Homogenization of attractors of non-linear evolutionary equations
The paper is devoted to the homogenization of non{linear evolutionary equations in domains with "traps".
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irk-123456789-1692752020-06-10T01:26:30Z Homogenization of attractors of non-linear evolutionary equations Khruslov, E. Pankratov, L. The paper is devoted to the homogenization of non{linear evolutionary equations in domains with "traps". 1999 Article Homogenization of attractors of non-linear evolutionary equations / E. Khruslov, L. Pankratov // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 75-83. — Бібліогр.: 14 назв. — англ. 0236-0497 http://dspace.nbuv.gov.ua/handle/123456789/169275 en Нелинейные граничные задачи Інститут прикладної математики і механіки НАН України |
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The paper is devoted to the homogenization of non{linear evolutionary equations in domains with "traps". |
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Article |
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Khruslov, E. Pankratov, L. |
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Khruslov, E. Pankratov, L. Homogenization of attractors of non-linear evolutionary equations Нелинейные граничные задачи |
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Khruslov, E. Pankratov, L. |
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Khruslov, E. |
| title |
Homogenization of attractors of non-linear evolutionary equations |
| title_short |
Homogenization of attractors of non-linear evolutionary equations |
| title_full |
Homogenization of attractors of non-linear evolutionary equations |
| title_fullStr |
Homogenization of attractors of non-linear evolutionary equations |
| title_full_unstemmed |
Homogenization of attractors of non-linear evolutionary equations |
| title_sort |
homogenization of attractors of non-linear evolutionary equations |
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Інститут прикладної математики і механіки НАН України |
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1999 |
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http://dspace.nbuv.gov.ua/handle/123456789/169275 |
| citation_txt |
Homogenization of attractors of non-linear evolutionary equations / E. Khruslov, L. Pankratov // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 75-83. — Бібліогр.: 14 назв. — англ. |
| series |
Нелинейные граничные задачи |
| work_keys_str_mv |
AT khruslove homogenizationofattractorsofnonlinearevolutionaryequations AT pankratovl homogenizationofattractorsofnonlinearevolutionaryequations |
| first_indexed |
2025-07-15T04:01:58Z |
| last_indexed |
2025-07-15T04:01:58Z |
| _version_ |
1837684091004125184 |
| fulltext |
HOMOGENIZATION OF ATTRACTORS OF
NON–LINEAR EVOLUTIONARY EQUATIONS
c© E.Khruslov, L.Pankratov
1. Introduction
The paper is devoted to the homogenization of non–linear evolutionary equations in
domains with ”traps”. Namely, we consider an initial boundary value problem for a
semilinear parabolic equation of the form :
∂u
∂t
− Lεu + f(u) = h, t > 0,
uε(x, 0) = uε
0(x),
(1.1)
where Lε is a linear second order differential operator with corresponding boundary
condition.
We study three different cases of the definition of this operator :
1) Lε is the Beltrami–Laplace operator ∆ε defined on the Riemmanian manifold Mε
of a special structure depending on a small parameter ε;
2) Lε is the usual Laplace operator ∆ defined in the domain Ωε with a large number of
perforated spheres, periodically with a period ε distributed in the domain; it is assumed
that the diameter of perforation is small with respect to the period of the structure;
3) Lε is an operator with coefficients asymptotically degenerating on a set of peri-
odically with a period ε distributed thin (with respect to the period of the structure)
spherical annuluses :
Lε =
n∑
i,j=1
∂
∂xi
(
aε
ij(x)
∂
∂xi
)
.
The structure of the operator Lε implies three different approa– ches to the defi-
nition of traps in the initial boundary value problem (1.1).
Following the papers [1–5] we assume that the function f(u) in (1.1) belongs to the
class C2(R) and has the following properties :
|f ′(u)| < C1; uf(u) ≥ C2u
2 − C3;
u∫
0
f(ξ)dξ ≥ C4u
2 − C5 (1.2)
with some positive constants Cj(j = 1, 2, ..., 5).
Using the classical technique developed, for instance, in [6, 7] we can prove the
existence and uniqueness theorem that allow us to construct an evolution operator Sε
t
in corresponding spaces by the formula Sε
t uε
0 = uε(t), where uε(t) = uε(x, t) is the
solution of problem (1.1) (for the details see [1, 3, 5]). Application of standard methods
(see, e.g. [8–11]) makes it possible to prove that the dynamical system in each case for
every ε > 0 has a compact global attractor Aε and this attractor has a finite Hausdorff
dimension. We study the asymptotical behavior of Aε in the cases 1)–3) as ε → 0. Our
principal goal is to learn how the transition to homogenization description reflects on
the long–time dynamics.
The asymptotic behaviour of the solutions uε(x, t) of problem (1.1) in the cases 1)–
3) as ε → 0 was studied for a finite time interval in [1, 2, 4]. It was shown that the
homogenization of these problems leads to a system of a semilinear parabolic equation
coupled with an ordinary differential equation :
∂u
∂t
−
n∑
i,j=1
aij
∂2u
∂xi∂xj
+ a1(u− v) + f(u) = h1(x), x ∈ Ω, t > 0;
∂v
∂t
+ a2(v − u) + f(v) = h2(x), x ∈ Ω, t > 0;
∂u
∂n
= 0, x ∈ ∂Ω, t > 0; u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ Ω;
(1.3)
where the coefficients aij(i, j = 1, 2, ..., n) and ak(k = 1, 2) are calculated from the
solutions of cellular problems and the parameters of the structure.
We consider the long–time dynamics of homogenized system (1.3) and show that it
possesses a finite–dimensional global attractor A. We investigate the properties of A
and prove that global attractors Aε tend to A in a suitable sense as ε → 0 (see [1, 3,
5]).
We also note that the homogenization problem (1.1) with uniformly non–degenerating
elliptic operator was studied by a number of authors (see, e.g. [12–14]).
2. Homogenization results
The aim of the Section 2 is to present the homogeniztion results we use for the proof
of the convergence of the global attractors of problem (1.1).
2.1. Homogenization of semilinear parabolic equations
on manifolds with complicated microstructure
We consider on n–dimensional (n ≥ 2) Riemannian manifold Mε of complicated
microstructure which depends on ε > 0 initial boundary value problem (1.1) with the
Neumann boundary condition on the boundary ∂Mε. We suppose that local structure
of the manifold Mε becames more and more complicated, as ε tends to zero. Now we
describe the structure of the manifold Mε. Let Ω be a smooth bounded domain in
Rn(n ≥ 2) and let
Fε =
⋃
j∈Nε
F (xi, aε)
be a union of balls F (xi, aε) of radius aε << ε (limε→0 aεε
−1 = 0) with centers in
xj = jε (j ∈ Zn) such that F (xi, aε) ∈ Ω. Here Nε stands for the corresponding set
of multiindexes j ∈Zn. In Rn+1 we consider the surfaces (below x = (x1, ..., xn) ∈
Rn, y ∈ R, (x, y) ∈ Rn+1):
Ωε = {(x; 0) ∈ Rn+1 : x ∈ Ω \ Fε}
and
Bj
ε = (jε; 0) + Bε, j ∈ Nε ⊂ Zn,
where
Bε = {(x; y) ∈ Rn+1 : |x|2 + (y −
√
b2ε2 − a2
ε)
2 = b2ε2, y ≤ 0}.
Here b is a parameter such that aεε
−1 < b < 1. We assume that
Mε = Ωε
⋃
⋃
j∈Nε
Bj
ε
,
i.e. Mε consists of a piece of flat submanifold in Rn+1 with bubbles Bj
ε . We de-
fine the Riemannian structure on Mε by C∞ metric tensor gε(x) = {gε
αβ(x); α, β =
1, 2, ..., n}, x ∈ Mε, and assume the following : (i) the metric coincides with euclidean
one on Ωε; (ii) the metric is the same for all bubbles Bj
ε , j ∈ Nε; (iii) there exist positive
constants C1 and C2 such that
C1ε
n|ξ|2 ≤
∑
αβ
gε
αβ(x)ξαξβ ≤ C2ε
n|ξ|2, ε > 0,
for all x ∈ Bj
ε , j ∈ Nε and for all ξ ∈ Rn.
We consider problem (1.1) on the Riemannian manifold (Mε, g
ε), which can be
treated as a model of diffusion in medium with traps. The operator Lε is the Beltrami–
Laplace operator ∆ε of the form
∆ε =
1√
|gε|
∑
α,β
∂
∂xα
(√
|gε|gαβ
ε
∂
∂xβ
)
,
where |gε| = det gε and gαβ
ε are the components of the inverse to gε tensor.
Now we introduce a parameter for description of asymptotical behaviour of manifolds.
Let 0 ∈ Ω for the simplicity. We denote
Gε = {(x; 0) ∈ Rn+1 : aε ≤ |x| < ε/2}; Dε = Bε
⋃
Gε.
From now on ‖ · ‖O is the norm in L2(O) and ‖ · ‖1,O is the norm in the Sobolev
space H1(O). We set
λε = inf
{‖∇εv‖2Dε
‖v‖2Dε
: v ∈ H1
0 (Dε)
}
. (2.1)
It is clear that λε is the first eigenvalue of the Dirichlet problem
∆εv + λεv = 0, x ∈ Dε; v = 0, x ∈ ∂Dε. (2.2)
Our main assumption concerning to the behaviour of the bubbles Bj
ε (and manifold
Mε) is the existence of the limits
λ = lim
ε→0
λε; µ = lim
ε→0
ε−nmε > 0, (2.3)
where
mε = V ol(Bε) =
∫
Bε
√
|gε|dx1...dxn.
Let Pε be a bounded operator from L2(Mε) into L2(Ω) defined by the formula
(Pε)(x) =
{
u(x), x ∈ Ωε;
0, x ∈ Ω \ Ωε.
Let {xα = αε, α ∈ Zn} be a lattice in Rn and let Qε be a linear interpolation operator
that is defined as follows. For each node of the sublattice {xα = αε, α ∈ Nε} we set
(Qεu)(xα) =
1
mε
∫
Bj
ε
u(x)dx, α ∈ Nε; (2.4a)
where mε is the volume of the ball Bj
ε . For each node {xα = αε, α 6∈ Nε} we set
(Qεu)(xα) being equal to a mean between the values of (Qεu) in the nearest nodes of
the lattice. In the whole (Qεu) is a poly–linear spline, i.e.
(Qεu)(x) =
∑
α
(Qεu)(xα)
n∏
j=1
χ
(xj
ε
− αj
)
, (2.4b)
where x = (x1, ..., xn) ∈ Ω, α = (α1, ..., αn) ∈ Zn and χ(τ) = 0 for |τ | > 1; χ(τ) = 1−|τ |
when |τ | ≤ 1. It is clear that Qε is linear bounded operator from L2(Mε) into H1(Ω)
for every ε > 0.
Let us assume that the following homogenization conditions hold true :
(A.1) for any ε ∈ (0, ε0)
‖uε
0‖1,Mε + ‖∇Qεu
ε
0‖Ω + ‖hε‖1,Mε + ‖∇Qεh
ε‖Ω ≤ C,
where the constant C is independent of ε;
(A.2) there exist functions u0, v0, h1, h2 from L2(Ω) such that Pεu
ε
0 → u0, Qεu
ε
0 →
v0, Pεh
ε → h1, Qεh
ε → h2 strongly in L2(Ω);
These homogenization conditions along with (2.3) imply that (for the proof see [1])
lim
ε→0
{max
[0,T ]
‖Pεu
ε(t)− u(t)‖Ω + max
[0,T ]
‖Qεu
ε(t)− v(t)‖Ω} = 0, (2.5)
for any T > 0, where U(x, t) = (u(x, t), v(x, t)) is the solution of the problem (1.3) in the
class W = {U(t) : U(t) ∈ C(R+,H1(Ω)×L2(Ω));
d
dt
U(t) ∈ L2(R+, L2(Ω)×L2(Ω))}.
The coefficients aij and ak in (1.3) are calculated as follows : aij = δij , a1 = λµ, a2 = λ.
2.2. Homogenization of semilinear parabolic equations
in domains with spherical traps
Let Ω be a smooth bounded domain in Rn (n ≥ 3) and let Γα
ε be (n−1)–dimensional
sphere centered at the point xα ∈ Ω with radius rε = rε (r < 1/5). We denote by Dα
ε
the connected subset of Γα
ε with the diameter dε = dεn/(n−2). Then Sα
ε = Γα
ε \ D̄α
ε is a
Rn−1 perforated sphere in the domain Ω and we denote Fε =
⋃
α∈Nε
Sα
ε the union of
such perforated spheres with centers at points xα = αε (α ∈ Zn). Here Nε stands for
the corresponding subset of multiindexes α ∈ Zn, such that Fε ⊂ Ω.
We consider initial boundary value problem (1.1) in the domain Ωε = Ω \ Fε with
the Neumann boundary condition on ∂Ωε. We set
νε = inf
{‖∇v‖2B(Rε)
‖v‖2B(Rε)
, v ∈ H1(B(Rε) \ Sε); v = 0, x ∈ ∂B(Rε)
}
,
where Rε = ε/3; ∂B(Rε) is the surface of the Rn open ball B(Rε); Sε is the Rn−1
perforated sphere of radius rε = rε strictly included in the ball B(Rε). We assume that
both the sphere and the ball are centered at 0 and 0 ∈ Ω for sake of simplicity. It is clear
that νε is the first eigenvalue of some mixed boundary value problem for the Laplace
operator in B(Rε) \ Sε (for details see [2]).
We will assume that there exists a limit :
ν = lim
ε→0
νε. (2.6)
The existence of the limit (2.6) is dicussed in [2].
Now we introduce the functions vi(x) (i = 1, 2, ..., n) which are the solutions of the
following auxiliary problems :
∆vi = 0, x ∈ P = K \ B̄;
∂vi
∂n
= (xi, n), x ∈ ∂B;
vi(x), Dvi(x) are K − periodic;
(2.7)
where n is the unitary normal to ∂B; K = {x ∈ Rn; |xi| < 1/2r; i = 1, 2, ..., n}; B
is a Rn unit open ball in K. It is known that this problem has a unique solution vi(x)
(up to a constant).
Let us introduce the notation :
Gε = Ω \
⋃
α
B̄α(rε); Bε =
⋃
α
Bα(rε).
Let Qε be a polylinear spline defined by (2.4a), (2.4b) with Bα(rε) instead of Bj
ε and
let Pε be a standard linear continuation operator from Gε to Ω. We assume that the
following homogenization conditions hold :
(B.1) for any ε ∈ (0, ε0) and some constant C independent of ε
‖∇uε
0‖Ωε
+ ‖∇Qεu
ε
0‖Ω + ‖∇hε‖Ωε
+ ‖∇Qεh
ε‖Ω ≤ C;
(B.2) there exist functions u0, v0, h1, h2 ∈ L2(Ω) such that Pεu
ε
0 → u0, Qεu
ε
0 → v0,
Pεh
ε → h1, Qεh
ε → h2 strongly in L2(Ω);
These homogenization conditions along with (2.6) imply that (for the proof see [2])
lim
ε→0
{max
[0,T ]
‖Pεu
ε(x, t)− u(x, t)‖Ω + max
[0,T ]
‖Qεu
ε(x, t)− v(x, t)‖Ω} = 0,
for any T > 0, where U(x, t) = (u(x, t), v(x, t)) is the solution of problem (1.3) in the
class W.
The coefficients aij and ak in (1.3) are calculated from cellular problem (2.7) solutions
by
aij = δij
1− rn
1− θ
∫
P
(∇vi,∇vj)dx
, a1 =
a2θ
1− θ
, a2 = ν.
Here
θ =
πn/2rn
Γ
(
n
2 + 1
) , (2.8)
δij is the Kronecker symbol and Γ is the Gamma function.
2.3. Homogenization of semilinear parabolic equations
with asymptotically degenerating coefficients
Let Ω be a smooth bounded domain from Rn, (n ≥ 2). Let us introduce the notation:
Gα
ε = {x ∈ Ω : rε − dε < |x− xα| < rε};
Bα(rε − dε) = {x ∈ Ω : |x− xα| < rε − dε};
Gε =
⋃
α∈Nε
Gα
ε ; Bε =
⋃
α∈Nε
Bα(rε − dε); Ωε = Ω \ (Gε
⋃
Bε);
where xα = αε (α ∈ Zn) and Nε is a set of multi–indices such that Gα
ε ⊂ Ω; rε = rε
(r < 1/4); dε = dε2+γ (0 ≤ γ < 1).
In the domain Ω we consider the boundary value problem (1.1) with
Lε =
n∑
i,j=1
∂
∂xi
(
aε
ij(x)
∂
∂xi
)
.
and the Neumann boundary condition on ∂Ω. The coefficients aε
ij(x) in Lε are defined
as follows :
aε
ij(x) =
{
δij , x ∈ Ω \ Gε;
aεδij ≡ aδijε
3+γ , (a > 0), x ∈ Gε.
(2.9)
This structure of the diffusion matrix allows us to interpret the problem (1.1) as a
rection–diffusion problem in a medium with traps Bα(rε − dε).
Let us introduce the notation :
Jε(uε) =
1
2
{‖∇uε‖2Ωε
+ aε‖∇uε‖2Gε
+ ‖∇uε‖2Bε
}
;
As in the previuos Sections we suppose for sake of simplicity that 0 ∈ Ω. Let Qε be
a polylinear spline defined by (2.4a), (2.4b) with Bα(rε − dε) instead of Bj
ε and let Pε
be a standard linear continuation operator from Ωε to Ω.
Let uε(x, t) be the solution of problem (1.1). We assume that the following conditions
hold :
(C.1) for any ε ∈ (0, ε0)
‖uε
0‖2Ω + Jε(uε
0) + ‖∇Qεu
ε
0‖2Ω + ‖hε‖21,Ω + ‖∇Qεh
ε‖2Ω ≤ C,
where C denotes any constant independent of ε;
(C.2) there exist functions u0, v0, h1, h2 ∈ L2(Ω) such that Pεu
ε
0 → u0, Qεu
ε
0 → v0,
Pεh
ε → h1, Qεh
ε → h2 strongly in L2(Ω).
Then we have
lim
ε→0
{max
[0,T ]
‖Pεu
ε(x, t)− u(x, t)‖Ω + max
[0,T ]
‖Qεu
ε(x, t)− v(x, t)‖Ω} = 0,
where the pair of functions U(x, t) = (u(x, t), v(x, t)) is the solution of problem (1.3) in
the class W. The coefficients aij and ak in (1.3) are calculated from cellular problem
(2.7) solutions and the structure parameters as follows :
aij = δij
1− rn
1− θ
∫
P
(∇vi,∇vj)dx
, a1 =
b2θ
1− θ
, a2 =
an
rd
.
Here δij is the Kronecker symbol, Γ is the Gamma function and the parameter θ is
defined by (2.8).
3. Convergence of attractors
It can be proved that if U0 ∈ F = H1(Ω)×H1(Ω) and h2 ∈ H1(Ω) then
U(t) ∈ C(R+,F);
d
dt
U(t) ∈ L2
loc(R+, L2(Ω)×H1(Ω)).
So we can define the evolution semigroup St in the space F by the formula StU0 =
U(t), where U(t) = (u(x, t), v(x, t)) is the solution of the problem (1.3) and U0 =
(u0, v0). We prove the following assertion on the existence of finite dimensional weak
global attractor for the semigroup St in F .
THEOREM 3.1. Assume that (1.2) is satisfied and
a2 + inf{f ′(u)} > 0, h2(x) ∈ H1(Ω), (3.1)
where inf in (3.1) is taken over the real functions u. Then the dynamical system (St,F)
has weak global attractor A. This attractor has finite Hausdorff dimension as a compact
set in L2(Ω)× L2(Ω).
Recall (see, [8, 9]) that weak global attractor A is a bounded weakly closed set in F
such that (i) StA = A for any t > 0 and (ii) for any weak neighbourhood O of A and
for any bounded set B ⊂ F we have StB ⊂ O, when t ≥ t0 (B,O). We also note that
assumption (3.1) is of prime importance for the existence of finite dimensional attractor
A.
At last using the methods developed in [1], (see also [3, 5]) and some estimates
borrowed from [1, 2, 4] we prove our main result.
THEOREM 3.2. Assume that (1.2), (3.1) and the homogenization conditions (A.1)–
(A.2) (conditions (B.1)–(B.2) or (C.1)–(C.2)) dealing with hε along with (2.3) (along
with (2.6) in the case 2)) are satisfied. Then we have
lim
ε→0
sup
uε∈Aε
{ inf
(u,v)∈A
(‖Pεu
ε − u‖Ω + ‖Qεu
ε − v‖Ω)} = 0,
where operators Pε and Qε are defined for the cases 1)–3) in the Sections 2.1–2.3.
Thus the space structure of the attractor Aε becomes more and more complicated in
homogenization process (ε → 0). Besides, in fact, we observe a limiting spliting of each
element from Aε into two–component function. In particular, the same effect is valid
for stationary solutions of (1.1) (for the details see [5]).
COROLLARY 3.1. Let the conditions of Theorem 3.2 are valid. Then for each
stationary solution uε(x) of (1.1) we have that
lim
ε→0
{ inf
(u,v)∈Z
(‖Pεu
ε − u‖Ω + ‖Qεu
ε − v‖Ω)} = 0,
where Z is the set of stationary solutions of (1.3).
References
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spherical traps, Applicable Analysis (1997), no. 64, 303–317.
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equations in domains with spherical traps, C.R. Acad. Sci. Paris (submitted for publication).
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