MTM2007-65088, Spain. 0378-4754/$36.00 © 2011 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2010.12.015

S. Antontsev, S. Shmarev / Mathematics and Computers in Simulation 81 (2011) 2018–2032

we write problem (1.1) in the following form: ⎧   ⎪ ∂t Φ0 (z, v) = div b(z, v)|∇v + B(z, v)|p(z)−2 (∇v + B(z, v) + f ⎪ ⎨ u0 |u0 |γ(x,0) ⎪ ⎪ in Ω, ⎩ v(x, 0) ≡ v0 (x) = 1 + γ(x, 0)

2019

in Q, (1.3)

v = 0 on Γ

with

 b(z, v) ≡ a(z, Φ0 (z, v)), B(z, v) = −∇γ(z)

u(z)

|s|γ(z) ln |s| ds.

0

Problem (1.3) will be the subject of the further study. Equations of the types (1.1) and (1.3) 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 [2,5,14,15,21,34,6,27,28] and the further references therein. The questions of existence and uniqueness of solutions to equations like (1.1) and (1.3) with constant exponents of nonlinearity α and p were studied by many authors – see [14,17,23,31,33] for equations of the type (1.1) and [14,15,20] for the equations of the type (1.3) with constant exponent p. In the present work we prove the existence theorem for the Dirichlet problem (1.3) in which the exponents α and p are allowed to be variable. 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 [1,3,4,8–10,12], see also [7] for a review of the results concerning elliptic equations with variable nonlinearity. The notion of solution to problem (1.3) is introduced in Definition 3.1 below: this is the so-called energy solution. Another notion of solution was adopted in [1] (see Remark 4.1 below). Relations between different definitions of solution to the evolution p(x,t)-Laplace equation in dependence of the regularity of the variable exponent p(x, t) are discussed in [1]. Existence and uniqueness of the renormalized and entropy solutions to elliptic equations of the p(x)Laplace type and the evolution p(x)-Laplace equation are proved in [13,35,36]. The question of existence of such solutions is still open for evolution p-Laplace equation with the exponent p depending on t, as well as for the doubly nonlinear elliptic and parabolic equations with variable exponents of nonlinearity. The paper is organized as follows. In Section 2 we collect some known facts from the theory of Orlicz–Sobolev spaces and prove several auxiliary assertions. In particular, we derive the formula of integration by parts, which is required to deal with the nonlinear term ∂t Φ0 (z, v). The precise assumptions on the data and the main result are given in Section 3. A solution to problem (1.3) is obtained as the limit of solutions of the regularized problems for the evolution p(z)-Laplace equations already studied in [1,10]. In Section 4 we define the sequence of regularized problems, whose solutions belong to suitable Orlicz–Sobolev spaces, and derive a series of a priori estimates. In Section 5 we prove, relying on the a priori estimates of Section 4, that the constructed sequence converges to a weak solution of problem (1.3). The authors would like to express their gratitude to Professor M. Chipot for stimulating discussions of this work. 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 [16,18,22,25]. The further references can be found in the review papers [19,29]. Let Ω ⊂ Rn be 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 < ∞. (2.1) |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  Ap(·) (f ) = |f (x)|p(x) dx < ∞. Ω

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The set Lp(·) (Ω) equipped with the norm

f p(·),Ω ≡ f Lp(·) (Ω) = inf λ > 0 : Ap(·) (f/λ) ≤ 1 (Ω) with p(x) ∈ [p− , p+ ] ⊂ (1, ∞ ) is defined by becomes a Banach space. The Banach space W0 ⎧

1,p(·) ⎪ (Ω) = f ∈ Lp(·) (Ω) : |∇f |p(x) ∈ L1 (Ω), u = 0 on ∂Ω , ⎨ W0  ||Di u||p(·),Ω + ||u||p(·),Ω . ⎪ ⎩ ||u||W01,p(·) (Ω) = 1,p(·)

(2.2)

i

1,p(·)

Throughout the paper we use the following properties of the functions from the spaces W0 • if condition (2.1) is fulfilled, then C0∞ (Ω) is dense in W0 (Ω), and the space W0 closure of C0∞ (Ω) with respect to the norm (2.2) – see [30,37–39]; • if p(x) ∈ C0 (Ω), then the space W1,p(·) (Ω) is separable and reflexive; • if 1 < q(x) ≤ sup q(x) < inf p∗ (x) with 1,p(·)

1,p(·)

(Ω):

(Ω) can be defined as the

Ω

Ω

⎧ ⎨ p(x)n p∗ (x) = n − p(x) ⎩ ∞

if p(x) < n,

(2.3)

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 f p(·) , f p(·) ≤ Ap(·) (f ) ≤ max f p(·) , f p(·) ;

(2.4)



p • for all f ∈ Lp(·) (Ω), g ∈ Lp (·) (Ω) with p(x) ∈ (1, ∞), p = p−1 Hölder’s inequality holds, 

 1 1 f p(·) gp (·) ≤ 2f p(·) gp (·) . |fg|dx ≤ + p− (p − ) Ω

(2.5)

2.2. Parabolic spaces Lp(,) (Q) and W(Q) Let p(z), z = (x, t) ∈ Q, satisfy condition (2.1) in the cylinder Q. For every fixed t ∈ [0, T] we introduce the Banach space Vt (Ω) = {u(x) : u(x) ∈ L2 (Ω) ∩ W01,1 (Ω), uVt (Ω) = u2,Ω + ∇up(·,t),Ω ,

|∇u(x)|p(x,t) ∈ L1 (Ω)},



and denote by Vt (Ω) its dual. By W(Q) we denote the Banach space ⎧ ⎨ W(Q) = {u : [0, T ] → Vt (Ω)|u ∈ L2 (Q), |∇u|p(z) ∈ L1 (Q), u = 0 on Γ }, ⎩ u

W(Q)

(2.6)

= ∇up(·),Q + u2,Q .

W (Q)

is the dual of W(Q) (the space of linear functionals over W(Q): ⎧ n  ⎪ ⎪ w = w + Di wi , w0 ∈ L2 (Q), wi ∈ Lp (·) (Q), ⎪ 0 ⎪ ⎨ i=1  w ∈ W (Q) ⇔   ⎪  ⎪ ⎪ ⎪ w0 φ + wi Di φ dz. ⎩ ∀φ ∈ W(Q)w, φ = QT

i

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2021

The norm in W (Q) is defined by

||v||W (Q) = sup v, φ|φ ∈ W(Q), φW(Q) ≤ 1 . Set

  + 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). Without loss of generality, we may assume that this system forms an orthonormal basis of L2 (Ω). Lemma 2.1 ([10]). Let conditions (2.1) hold. Then the set {ψk } is dense in Vt (Ω) for every t ∈ [0, T]. Lemma 2.2 ([10]). For every u ∈ W(Q) there is a sequence {dk (t)}, dk (t) ∈ C1 [0, T], such that ||u −

m 

dk (t)ψk (x)W(QT ) → 0

as m → ∞.

k=1

2.3. Formulas of integration by parts Let ρ be the Friedrichs mollifying kernel

 ⎧ ⎨ κ exp − 1 if |s| < 1, 1 − |s|2 ρ(s) = ⎩ 0 if |s| > 1,

 κ = const :

Rn+1

ρ(z)dz = 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  s 1 , h > 0. v(s)ρh (z − s)ds with ρh (s) = n+1 ρ vh (z) = h h Rn+1 Lemma 2.3. If u ∈ W(QT ) with the exponent p(z) satisfying (2.1) in Q, then   uh W(Q) ≤ C 1 + uW(Q) and uh − uW(Q) → 0 as h → 0.

Lemma 2.3 is an immediate byproduct of [39,Theorem 2.1]. Lemma 2.4 ([10]). Let in the conditions of Proposition ut ∈ W (Q). Then (uh )t ∈ W (Q), and for every ψ ∈ W(Q) (uh )t , ψ → ut , ψ as h → 0. Lemma 2.5 (Integration by parts). Let v, w ∈ W(Q) and vt , wt ∈ W (Q) with the exponent p(z) satisfying (2.1) in Q. Then  t2   t2   2 ∀a.e. t1 , t2 ∈ (0, T ] vwt dz + vt wdz = vwdx|t=t t=t1 . t1

Ω

t1

Ω

Ω

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Proof. Let t1 < t2 . Take ⎧ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k(t − t1 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ χk (t) = 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k(t2 − t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0

for t ≤ t1 , 1 for t1 ≤ t ≤ t1 + , k for t1 +

1 1 ≤ t ≤ t2 − , k k

for t2 −

1 ≤ t ≤ t2 , k

(2.7)

for t ≥ t2 .

For every k ∈ N and h > 0    (vh wh χk )t dz ≡ (vh wh )t χk dz − k 0= QT

θ



θ− 1k

Q

Ω

2 vh wh dz|θ=t θ=t1 .

The last two integrals on the right-hand side exist because vh , wh ∈ L2 (QT ). Letting h → 0, we obtain the equality   t2   t1 + 1  k lim (vh (wh )t + (vh )t wh )χk (t)dz = k vwdz − k vwdz. h→0 Q

t2 − 1k

Ω

t1

Ω

to Lemmas 2.3 and 2.4 vh → v in W(Q), (wh )t = (wt )h → wt weakly in W (QT ) as h → 0, and vW ,   According (wh )t  are uniformly bounded. It follows that W   lim vh (wh )t χk (t)dz = lim (vh − v)(wh )t χk (t)dz h→0 h→0 Q Q    v((wh )t − wt )χk (t)dz + vwt χk (t)dz = vwt χk (t)dz. + lim h→0 Q

Q

Q

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

Q

By the Lebesgue differentiation theorem   θ   ∀a.e.θ > 0 lim k vwdx dt = vwdx, k→0

θ− 1k

Ω

Ω

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

Ω

k→∞ Q

k→∞

θ−





θ

1 k

Ω

2 vwdx|t=t θ=t1

Corollary 2.1. Let u ∈ W(Q) and ut ∈ W (Q) with the exponent p(z) satisfying (2.1). Then  t2  1 2 uut dz = u22,Ω |t=t ∀a.e. t1 , t2 ∈ (0, T ] t=t1 . 2 t1 Ω We will need two elementary inequalities. Proposition 2.1 ([14]). For every p ≥ 2, |a| ≥ |b| ≥ 0 ||a|p−2 a − |b|p−2 b| ≤ C(p)|a − b|(|a| + |b|)p−1 .

= Ω

2 vwdx|t=t θ=t1 .

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2023

This proposition is an immediate byproduct of the easily verified relation 1 − t p−1 ≤ C(p)(1 − t)(1 + t)p−1

∀p ≥ 2,

t ∈ [0, 1].

Proposition 2.2 ([14]). 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 − t p−1 ≤ C(p)(1 − t)1−β (1 + t)p−2+β ,

t ∈ [0, 1]

with the same p and β. Lemma 2.6. Let u ∈ W(Q) ∩ L∞ (Q), ut ∈ W (Q), and let the exponent p(z) satisfy (2.1). Introduce the function  u (ε + |s|)γ(z) ds, ε > 0, v= 0

with the exponent γ(z) ≥ γ − > − 1 such that γ t ∈ L2 (Q) and | ∇ γ(z)|p(z) ∈ L1 (Q). For a.e. t1 , t2 ∈ [0, T]  t2    t2   uv uv v t=t2 ut vdz = dx|t=t1 + γt dz + ε dx|t=t2 γ + 2 γ + 2 γ + 2 t=t1 t1 Ω Ω t1 Ω Ω (2.8)  t2   u  t2  γt γt γ +ε (ε + |s|) ln(ε + |s|)dsdz − ε v dz ≡ με (u, v). 2 t1 Ω γ +2 0 t1 Ω (γ + 2) Proof. Let uh ∈ C∞ (Q) be the mollification of u ∈ W(Q) and  uh  signuh  (ε + |s|)γ(z) ds ≡ vh = (ε + |uh |)γ+1 − εγ+1 . γ +1 0 Since u and uh are bounded by a constant K0 , and γ(z) ≥ γ − > − 1, it follows from Lemmas 2.3 and 2.4 that   − |vh − v| ≤ C max |uh − u|, |uh − u|1+min{0,γ } , C ≡ C(ε, p± , α± , K0 ). The inclusion u ∈ L∞ (Q) entails the convergence ||vh − v||Ls (Q) → 0 as h → 0 for every s > 1. Explicitly calculating the primitive, in the same way we check that for every s > 1  u   h  γ(z)   (ε + |s|) ln(ε + |s|)ds → 0 as h → 0.   Ls (Q)

u

k (t) Let ψk (z) = χγ+2 with the function χk introduced in (2.7). Following the proof of Lemma 2.5, we find:  t2   t2   θ  uh v h uh v h θ=t2 dx| γt χk (t)dz = χ (t)(u ) v dz − k dt k h t h 1 θ=t1 γ + 2 γ +2 θ− Ω t1 Ω t1 Ω k  t2   uh γt −ε χk (t) (ε + |s|)γ ln(ε + |s|)dsdz γ +2 0 t1 Ω

 +ε

t2

 χk (t)vh

t1

Ω

γt dz + εk (γ + 2)2





θ

θ−

1 dt k

Ω

(2.9)

vh dx|θ=t2 . γ + 2 θ=t1

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



|v| ≤ M1 (γ , M), 0

|u|

(ε + |s|)γ |ln(ε + |s|)|ds ≤ M2 (γ ± , M),

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which provide the inclusion



|∇v| ≤ (ε + |u|)γ(z) |∇u| + |∇γ|

|u|

(ε + |s|)γ(z) | ln(ε + |s|)|ds ∈ Lp(·) (Q).

0

By Lemma 3 vh W(Q) ≤ C(1 + vW(Q) ) and vh − vW(Q) → 0

as h → 0.

We may now pass to the limit as h → 0 in every term of (2.9), following the proof of Lemma 2.5:  θ  t2   t2   uv uv 2 k dx|θ=t γt χk (t)dz = χ (t)u vdz − t k 1 dt θ=t1 θ− Ω γ +2 Ω Ω γ +2 t1 t1 k  t2   u γt −ε χk (t) (ε + |s|)γ ln(ε + |s|)dsdz γ + 2 t1 0 Ω  θ  t2   γt v 2 χk (t)v dz + εk dx|θ=t +ε 1 dt θ=t1 . 2 γ + 2 (γ + 2) θ− t1 Ω Ω k Letting k → ∞ and applying the Lebesgue differentiation theorem, we arrive at (2.8). γ

Remark 2.1. Let ε = 0, u ∈ W(Q), ut ∈ W (Q), and let v = u|u| γ+1 ∈ W(Q). Under the foregoing conditions on the exponents p(z) and γ(z) the following formula of integration by parts holds:  t2    t2  uv uv 2 + ∀a.e. t1 , t2 ∈ [0, T ] dx|t=t γt dz ≡ μ(u, v). ut vdz = (2.10) t=t1 Ω Ω γ +2 Ω γ +2 t1 t1 3. Assumptions and results The existence result is established for the problem ⎧   ⎨ ∂t Φ0 (z, v) = div b(z, v)|∇v + B(z, v)|p(z)−2 (∇v + B(z, v) + f ⎩ v(x, 0) in Ω,

v=0

in Q, (3.1)

on Γ,

which is formally equivalent to problem (1.1). 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 Carathéodory function, there exists positive constants a± such that ∀z ∈ Q, r ∈ Z a− ≤ a(z, r) ≤ a+ < ∞, • α(z), p(z) are measurable and bounded in Q, there exist constants

(3.2) α± , p±

such that

−1 < α− ≤ α(z) ≤ α+ < ∞, 1 < p− ≤ p(z) ≤ p+ < ∞, α− + p− > 1, • the exponent γ(z) =

α(z) p(z)−1

|∇γ(z)|p(z) ∈ L1 (Q),

(3.3)

satisfies ∂t γ(z) ∈ L2 (Q),

(3.4)

The solution of problem (3.1) is understood in the following sense. Definition 3.1. A function v(z) is called weak solution of problem (3.1) if 1. 1. v ∈ W(Q) ∩ L∞ (Q), ∂t Φ0 (z, v) ∈ W (Q), 2. for every φ ∈ W(Q)    φ∂t Φ0 (z, v) + b(z, v)|∇v + B(z, v)|p−2 (∇v + B(z, v)) · ∇φ − fφ dz = 0, Q

(3.5)

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3. ∀φ(x) ∈ C0∞ (Ω)



Ω Φ0 (z, v(z))φ(x)dx

→



Ω Φ0 ((x, 0), v0 (x))φ(x)dx

2025

as t → 0.

The main result of the paper is given in the following theorem. Theorem 3.1. Let the exponent p(z) satisfy the log-continuity condition conditions (2.1) in Q, and let conditions (3.2), (3.3), (3.4) be fulfilled. Then for every f ∈ L1 (0, T ; L∞ (Ω)), u0 , v0 ∈ L∞ (Ω) problem (3.1) has at least one weak solution v(z) in the sense of Definition 3.1. 4. Regularization of problem (3.1) A solution of problem (3.1) has the form v = Φ−1 0 (z, u) where u is obtained as the limit of the sequence of solutions of the regularized problems ⎧   ⎨ ut = div Aε,K (z, u)|∇u|p(z)−2 ∇u + f (z) in Q, (4.1) ⎩ u(x, 0) = u in Ω, u = 0 on Γ 0 with the coefficient Aε,K (z, u) = a(z, u)(ε + min{K, |u|})α(z) , depending on the given parameters ε ∈ (0, 1), K > 0. For every ε ∈ (0, 1) and 1 < K < ∞ the coefficient Aε,K (z, u) is separated away from zero and infinity which allows us to treat problem (4.1) as the Dirichlet problem for the evolutional p(z)-Laplacian. Definition 4.1. ([10]). A function u ∈ L∞ (0, T ; L2 (Ω)) ∩ W(Q) is called weak solution of problem (4.1) if for every test-function φ ∈ L∞ (0, T ; L2 (Ω)) ∩ W(Q) such that φt ∈ W (Q) and for every t1 , t2 ∈ [0, T]   t2    t=t2 uφt − Aε,K (z, u)|∇u|p(z)−2 ∇u · ∇φ + fφ dz. uφdx|t=t1 = (4.2) Ω

t1

Ω

Theorem 4.1. For every u0 ∈ L2 (Ω), f ∈ L2 (Q), ε > 0, K > 0 problem (4.1) has at least one weak solution u ∈ L∞ (0, T ; L2 (Ω)) ∩ W(Q) such that ut ∈ W (Q). Moreover, if u0 ∈ L∞ (Ω), f ∈ L1 (0, T ; L∞ (Ω)), then this solution belongs to L∞ (Q) and obeys the estimate T u∞,Q ≤ u0 ∞,Ω +

f (·, s)∞,Ω ds ≡ K0 .

(4.3)

0

Sketch of proof. The detailed proof of Theorem  4.1 can be found in [10]. The solution is thought as the limit of the sequence of Galerkin’s approximations u(m) = m 1 ci,m (t)ψi (x), where {ψ i } is a basis of the space V+ (Ω). We may assume that the system {ψi } is orthogonal in L2 (Ω). Substituting u(m) into (4.2) with the test-functions ψi , i = 1, . . ., m, we obtain a system of ordinary differential equations for the coefficients ci,m (t). This system can be solved on the interval [0, T], and the functions u(m) admit the uniform estimate    ||u(m) ||L∞ (0,T ;L2 (Ω)) + |∇u(m) |p(z) dz ≤ C ||u0 ||22,Ω + ||f ||22,Q , Q

which allows one to pass to the limit in the sequence {u(m) }. The justification of the limit passage as m → ∞ is based on the monotonicity of the operator |s|p−2 s : Rn → Rn and the formulas of integration by parts in t for the elements of the spaces W(Q), W (Q) (see Lemma 2.5). Corollary 4.1. According to (4.3), for u0 ∈ L∞ (Ω), f ∈ L1 (0, T ; L∞ (Ω)) and K > K0 the solutions of the regularized problem (4.1) do not depend on the parameter K and, in fact, are solutions of the problem ⎧   ⎪ ∂t uε = div Aε (z, uε )|∇uε |p(z)−2 ∇uε + f (z), for z ∈ Q, ⎪ ⎪ ⎨ uε (x, 0) = u0 in Ω, uε = 0 on Γ, (4.4) ⎪ α(z) ⎪ γ(z)(p(z)−1) ⎪ , ε > 0, γ(z) = . ⎩ Aε (z, uε ) ≡ Aε,∞ = a(z, uε )(ε + |uε |) p(z) − 1

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Corollary 4.2. It follows from Theorem 4.1 and Lemma 2.5 that if a function u ∈ L∞ (0, T ; L2 (Ω)) ∩ W(Q) is a weak solution of problem (4.1) in the sense of Definition 4.1, then ∂t u ∈ W ’ (Q) and for every test-function φ ∈ W(Q)    φ∂t u + Aε,K (z, u)|∇u|p(z)−2 ∇u · ∇φ − fφ dz = 0. (4.5) Q

Taking for the test-function φ(x) ∈ C0∞ (Ω) and applying Lemma 2.5 to the first term on the left-hand side of (4.5), we find that for every t1 , t2 ∈ [0, T]   t2    Aε,K (z, u)|∇u|p(z)−2 ∇u · ∇φ(x) − fφ(x) dz, uφ(x)dx|tt21 = Ω

t1

Ω

which yields   lim uφ(x)dx = u0 (x)φ(x)dx t→0 Ω

∀φ(x) ∈ C0∞ (Ω)

Ω

by virtue of the absolute continuity of the integral. Remark 4.1. Existence of solutions to the evolutional p(x,t)-Laplace equation is proved in [1] under weaker assumptions on the exponent p(z): it is assumed that p(x, t) is log-continuous with respect to t and p(·, t) ∈ C0 (Ω). In this case a solution to the Dirichlet problem for the equation ut = divA(∇u), A(∇u) = |∇u|p(z)−2 ∇u : W(Q) → W (Q), is obtained as the limit of the sequence of solutions of the regularized equations + −2

ut = divAε (∇u), Aε (∇u) = ε|∇u|p

∇u + |∇u|p(z)−2 ∇u, ε > 0.

It is shown that +

Lp (0, T ; V+ (Ω)) ∈ uε → u ∈ W(Q),

+)

L(p





(0, T ; V+ (Ω)) ∈ Aε (∇uε ) → A(∇u) ∈ W (Q).

Remark 4.2. This note does not address the question of uniqueness of solution to problem (3.1). Uniqueness of energy solutions to the stationary counterpart of equation (3.1) with variable exponents p(x) and γ(x) is proved in [11] by a method based on the ideas of paper [4]. In the case of constant exponents γ, p, and Φ0 (z, v) ≡ φ(v), uniqueness of solution to problem (3.1) is proved in [14,15] for the class of solutions satisfying ∂t φ(v) ∈ L1 (Q) and, under less restrictive assumptions, in [26] (see also [20].) Uniqueness of energy solutions v ∈ W(Q) ∩ L∞ (Q) for equation (3.1) with variable p(z) and γ(z) does not follow directly from the results of the cited works. The difficulty of this issue seems to be not merely technical, which is why we postpone its discussion for further publications. 4.1. A priori estimates Due Corollary 4.1, from now on we may consider problem (4.1) as a problem with the unique regularization parameter ε. We devote this subsection to derive several independent of ε a priori estimates for the solutions of problem (4.4). It is always assumed that the conditions of Theorem 4.1 are fulfilled. Let us introduce the vector-valued function Gε = Aε (z, uε )|∇uε |p(z)−2 ∇uε and write equation (4.4) in the form ∂t uε = div Gε + f (z). Lemma 4.1. The functions Gε satisfy the estimate  p(z) |Gε |p (z) dz ≤ C, p (z) = , p(z) −1 Q with an independent of ε constant C = C(a± , p± , α± , K0 , ||f||2,Q , || ∇ γ||p(·),Q ).

(4.6)

(4.7)

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Proof. Since uε ∈ W(Q), then  uε φε = (ε + |s|)γ(z) ds ∈ W(Q)

2027

for every ε > 0

0

and can be taken for the test-function in (4.5) – see the proof of Lemma 6. Applying Lemma 6 to (4.5) with the test-function φε , we arrive at the inequality     |uε | a− |Gε |p (z) dz ≤ C + |fφ|dz + |Gε ||∇γ| (ε + |s|)γ | ln(ε + |s|)|dsdz Q

Q

Q

in which the constant C depends only on K0 ,   |u0 | I0 = |u0 | (ε + |s|)γ(x,0) ds. Ω

γ ±,

0

||γ t ||2,Q and

0

Estimating the last term on the right-hand side by Young’s inequality we obtain

   a− |Gε |p (z) dz ≤ C 1 + |∇γ(z)|p(z) dz 2 Q Q with a known constant C , depending on γ ± , K0 , p± , ||f||2,Q , ∇γp(·),Q , ||γ t ||2,Q and I0 . Lemma 4.2. Under the conditions of Theorem 4.1 || ∂ t uε ||W’(Q) ≤ C with a constant C independent of ε. Proof. It is sufficient to check that |(∂t uε , φ)2,Q | ≤ C||φ||W(Q) with a constant C independent of φ and ε for every φ ∈ W(Q) such that φ(x, 0) = φ(x, T) = 0. By virtue of (4.6)        φ∂t uε dz ≤ |G ||∇φ|dz + |f ||φ|dz ≤ ||Gε ||p (·),Q ||∇φ||p(·),Q + ||f ||2,Q ||φ||2,Q ε   Q Q Q   ≤ C ||Gε ||p (·),Q + ||f ||2,Q ||φ||W(Q) and the needed estimate follows from Lemma 4.1. Lemma 4.3. Under the foregoing assumptions the sequence of solutions of problem (4.4) is relatively compact in Ls (Q) with some 1 < s < ∞. Proof. Since the solutions uε are uniformly bounded, estimate (4.7) entails the inequality   − − − ν−1 p (ε + |uε |)(ν−1)p |∇uε |p dz |∇(|uε | uε )| dz ≤ C1 Q

Q



 (ε+|uε |)γ(z)p(z) |∇uε |p(z) dz + C3

≤ C2 Q

provided that

((ν−1)−γ)

(ε+|uε |) Q

p(z)p− p(z) − p− dz ≤ C4 , (4.8)

ν ≥ 1 + γ +.

||∂t uε ||W (Q)

The uniform estimates   ≤ C, ||∇ |uε |ν−1 uε ||p− ,Q ≤ C,

and the results of [32, Sec. 8] yield relative compactness of the sequence {uε } in Ls (Q) with some s ∈ (1, ∞ ). Gathering the above estimates we may find a function u ∈ L∞ (Q) ∩ Ls (Q) and a subsequence of {uε } (we conserve for this subsequence the same notation uε ) such that ⎧ ||uε ||∞,Q ≤ K0 , K0 = const independent of ε, ⎪ ⎪ ⎪ ⎨ u → u in Ls (Q) (1 < q < ∞), ε (4.9) ⎪ u → u a.e. in Q, ε ⎪ ⎪ ⎩ ∂t uε → ∂t u weakly in W (Q).

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Let us introduce the functions  uε (z) (ε + |s|)γ(z) ds, vε (z) =

γ(z) =

0

α(z) ≥ γ − > −1. p(z) − 1

(4.10)

Since the mapping uε → vε is monotone, there exists the inverse function uε = Φε (z, vε ). The following formulas hold:  uε γ (ε + |s|)γ ln (ε + s) ds, (a) ∇vε = (ε + |uε |) ∇uε + ∇γ 0  uε (4.11) (ε + |s|)γ ln (ε + s) ds. (b) ∂t vε = (ε + |uε |)γ ∂t uε + γt 0

Due to Lemma 4.1, uniform in ␧ boundedness of uε yields  ≤ C(K0 ) |Gε |p (z) + |∇γ|p(z) ∈ L1 (Q),

By the assumption |∇vε |p(z)

γ − > − 1. 

which gives the estimate ||∇vε ||p(·),Q ≤ C with an independent of ε constant C. It follows that there exist a subsequence of {vε } and a function V ∈ Lp (·) (Q) such that

|∇vε |p(z)−2 ∇vε → V

weakly in Lp (·) (Q).

(4.12)

On the other hand, the continuity of the mapping uε → vε and the convergence uε → u (see (4.9)) give ⎧ |vε | ≤ C(K0 ), ⎪ ⎪ ⎪ ⎨ vε → v in Lq (Q) for some q ∈ (1, ∞), ⎪ vε → v a.e. in Q, ⎪ ⎪ ⎩ ∇vε → ∇v weakly in Lp(z) (Q), with

 v= 0

u

|s|γ ds =

(4.13)

u|u|γ ∈ W(Q). γ +1

The next step is to check that the limit function u = lim Φε (z, vε ) is a solution of problem (3.1). ε→0

5. Existence of weak solution to problem (3.1). Proof of Theorem 3.1 Let us write the regularized problems (4.4) as the problem for the functions vε defined by (4.10):    ∂t Φε (z, vε ) = div b(z, vε )|∇vε + B(z, vε )|p(z)−2 (∇vε + B(z, vε ) + f in Q, vε = 0 on Γ, vε (x, 0) = v0 (x) in Ω, where



B(z, vε ) = −∇γ(z)

uε

(5.1)

(ε + |s|)γ(z) ln (ε + |s|) ds, b(z, vε ) ≡ a(z, uε ), uε = Φε (z, vε ).

0

For every ε > 0 problem (4.4) has at least one bounded solution uε = Φε (z, vε ) ∈ W(Q). The corresponding function vε = Φ−1 ε (z, uε ) is a solution of problem (5.1) in the sense of Definition 3.1. By virtue of (4.13) and (4.9) there

exist functions u ∈ L2 (Q) and Λ ∈ Lp (·) (Q) such that for a subsequence of solutions of problem (5.1) {vε } (for this subsequence we conserve the same notation {vε }) ⎧ 1 ⎪ ⎪ v →v≡ |u|γ u, Φε (z, vε ) → Φ0 (z, v) ≡ u, ⎪ ⎪ ε γ + 1 ⎪ ⎨

 ∇γ u|u|γ ∇γ γ (5.2) B(z, vε ) → B(z, v) = v (1 − (1 + γ) ln |v|) = − u|u| ln |u| a.e.inQ, ⎪ ⎪ γ + 1 γ + 1 γ + 1 ⎪ ⎪ ⎪ ⎩ b(z, vε )|∇vε + B(z, vε )|p(z)−2 (∇vε + B(z, vε )) → Λ weakly in Lp (·) (Q).

S. Antontsev, S. Shmarev / Mathematics and Computers in Simulation 81 (2011) 2018–2032

Letting ε → 0 in (3.5) for uε , we find that for every φ ∈ W(Q)  (ut φ + Λ∇φ − fφ) = 0, u = Φ0 (z, v).

2029

(5.3)

Q

To complete the proof of existence amounts to check that   ∀φ ∈ W(Q) Λ(z) · ∇φdz = b(z, v)|∇v + B(z, v)|p(z)−2 (∇v + B(z, v)) · ∇φdz. Q

Q

Let us define the function F (ξ, η) = |∇ξ + B(z, η)|p(z)−2 (∇ξ + B(z, η)) and make use of the representation F (vε , vε ) − F (v, v) ≡ |∇vε + B(z, vε )|p(z)−2 (∇vε + B(z, vε )) − |∇v + B(z, v)|p(z)−2 (∇v + B(z, v)) (ε)

(ε)

= [F (vε , vε ) − F (vε , v)] + [(F (vε , v)) − F (v, v)] ≡ J1 + J2 (ε)

5.1. Step 1: J1 → 0 as ε → 0 Let us assume that p(z) ≥ 2 at a point z ∈ Q. According to Proposition 2.1 with a = ∇vε + B(z, vε ), b = ∇vε + B(z, v) and p = p(z)   (ε) |J1 | ≤ C(p+ , p− )|B(z, vε ) − B(z, v)| |B(z, vε )|p(z)−1 + |B(v)|p(z)−1 + |∇vε |p(z)−1 . If p(z) ∈ (1, 2), we apply Proposition 2.2 to obtain   (ε) |J1 | ≤ C(p+ , p− )|B(z, vε ) − B(z, v)|1−α |B(z, vε )|p(z)−2+α + |B(z, v)|p(z)−2+α + |∇vε |p(z)−2+α . Set q+ (z) = max{p(z), 2}, q− (z) = min{p(z), 2}, Q+ = Q ∩{z : p(z) ≥ 2}, Q− = Q ∩{z : p(z) ∈ (1, 2)}. For every testfunction φ ∈ W(Q)              J (ε) · ∇φdz ≤     1    + . . . dz +  − . . . dz Q Q Q    ± ≤ C1 (p ) |B(z, vε ) − B(z, v)| |B(z, vε )|p(z)−1 + |B(z, v)|p(z)−1 + |∇vε |p(z)−1 dz (5.4) + Q   |B(z, vε ) − B(z, v)|1−β |B(z, vε )|p(z)−2+β + |B(z, v)|p(z)−2+β + |∇vε |p(z)−2+β dz + C2 (p± ) Q−

with 2 − p < β < 1. Applying Hölder’s inequality we conclude that the right-hand side of this inequality tends to zero as ε → 0. (ε)

5.2. Step 2: J2 → F (v, v) as ε → 0 Taking vε for the test-function in identity (3.5) for uε and applying the integration-by-parts formula (2.8) we obtain the energy relation   με (uε , vε ) + b(z, vε )F (vε , vε ) · ∇vε = fvε dz (5.5) Q

Q

with με ( · , · ) defined in (2.8). Let us recall that |με | are bounded uniformly with respect to ε. Lemma 5.1. For every ∀φ ∈ W(Q)   lim b(z, vε )F (vε , v) · ∇φdz = ε→0

QT

QT

b(z, v)F (v, v) · ∇φdz.

(5.6)

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Proof. The proof is an adaptation of the arguments of [24, Chapter 2, Section 1.2.2]. By monotonicity  0≤ b(z, vε ) (F (vε , v) − F (η, v)) · ∇(vε − η))dz.

(5.7)

Q

Subtracting (5.5) from (5.7), we get  0 ≤ −με (uε , vε ) − b(z, vε )F (vε , vε ) · ∇vε dz Q   b(z, vε )(F (vε , v) − F (η, v)) · ∇(vε − η)dz + fvε dz + QT

Q

≡ −με (uε , vε ) +

5 

(5.8)

Ii (ε)

i=1

where



I1 (ε) =

b(z, vε )(F (vε , v) − F (vε , vε )) · ∇vε dz, QT



I2 (ε) = −

b(z, vε )F (η, v) · ∇vε dz, QT

 b(z, vε )F (vε , v) · ∇ηdz, I3 (ε) = −  QT b(z, vε )F (η, v) · ∇ηdz, I4 (ε) = QT fvε dz. I5 (ε) = Q

Using the continuity of b(z, s) with respect to s and the weak convergence ∇vε → ∇v,

b(z, vε )F (vε , vε ) = (b(z, vε ) − b(z, v))F (vε , vε ) + b(z, v)F (vε , vε ) → Λ,

it is easy to see that  lim I2 (ε) = − b(z, v)F (η, v) · ∇vdz, Q Λ · ∇ηdz, lim I3 (ε) = −  Q b(z, v)F (η, v) · ∇ηdz. lim I4 (ε) = Q

Let us consider the term I1 (ε). Since  B(z, vε ) → A(z, v) a.e. in Q, |∇vε |p(z) dz ≤ C, Q

we find, repeating the proof of (5.4), that |I1 | → 0 as ε → 0. Letting ε → 0 in (5.8) and using the inequality  lim inf με (uε , vε ) ≥ μ(u, v) = ut vdz ε→0

Q

with μ given in (2.10), we obtain the inequality   0 ≤ −μ(u, v) − b(z, v)F (v, v) · ∇vdz − Λ · ∇ηdz Q Q b(z, v)F (η, v) · ∇ηdz + fvdz. + Q

Q

(5.9)

S. Antontsev, S. Shmarev / Mathematics and Computers in Simulation 81 (2011) 2018–2032

Combining (5.9) with (2.10) we have  (Λ − b(z, v)F (η, v) · ∇(v − η)) dz. 0≤

2031

(5.10)

Q

Let us take η = v − λw with λ = const and w ∈ W(Q). Under this choice of the test-function  (Λ − b(z, v)F (v − λw, v)) · ∇wdz. 0≤λ

(5.11)

Q

Simplifying and letting λ → 0 we find that  (Λ − b(z, v)F (v, v)) · ∇wdz. 0≤ Q

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