Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2010, Article ID 173408, 25 pages
doi:10.1155/2010/173408
Research Article
Quantitative Homogenization of Attractors of
Non-Newtonian Filtration Equations
Emil Novruzov
Department of Mathematics, Hacettepe University, 06532 Beytepe, Turkey
Correspondence should be addressed to Emil Novruzov, novruzov emil@yahoo.com
Received 8 December 2009; Accepted 11 April 2010
Academic Editor: Saad A. Ragab
Copyright q 2010 Emil Novruzov. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
For a rapidly spatially oscillating nonlinearity g we compare solutions u of non-Newtonian
filtration equation ∂t βu − D|Du |p−2 Du ϕu Du gx, x/, u fx, x/ with solutions
u0 of the homogenized, spatially averaged equation ∂t βu0 − D|Du0 |p−2 Du0 ϕu0 Du0 g 0 x, u0 f 0 x. Based on an ε-independent a priori estimate, we prove that ||βu −βu0 ||L1 Ω ≤
Ceρt uniformly for all t ≥ 0. Besides, we give explicit estimate for the distance between the
nonhomogenized A and the homogenized A0 attractors in terms of the parameter ; precisely, we
show fractional-order semicontinuity of the global attractors for 0 : distL1 Ω A , A0 ≤ Cγ .
1. Introduction
This paper is devoted to the study of nonlinear parabolic equations related to the p-Laplacian
operator. The problems related to such type of equation arise in many applications in the
fields of mechanics, physics, and biology non-Newtonian fluids, gas flow in porous media,
spread of biological populations, etc..
For example, in the study of the water in filtration through porous media, Darcy’s
linear relation
V −Kτωx
∗
satisfactorily describes flow conditions provided that the velocities are small. Here V
represents the velocity of the water, τ is the volumetric water moisture content, Kτ is
the hydraulic conductivity, and ω is the total potential, which can be expressed as the sum
of a hydrostatic potential ψτ and a gravitational potential z: ωτ ψτ z. However,
from the physical point of view, ∗ fails to describe the flow for large velocities. To get a
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Mathematical Problems in Engineering
more accurate description of the flow in this case, several nonlinear versions of ∗ have been
proposed. One of these versions is
V α −Kτωx
1.1
ut − div |u|m |∇u|p−2 ∇u fx, t, u.
∗∗
which leads us to the equation
For the first time, nonlinear relationship instead of Darcy‘s relation are suggested by Dupuit
and Forchheimer. This modification, known as Darcy-Forchheimer’s relation, has led to
much research from experimental, theoretical, and numerical points of view. For example,
Darcy-Forchheimer equations are widely used in reservoir engineering and other subsurface
applications.
One-dimensional by spatial variable variant of ∗∗ with p ∈ 3/2, 2 and m > p − 1
arises in the study of a turbulent flow of a gas in a one-dimensional porous medium. This
phenomenon was first described by Leibenson in 1. One of the first papers devoted to the
existence problem for such type of equation was an initiating paper by Raviart 2. In 3, the
author investigated the regularity problem for more general case which includes Leibenson’s
equation after changing |u|σ u v of non-Newtonian equation as
→
−
∂t βv div Bx, t, v, ∇v B0 x, t, v, ∇v 0
1.2
→
−
with some restriction on the functions B and B0 . Earlier, in 4 authors have investigated
existence and the stabilization of solutions as t ∞ of nonlinear parabolic equation of
mentioned type describing certain models related to turbulent flows. Also note that there
is an extensive literature devoted to the existence, regularity, and the large-time behavior
→
−
of solutions of 1.2 under various conditions on functions β, B, and B0 see 2–10 and
references therein.
In this paper we show that, under natural assumptions on the terms f and g, the
longtime behavior of solutions of the equation
p−2
∂t β u0 − D Du0 Du0 ϕ u0 Du0 g 0 x, u0 f 0 x
1.3
can be described in terms of the global attractor A of the associated dynamical system related
to the equation
p−2
∂t βu − D |Du |
Du ϕu Du
x
g x, , u
x
,
f x,
1.4
where functions gx, x/, u and fx, x/ are constructed according to some ideas
previously presented in 11, where authors carry out a quantitative comparison with the
averaged or homogenized equations, in particular for quasiperiodic inhomogeneities with
Mathematical Problems in Engineering
3
Diophantine frequencies see 11, pages 176–180. Note that a quantitative homogenization
aims at determining a specific rate of convergence. Method of the construction of the
homogenized equation allows us to assert that ||βu − βu0 ||L1 Ω ≤ Ceρt Theorem 3.1
below uniformly for all t ≥ 0. As we mentioned above, this result requires g and f to
depend quasiperiodically on the rapid space variable z x/. At the same time, being
interested in quantitative strong convergence not only of individual trajectories but also of
global attractors, we also show that global attractors A tend to attractor A0 in a suitable
sense providing fractional-order semicontinuity of the global attractors for 0. On a related
note, the author would like to mention several results on homogenization of the attractor for
nonlinear parabolic equations.
In 12 the Cauchy problem for parabolic equations on Riemannian manifolds with
complicated microstructure has been considered and a connection between global attractors
of the initial problem of the homogenized one has been established. The asymptotical
behavior of the global attractor of the boundary value problem for semilinear equation
ut − Δu fu h x
1.5
was investigated in 13 and it was shown that this tended in a suitable sense to the finitedimensional weak global attractor of some system of a parabolic p.d.e. coupled with an o.d.e.
Also it is necessary to note the papers in 14–16, where the media properties are
assumed to be oscillatory, focusing on the homogenization of the attractor for semilinear
parabolic equations see 14 and systems see 15, and of the quasilinear parabolic
equations see 16.
Note that previous approaches on homogenization of attractors are limited to certain
types of nonlinear equations. In particular, they assume the main part of the equation to be
linear or monotone. The consideration of equation 1.4 is a first step in the investigation of
the more general equation
x x
x
p−2
,
∂t βu − D a
|Du | ϕu Du g x, , u f x,
1.6
where the media properties are assumed to be oscillatory. This often arises in porous media
flows 17. We also note the paper in 18 where authors deal with the homogenization
problem for a one-dimensional parabolic PDE with random stationary mixing coefficients
in the presence of a large zero-order term and show that the family of solutions of the studied
problem converges in law.
Our work is inspired, on one hand, by results of the studies in 7, 8 which are devoted
to the existence and regularity of the attractor and, on the other hand, by the paper in 11
that is related to quantitative homogenization of the global attractor for reaction-diffusion
systems. For proof of the existence we use sketch of the proof of corresponding result;
however, compared to the studies in 7, 8 we consider the equation with an additional
nonlinear term ϕs. Note that this term prevents us from applying the result from the paper
in 4 which is often used to prove the uniqueness of the solution. Here we use a technique
see 19 which is based on the fact that the interval a, b may be divided into subintervals
where the sign of the “difference” ux, t − vx, t ux, t and vx, t are solutions with the
same initial condition does not change for fixed t. Also note that the results of the paper in
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11 cannot be directly applied to prove homogenization of the attractor for 1.4 because of
nonlinear terms β and ϕ.
Our paper is organized as follows. In Section 2 we consider the Dirichlet problem
∂t βu − D |Du|p−2 Du ϕuDu gx, u fx,
u|Γ 0,
Γ ∂Ω × 0, T ,
ux, 0 u0 x,
1.7
1.8
1.9
where D ∂x , Ω a, b under the following hypotheses
H1 u0 x and βu0 are in L2 Ω,
H2 βζ is an increasing locally Lipschitzian function from R to R, with β0 0.
H3 g·, ζζ c1 |ζ|k c2 , ∂ζ g·, ζ > −λ, without loss of generality we suppose that
g·, 0 0 and there exist positive constants c3 and c4 such that signζgx, ζ ≥
c3 |βζ|q−1 − c4 for a. e. x, t ∈ Ω × R and q > 2.
H4 ϕζ is a function from the space C1 such that ϕζ ≥ 0, ϕ0 0, and, for each M
α
and |ζ| ≤ M, |ϕ ζ| ≤ |ζ|l c β ζ almost everywhere, for some l, c ≥ 0 and
α ≥ 1.
H5 fx ∈ L∞ Ω, p ≥ 4.
Under the mentioned condition we show the existence and uniqueness of the solution and
the existence of the L2 -global attractor.
Further, in order to show more regularity of the solution we suppose the following condition.
α1
H6 For each M and |ζ| ≤ M, |gx, ζ − fx| ≤ |ζ|l1 c β ζ almost everywhere,
for some l1 , c ≥ 0 and α1 ≥ 1.
Finally in Section 3, we derive, under conditions H1–H5, the following estimate:
βu − β u0 L1 Ω
≤ Ceρt .
1.10
Also, under additional condition on g we prove fractional-order semicontinuity of the
global attractors for 0, namely, the validity of the estimation distL1 Ω A , A0 ≤ Cγ ,
where A and A0 are global attractors of the semigroups generated by the problems 1.4,
1.8, 1.9 and 1.3, 1.8, 1.9, respectively.
2. Existence and Uniqueness
First, we suppose that in equation 1.4 is a constant. This leads us to the problem 1.7, 1.8,
1.9 where gx, u and fx are denoted as gx, x/, u and fx, x/ correspondingly.
Mathematical Problems in Engineering
5
We use the standard regularization of the problem 1.7, 1.8, 1.9:
∂t βη u − D
|Du|p−2 ϕu η Du gx, u fx,
2.1
u|Γ 0,
1.8
Γ ∂Ω × 0, T ,
ux, 0 u0 x,
1.9
where the sequence βη ∈ C1 R is such that βη 0 0, βη → β in Cloc R, Mη > βη > η, and
|βη | ≤ |β|.
Let u0η η>0 be a sequence in C0∞ Ω such that u0η → u0 almost everywhere in Ω and
||u0η ||L2 Ω , ||βη u0η ||L2 Ω ≤ c, with constant c > 0. To show the solvability of the problem 1.8,
1.9, 2.1 we use the result for the classical solvability in the large given by Ladyžhenskaya
et al. see 20, Theorem 4.15.2, ch. VI.
Our proof is based on a priori estimates and is similar to that in 7. First, using the
properties of ϕ and βη , and then multiplying the equation by |βη uη |k βη uη , we deduce,
analogously to 7, 8, the following lemma.
Lemma 2.1. Under the hypotheses H1–H5 for any η ∈ 0, 1 the following estimates hold:
||uη ||L∞ τ,T ;L∞ Ω cτ, T ,
βη uη L∞ 0,T ;L2 Ω∩Lq QT cT ,
uη 1,p
Lp 0,T ;W0 Ω
cT .
2.2
Proof. Multiplying equation 2.1 by |βη uη |k βη uη and integrating by parts, we get
1 d
k 2 dt
Ω
k2
βη uη dx k 1
2
× Duη dx Duη p−2 ϕ uη η βη uη k βη uη
Ω
k g x, uη βη uη βη uη dx Ω
Ω
k fxβη uη βη uη dx.
2.3
By virtue of the positivity of the βη uη and ϕuη we have
1 d
k 2 dt
Ω
k2
βη uη dx k g x, uη βη uη βη uη dx ≤
Ω
Ω
k fxβη uη βη uη dx.
2.4
By conditions H2 and H3 we derive
k2
kq
1 d
βη uη dx c3
βη uη dx
k 2 dt Ω
Ω
k k βη uη βη uη dx.
≤
fx βη uη βη uη dx c4
Ω
Ω
2.5
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Mathematical Problems in Engineering
Consequently, from H5 we obtain
1 d
k 2 dt
Ω
k2
βη uη dx c3
Ω
kq
βη uη dx ≤ c f c4
Ω
k1
βη uη dx.
2.6
Further, using Holder’s inequality on both sides of the latter inequality, we deduce
that there exist two constants α0 > 0 and λ0 > 0 such that
q−1
d βη uη Lk2 Ω λ0 βη uη Lk2 Ω ≤ α0 ,
dt
2.7
which implies from Ghidaglia’s lemma 9 that
βn uη k2
≤
L Ω
α0
λ0
1/q−1
1
1/q−2 ct.
λ0 q − 2 t
2.8
As k → ∞ in 2.8, and for t ≥ τ > 0, we have
βη uη t
L∞ Ω
≤ cτ,
2.9
which implies that
uη t
L∞ Ω
≤ max βη−1 cτ, βη−1 −cτ δη .
2.10
Since βη converges to β in Cloc R, then the sequence δη is bounded in R as
η → ∞. Thus δη is bounded by maxβη−1 cτ, |βη−1 −cτ|, which is finite. Whence
||uη ||L∞ τ,T ;L∞ Ω cτ.
On the other hand, taking k 0 in 2.3, using Holder’s inequality, and integrating
over 0, T , we obtain the second estimate of the statement of Lemma 2.1 as
βη uη L∞ 0,T ;L2 Ω∩Lq QT cT .
2.11
Further, in order to prove the latter estimate of Lemma 2.1, we multiply 2.1 by uη
and integrate over Qt :
1
k2
t 0
Ω
∂t βη uη uη dxdθ k 1
t 0
Ω
g x, uη uη dxdθ t Duη p−2 ϕ uη η Duη 2 dxdθ
t 0
0
Ω
Ω
fxuη dxdθ.
2.12
Mathematical Problems in Engineering
7
Therefore,
t 1
k2
0
Ω
∂t βη uη uη dxdθ k 1
t Ω
0
g x, uη uη dxdθ t 0
t Duη p−2 ϕ uη η Duη 2 dxdθ
0
Ω
1
fxuη dxdθ k
2
Ω
t Ω
0
2.13
βη uη ∂t uη dxdθ.
Hence,
1
k2
Ω
βη uη t uη tdx −
1
k2
Ω
βη u0η u0η dx
t t p−2
2
k 1
Duη
ϕ uη η Duη dxdθ g x, uη uη dxdθ
0
t Ω
0
Ω
fxuη dxdθ 1
k2
t 0
Ω
0
Ω
2.14
dBη uη
dxdθ,
dτ
s
where Bη s 0 βη θdθ.
Now, taking into account that βη 0 0 and βη > η > 0, we conclude that Bη s ≤
βη ss. Thus,
1
k2
1
βη uη tuη tdx −
βη u0η u0η dx
k
2
Ω
Ω
t t Duη p−2 ϕ uη η Duη 2 dxdθ k 1
g x, uη uη dxdθ
0
t Ω
1
1
fxuη dxdθ Bη uη t dx −
≤
k
2
k
2
0 Ω
Ω
1
βη uη tuη tdx,
k2 Ω
Ω
0
Ω
Bη u0η dx ≤
t 0
Ω
fxuη dxdθ
2.15
or
k 1
≤
t t Duη p−2 ϕ uη η Duη 2 dxdθ g x, uη uη dxdθ
0
Ω
0
1
fxuη dxdθ βη u0η u0η dx.
k
2
Ω
t Ω
0
Ω
2.16
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Mathematical Problems in Engineering
From condition ∂ζ g > −λ we obtain that g·, ζ > −λζ and, consequently,
t Duη p−2 ϕ uη η Duη 2 dxdθ
k 1
0
≤
t 0
Ω
2
uη λ 1dxdθ
Ω
t
0
2
f dxdθ 1
k2
Ω
2
βη u0η dx 1
k2
Ω
Ω
2
u0η dx.
2.17
Therefore,
T 0
Ω
Duη p dxdθ ≤ cT .
2.18
Thereby assertion follows.
Corollary 2.2. Under condition of Lemma 2.1 there exists c such that
1 tτ
Duη p dxdτ ≤ c for arbitrary τ.
τ t
Ω
2.19
It is sufficient to carry out the proof of Lemma 2.1 using integration on the interval
t, t τ instead of 0, t and taking into account 2.8.
Lemma 2.3. Under the hypotheses H1–H5 there exist constants ci such that for any η ∈ 0, 1
the following estimates hold: ||uη ||L∞ τ,T ;W 1,p Ω cτ, T ,
0
T 2 ∂t uη 2 dxdθ ≤ cτ, T ,
βη uη
tτ Ω
τ
t
Ω
2
βη uη ∂t uη dxdθ ≤ cτ
t τ.
2.20
Proof. Multiplying 2.1 by ∂t uη and integrating over s, τ t, where τ ≤ t ≤ s, we get
tτ s
βη
Ω
2
uη ∂t uη dxdθ −
s
tτ Ω
s
tτ Ω
g x, uη ∂t uη dxdθ D
Duη p−2 ϕ uη η Duη ∂t uη dxdθ
2.21
tτ Ω
s
fx∂t uη dxdθ.
Then, using integration by parts, we derive
tτ s
Ω
βη
2
uη ∂t uη dxdθ s
tτ Ω
s
where G·, s s
0
tτ Duη p−2 ϕ uη η Duη D∂t uη dxdθ
∂t Gx, udxdθ g·, ζdζ.
Ω
tτ s
Ω
∂t fxuη dxdθ,
2.22
Mathematical Problems in Engineering
9
Hence, using condition H4, we get
tτ 2
βη uη ∂t uη dxdθ
Ω
s
tτ
d
dθ
s
2
2
1
1
Duη θp dx 1
ϕ uη θ Duη θ dx ηDuη θ dx
p2 Ω
2 Ω
2 Ω
G x, uη θ dx −
fxuη θ dθ
Ω
≤
1
2
1
2
tr Ω
s
Ω
2
ϕ uη θ × Duη θ ∂t uη dxdθ
tr α
l
Duη 2 ∂t uη dxdθ.
uη c
β s
Ω
s
2.23
Further, taking into account that βη uη is uniformly bounded by η uη ∈ −δ, δ, where
δ is the bound in the proof of Lemma 2.1, it is possible to choose βη so that βη ≤ L, where L
is the Lipschitz constant of βζ on −δ, δ. Therefore,
1
2
tτ
α
2 l
uη c
β s Duη ∂t uη dxdθ
Ω
s
≤
2
≤
tτ 2 2
l
α 1 tτ
uη c Duη 4 dxdθ
∂t uη dxdθ βη uη
2 s
s
Ω
Ω
α−1
L
2
tτ 2
1
βη uη ∂t uη dxdθ 2
Ω
s
2.24
tτ 2 l
uη c Duη 4 dxdθ.
s
Ω
Consequently,
tτ Ω
s
2
βη uη ∂t uη dxdθ
tτ
s
d
dθ
2
2
1
1
Duη θp dx 1
ϕ uη θ Duη θ dx ηDuη θ dx
p2 Ω
2 Ω
2 Ω
G x, uη θ dx −
fxuη θdx dθ
Ω
≤
α−1
L
2
tτ s
Ω
2
1
βη uη ∂t uη dxdθ 2
Ω
tτ 2 l
uη c Duη p 1 dxdθ.
s
Ω
2.25
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Mathematical Problems in Engineering
Now, using Lemma 2.1 and Corollary 2.2, we easily deduce
1
2
tτ s
2
1
Duη p t τdx ϕ uη Duη t τdx
p2 Ω
Ω
Ω
2
1
ηDuη t τdx G x, uη t τ dx −
fxuη t τdx
2 Ω
Ω
Ω
2.26
p
2
2
1
1
Duη sdx ≤
ϕ uη Duη sdx η Duη sdx
p2 Ω
2 Ω
Ω
1
G x, uη s dx −
fxuη sdx cτ.
2
Ω
Ω
βη
2
uη ∂t uη dxdθ After integrating over t, t τ, we derive
1
τ
p2
Ω
≤
2
2
1
ϕ uη Duη t τdx τ
ηDuη t τdx
2
Ω
Ω
tτ tτ 2
Duη p sdxds ϕ uη Duη sdxds
Duη p t τdx τ
1
p2 t
t
Ω
Ω
tτ tτ tτ 2
1
η Duη sdxds G x, uη s dxds fxuη sdxds
2 t
t
t
Ω
Ω
Ω
−τ
fxuη t τdx − τ
G x, uη t τ dx cττ.
Ω
Ω
2.27
Since, by virtue of conditions imposed on the function g, −λ/2|ζ|2 < G·, ζ ≤ c|ζ|k λ/2|ζ|2 c, then
2
1
Duη p t τdx ϕ uη Duη t τdx
p2 Ω
Ω
tτ tτ 2
1
1
Duη p sdxds ≤
ϕ uη Duη sdxds
τ p2 t
t
Ω
Ω
tτ k
2
1 tτ
λ 2
uη c dxds
ηDuη sdxds cuη 2 t
2
t
Ω
Ω
tτ k
λ 2
uη c tdx
fxuη sdxds cuη 2
t
Ω
Ω
Ω
fxuη tdx cτ
Mathematical Problems in Engineering
tτ p
1
1
1 tτ
Duη 4 sdxds
Duη sdxds ε
≤
τ p2 t
2
t
Ω
Ω
11
2
1 −1 tτ
1 tτ
2
ε
ϕ uη s dxds ηDuη sdxds
2
2
t
t
Ω
Ω
tτ tτ k
λ
2
uη s c dxds fxuη sdxds cτ.
cuη s 2
t
t
Ω
Ω
2.28
As we noted above, βη ≤ L on −δ, δ. Hence,
tτ Ω
t
≤
ϕ2 uη dxds ≤
t
tτ uη Ω
t
tτ uη
Ω
2
ϕ sds dxds
0
L∞ τ,T ;L∞ Ω
uη l ∞
L τ,T ;L∞ Ω
2.29
c Lα dxds.
Thus, we obtain that
1
p
Ω
Duη p t τdx Ω
2
ϕ uη Duη t τdx ≤ cτ
2.30
for t ≥ τ.
Now returning to 2.23, we have
tτ Ω
t
2
βη uη ∂t uη dxdθ ≤ cτ.
2.31
Also, choosing βη so that βη ≤ L, we derive
T 2 ∂t uη 2 dxdθ ≤ cτ, T .
βη uη
τ
Ω
2.32
Thus, the lemma is proved.
Passage to the Limit in 2.1.
Analogously to 7, 8, by estimates from Lemmas 2.1 and 2.3, we deduce that there exist that
a subsequence uη such that
1,p
uη → u weakly in Lp 0, T ; W0 Ω and a.e.,
βη uη → βu strongly in C0, T ; L2 Ω,
∂t βη uη → ∂t βu weakly in L2 Q,
D|Duη |p−2 Duη → χ weakly in Lp 0, T ; W −1,p Ω.
12
Mathematical Problems in Engineering
Further, it is easy to see that
g x, uη −→ gx, u
weakly in L2 Q.
2.33
Indeed, we know that uη → u a.e. Besides, by virtue of the embedding RQ ⊂ CQ,
where RQ is a domain of the solvability, the operator g : RQ → L2 Q generated by
the expression gx, u is bounded. Hence, by the continuity of g we obtain that gx, uη →
gx, u a.e. By applying a known lemma from 10, 1.1, Lemma 1.3 we can conclude that
g x, uη −→ gx, u
weakly in L2 Q.
2.34
Arguing similarly, by help of the lemma from 10, 1.1, Lemma 1.3 and conditions
imposed on ϕζ, we obtain
ϕ uη Duη −→ ϕuDu weakly in L2 Q,
ϕ uη Duη −→ ϕuDu
weakly in L2 Q,
D ϕ uη Duη −→ D ϕuDu weakly in L2 0, T ; W −1,2 Ω .
2.35
2.36
Observe now that from 2.35, in view of the known theorem,
Q
2
ϕ uη Duη dxdt ϕuDu
L2 Q
≤ lim inf ϕ uη Duη n∞
L2 Q
2
ϕ uη Duη dxdt.
2.37
Q
Consequently, using standard monotonicity argument 10, we derive that χ D|Du|p−2 Du.
Therefore, the following theorems hold.
Theorem 2.4. Under hypotheses H1–H5 the problem 1.7, 1.8, 1.9 has a weak solution
1,p
1,p
ux, t such that u ∈ Lp 0, T ; W0 Ω ∩ L∞ 0, T ; L2 Ω ∩ L∞ τ, T ; W0 Ω, for all τ > 0 and
βu ∈ C0, T ; L2 Ω ∩ Lq Q.
Now, we prove that the solution of the problem is unique.
Lemma 2.5. Suppose that the assumptions of Theorem 2.4 are fulfilled and there exists a constant c
such that the function ζ → gx, ζ cβζ is increasing for a. e. x ∈ Ω. Then the solution of the
problem 1.7, 1.8, 1.9 is unique.
Proof. Let ux, t and vx, t be two solutions of the problem 1.7, 1.8, 1.9 with the same
initial condition: ux, 0 vx, 0 u0 x. Consider the “difference” ux, t0 − vx, t0 , where
t0 is an arbitrary point from 0, T . The above “difference” is a continuously differentiable
Mathematical Problems in Engineering
13
function for almost all t > 0, because by virtue of estimation
T τ
Ω
βη uη 2 |∂t uη |2 dxdt ≤
c2 τ, T , using the equation, we can conclude that |Du|p−2 Du ∈ L2 τ, T ; W01,2 Ω and
consequently Dux, t ∈ CΩ for almost all t. We choose t0 such that ux, t0 ∈ C1 Ω.
Hence, the interval a, b may be divided into the subintervals where sign of “difference”
ux, t0 − vx, t0 does not change. Let x1 , x2 be an interval such that ux, t0 − vx, t0 > 0
and uxi , t vxi , ti 1, 2. Then from 1.7 we obtain that
x2
x2
∂t βu − βv dx −
x1
x2
x2
D |Du|p−2 Du − |Dv|p−2 Dv dx −
D ϕuDu − ϕvDv dx
x1
x1
gx, u − gx, v dx 0.
x1
2.38
By applying Newton-Leibniz formula we have
d
dt
x2
βu − βv dx − |Du|p−2 Dux2 , t |Dv|p−2 Dvx2 , t |Du|p−2 Dux1 , t
x1
− |Dv|p−2 Dvx1 , t − ϕuDux2 , t ϕvDvx2 , t ϕuDux1 , t − ϕvDvx1 , t
x
x
2
2
gx, u cβu − gx, v − cβvdx −
x1
cβu − cβvdx
x1
0.
tt0
2.39
Since Dux1 , t0 ≥ Dvx1 , t0 , Dux2 , t0 ≤ Dvx2 , t0 , uxi , t vxi , ti 1, 2, it
follows that
d
dt
x2
βu − βv dx
x1
tt0
< c
βu − βv dx
x1
x2
.
2.40
tt0
x
x
x12 |βu−βv|dx are absolutely
Note that the functions ψt x12 βu−βvdx and ψt
continuous by virtue of the inclusion ∂t βu ∈ L2 τ, T, L2 Ω. Hence, these functions possess
a derivate for almost all t. Besides, it is obvious that ψt
≥ ψt, and if βux, t −βvx, t ψt . Consequently,
0 for x ∈ x1 , x2 , then, ψt
d
dt
x2
x1
|βu − βv|dx
tt
d
≤
dt
βu − βvdx
x1
x2
2.41
tt
for almost all t . From here, without loss of generality, we can assume that
d
dt
x2
x1
|βu − βv|dx
tt0
d
≤
dt
βu − βvdx
x1
x2
tt0
.
2.42
14
Mathematical Problems in Engineering
It follows that
d
dt
βu − βvdx
x1
x2
βu − βvdx
< c
x1
x2
tt0
2.43
.
tt0
The same estimation holds for an arbitrary interval on which ux, t0 −vx, t0 does not change
its sign. Summing up similar inequalities over subintervals, we get
d
dt
|βu − βv|dx
a
b
< c βu − βvdx
a
b
tt0
2.44
tt0
almost everywhere. In view of integrability of both sides of the latter inequality βu ∈
W 1,2 τ, T ; L2 Ω we have
t
t
d
dt
b
βu − βvdxdt < c
t b
t
a
βu − βvdxdt,
2.45
a
where t > t > τ, or
|βu − βv|dx
a
b
tt
|βu − βv|dx
a
b
<
c
tt
t b
t
βu − βvdxdt.
2.46
a
Thus, by Gronwall’s lemma,
βu − βvdx
a
b
tt
≤ ect −t βu − βvdx
a
b
.
2.47
tt
Now, taking into account that βu ∈ C0, T ; L2 Ω and ux, 0 vx, 0 u0 x, we
obtain ux, t vx, t, which finishes the proof of uniqueness.
Remark 2.6. Note that the condition on the function ζ → gx, ζ cβζ from the lemma can
be changed to the following.
H2 The function β−1 s is a locally Lipschitzian function from R to R it follows
directly from the proof.
In this case, we can exclude the condition H4 of Theorem 2.4 and conditions H4
and H6 of Theorem 2.8 below.
So, analogously to the corresponding result from 8, we obtain that problem 1.7,
1.8, 1.9 generates a continuous semigroup St : L2 Ω → L2 ΩSt u0 ut, · and the
following theorem holds.
Theorem 2.7. Assume that assumptions of Lemma 2.5 are satisfied. Then the semigroup St associated
with the boundary value problem 1.7, 1.8, 1.9 possesses a maximal attractor A which is bounded
1,p
in W0 Ω ∩ L∞ Ω, compact, and connected in L2 Ω.
Mathematical Problems in Engineering
15
For the concepts of absorbing sets and global attractors used here, we refer the reader
to 9.
Under an additional condition we can obtain more regularity for ux, t.
Theorem 2.8. Assume that u0 x ∈ W 1,∞ Ω, βu0 ∈ L∞ Ω, and the conditions H2–H6 are
fulfilled. Then Du ∈ L∞ Q.
Proof. We prove this fact by multiplying 2.1 on expression |Duη |σ ∂t uη , taking into account
of the arbitrarity of σ
T 0
Ω
βη
2 σ
uη ∂t uη Duη dxdt −
−η
0
Ω
σ
D uη ∂t uη Duη dxdt −
2
Ω
T Ω
0
T 0
T p−2
σ
D Duη Duη ∂t uη Duη dxdt
T 0
Ω
σ
D ϕ uη Duη ∂t uη Duη dxdt
2.48
σ
g x, uη − fx ∂t uη Duη dxdt 0.
Let us estimate the terms of 2.48 separately.
Integrating by parts the second integral of 2.48, we get
−
T 0
Ω
p−2
σ
D Duη Duη ∂t uη Duη dxdt
− p−1
T 0
− p−1
T 0
Ω
Ω
Duη p−2 D2 uη ∂t uη Duη σ dxdt
Duη pσ−2 D2 uη ∂t uη dxdt
T pσ−2
p−1
−
D Duη Duη ∂t uη dxdt
pσ−1 0 Ω
T p−1
Duη pσ−2 Duη D∂t uη dxdt
pσ−1 0 Ω
T
p−1
d
Duη pσ dxdt.
p σ − 1 p σ 0 dt Ω
2.49
Arguing similarly, we obtain
−η
T 0
T σ
σ
η
D2 uη ∂t uη Duη dxdt −
D Duη Duη ∂t uη dxdt
σ1 0 Ω
Ω
T η
Duη σ Duη D∂t uη dxdt
σ1 0 Ω
T
η
d
Duη σ2 dxdt.
σ 1σ 2 0 dt Ω
2.50
16
Mathematical Problems in Engineering
A series simple calculations give us the estimate of the third term of 2.48:
−
T Ω
0
1
σ2
1
σ2
σ
D ϕ uη Duη ∂t uη Duη dxdt
T Ω
0
T Ω
0
σ
−
σ1
σ2
ϕ uη ∂t Duη dxdt σ2
ϕ uη ∂t Duη dxdt −
T 0
σ1
≥−
σ2
×
×
T 0
Ω
σ
ϕ uη D∂t uη Duη Duη dxdt
T 1
σ 1σ 2
σ2 dxdt −
∂t ϕ uη Duη σ2
ϕ uη ∂t uη Duη dxdt Ω
Ω
0
Ω
σ
σ1
1
σ 1σ 2
Ω
0
T Ω
0
T 0
Ω
T α1 l1
∂t uη Duη σ2 dxdt uη c
βη uη
0
Ω
σ2
ϕ uη ∂t uη Duη dxdt
σ2
ϕ uη ∂t uη Duη dxdt
σ2 dxdt
∂t ϕ uη Duη 1
σ 1σ 2
σ2 α1 σ 1 T
∂t uη 2 Duη σ dxdt
dxdt ≥ −
βη uη
∂t ϕ uη Duη σ 22 0 Ω
Ω
σ 1
2σ 2
T 2 l1
uη c Duη σ4 dxdt 0
Ω
T 0
−
0
0
σ
ϕ uη ∂t uη D Duη Duη dxdt
T 0
−
T T 1
σ 1σ 2
σ1
−
σ2
σ
σ1
T σ2
ϕ uη ∂t uη Duη dxdt −
Ω
σ
σ1
σ2 σ 1 α1 −1
dxdt ≥ −
∂t ϕ uη Duη L
σ 22
Ω
σ 1
2σ 2
Ω
1
σ 1σ 2
T 0
Ω
2 σ
βη uη ∂t uη Duη dxdt
T T 2 2
l1
l1
uη c Duη σp2 dxdt − σ 1
uη c dxdt
2σ 2 0 Ω
0 Ω
σ2
ϕ uη Duη dx −
Ω
ϕu0 |Du0 |σ2 dx.
2.51
Further, taking into account that βη uη is uniformly bounded by η uη ∈ −δ, δ, where
δ is the bound in the proof of Lemma 2.1, it is possible to choose βη so that βη ≤ L, where L
Mathematical Problems in Engineering
17
is the Lipschitz constant of βζ on −δ, δ. Therefore,
T σ
g x, uη − fx ∂t uη Duη dxdt
0 Ω
T α1 σ
l1
≤
∂t uη Duη dxdt
uη c
βη uη
0 Ω
T 2 2 σ
l1
α1 1 T
uη c Duη σ dxdt
∂t uη Duη dxdt βη uη
2 0 Ω
0 Ω
T 2 2 σ
l1
α1 −1
1 T
uη c Duη σ dxdt.
≤ L
βη uη ∂t uη Duη dxdt 2
2 0 Ω
0 Ω
2.52
≤
2
Also note that if βu0 ∈ L∞ Ω then using the same arguments as in proof of 2.8
we conclude that ||uη ||L∞ QT c. Thus, combining 2.49–2.52 and choosing sufficiently
small, we rewrite 2.48 in the form
Ω
Duη pσ dx ≤ c1
T 0
Ω
Duη pσ dxdt c2 T c3
Ω
|Du0 |pσ dx.
2.53
Applying Gronwall ’s lemma, we get
Ω
Duη pσ dx c 2 T c3
Ω
|Du0 |
pσ
dx ec1 T ,
2.54
where the constant ci does not depend on σ. Consequently,
1
|Ω|
1/pσ
c2
c3
ec1 T/pσ
T
|Du0 |pσ dx |Ω|
|Ω|
Ω
Ω
1/pσ
1/pσ
c3
c2
ec1 T/pσ
ec1 T/pσ
.
T
|Du0 |pσ dx
|Ω| Ω
|Ω|
Duη pσ dx
1/pσ
2.55
By letting σ tend to ∞, we get necessary estimation. Theorem is proved.
3. Quantitative Homogenization
As we mentioned earlier, our goal is to compare the global behavior of solutions u x, t of
1.4 for → 0 with solutions u u0 x, t of the homogenized equation
∂t βu − D |Du|p−2 Du ϕuDu g 0 x, u f 0 x,
3.1
where 3.1 and 1.4 are supplied with the same initial data u x, 0 u0 x, 0 u0 x, and
homogenized nonlinearity g 0 and inhomogeneity f 0 are defined according to assumption
from 11, pages 172-174.
18
Mathematical Problems in Engineering
We suppose that the function gx, z, ζ has the following structure:
gx, z, ζ M
bj x, zgj ζ,
3.2
j1
where gj ∈ C1 is supposed to satisfy a condition H3.
For all j 1, we suppose that bj x, z are bounded:
j
b x, z ≤ C,
3.3
and the average b0j x of bj x, x/ exist in L∞
w∗ Ω, for → 0 :
x
bj x,
, vx −→ b0j x, vx ,
−→0
3.4
for vx ∈ L1 Ω, where ·, · indicates duality.
We also suppose that
x
, vx −→ f 0 x, vx ,
f x,
−→0
3.5
for any vx ∈ L2 Ω.
The equation ∂t βu0 − D|Du0 |p−2 Du0 ϕu0 Du0 g 0 x, u0 f 0 x is called the
homogenization of equation if
M
b0j xgj u0 .
g 0 x, u0 3.6
j1
Denoting bj x, z bj x, z − b0j x, for x ∈ Ω, z ∈ R, we assume that there exist
functions B j x, z which are uniformly bounded for all x ∈ Ω, z ∈ R given as
j
B x, z ≤ C
3.7
and which represent bj such that bj x, z ∂z Bj x, z. With respect to the x-derivatives we
assume -independent L1 -bound
∂x Bj ·, · L1 Ω
≤ C,
3.8
uniformly for all j 1, . . . , M. Here ∂x is a partial’s derivatives with respect to the first
argument x of the function Bj x, z.
Mathematical Problems in Engineering
19
z fx, z−f 0 x and require the existence of a function
Analogously, we denote fx,
z admits a divergence representation fx,
z ∂z Jx, z.
Jx, z such that fx,
Besides, we assume bounds
|Jx, z| ≤ C,
∂x J ·, · L1 Ω
≤ C.
3.9
Note that sufficient conditions which guarantee the existence of divergence represen z by help of Bj x, z and Jx, z, respectively, are established in
tations for bj x, z and fx,
11, Theorem 3.2, pages 176-180.
The following theorem holds.
Theorem 3.1. Let gx, z, w and fx, z satisfy conditions 3.2–3.9 and let assumptions of
Lemma 2.5 be fulfilled. Then there exists a positive constant ρ such that the solutions u x, t and
u0 u, x of the respective problems 1.4, 1.8, 1.9 and 1.3, 1.8, 1.9 with equal initial data
u0 x ∈ L2 Ω satisfy the quantitative homogenization estimation
|βu − βu0 |
L1 Ω
≤ Ceρt
uniformly for 0 ≤ t < ∞.
3.10
Proof. Consider the “difference” u x, t0 − u0 x, t0 , where t0 is an arbitrary point from 0, T .
The above “difference” is a continuously differentiable function, in virtue of Lemma 2.3.
Hence, the interval a, b may be divided into the subintervals where sign of the “difference”
u x, t0 − u0 x, t0 does not change. Let x1 , x2 be an interval such that u x, t0 − u0 x, t0 > 0
and u xi , t0 u0 xi , t0 i 1, 2. Then from equations 1.3 and 1.4 we obtain that
d
dt
x2
x2 p−2
βu − β u0 dx −
D |Du |p−2 Du − Du0 Du0 dx
x1
−
x2
x1
x2 0
0
g 0 x, u − g 0 x, u0 dx
D ϕu Du − ϕ u Du dx x1
−
3.11
x1
x2 x2 x
x
− f 0 x dx.
g x, , u − g 0 x, u dx f x,
x1
x1
By applying Newton-Leibniz formula we have
d
dt
x2 p−2
βu − β u0 dx − |Du |p−2 Du x2 , t Du0 Du0 x2 , t
x1
p−2
|Du |p−2 Du x1 , t − Du0 Du0 x1 , t ϕu Du x2 , t − ϕ u0 Du0 x2 , t
x2 0
0
− ϕu Du x1 , t ϕ u Du x1 , t g 0 x, u − g 0 x, u0 dx
3.12
x1
x2 x2 x x
0
0
− f x dx
g x, , u − g x, u dx −
f x,
x1
x1
tt0
0.
20
Mathematical Problems in Engineering
Since Du x1 , t0 ≥ Du0 x1 , t0 , Du x2 , t0 ≤ Du0 x2 , t0 , and u xi , t u0 xi , t i 1, 2, it follows that
d
dt
x2 x1
0
βu − β u
dx
<−
x2 g 0 x, u − g 0 x, u0 dx
x1
tt0
3.13
x2 x2 x x
0
0
−
g x, , u − g x, u dx f x,
− f x dx
x1
x1
.
tt0
Note that
x2 x 0
g x, , u − g x, u dx
−
x1
tt0
M x2 x
gj u − b0j xgj u dx
−
bj x,
j1 x1
−
M x2
j1
x1
bj
x
gj u dx
x,
3.14
tt0
.
tt0
Obviously,
d
x
x
x
Bj x,
− ∂x Bj x,
,
∂z Bj x,
dx
3.15
where, as we mentioned above, ∂x indicate partial derivatives with respect to the first
argument x of the function Bj x, z.
Consequently,
x2 x 0
g x, , u − g x, u dx
−
x1
tt0
−
M x2 j1
x1
∂z Bj
x
x,
gj u dx
3.16
tt0
M x2
M x2
d j
x
x
j
B x,
−
∂x B x,
gj u dx gj u dx
dx
j1 x1
j1 x1
tt0
.
Mathematical Problems in Engineering
21
Therefore, in view of condition H3 and 3.7
x2 x 0
−
g x, , u − g x, u dx
x1
tt0
M x2 M
j
B x, x ∂x gj u dx ∂x Bj ., . ≤
x1
j1
≤ C
j1
L1 x1 ,x2 gj u L∞ x1 ,x2 tt0
M x2
j1
M
∂x gj u dx ∂x Bj ·, · gj u ∞
.
1
L x1 ,x2 tt
0
x1
L x1 ,x2 j1
3.17
Observe now that the approximate solution uη is bounded
∞
in the space
1,p
τ, T ; W0 Ω
L
uniformly with respect to , because in the proof of this Lemma 2.3 we
use the same constant ci and λ condition H3 to estimate every gj ω from 3.2, and from
condition 3.5 and H5 using known theorem on boundedness of the weakly convergence
sequence in normed space we can conclude that |fx, x/| ≤ C. Thus, ||u ||L∞ τ,T ;W 1,p Ω ≤ C.
0
x2
Consequently, M
j1 x1 |∂x gj u |dx ≤ C and 3.17 becomes
x2 x 0
−
g x, , u − g x, u dx
x1
tt0
L1 x1 ,x2 M
· j
≤ C C ∂x B ·,
j1
.
3.18
tt0
Analogously, by 3.9, using the same arguments as in proof of 3.18, we deduce that
x2 x
− f 0 x dx ≤ C.
f x,
x1
3.19
Now, bearing in mind that the function ζ → g 0 x, ζ cβζ is increasing and
combining 3.18 and 3.19, we rewrite 3.13 in the form
d
dt
0
βu − β u
dx
x2 x1
tt0
x2 0 <C
βu − β u dx
x1
tt0
M
· j
C C ∂x B ·,
j1
L1 x1 ,x2 3.20
.
tt0
22
Mathematical Problems in Engineering
x
x
x12 |βu −
Further, note that the functions ψt x12 βu − βu0 dx and ψt
βu0 |dx are absolutely continuous βu t ∈ L2 τ, T, L2 Ω. Besides, it is obvious that ψt
≥
ψt and ψt
0 ψt0 . Consequently,
d
dt
x2 0 βu − β u dx
x1
tt0
d
≤
dt
x2 0 ≤ C
βu − β u dx
x1
0
βu − β u
dx
x2 tt0
x1
tt0
L1 x1 ,x2 3.21
M
· j
C C ∂x B ·,
j1
.
tt0
Hence, by condition 3.8
d
dt
x2 0 βu − β u dx
x1
tt0
x2 0 ≤ C
βu − β u dx
x1
C.
3.22
tt0
The same estimation holds for an arbitrary interval on which u x, t − u0 u, x does
not change its sign. Summing up similar inequalities over subintervals, we get
d
dt
b
0 βu − β u dx
a
tt0
b
0 < C βu − β u dx
a
C.
3.23
tt0
Consequently,
b
0 βu − β u dx
a
≤ Ceρt −t t > t > τ .
3.24
tt
Thus, taking into account that βu ∈ C0, T ; L2 Ω,
|βu − βu |dx ≤ Ceρt .
a
b
0
3.25
t
Thereby, assertion follows.
Remark 3.2. It is easy to see that the condition on the function ζ → gx, ζ cβζ in
Lemma 2.5, which we use in Theorem 3.1, can be changed to the H2 . In this case we can
exclude condition H4.
M 0j
< 0 and condition H2 is fulfilled then we derive,
Now, note that if
j1 b x < c
using simple reasoning, the following exponential attraction with exponential rate ν > 0:
distL1 Ω S0t u0 ; A0 ce−νt .
Indeed, we know that solution of the corresponding stationary problem belongs to the
attractor, that is, it belongs to A0 . Denoting this solution by v, we will use the same arguments
as in proof of Lemma 2.5.
Mathematical Problems in Engineering
23
Defining u S0t u0 , we have
d
dt
x2
βu − βvdx −
x1
x2 x2
p−2
D |Du|
p−2
Du − |Dv|
Dv dx −
x2
x1
D ϕuDu − ϕvDv dx
x1
g 0 x, u − g 0 x, v dx 0,
x1
3.26
where x1 and x2 are such that βu − βv ≥ 0 on the interval x1 , x2 . Hence,
d
dt
x2
x2
βu − βvdx −
x1
D |Du|p−2 Du − |Dv|p−2 Dv dx
x1
−
x2
D ϕuDu − ϕvDv dx x1
M
b x
0j
x2
3.27
gj u − gj v dx 0.
x1
j1
Bearing into mind that gj w is supposed to satisfy condition H3 ∂ζ gj > −λ, we have
x2
d
dt
βu − βvdx ≤ −λ
c
x2
u − vdx.
3.28
βu − βv dx.
3.29
βu − βvdx.
3.30
x1
x1
Using H2 , we derive
x2
d
dt
βu − βvdx ≤ −λ
c
x2
x1
x1
Hence, from2.42
d
dt
x2
βu − βvdx ≤ −λ
c
x1
x2
x1
Further, arguing similarly to the proof of Lemma 2.5, we arrive to
d
dt
b
βu − βvdx < −λ
c
a
b
βu − βvdx,
3.31
a
or
b
βu − βvdx < cu0 , ve−λct .
3.32
a
Consequently, by condition H2
x2
x1
|u − v|dx ≤ L1
x2
x1
βu − βvdx ≤ L1 cu0 , ve−λct ce−νt .
3.33
24
Mathematical Problems in Engineering
c, L1 is the Lipschitz
Thus, we obtain that distL1 Ω S0t u0 ; A0 ce−νt , where ν λ
constant.
Hence, using Remarks 2.6 and 3.2, with the help of Lemma 4.1 of the study in 11, we
obtain the estimate for the distance between the nonhomogenized A and the homogenized
A0 attractors in terms of the parameter distL1 Ω A , A0 ≤ Cγ . So, the following theorem
holds.
Theorem 3.3. Let gx, z, w and fx, z satisfy conditions 3.2–3.9, and assumptions
0j
< 0. Then the global
H1–H3, H2 , and H5 are fulfilled. Also suppose that M
j1 b x < c
attractors A of the problem 1.4, 1.8, 1.9 satisfy an upper semicontinuity distance estimate of
the form distL1 Ω A , A0 ≤ Cγ .
References
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Izvestiia Akademii nauk Kirgizskoı̆ SSR. Geography and Geophysics, vol. 9, pp. 7–10, 1945.
2 P. A. Raviart, “Sur la résolution de certaines équations paraboliques non linéaires,” Journal of
Functional Analysis, vol. 5, pp. 299–328, 1970.
3 A. V. Ivanov, “Regularity for doubly nonlinear parabolic equations,” Journal of Mathematical Sciences,
vol. 83, no. 1, pp. 22–37, 1997.
4 J. I. Diaz and F. De Thélin, “On a nonlinear parabolic problem arising in some models related to
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