On Singular Limit and Upper Semicontinuous Family of Attractors of Thermoviscoelastic Berger Plate
A system of partial differential equations with integral terms which take into account hereditary effects is considered. The system describes a behaviour of thermoviscoelastic plate with Berger's type of nonlinearity. The hereditary effect is taken into account both in the temperature variable...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1066482016-10-02T03:02:56Z On Singular Limit and Upper Semicontinuous Family of Attractors of Thermoviscoelastic Berger Plate Potomkin, M. A system of partial differential equations with integral terms which take into account hereditary effects is considered. The system describes a behaviour of thermoviscoelastic plate with Berger's type of nonlinearity. The hereditary effect is taken into account both in the temperature variable and in the bending one. The main goal of the paper is to analyze the passage to the singular limit when memory kernels collapse into the Dirac mass. In particular, it is proved that the solutions to the system with memory are close in some sense to the solutions to the corresponding memory-free limiting system. Besides, the upper semicontinuity of the family of attractors with respect to the singular limit is obtained. 2010 Article On Singular Limit and Upper Semicontinuous Family of Attractors of Thermoviscoelastic Berger Plate / M. Potomkin // Журнал математической физики, анализа, геометрии. — 2010. — Т. 6, № 3. — С. 305-336. — Бібліогр.: 25 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106648 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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A system of partial differential equations with integral terms which take into account hereditary effects is considered. The system describes a behaviour of thermoviscoelastic plate with Berger's type of nonlinearity. The hereditary effect is taken into account both in the temperature variable and in the bending one. The main goal of the paper is to analyze the passage to the singular limit when memory kernels collapse into the Dirac mass. In particular, it is proved that the solutions to the system with memory are close in some sense to the solutions to the corresponding memory-free limiting system. Besides, the upper semicontinuity of the family of attractors with respect to the singular limit is obtained. |
| format |
Article |
| author |
Potomkin, M. |
| spellingShingle |
Potomkin, M. On Singular Limit and Upper Semicontinuous Family of Attractors of Thermoviscoelastic Berger Plate Журнал математической физики, анализа, геометрии |
| author_facet |
Potomkin, M. |
| author_sort |
Potomkin, M. |
| title |
On Singular Limit and Upper Semicontinuous Family of Attractors of Thermoviscoelastic Berger Plate |
| title_short |
On Singular Limit and Upper Semicontinuous Family of Attractors of Thermoviscoelastic Berger Plate |
| title_full |
On Singular Limit and Upper Semicontinuous Family of Attractors of Thermoviscoelastic Berger Plate |
| title_fullStr |
On Singular Limit and Upper Semicontinuous Family of Attractors of Thermoviscoelastic Berger Plate |
| title_full_unstemmed |
On Singular Limit and Upper Semicontinuous Family of Attractors of Thermoviscoelastic Berger Plate |
| title_sort |
on singular limit and upper semicontinuous family of attractors of thermoviscoelastic berger plate |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2010 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/106648 |
| citation_txt |
On Singular Limit and Upper Semicontinuous Family of Attractors of Thermoviscoelastic Berger Plate / M. Potomkin // Журнал математической физики, анализа, геометрии. — 2010. — Т. 6, № 3. — С. 305-336. — Бібліогр.: 25 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT potomkinm onsingularlimitanduppersemicontinuousfamilyofattractorsofthermoviscoelasticbergerplate |
| first_indexed |
2025-07-07T18:49:02Z |
| last_indexed |
2025-07-07T18:49:02Z |
| _version_ |
1837015124197507072 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2010, vol. 6, No. 3, pp. 305–336
On Singular Limit and Upper Semicontinuous Family
of Attractors of Thermoviscoelastic Berger Plate
M. Potomkin
Department of Mechanics and Mathematics, V.N. Karazin Kharkiv National University
4 Svobody Sq., Kharkiv, 61077, Ukraine
E-mail:mika potemkin@mail.ru
Received June 11, 2009
A system of partial differential equations with integral terms which
take into account hereditary effects is considered. The system describes
a behaviour of thermoviscoelastic plate with Berger’s type of nonlinearity.
The hereditary effect is taken into account both in the temperature variable
and in the bending one. The main goal of the paper is to analyze the passage
to the singular limit when memory kernels collapse into the Dirac mass.
In particular, it is proved that the solutions to the system with memory
are close in some sense to the solutions to the corresponding memory-free
limiting system. Besides, the upper semicontinuity of the family of attractors
with respect to the singular limit is obtained.
Key words: materials with memory, attractors, upper semicontinuity.
Mathematics Subject Classification 2000: 35B41, 35B35.
1. Introduction
Let Ω be a bounded domain in R2 with smooth boundary ∂Ω. Our main
goal in this paper is to study asymptotic behaviour of the following system of
integrodifferential equations arising in the plate theory:
utt + (1 + h(0))∆2u +
∫∞
0 h′(s)∆2u(t− s)ds
+
(
Γ− ∫
Ω |∇u|2 dx
)
∆u + ∆v = p(x),
vt −
∫∞
0 k(s)∆v(t− s)ds−∆ut = 0,
x = (x1, x2) ∈ Ω ⊂ R2, t > 0
(1.1)
with initial data
u(t,x)|t≤0 = u0(−t,x), v(t,x)|t≤0 = v0(−t,x).
c© M. Potomkin, 2010
M. Potomkin
Here we consider a thin plate of uniform thickness. When the plate is un-
loaded and in null equilibrium, its middle surface occupies a region Ω ⊂ R2 of the
plane {x3 = 0}; u(t,x) is a vertical component of displacement of the correspond-
ing point in the middle surface. The presence of nonlocal term
(
Γ− ∫
Ω |∇u|2 dx
)
is explained by peculiarities of derivation of equation due to Berger’s approach
(see [2]). In the first equation it is taken into account that the material is ho-
mogeneous, isotropic and viscous, so a convolution integral with scalar kernel
h(s) appears. The function v(t,x) is a temperature variation field and thus it
satisfies one of the variants of heat equation. Here we consider the heat equation
according to Gurtin–Pipkin Law (see [16]), with the convolution integral with the
scalar kernel k(s) instead of usual Furier Law, which has two main shortcomings.
First, it is unable to take into account the memory effects. Second, it predicts
that a thermal disturbance at one point of the body is instantly felt everywhere
in the body. Also, we note that for the sake of simplicity we put all the other
physical constants equal to one.
The functions h(s) and k(s) are kernels which are the smooth, decreasing
and convex functions defined on [0, +∞). The functions h(s) and k(s) vanish at
infinity. In what follows we study the properties of the problem when h(s) and
k(s) collapse to a Dirac mass (as in [1, 4, 23], see also [9, 14] for models with
memory). Below in the paper this limiting procedure will be frequently called
a singular limit. We consider any fixed small parameters 0 < σ, ε ≤ 1 and
hσ(s) =
1
σ
h
( s
σ
)
, kε(s) =
1
ε
k
(s
ε
)
, s ∈ R+, (1.2)
where R+ = (0, +∞). Then we consider the system
utt + (hσ(0) + 1) ∆2u +
∫∞
0 h′σ(s)∆2u(t− s)ds
+
(
Γ− ∫
Ω |∇u|2 dx
)
∆u + ∆v = p(x),
vt −
∫∞
0 kε(s)∆v(t− s)ds−∆ut = 0
(1.3)
instead of system (1.1). If we formally pass to the limit σ, ε → 0+, the system
above collapses into the following one:
{
utt + ∆2ut + ∆2u +
(
Γ− ∫
Ω |∇u|2 dx
)
∆u + ∆v = p(x),
vt −∆v −∆ut = 0.
(1.4)
System (1.4) preserves a viscous dissipation effect expressed by the term ∆2ut
which replaces hσ(0)∆2u +
∫∞
0 h′σ(s)∆2u(t− s)ds in the first equation of system
(1.3). The limiting heat process is described by usual heat equation. In this case
the corresponding integral term
∫∞
0 kε(s)∆v(t − s)ds in (1.3) is replaced by ∆v
in (1.4).
306 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3
On Singular Limit and Upper Semicontinuous Family of Attractors
The boundary conditions for (1.3) are
u(t) = 0, x ∈ ∂Ω, t ≥ 0,
∆
[
(hσ(0) + 1)u(t) +
∫∞
0 h′σ(s)u(t− s)ds
]
= 0, x ∈ ∂Ω, t ≥ 0,
v(t) = 0, x ∈ ∂Ω, t ≥ 0
(1.5)
and for (1.4),
{
u(t) = ∆ [ut(t) + u(t)] = 0, x ∈ ∂Ω, t ≥ 0,
v(t) = 0, x ∈ ∂Ω, t ≥ 0.
(1.6)
The boundary conditions on function u are widely used as simplified hinged
(or edge-free) boundary conditions (e.g., the similar conditions were considered
in [4, 5, 14, 17, 19, 22]).
As in a plenty of previous works on systems with memory (e.g., see [9, 13,
14, 20] and references therein), by following [10], we will introduce new auxiliary
variables which replace the convolution integrals in original equation by a func-
tional operator applied to one of the added variables. It makes possible to apply
the asymptotic theory of semigroups.
The linear versions of the model considered and the related ones were studied
in [13, 14, 17]. In [13], the model was linear and the memory effects were taken
into account in thermal variable only (i.e., the kernel h(s) satisfied h(0) =
h(∞) = 0). The convergence of solutions to zero point was obtained there. The
linear version of the problem considered in this paper was studied in [14]. Uniform
exponential decay to zero point with respect to the parameters σ and ε, which
played the same role as in this paper, and the singular limit result were obtained in
[14]. In this paper, the model has a nonlinear term, so additional question about
the upper semicontinuity of attractors with respect to the parameters σ and ε is
answered here. The similar result on the upper semicontinuity of attractors for
nonlinear thermoviscoelastic Mindlin–Timoshenko model was obtained in [12].
Isothermal Berger model of oscillations of a plate without memory effects with
emphasis on its asymptotic behaviour was studied in [4, 6]. The thermoelastic
model with nonlinearity of Berger type and different boundary conditions was
considered in [3, 15].
This work is a continuation of [22] where the model was considered with the
fixed parameters σ and ε. The existence of the compact global attractor, its finite
dimensionality and boundedness with respect to topology stronger than topology
of the phase space were obtained. The main technique for treating the model is
the so-called stabilizability inequality (in our paper this inequality is formulated
in Th. 5.3). Similar inequalities were also obtained in various problems on the dis-
sipative wave dynamics and have become an important part of the studying of the
existence, smoothness and finite dimensionality of attractors (see [5–8] and refe-
rences therein). One should notice that these estimates are not the consequences
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3 307
M. Potomkin
of some common abstract results but essentially depend on the peculiarities of
the model under consideration. In works [5–8] the authors proposed how to use
stabilizability estimates to obtain finite dimensionality and smoothness of the
attractor and how to construct exponential attractors and determining function-
als. In this paper we will show that the coefficients of stabilizability inequality
obtained for our problem are uniform with respect to the parameters σ and ε.
It will help to prove the upper semicontinuity of attractors (Th. 6.2).
There are two main results obtained in this paper. The first one is the close-
ness between the corresponding solutions of the model with memory and the
memory-free limiting model on finite time intervals (singular limit). The second
result is the closeness between the attractors of the model with memory and the
memory-free limiting one. To get the result on the attractors we use a uniform
stabilizability estimate. Besides, we provide analysis (well-posedness and exis-
tence of the compact global attractor) for limiting problem (1.4). Up to our
knowledge, problem (1.4) has not been considered before.
For simplicity, we do not consider the limits for σ → 0 with ε = 0 (ε → 0
with σ = 0) separately, so below in the paper both parameters are either strictly
larger than 0 or equal to 0 simultaneously.
We conclude the introduction with a brief plan of the paper.
In Section 2 we rewrite the system in abstract form. Besides, the assumptions
to be used in the sequel are given. Section 3 collects the results obtained in [22]
on the considered semigroup. The question about the singular limit on finite
time intervals is answered in Section 4. Section 5 includes the assertions of the
existence of global attractor and of the uniform stabilizability estimate for both
(1.3) and (1.4). Section 6 contains the theorem on the upper semicontinuity of the
family of attractors when (σ, ε) → (0, 0). The proofs are relegated to Section 7.
Except the proofs of stabilizability estimates, Section 7 also contains the proof
of the smoothness of the family of attractors which is used in Theorem 6.2.
For reader’s convenience we note that the main results are formulated in
Theorems 4.1 and 6.2.
2. Main Settings
2.1. Kernels
The conditions on the kernels h(s) and k(s) imposed below are similar to
those in [9, 13, 14].
First, we assume that h, k : [0, +∞) → R+ are smooth, decreasing and
summable functions. For the sake of simplicity, we assume that h(0) = k(0) = 1,
moreover,
∞∫
0
h(s)ds =
∞∫
0
k(s)ds = 1.
308 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3
On Singular Limit and Upper Semicontinuous Family of Attractors
We set β(s) = −h′(s) and µ(s) = −k′(s), where β and µ are supposed to
satisfy
β, µ ∈ C1(R+) ∩ L1(R+) ∩ C[0,∞), β(s) ≥ 0, µ(s) ≥ 0, (2.1)
and there exists a constant δ > 0 such that
β′(s) + δβ(s) ≤ 0 and µ′(s) + δµ(s) ≤ 0. (2.2)
Consider any σ ∈ (0, 1] and ε ∈ (0, 1]. We set
βσ(s) =
1
σ2
β
( s
σ
)
= −h′σ(s) and µε(s) =
1
ε2
µ
(s
ε
)
= −k′ε(s), (2.3)
where hσ(s) and kε(s) are defined in (1.2).
We note that the properties of (2.1) for βσ(s) and µε(s) are preserved.
Assumptions (2.2) turn into
β′σ(s) +
δ
σ
βσ(s) ≤ 0 and µ′ε(s) +
δ
ε
µε(s) ≤ 0. (2.4)
In the sequel we will frequently use the following equalities:
∞∫
0
βσ(s)ds =
1
σ
,
∞∫
0
µε(s)ds =
1
ε
,
∞∫
0
sβσ(s)ds =
∞∫
0
sµε(s)ds = 1.
2.2. Spaces
We denote by H a separable Hilbert space with inner product (·, ·) and the
corresponding norm ‖·‖. Let A be a selfadjoint positive linear operator defined
on a domain D(A) ⊂ H. Assume that there exists an eigenbasis {ek}∞k=1 of the
operator A such that
(ek, ej) = δkj , Aek = λkek, k, j = 1, 2, . . . ,
and 0 < λ1 ≤ λ2 ≤ . . . , lim
k→∞
λk = ∞, where λk is a corresponding eigenvalue of
the operator A.
We introduce the scale of Hilbert spaces Hs = D(As/2) with s ∈ R endowed
with usual inner products (v, w)2s = (Asv, Asw).
We introduce the weighted Hilbert spaces L2
βσ
(R+;H2) and L2
µε
(R+; H1) of
measurable functions ξ with values in H2 or H1, respectively, such that
‖ξ‖2
L2
βσ
(R+;H2) ≡
+∞∫
0
βσ(s) ‖ξ(s)‖2
2 ds < ∞
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3 309
M. Potomkin
and
‖ξ‖2
L2
µε
(R+;H1) ≡
+∞∫
0
µε(s) ‖ξ(s)‖2
1 ds < ∞.
The following Cartesian product of Hilbert spaces will play the role of a phase
space for the considered model (m = 0):
Hm
σ,ε =
{
Hm+2 ×Hm ×Hm × L2
βσ
(R+; Hm+2)× L2
µε
(R+; Hm+1), if σ, ε > 0,
Hm+2 ×Hm ×Hm, if σ = ε = 0.
We note also that if the index is equal to null, we may omit it (e.g., use H
instead of H0
0,0).
To perform a comparison between a five-component vector from Hm
σ,ε and
a three-component vector from Hm, we have to introduce the following lifting
and projection maps (we preserve the notations introduced formerly in [14]):
Lσ,ε : Hm → Hm
σ,ε, Qσ : Hm
σ,ε → L2
βσ
(R+;Hm+2),
P : Hm
σ,ε → Hm, Qε : Hm
σ,ε → L2
µε
(R+;Hm+1)
defined by
Lσ,ε(u, ut, v) =
{
(u, ut, v, 0, 0), if σ, ε > 0,
(u, ut, v), if σ = ε = 0,
P(u, ut, v, ξ, η) = (u, ut, v), Qσ(u, ut, v, ξ, η) = ξ and Qε(u, ut, v, ξ, η) = η,
respectively.
2.3. Memory Variables
Following the ideas of Dafermos (see [9, 10, 14, 20] and references therein),
we introduce additional variables, namely, the summed past history of u and v
defined as
ξt(s) = u(t)− u(t− s), ηt(s) =
s∫
0
v(t− y)dy.
They formally satisfy the linear equations
∂
∂tξ
t + ∂
∂sξ
t = ut(t), ∂
∂tη
t + ∂
∂sη
t = v(t)
and ξt(0) = ηt(0) = 0, whereas
ξ0(s) = ξ0(s) ≡ u0(0)− u0(s), η0(s) = η0(s) ≡
s∫
0
v0(y)dy.
310 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3
On Singular Limit and Upper Semicontinuous Family of Attractors
Let Tσ, Tε be the linear operators in L2
βσ
(R+;H2) and L2
µε
(R+;H1), respec-
tively, with the domains
D(Tσ) =
{
ξ ∈ L2
βσ
(R+;H2)
∣∣∣ξs ∈ L2
βσ
(R+; H2), ξ(0) = 0
}
,
D(Tε) =
{
η ∈ L2
µε
(R+; H1)
∣∣ηs ∈ L2
µε
(R+;H1), η(0) = 0
}
,
defined by Tσξ = −ξs and Tεη = −ηs for all admissible ξ and η. Here ηs denotes
the distributional derivative with respect to the variable s.
These operators satisfy the following inequalities (see, e.g., [20]):
(Tσξ, ξ)L2
βσ
(R+;H2) ≤ − δ
2σ
‖ξ‖2
L2
βσ
(R+;H2) , ∀ξ ∈ D(Tσ), (2.5)
(Tεη, η)L2
µε (R+;H1) ≤ − δ
2ε
‖η‖2
L2
µε (R+;H1) , ∀η ∈ D(Tε). (2.6)
2.4. Requirements on Nonlinearity
We impose conditions on function M(·)
M(z) ≡
z∫
0
M(ξ)dξ ≥ −az − b, a ∈ (0, λ1), b ∈ R,
M(z) ∈ C2(R+).
(2.7)
We note that M(z) = z − Γ satisfies (2.7). This M(z) corresponds to the
standard Berger nonlinearity.
2.5. Abstract Form of the Problem
In view of introduced settings, when σ, ε > 0, original system (1.3) may be
rewritten as follows:
utt + A2u +
+∞∫
0
βσ(s)A2ξt(s)ds−Av = p−M
(∥∥A1/2u
∥∥2
)
Au,
vt +
+∞∫
0
µε(s)Aηt(s)ds + Aut = 0,
ξt
t = Tσξt + ut(t), ηt
t = Tεη
t + v(t),
u|t=0 = u0, ut|t=0 = u1, v|t=0 = v0, ξt|t=0 = ξ0, ηt|t=0 = η0.
(2.8)
Limiting memory-free system (1.4) has the form
utt + A2u + A2ut −Av = p−M
(∥∥A1/2u
∥∥2
)
Au,
vt + Av + Aut = 0,
u|t=0 = u0, ut|t=0 = u1, v|t=0 = v0.
(2.9)
To formulate the existence and uniqueness results, we have to define various
types of solutions following the theory of semigroups of linear operators
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3 311
M. Potomkin
(see [21]). For this, we write the linear part of equation (2.8) as the linear operator
Lσ,ε : D(Lσ,ε) ⊂ Hσ,ε → Hσ,ε given by
Lσ,εU =
w
−A2u−
∞∫
0
βσ(s)A2ξ(s)ds + Av
−
∞∫
0
µε(s)Aη(s)ds−Aut
Tσξ + w
Tεη + v
, U =
u
w
v
ξ
η
∈ Hσ,ε
and equipped with the domain
D(Lσ,ε) =
U =
u
w
v
ξ
η
∈ Hσ,ε
∣∣∣∣∣∣∣∣∣∣∣∣
ξ ∈ D(Tσ), η ∈ D(Tε),
w ∈ H2, v ∈ H1,
A2u +
∞∫
0
βσ(s)A2ξ(s)ds−Av ∈ H,
∞∫
0
µε(s)Aη(s)ds ∈ H.
.
For problem (2.9), the linear part is represented by L0,0 : D(L0,0) ⊂ H → H
given as follows
L0,0 =
w
−A2 [w + u] + Av
−Av −Aw
, U =
u
w
v
∈ H
and equipped with the domain
D(L0,0) =
U =
u
w
v
∈ H
∣∣∣∣
u + w ∈ H4,
w, v ∈ H2
.
Using the standard method, it is possible to prove that Lσ,ε is an infinitesimal
operator of a strongly continuous linear semigroup. For the case σ, ε > 0 we refer
to [13, 22]. To prove the same statement for the operator L0,0 it is sufficient to
note that (i) the point 0 ∈ C belongs to the resolvent set of the operator L0,0, and
(ii) the operator L0,0 is dissipative, i.e., for all U ∈ D(L0,0) : (L0,0U,U)H ≤ 0.
Therefore, Theorem 1.4.3 in [21] (the Lumer–Phillips Theorem) is applicable, and
L0,0 is also an infinitesimal operator of a strongly continuous linear semigroup.
After having made final notations for the nonlinear term, namely,
f(U) = (0,−M
(∥∥∥A1/2u
∥∥∥
2
)
Au + p, 0, 0, 0)T,
312 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3
On Singular Limit and Upper Semicontinuous Family of Attractors
we may rewrite the nonlinear problem (2.8) (or (2.9)) as a first order problem of
the form {
U̇(t) = Lσ,εU(t) + f(U(t))
U(0) = U0 ∈ Hσ,ε.
(2.10)
We recall that according to [21], U(t) is a mild solution of (2.10) if U(t)
satisfies the following equality:
U(t) = etLσ,ε
U0 +
t∫
0
e(t−τ)Lσ,ε
f(U(τ))dτ,
where etLσ,ε
is the linear semigroup on Hσ,ε whose infinitesimal operator is Lσ,ε.
We say that U(t) is a strong solution on the interval [0, T ) if it is continuously
differentiable, its values lie in D(Lσ,ε) and it satisfies (2.10).
3. Nonlinear Semigroup
We collect here some results on the generation of nonlinear semigroup and
some of its properties. All results (Th. 3.1, Props. 3.2, 3.4, 3.5) are shown in [22].
3.1. Well-Posedness
We recall that Sσ,ε(t) : Hσ,ε → Hσ,ε is called a continuous semigroup of op-
erators if (i) Sσ,ε(0) = I, Sσ,ε(t + τ) = Sσ,ε(t)Sσ,ε(τ), t, τ ≥ 0, (ii) the map-
ping t → Sσ,ε(t)U0 is continuous for any U0 ∈ Hσ,ε, and (iii) the mapping
U → Sσ,ε(t)U is continuous for any t ≥ 0. Also the couple (Hσ,ε, Sσ,ε) is called
a dynamical system on the phase space Hσ,ε.
Theorem 3.1. Let assumptions (2.1), (2.4) and (2.7) hold true and σ, ε > 0
or σ = ε = 0. Assume also that p ∈ H. Then for all U0 ∈ Hσ,ε and T > 0
there exists a unique mild solution U(t) ∈ C(0, T ;Hσ,ε). Therefore, we may
set Sσ,ε(t)U0 = U(t) as a nonlinear semigroup corresponding to problem (2.8)
(σ, ε > 0) or to problem (2.9) (σ = ε = 0).
Besides, if U1, U2 ∈ Hσ,ε and ‖Ui‖Hσ,ε
≤ R, then there exists a positive
constant CR,T such that
‖Sσ,ε(t)U1 − Sσ,ε(t)U2‖Hσ,ε
≤ CR,T ‖U1 − U2‖Hσ,ε
, t ∈ [0, T ] . (3.1)
Therefore, Sσ,ε(t) is a continuous semigroup of operators. And if U0 ∈
D(Lσ,ε), then the corresponding mild solution U(t) is a strong solution.
The proof is based on the abstract result from [21] on the perturbation of s.c.
linear semigroup. For details we refer to [22, Th. 2.1].
Further analysis requires additional information about the considered semi-
group.
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3 313
M. Potomkin
Consider the functional
Φ(U) =
1
2
‖U‖2
Hσ,ε
+
1
2
M
(∥∥∥A1/2u
∥∥∥
2
)
− (p, u),
which we call the Lyapunov function after having substituted a mild solution
U(t) instead of U .
It is shown in [22, Subsect. 2.5] that the Lyapunov function possesses the
properties that can be formulated as follows (if compared with [22], here we add
the obvious uniformity with respect to σ and ε of constants C and α)
Proposition 3.2. There holds (i) if ‖U‖Hσ,ε
≤ R, then
α ‖U‖2
Hσ,ε
− C ≤ Φ(U) ≤ max
s∈[0,R2]
|M(s)|+ ‖U‖2
Hσ,ε
+ C, (3.2)
where α and C are strictly positive and do not depend on σ, ε and R, (ii) for each
mild solution U(t) of problem (2.8) the function Φ(U(t)) is nonincreasing with
respect to t, and (iii) if Φ(Sσ,ε(t)U0) = Φ(U0) for any t > 0, then Sσ,ε(t)U0 = U0
for any t > 0.
The lemma below will be needed in Section 4. Besides, it asserts that spaces
Hm
σ,ε are positively invariant (i.e., Sσ,ε(t)Hm
σ,ε ⊆ Hm
σ,ε).
Lemma 3.3. Let m ≥ 0 and U0 ∈ BHm
σ,ε
(R). Then there exist positive con-
stants CR, LR which do not depend on the parameters σ and ε such that
‖U(t)‖2
Hm
σ,ε
≤ CReLRt,
where U(t) = (u(t), ut(t), v(t), ξt, ηt) is a solution to (2.8) (or to (2.9) if σ = 0
and ε = 0) with U(0) = U0.
P r o o f. The case m = 0 is obtained, e.g., if we apply the properties of
Φ ((i) and (ii) from Prop. 3.2). Namely, for all U0 ∈ BHm
σ,ε
(R) there exists CR
uniform with respect to σ and ε such that ‖Sσ,ε(t)U0‖2
Hσ,ε
≤ CR.
Now we turn to m > 0. Here we need to introduce the projector
PN : H −→ Lin {e1, . . . , eN} ,
where each ek is an eigenvector of the operator A.
Multiplying the equations in (2.8), or (2.9)), by AmPNut in H, AmPNv in H,
PNξt in L2
βσ
(R+; Hm+2), PNηt in L2
µε
(R+; Hm+1), we obtain (for similar calcu-
lations we refer to [4, Ch. 4])
1
2
d
dt
‖PNU(t)‖2
Hm
ε,σ
≤
(
M
(∥∥∥A1/2u
∥∥∥
2
)
Au− p, PNut
)
m
≤ C∗
R(1+‖PNU(t)‖2
Hm
ε,σ
).
314 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3
On Singular Limit and Upper Semicontinuous Family of Attractors
We used above the desired estimate for m = 0, and PNU(t) denoted the vector
(PNu(t), PNut(t), PNv(t), PNξt, PNηt). The Gronwall Lemma implies
‖PNU(t)‖2
Hm
ε,σ
≤ C∗∗
R (1 + ‖PNU(0)‖2
Hm
ε,σ
)eLRt.
Now we may extend this estimate to the case of any m ≥ 0 by passing to the limit
with respect to N what is possible due to the fact that initial data lies in Hm
σ,ε.
3.2. Set of Stationary Points
We define
N = {U0 ∈ Hσ,ε |Sσ,ε(t)U0 = U0 for all t > 0} ,
which is called the set of stationary points to problem (2.8). It turns out that
this set depends neither on σ nor on ε, i.e., the following proposition holds.
Proposition 3.4. The set N can be represented as follows:
N =
{
U = (u; 0; 0; 0; 0) : A2u + M
(∥∥∥A1/2u
∥∥∥
2
)
Au = p
}
,
and there exists such R > 0 that ‖u‖2 ≤ R, ∀u ∈ H2 : (u; 0; 0; 0; 0) ∈ N and R
apparently depends neither on σ nor on ε.
3.3. Explicit Representation Formulas
In the sequel we need the typical for equations with infinite memory explicit
representation formulas (similar to those considered in [9, 13, 14, 22]).
Proposition 3.5. Let U(t) = (u(t);w(t); v(t); ξt; ηt) be a mild solution of
(2.10) with initial data U0 = (u0;w0; v0; ξ0; η0). Then the following representa-
tions hold true:
ξt(s) =
{
u(t)− u(t− s), t > s > 0,
ξ0(s− t) + u(t)− u(0), t ≤ s,
(3.3)
and
ηt(s) =
s∫
0
v(t− y)dy, t > s > 0,
η0(s− t) +
t∫
0
v(t− y)dy, t ≤ s.
(3.4)
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3 315
M. Potomkin
4. A Singular Limit on Finite Time Intervals
In this section we prove our first main result — the closeness between the
solutions of (2.8) and (2.9) as ε and σ tend to zero on finite time intervals and
with sufficiently smooth initial data. In the proof we rely on the ideas applied
in [9, 14]. We note here that the model in [14] is linear and exponential stable,
so the analysis there is simpler and allows to state the singular limit result on
the infinite time interval. The work [9] deals only with the thermal convolution
integral collapsing into −∆v and with another type of nonlinearity. Traditionally,
C will denote a generic positive constant. In further proof C is allowed to depend
on R and T , positive parameters that will be introduced in theorem below.
Theorem 4.1. Let R > 0 and T > 0. Assume also that U0 ∈ BH2
σ,ε
(R). Then
for all t ∈ [0, T ] there hold
‖PSσ,ε(t)U0 − S0,0(t)PU0‖H ≤ C( 8
√
σ + 8
√
ε), (4.1)
‖QσSσ,ε(t)U0‖L2
βσ
(R+;H2) + ‖QεSσ,ε(t)U0‖L2
µε (R+;H1)
≤ e−
δt
4σ ‖ξ0‖L2
βσ
(R+;H2) + e−
δt
4ε ‖η0‖L2
µε (R+;H1) + C(
√
σ +
√
ε). (4.2)
R e m a r k. If we collect both above estimates, we may formulate the
statement of the theorem as follows:
For all R > 0 and T > τ > 0 there holds
lim
ε → 0+
σ → 0+
sup
U0∈BH2
σ,ε
(R)
sup
t∈[τ,T ]
‖Sσ,ε(t)U0 − Lσ,εS0,0(t)PU0‖Hε,σ
= 0. (4.3)
The necessity of introducing τ > 0 in (4.3) is caused by the presence of the
terms with exponents of the form e−
δt
4σ multiplied by the norm of initial memory
in (4.2).
P r o o f. Let us assume that (û(t), ût(t), v̂(t), ξ̂t, η̂t) is a solution to (2.8)
with initial data U0 = (u0, u1, v0, ξ0, η0), and first three components of the time-
dependent vector (u(t), ut(t), v(t), ξ(t); η(t)) stand for the solution to (2.9) with
initial data (u0, u1, v0). Two other components satisfy the following two Cauchy
problems in L2
βσ
(R+; H2) and L2
µε
(R+; H1):
{
ξt
t = Tσξt + ut(t), t > 0,
ξ0 = ξ0,
{
ηt
t = Tσηt + v(t), t > 0,
η0 = η0.
316 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3
On Singular Limit and Upper Semicontinuous Family of Attractors
Besides, we introduce the functions u(t) = û(t) − u(t), ut(t) = ût(t) − ut(t),
v(t) = v̂(t)− v(t), ξ
t = ξ̂t − ξt and ηt = η̂t − ηt, which satisfy the system
utt + A2u +
∞∫
0
βσ(s)A2ξ̂t(s)ds−A2ut −Av
= M
(∥∥A1/2u
∥∥2
)
Au−M
(∥∥A1/2û
∥∥2
)
Aû,
vt +
∞∫
0
µε(s)Aη̂t(s)ds−Av + Aut = 0,
ξ
t
t = Tσξ
t + ut(t), ηt
t = Tεη + v(t)
(4.4)
with the null initial data.
For further analysis one more lemma is required.
Lemma 4.2. Assume that the conditions of Theorem 4.1 hold. Then for any
t ∈ [0, T ] there holds
max
{∥∥∥ξ̂t
∥∥∥
2
L2
βσ
(R+;H2)
,
∥∥ξt
∥∥2
L2
βσ
(R+;H2)
}
≤ ‖ξ0‖2
L2
βσ
(R+;H2) e−
δt
2σ + Cσ, (4.5)
and
max
{∥∥η̂t
∥∥2
L2
µε (R+;H1)
,
∥∥ηt
∥∥2
L2
µε (R+;H1)
}
≤ ‖η0‖2
L2
µε (R+;H1) e−
δt
2ε + Cε. (4.6)
P r o o f. The result can be obtained by multiplying the equation of ξ (or η)
by ξ (or η) in L2
βσ
(R+; H2) (or L2
µε
(R+;H1)). Taking (2.5), (2.6) and Lemma 3.3,
we have
d
dt ‖ξ‖2
L2
βσ
(R+;H2) + δ
σ ‖ξ‖2
L2
βσ
(R+;H2)
≤ C
∞∫
0
βσ(s) ‖Aξ(s)‖ ds
≤ C
(∞∫
0
βσ(s)ds
)1/2 (∞∫
0
βσ(s) ‖Aξ(s)‖2 ds
)1/2
≤ C√
σ
‖ξ‖L2
βσ
(R+;H2)
≤ δ
2σ ‖ξ‖2
L2
βσ
(R+;H2) + C.
And the Gronwall Lemma leads to inequality (4.5). Inequality (4.6) is ob-
tained in the same manner.
We proceed the proof of Theorem 4.1 by considering system (4.4). Multi-
plying the first equation by ut in H, the second, by v in H, the third, by ξ in
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3 317
M. Potomkin
L2
βσ
(R+; H2), the fourth, by η in L2
µε
(R+;H1), and summerizing the four obtained
equalities we get
1
2
d
dt
(
‖Au‖2 + ‖ut‖2 + ‖v‖2 +
∥∥ξ
∥∥2
L2
βσ
(R+;H2)
+ ‖η‖2
L2
µε (R+;H1)
)
≤ Jσ(t) + Iε(t) + (M
(∥∥A1/2u
∥∥2
)
Au−M
(∥∥A1/2û
∥∥2
)
Aû, ut),
where
Jσ(t) = −
∞∫
0
βσ(s)(Aξt(s), Aut(t))ds + (Aut(t), Aut),
Iε(t) = −
∞∫
0
µε(s)(A1/2ηt(s), A1/2v(t))ds + (A1/2v(t), A1/2v(t)).
We write each of introduced terms, Jσ and Iε, as follows:
Jσ =
5∑
j=1
Jj(t), Iε =
5∑
j=1
Ij(t),
where the terms Jj are defined by
J1(t) =
∫∞√
σ sβσ(s)(Aut(t), Aut(t))ds,
J2(t) = − ∫∞√
σ βσ(s)(Aξt(s), Aut(t))ds,
J3(t) = − ∫ √σ
min{√σ,t} βσ(s)(Aξ0(s− t), Aut(t))ds,
J4(t) =
∫ √σ
min{√σ,t}(s− t)βσ(s)(Aut, Aut(t))ds,
J5(t) =
∫ √σ
0 βσ(s)
[∫ min{s,t}
0 (A(ut(t)− ut(t− y)), Aut(t))dy
]
ds,
and the terms Ii are defined by
I1(t) =
∫∞√
ε sµε(s)(A1/2v(t), A1/2v(t))ds,
I2(t) = − ∫∞√
ε µε(s)(A1/2ηt(s), A1/2v(t))ds,
I3(t) = − ∫ √ε
min{√ε,t} µε(s)(A1/2η0(s− t), A1/2v(t))ds,
I4(t) =
∫ √ε
min{√ε,t}(s− t)µε(s)(A1/2v, A1/2v(t))ds,
I5(t) =
∫ √ε
0 µε(s)
[∫ min{s,t}
0 (A1/2(v(t)− v(t− y)), A1/2v(t))dy
]
ds.
We also note that the inequalities below hold
∞∫
√
σ
sβσ(s)ds ≤ Cσ,
∞∫
√
σ
βσ(s)ds ≤ C
√
σ. (4.7)
318 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3
On Singular Limit and Upper Semicontinuous Family of Attractors
The first relation (for σ ∈ (0, 1]) can be obtained as follows:
∞∫
√
σ
sβσ(s)ds =
∞∫
1/
√
σ
sβ(s)ds ≤ eδβ(1)
δ2
(
1 +
δ√
σ
)
e
− δ√
σ ≤ Cασα, ∀α ∈ R,
where (2.4) is used. The second one holds due to
∞∫
√
σ
βσ(s)ds ≤ 1√
σ
∞∫
√
σ
sβσ(s)ds ≤ Cασα, ∀α ∈ R.
It is apparent that the same inequalities take place for µε
∞∫
√
ε
sµε(s)ds ≤ Cε,
∞∫
√
ε
µε(s)ds ≤ C
√
ε. (4.8)
Now we will estimate Jj and Ij by using
∥∥A2û(t)
∥∥2 + ‖Aût(t)‖2 +
∥∥∥A1/2v̂(t)
∥∥∥
2
≤ C, (4.9)
∥∥A2u(t)
∥∥2 +
t∫
0
∥∥A2ut(τ)
∥∥2
dτ +
t∫
0
‖Av(τ)‖2 dτ
+ ‖Aut(t)‖2 +
∥∥∥A1/2v(t)
∥∥∥
2
≤ C, (4.10)
which follow from Lemma 3.3 and the condition that the initial data U0 is taken
from a closed ball in H4
σ,ε of radius R.
1. Using (4.9), (4.10) and the properties of the kernels (4.7) and (4.8), we
immediately have
J1(t) =
∫∞√
σ sβσ(s)(Aut(t), Aut(t))ds ≤ Cσ,
I1(t) =
∫∞√
ε sµε(s)(A1/2v(t), A1/2v(t))ds ≤ Cε.
2. To estimate J2(t) (and I2(t)) we apply Lemma 4.2
J2(t) = −
∞∫
√
σ
βσ(s)(Aξt(s), Aut(t))ds ≤ C
∞∫
√
σ
βσ(s)
∥∥Aξt(s)
∥∥ ds
≤ C
∞∫
√
σ
βσ(s)
∥∥Aξt(s)
∥∥2
ds + C
∞∫
√
σ
βσ(s)ds.
And application of Lemma 4.2 finally gives J2(t) ≤ Ce−
δt
2σ + Cσ.
The inequality I2(t) ≤ Ce−
δt
2ε + Cε can be obtained in a similar way.
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3 319
M. Potomkin
3. If t ≥ √
σ, then J3(t) vanishes. If we consider the case t <
√
σ, then
J3(t) = −
√
σ∫
t
βσ(s)(Aξ0(s− t), Aut(t))ds ≤ C
√
σ−t∫
0
βσ(y + t) ‖Aξ0(y)‖ dy
≤ Ce−
δt
σ
∞∫
0
βσ(s) ‖Aξ0(s)‖ ds ≤ Ce−
δt
σ
(∞∫
0
βσ(s)ds
)1/2
= C√
σ
e−
δt
σ ,
and, similarly, I3(t) ≤ C√
ε
e−
δt
ε .
4. J4(t) is treated in the same way as J3(t), namely,
J4(t) =
√
σ∫
t
(s− t)βσ(s)(Aut(t), Aut(t))ds ≤ C
√
σ−t∫
0
sβσ(s + t)ds ≤ Ce−
δt
σ ,
and, similarly, I4(t) ≤ Ce−
δt
σ .
5. Before estimating J5(t), we note that the first equation in (2.9) gives
t∫
t−y
‖utt(ς)‖2 dς ≤ 5
[
t∫
t−y
∥∥A2ut(ς)
∥∥2
dς +
t∫
t−y
∥∥A2u(ς)
∥∥2
dς
+
t∫
t−y
(M
(∥∥A1/2u
∥∥2
)
‖Au‖)2dς +
t∫
t−y
‖Av(ς)‖2 dς +
t∫
t−y
‖p‖2 dς
]
≤ C,
which implies
‖ut(t)− ut(t− y)‖ ≤
t∫
t−y
‖utt(ς)‖ dς ≤ √
y
t∫
t−y
‖utt(ς)‖2 dς
1/2
≤ C
√
y.
Moreover,
‖Aut(t)−Aut(t− y)‖ ≤ ‖ut(t)− ut(t− y)‖1/2
∥∥A2ut(t)−A2ut(t− y)
∥∥1/2
≤ ‖ut(t)− ut(t− y)‖1/2
(∥∥A2ut(t)
∥∥1/2 +
∥∥A2ut(t− y)
∥∥1/2
)
≤ C 4
√
y
(∥∥A2ut(t)
∥∥1/2 +
∥∥A2ut(t− y)
∥∥1/2
)
.
Hence,
J5(t) =
√
σ∫
0
βσ(s)
[∫ min{s,t}
0 (Aut(t)−Aut(t− y), Aut(t))dy
]
ds
≤ C
√
σ∫
0
βσ(s)
[∫ min{s,t}
0
4
√
y
(∥∥A2ut(t)
∥∥1/2 +
∥∥A2ut(t− y)
∥∥1/2
)
dy
]
ds
≤ C 8
√
σ
√
σ∫
0
βσ(s)
[∫ min{s,t}
0
(∥∥A2ut(t)
∥∥1/2+
∥∥A2ut(t− y)
∥∥1/2
)
dy
]
ds ≡ C 8
√
σg2,σ(t),
320 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3
On Singular Limit and Upper Semicontinuous Family of Attractors
and, similarly, I5(t) ≤ C 8
√
εg2,ε with g2,ε defined by
g2,ε(t) =
√
ε∫
0
µε(s)
[∫ min{s,t}
0
(
‖Av(t)‖1/2 + ‖Av(t− y)‖1/2
)
dy
]
ds.
Functions g2,σ and g2,ε possess the property
T∫
0
[g2,σ(t) + g2,ε(t)] dt ≤ C. (4.11)
Collecting all above estimates, including
(M
(∥∥∥A1/2u
∥∥∥
2
)
Au−M
(∥∥∥A1/2û
∥∥∥
2
)
Aû, ut) ≤ C
(
‖Au‖2 + ‖ut‖2
)
,
we finally obtain
1
2
d
dt
(
‖Au‖2 + ‖ut‖2 + ‖v‖2 +
∥∥ξ
∥∥2
L2
βσ
(R+;H2)
+ ‖η‖2
L2
µε (R+;H1)
)
≤ g1,σ + g1,ε + C 8
√
σg2,σ(t) + C 8
√
εg2,ε(t) + C
(
‖Au‖2 + ‖ut‖2
)
,
where gi,ς = C
(√
ς + 1√
ς
e−
δt
2ς
)
, i = 1, 2, for which the following inequality holds:
t∫
0
gi,ς(y)dy ≤ C
√
ς, t ∈ [0, T ].
Applying the Gronwall Lemma and taking into account (4.11), we get the
result
1
2
(
‖Au‖2 + ‖ut‖2 + ‖v‖2 +
∥∥ξ
∥∥2
L2
βσ
(R+;H2)
+ ‖η‖2
L2
µε (R+;H1)
)
≤ C( 8
√
σ + 8
√
ε).
5. Attractors and Their Properties
First, we recall some definitions.
Definition 5.1. Aσ,ε ⊂ Hσ,ε is called a global attractor if (i) Aσ,ε is a closed
bounded strictly invariant set (Sσ,ε(t)Aσ,ε = Aσ,ε ∀t ≥ 0) and (ii) Aσ,ε possesses
the uniform attraction property, i.e., for any bounded set B ⊂ Hσ,ε the following
equality holds true:
lim
t→+∞ sup
U∈B
distHσ,ε (Sσ,ε(t)U,Aσ,ε) = 0.
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3 321
M. Potomkin
We also define (for details we refer to [4, 25]) the unstable manifold Mu (N )
emanating from N as a set of all U ∈ Hσ,ε such that there exists a full trajectory
γ = {U(t) : t ∈ R} with the properties
U(0) = U and lim
t→−∞distHσ,ε(U(t),N ) = 0.
5.1. Case σ, ε > 0
The result [22, Th. 3.9] obtained on the attractors of the case σ, ε > 0 and
some of their properties is stated as follows
Theorem 5.2. Assume that all conditions of Theorem 3.1 hold. Then for
any positive values of parameters σ and ε the dynamical system (Hσ,ε, Sσ,ε(t))
possesses a compact global attractor Aσ,ε of finite fractal dimension.
Moreover, the attractor Aσ,ε consists of full trajectories belonging to the
domain D(Lσ,ε) and, moreover, Aσ,ε = Mu (N ), where N denotes the set of
stationary points.
To provide the standard procedure of the proof of upper semicontinuity of
attractors in Section 6 (see [1, 23]) we need the two theorems below. We refer
to [22, Th. 3.11 and estimate (4.2)], where these theorems are stated and proved
with fixed σ > 0 and ε > 0. Here we need a uniform (with respect to σ and
ε) variant of these theorems. Section 7 is devoted to the verification that all
constants appeared in Theorems 5.3 and 5.4 are uniform with respect to σ and ε.
The first auxiliary result is the stabilizability estimate. For the similar esti-
mates and their application to asymptotic analysis we refer to [3, 5–8].
Theorem 5.3. Assume σ, ε > 0. Let (u1(t); v1(t); ξ1,t; η1,t) and (u2(t); v2(t);
ξ2,t; η2,t) be two solutions of problem (2.10) with initial data U i = (ui
0;u
i
1; v
i
0; ξ
i
0; η
i
0),
i = 1, 2. Assume that
∥∥Aui(t)
∥∥2 +
∥∥ui
t(t)
∥∥2 +
∥∥vi(t)
∥∥2 +
∥∥ξi,t
∥∥2
L2
βσ
(R+;H2)
+
∥∥ηi,t
∥∥2
L2
µε (R+;H1)
≤ R2
for all t ≥ 0. Let
Z(t) =
(
u1(t)− u2(t);u1
t (t)− u2
t (t); v
1(t)− v2(t); ξ1,t − ξ2,t; η1,t − η2,t
)
and z(t) = u1(t)− u2(t).
Then there exist positive constants CR and γ depending neither on σ nor on
ε such that
‖Z(t)‖2
Hσ,ε
≤ CR ‖Z(0)‖2
Hσ,ε
e−γt + CR sup
0≤τ≤t
‖z(τ)‖2 . (5.1)
322 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3
On Singular Limit and Upper Semicontinuous Family of Attractors
We note that this theorem also implies that the fractal dimension of Aσ,ε can
be estimated by constant uniform with respect to σ and ε (we refer to the similar
uniform results of [3, 5, 6]).
The second auxiliary result is the theorem stating that the attractor
is a bounded subset of some ”smoother” space.
Theorem 5.4. There exists a positive constant C not depending on t, σ and
ε such that for any trajectory U(t) = (u(t);ut(t); v(t); ξt; ηt) lying in the attractor
we have
‖utt(t)‖2 + ‖Aut(t)‖2 + ‖vt(t)‖2 +
∥∥ξt
t
∥∥2
L2
βσ
(R+;H2)
+
∥∥ηt
t
∥∥2
L2
µε (R+;H1)
+
∥∥∥A3/2u(t)
∥∥∥
2
+
∥∥∥A1/2v(t)
∥∥∥
2
+
∥∥Tσξt
∥∥2
L2
βσ
(R+;H2)
+
∥∥Tεη
t
∥∥2
L2
µε (R+;H1)
≤ C2(5.2)
for all t ∈ R.
5.2. Case σ = ε = 0
Up to our knowledge, the following theorem is new. Its statement follows
from stabilizability estimate (5.3) in the same way as in [3, 5–8] for the gradient
dynamical systems possessing stabilizability estimates.
Theorem 5.5. Assume that condition (2.7) holds and p ∈ H. Then the
dynamical system (H, S0,0(t)) possesses a compact global attractor A0,0 of finite
fractal dimension.
And the result on the stabilizability estimate to problem (2.9) is formulated
as follows
Theorem 5.6. Let (u1(t); v1(t)) and (u2(t); v2(t)) be two solutions of problem
(2.9) with initial data U i = (ui
0; u
i
1; v
i
0), i = 1, 2. Assume that
∥∥Aui(t)
∥∥2 +∥∥ui
t(t)
∥∥2 +
∥∥vi(t)
∥∥2 ≤ R2 for all t ≥ 0. Let
Z(t) =
(
u1(t)− u2(t);u1
t (t)− u2
t (t); v
1(t)− v2(t)
)
and z(t) = u1(t)− u2(t).
Then there exist positive constants CR and γ such that
‖Z(t)‖2
H ≤ CR ‖Z(0)‖2
H e−γt + CR sup
0≤τ≤t
‖z(τ)‖2 . (5.3)
The proof of Theorem 5.6 is relegated to Subsection 7.2.
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3 323
M. Potomkin
6. Upper Semicontinuous Family of Attractors
Our second main result is the closeness between Aσ,ε and A0,0. To state the
corresponding theorem we need the following definition.
Definition 6.1. Let X be a Banach space, and B1,B2 ⊂ X. We denote by
distX(B1,B2) = sup
z1∈B1
inf
z2∈B2
‖z1 − z2‖X
the Hausdorff semidistance in X from B1 to B2.
Theorem 6.2. Let Aσ,ε be an attractor of the dynamical system (Hσ,ε, Sσ,ε(t)).
A0,0 denotes an attractor of (H, S0,0(t)). Then
lim
σ → 0+
ε → 0+
[
distH
(
PAσ,ε,A0,0
)
+ sup
U∈Aσ,ε
(
‖QσU‖L2
βσ
(R+;H2) + ‖QεU‖L2
µε (R+;H1)
)]
= 0. (6.1)
P r o o f. In this proof we need Theorems 5.3, 5.4 and 5.6. We should notice
that their proofs do not depend on the arguments used in this section.
For further arguments we note that Theorem 5.2 implies that Aσ,ε consists
of full trajectories {U(t) = Sσ,ε(t)U0}t∈R. Moreover, each of these trajectories
corresponds to strong solutions to problem (2.10) (it is also stated in Th. 5.2).
We split our arguments in steps.
Step I. We consider any {U(t) = Sσ,ε(t)U0}t∈R ⊂ Aσ,ε. The following estimate
holds true:
‖Ut(t)‖Hσ,ε
+ ‖PU(t)‖H1 ≤ C, (6.2)
where constant C from (5.2) is independent of σ, ε.
In particular, it means that the set
{
PU ∈ C(R;H1)| U(t) ∈ Aσ,ε, σ, ε ∈ (0, 1]
}
is bounded in C(0, T ;H1). Analogously, the set
{PUt ∈ C(R;H)| U(t) ∈ Aσ,ε, σ, ε ∈ (0, 1]}
is bounded in C(0, T ;H).
Step II. Assume by contradiction that there exists ρ > 0, sequences σn → 0+,
εn → 0+ and the corresponding elements Un ∈ Aσn,εn such that
‖Un − Lσn,εnU0‖Hσn,εn
≥ ρ > 0 ∀U0 ∈ A0.
324 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3
On Singular Limit and Upper Semicontinuous Family of Attractors
Let Un(t) = (un(t)ut,n(t), vn(t), ξn,t, ηn,t) ⊂ Hσn,εn be a full trajectory in attrac-
tor Aσn,εn with initial data equal to Un, i.e., Un(t) = Sσn,εn(t)Un. Then
{PUn}∞n=1 − bounded in C([−T, T ];H1),
{PUt,n}∞n=1 − bounded in C([−T, T ];H)
for arbitrary T > 0. That implies the existence of a triple
U∗(t) = (u∗(t), u∗t (t), v
∗(t)) ∈ C([−T, T ];H)
such that
lim
n→∞ max
t∈[−T,T ]
[‖un(t)− u∗n(t)‖2 +
∥∥ut,n(t)− u∗t,n(t)
∥∥ + ‖vn(t)− v∗n(t)‖] = 0
and, moreover,
sup
t∈R
‖U∗(t)‖H ≤ C. (6.3)
Now we will show that ξn,t and ηn,t vanish as n → +∞. For this we apply
Lemma 4.2 application of which is justified because of
‖Aut,n(t)‖+
∥∥∥A1/2vn(t)
∥∥∥ ≤ C,
where the constant C is taken from (5.2). Then the convergences
lim
n→∞
∥∥ξ0
n
∥∥
L2
βσn
(R+;H2)
= lim
n→∞
∥∥η0
n
∥∥
L2
µεn
(R+;H1)
= 0
follow from the invariance of attractor. (We can take Un(−τ), where τ > 0 as
the initial data in (4.5) and (4.6), and then pass to the limit n → +∞.)
We meet a contradiction if we show that U∗(0) ∈ A0,0 that occurs if and only
if U∗(t) is a full bounded trajectory of S0,0(t). The boundedness follows from
(6.3). The fact that U∗(t) is a solution to (2.9) will be shown in the next step.
Step III. Functions Un(t) satisfy (2.10), i.e.,
{
U̇n(t) = Lσn,εnUn(t) + f(Un(t)),
Un(0) = Un.
We need to show that
{
U̇∗(t) = L0,0U∗(t) + f(U∗(t)),
U∗(0) = U∗
0 .
(6.4)
For this we note that
∞∫
0
βσn(s)ξn,t(s)ds → u∗t (t) in C([−T, T ];H) (6.5)
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3 325
M. Potomkin
and, similarly,
∞∫
0
µεn(s)ηn,t(s)ds → v∗(t) in C([−T, T ]; H). (6.6)
To obtain (6.5), we rewrite the memory variable ξn in terms of un
t
∞∫
0
βσn(s)ξn,t(s)ds =
∞∫
0
hσn(s)un
t (t− s)ds.
Then
∥∥∥∥
∞∫
0
hσn(s)un
t (t− s)ds− un
t (t)
∥∥∥∥ =
∥∥∥∥
∞∫
0
hσn(s) [un
t (t− s)− un
t (t)] ds
∥∥∥∥
=
∥∥∥∥
∞∫
0
hσn(s)
s∫
0
un
tt(t− y)dyds
∥∥∥∥ ≤ Cσn,
which implies (6.5), since
∞∫
0
hσn(s)un
t (t− s)ds− u∗t (t) =
∞∫
0
hσn(s)un
t (t− s)ds− un
t (t) + un
t (t)− u∗t (t).
Convergence (6.6) is obtained in a similar way.
Thus,
P (Lσn,εnUn(t) + f(Un(t))) → L0,0U∗(t) + f(U∗(t)) in C([−T, T ];H−4),
and in view that
‖P (Lσn,εnUn(t) + f(Un(t)))‖C([−T,T ];H) ≤ C,
the function W (t) ≡ L0,0U∗(t) + f(U∗(t)) belongs to C([−T, T ];H). If we pass
to the limit in the relation
Un(t)− Un(0) =
t∫
0
Un,t(τ)dτ,
we obtain that W (t) = d
dtU
∗(t) and this ends the proof of Theorem 6.2.
326 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3
On Singular Limit and Upper Semicontinuous Family of Attractors
7. Proofs
In what follows we check whether the arguments provided in [22] allow to
obtain the necessary estimates uniform with respect to σ and ε. In further proofs
all generic constants CR and C are supposed to be uniform with respect to σ
and ε. Also, we consider the case σ = ε = 0 since it was not considered in [22].
7.1. Proof of Theorem 5.3
Let (u1, v1, ξ1, η1) and (u2, v2, ξ2, η2) be two strong solutions of problem (2.10)
with initial data U i = (ui
0, u
i
1, v
i
0, ξ
i
0, η
i
0), i = 1, 2, and assume that
∥∥Aui(t)
∥∥2 +
∥∥ui
t(t)
∥∥2 +
∥∥vi(t)
∥∥2 +
∥∥ξi,t
∥∥2
L2
βσ
(R+;H2)
+
∥∥ηi,t
∥∥2
L2
µε (R+;H1)
≤ R2 (7.1)
for ∀t ≥ 0.
The components of Z(t) (Z(t) was introduced in the statement of Th. 5.3)
satisfy the following equation
ztt + A2z +
∞∫
0
βσ(s)A2ξt(s)ds−Aϑ = F (t),
ϑt +
∞∫
0
µε(s)Aηt(s)ds + Azt = 0,
ξt
t + ξt
s = zt, ηt
t + ηt
s = ϑ,
(7.2)
where F (t) = M
(∥∥A1/2u2
∥∥2
)
Au2 −M
(∥∥A1/2u1
∥∥2
)
Au1.
To obtain an appropriate form of energy relation from (7.2), we first transform
the term (F (t), zt).
Lemma 7.1. Let (u1(t), v1(t), ξ1,t, η1,t) and (u2(t), v2(t), ξ2,t, η2,t) be strong
solutions to problem (2.10) satisfying (7.1). Then the following representation
holds:
(F (t), zt) =
d
dt
Q(t) + P (t), (7.3)
where the functions Q(t) ∈ C1(R+) and P (t) ∈ C(R+) satisfy the relations:
|Q(t)| ≤ CR ‖Az‖ ‖z‖ , (7.4)
|P (t)| ≤ CR
∣∣∣∣
(
Tσξ2,t, ξ2,t
)
L2
βσ
(R+;H2)
∣∣∣∣
1/2 (
‖Az‖2 + ‖zt‖2
)
(7.5)
with the constant CR chosen to be independent of σ and ε.
P r o o f. In [22], it is proved that the following representation holds:
(F (t), zt) =
d
dt
Q(t) + P (t),
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3 327
M. Potomkin
where
Q(t) = Q0(t) + σQ1(t), P (t) = σ [P1(t)− P2(t)] ,
and
Q0(t) = −
1∫
0
I1[u2 + λz, u2]dλ
(
Au2, z
)−
1∫
0
λM
(∥∥A1/2(u2 + λz)
∥∥2
)
dλ (Az, z) ,
Q1(t) =
∞∫
0
βσ(s)
(
ξ2,t(s), I2 ·Au2 + I1[u1, u2] ·Az
)
ds,
P1(t) =
∞∫
0
βσ(s)
(
ξ2,t
s (s), I2 ·Au2 + I1[u1, u2] ·Az
)
ds,
P2(t) =
∞∫
0
βσ(s)
(
ξ2,t(s), I4 ·Au2 + I2 ·Au2
t + 2I3 ·Az + I1[u1, u2] ·Azt
)
ds.
Terms Ii admit the following estimates:
|I1 [u1, u2]| ≤ CR ‖u1 − u2‖ , |I2| ≤ CR ‖Az‖ ‖z‖ ,
|I3| ≤ CR (‖Az‖+ ‖zt‖) , |I4| ≤ CR
(
‖Az‖2 + ‖zt‖2
)
.
A generic constant CR in the estimates above and in further arguments is
independent of σ and ε.
We estimate Qi(t) and Pi(t) as follows:
1. For Q(t) = Q0(t) + σQ1(t) we have |Q0(t)| ≤ CR ‖Az‖ ‖z‖ and
|Q1(t)| ≤ CR
∞∫
0
βσ(s)
∥∥ξ2,t(s)
∥∥ ds ‖Az‖ ‖z‖ ≤ CR
σ1/2 ‖Az‖ ‖z‖ .
Therefore, |Q(t)| ≤ CR(1 + σ1/2) ‖Az‖ ‖z‖.
2. For P (t) = σ [P1(t)− P2(t)] we have
|P1(t)| ≤ CR
∞∫
0
(−β′σ(s))
∥∥ξ2,t(s)
∥∥
2
ds ‖Az‖2
≤ CR
(∞∫
0
(−β′σ(s))ds
)1/2 √
− (Tσξ2,t, ξ2,t)L2
σ(R+;H2) ‖Az‖2
≤ CR
σ
√
− (Tσξ2,t, ξ2,t)L2
σ(R+;H2) ‖Az‖2 .
To estimate P2(t) we apply (2.5)
|P2(t)| ≤ CR
∞∫
0
βσ(s)
∥∥ξ2,t(s)
∥∥
2
ds
(
‖Az‖2 + ‖zt‖2
)
≤ CR
σ1/2
∥∥ξ2,t
∥∥
L2
βσ
(R+;H2)
(
‖Az‖2 + ‖zt‖2
)
≤ CR
√
− (Tσξ2,t, ξ2,t)L2
σ(R+;H2)
(
‖Az‖2 + ‖zt‖2
)
.
328 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3
On Singular Limit and Upper Semicontinuous Family of Attractors
And therefore, |P (t)| ≤ CR(1+σ)
√
− (Tσξ2,t, ξ2,t)L2
σ(R+;H2)
(
‖Az‖2 + ‖zt‖2
)
.
Thus, all necessary estimates are obtained and this concludes the proof.
Now we return to the proof of Theorem 5.3.
By (7.3), for these solutions we have the energy relation
d
dt
E0(t) =
(
Tσξt, ξt
)
L2
βσ
(R+;H2)
+
(
Tηt, ηt
)
L2
µε (R+;H1)
+ P (t), (7.6)
where
E0(t) = 1
2
[
‖zt(t)‖2 + ‖Az(t)‖2 + ‖ϑ(t)‖2
+
∥∥ξt
∥∥2
L2
βσ
(R+;H2)
+
∥∥ηt
∥∥2
L2
µε (R+;H1)
− 2Q(t)
]
.
It follows from (7.4) that
3
8
‖Z(t)‖2
H − CR ‖z(t)‖2 ≤ E0(t) ≤ 5
8
‖Z(t)‖2
H + CR ‖z(t)‖2 . (7.7)
Now we consider V (t) ≡ E0(t) + ω
3∑
i=1
Φi(t), where we set Φ1(t) = (zt, z),
Φ2(t) = − σ(A−2zt, ξ
t)L2
βσ
(R+;H2) and Φ3(t) = −ε(z + A−1ϑ, ηt)L2
µε (R+;H1).
A positive constant ω will be chosen in the sequel and it will be independent
of σ and ε. For V (t) we have an estimate similar to (7.7)
1
4
‖Z(t)‖2
H − CR ‖z(t)‖2 ≤ V (t) ≤ ‖Z(t)‖2
H + CR ‖z(t)‖2 (7.8)
as soon as ω is sufficiently small.
Now we compute the derivatives of Φi(t)
d
dtΦ1(t) = −‖Az‖2 −
∞∫
0
βσ(s)(ξt(s), z)1ds + (ϑ,Az) + (F (t), z) + ‖zt‖2 ,
d
dtΦ2(t) = σ
∞∫
0
βσ(s)
(
A2z +
∞∫
0
βσ(τ)A2ξt(τ)dτ −Aϑ− F (t), ξt(s)
)
ds
+σ
∞∫
0
βσ(s)(zt, ξ
t
s)ds− ‖zt‖2 .
d
dtΦ3(t) = ε
(∞∫
0
µε(τ)ηt(τ)dτ, ηt
)
L2
µε (R+;H1)
− (Az, ϑ)− ‖ϑ‖2
+ε(z + A−1ϑ, ηt
s)L2
µε (R+;H1).
Using the auxiliary inequalities below, we will be able to estimate Φi
in appropriate way.
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3 329
M. Potomkin
First, ∥∥∥∥
∞∫
0
βσ(s)ξt(s)ds
∥∥∥∥
2
≤
(∞∫
0
βσ(s)
∥∥ξt(s)
∥∥ ds
)2
≤ 1
σ ‖ξ‖2
L2
βσ(R+;H2)
≤ C
∣∣∣∣
(
Tσξt, ξt
)
L2
βσ(R+;H2)
∣∣∣∣ .
Second,
∞∫
0
βσ(s)(ξt, z)1ds ≤
∞∫
0
βσ(s)
(
1
σ
∥∥ξt
∥∥2
1
+ σ ‖z‖2
1
)
ds
= 1
σ
∥∥ξt
∥∥2
L2
βσ
(R+;H2)
+ ‖z‖2
1 ≤ C
∣∣∣∣
(
Tσξt, ξt
)
L2
βσ(R+;H2)
∣∣∣∣ + ‖z‖2
1 .
Third,
∞∫
0
βσ(s)(zt, ξ
t
s(s))ds =
∞∫
0
(−β′σ(s))(zt, ξ
t(s))ds
≤
∞∫
0
(−β′σ(s))
∥∥ξt(s)
∥∥ ds ‖zt‖ ≤ C
σ
(∞∫
0
(−β′σ(s))
∥∥ξt(s)
∥∥ ds
)1/2
‖zt‖
≤ C
σ
(∣∣∣∣
(
Tσξt, ξt
)
L2
βσ(R+;H2)
∣∣∣∣ + ‖zt‖2
)
.
We are in position to write the estimates for d
dt
3∑
i=1
Φi:
d
dt
3∑
i=1
Φi(t) = CR
∣∣∣∣
(
Tσξt, ξt
)
L2
βσ(R+;H2)
∣∣∣∣ + CR
∣∣∣∣
(
Tεη
t, ηt
)
L2
µε(R+;H1)
∣∣∣∣
−1
2
[
‖Az‖2 + ‖zt‖2 + ‖ϑ‖2
]
+ CR ‖z‖2 .
Choosing sufficiently small ω, we obtain that for some positive α∗
d
dt
V (t) + α∗ ‖Z(t)‖Hε,σ
≤ P (t) + CR ‖z(t)‖2 ,
and then the application of (7.5) and (7.8) implies the existence of γ > 0 such
that
d
dt
V (t) + γV (t) ≤ CR ‖z(t)‖2 + CR
∣∣∣∣
(
Tσξ2,t, ξ2,t
)
L2
βσ
(R+;H2)
∣∣∣∣
(
‖Az‖2 + ‖zt‖2
)
.
Using the Gronwall Lemma, we obtain
‖Z(t)‖2
Hσ,ε
≤ CR ‖Z(0)‖2
Hσ,ε
e−γt + CR max
τ∈[0,t]
‖z(t)‖2
+CR
t∫
0
e−γ(t−τ)
∣∣∣∣
(
Tσξ2,τ , ξ2,τ
)
L2
βσ
(R+;H2)
∣∣∣∣ ‖Z(τ)‖2
Hσ,ε
dτ.
330 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3
On Singular Limit and Upper Semicontinuous Family of Attractors
Now using the fact that
+∞∫
0
∣∣∣∣
(
Tσξ2,t, ξ2,t
)
L2
βσ
(R+;H2)
∣∣∣∣ dt ≤ CR, which follows
from the energy relation and inequality (7.1), we obtain the stabilizability esti-
mate with uniform (with respect to parameters σ and ε) coefficients.
7.2. Proof of Theorem 5.6
We use the same procedure as in the previous subsection. So, we need to
replace (7.1) by
∥∥Aui(t)
∥∥2 +
∥∥ui
t(t)
∥∥2 +
∥∥vi(t)
∥∥2 ≤ R2, t ≥ 0, (7.9)
and (7.2) by {
ztt + A2zt + A2z −Aϑ = F (t),
ϑt + Aϑ + Azt = 0,
(7.10)
where F (t) is the same as in (7.2). Next lemma holds
Lemma 7.2. Let (u1(t), v1(t)) and (u2(t), v2(t)) be strong solutions to problem
(2.9) satisfying (7.9). Then the representation (F (t), zt) = d
dtQ(t) + P (t) holds,
where the functions Q(t) ∈ C1(R+) and P (t) ∈ C(R+) satisfy the relations
|Q(t)| ≤ CR ‖Az‖ ‖z‖ , |P (t)| ≤ CR
∥∥u2
t
∥∥
(
‖Az‖2 + ‖zt‖2
)
.
This lemma can be proved in the same way as Lemma 7.1 (see also [3]).
Following the same procedure as in the previous subsection, we consider the
auxiliary function V (t) given by V (t) = E0(t)+ωΦ(t), where Φ(t) = (zt, z). We do
not repeat all arguments, since the proof is similar to [5, Th. 5.6], [22, Th. 3.11]
and Subsection 7.1, and give just a sketch of the proof.
As before
1
4
‖Z(t)‖2
H − C ‖z(t)‖2 ≤ V (t) ≤ ‖Z(t)‖2
H + C ‖z(t)‖2 .
Then
d
dtΦ(t) = −(Azt, Az)− ‖Az‖2 + (Az, ϑ) + (F (t), z) + ‖zt‖2
≤ C ‖Azt‖2 − 1
2 ‖Az‖2 + C
∥∥A1/2ϑ
∥∥2
+ C ‖z‖2 .
Taking ω sufficiently small, the following inequality holds with some strictly
positive constant γ
V (t) ≤ V (0)e−γt + C max
τ∈[0,t]
‖z(τ)‖2 + C
t∫
0
e−γ(t−τ)
∥∥u2
t (τ)
∥∥2
2
‖Z(τ)‖2
H dτ.
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3 331
M. Potomkin
And we get the desired conclusion in the same way as in the previous case
(Subsect. 7.1) by using the estimate
∞∫
0
∥∥u2
t (t)
∥∥2
2
dt < C.
7.3. Proof of Theorem 5.4
Step I. Theorem 5.2 states that Aσ,ε = Mu(N ).
We also know that the functional Φ(U) introduced in Subsection 3.2 decreases
on nonstationary trajectories. Since N ⊂ Aσ,ε, it implies that
max {Φ(U) : U ∈ Aσ,ε} = max {Φ(U) : U ∈ N} ,
where the right-hand side does not depend on σ and ε because of the form of N
(see Prop. 3.4). If in addition we notice that (see Prop. 3.2)
α ‖U‖2
Hσ,ε
− C ≤ Φ(U),
we may assert that there exists positive R independent of σ, ε such that
‖U‖Hσ,ε
≤ R ∀σ ∀ε ∀U ∈ Aσ,ε. (7.11)
In the steps below we will emphasize the dependence of generic constants
on R. But this dependence is formal since R is independent of σ, ε.
Step II. Current step follows the arguments of Step II from [22, Th. 4.1].
We also note that the main idea of this step is borrowed from [5].
Due to the uniform (with respect to σ,ε) feature of stabilizability inequality
(5.1) all constants below are uniform with respect to σ and ε.
Let
{
U(t) ≡ (u(t);ut(t); v(t); ξt; ηt)
} ⊂ Hσ,ε be a full trajectory from the
attractor Aσ,ε. Let |ω| < 1. Applying Theorem 5.3 with U1 = U(s + ω), U2 =
U(s) (and, accordingly, the interval [s, t] in place of [0, t]), we have
‖U(t + ω)− U(t)‖2
Hσ,ε
≤ C1e
−γ(t−s) ‖U(s + ω)− U(s)‖2
Hσ,ε
+C2 max
τ∈[s,t]
‖u(τ + ω)− u(τ)‖2 (7.12)
for any t, s ∈ R such that s ≤ t and for any ω with |ω| < 1. Taking the limit
s → −∞, (7.12) gives
‖U(t + ω)− U(t)‖2
Hσ,ε
≤ C2 max
τ∈(−∞,t]
‖u(τ + ω)− u(τ)‖2
for any t ∈ R and |ω| < 1. We obviously have
1
ω
‖u(τ + ω)− u(τ)‖ ≤ 1
ω
∫ ω
0
‖ut(τ + t)‖ dt, τ ∈ R.
332 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3
On Singular Limit and Upper Semicontinuous Family of Attractors
Therefore, by (7.11) we obtain
max
τ∈R
∥∥∥∥
U(τ + ω)− U(τ)
ω
∥∥∥∥
Hσ,ε
≤ CR for |ω| < 1.
The last estimate implies that the function U(t) is absolutely continuous and
thus possesses a derivative almost everywhere which satisfies ‖Ut(t)‖Hσ,ε
≤ CR.
And as it is stated in Theorem 5.2, U(t) is a strong solution to (2.10) and,
besides,
‖Lσ,εU‖Hσ,ε
≤ CR. (7.13)
Step III. Now we verify that
∥∥A1/2v
∥∥ ≤ CR.
It follows from the second and the fourth equalities of (2.8) and estimate
(7.13) that
∞∫
0
µε(s)Aη(s)ds = −vt −Aut ≡ v∗, ‖v∗‖ ≤ CR,
ηs − v = −ηt ≡ η∗, ‖η∗‖L2
µε (R+;H1) ≤ CR.
Since η ∈ D(Tε), the second equality gives
η(s) = sv +
s∫
0
η∗(y)dy.
Now we substitute this to the first equality
Av = −
∞∫
0
µε(s)
s∫
0
Aη∗(y)dyds + v∗,
where the right-hand side is estimated by generic constant CR in space H−1 since
∥∥∥∥
∞∫
0
µε(s)
s∫
0
Aη∗(y)dyds
∥∥∥∥
−1
≤
∞∫
0
µε(s)
s∫
0
∥∥A1/2η∗(y)
∥∥ dyds
=
∞∫
0
∥∥A1/2η∗(y)
∥∥ ∞∫
y
µε(s)dsdy ≤ ε
δ
∞∫
0
µε(y)
∥∥A1/2η∗(y)
∥∥ dy
≤ CR
√
ε
∞∫
0
µε(y) ‖η∗(y)‖2
1 dy ≤ CR
√
ε.
Step IV. Our purpose is to obtain ‖u(t)‖3 ≤ CR.
Let us consider the following Volterra equation:
u(t)−
t∫
−∞
βσ(t− y)
1 + 1
σ
u(y)dy =
h(t)
1 + 1
σ
,
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3 333
M. Potomkin
where
h(t) ≡ −A−2utt −A−2M
(∥∥∥A1/2u
∥∥∥
2
)
Au + A−2p + A−1v, ‖h(t)‖3 ≤ CR.
We will use the standard iteration method. Namely, w0(t) ≡ 0 and
wn(t) = h(t)
1+ 1
σ
+
t∫
−∞
βσ(t−y)
1+ 1
σ
wn−1(y)dy, n = 1, 2, 3, . . . .
The sequence introduced converges to u(t) in C([−T, T ];H3) for arbitrary
T > 0 in view that
sup
t∈R
‖wn+1(t)− wn(t)‖3 ≤ 1
1+σ sup
t∈R
‖wn(t)− wn−1(t)‖3
≤
(
1
1+σ
)n
1
1+ 1
σ
sup
t∈R
‖h(t)‖3 .
And, in particular,
sup
t∈[−T,T ]
‖wn+1(t)− wn(t)‖3 ≤ CR
(
1
1+σ
)n
1
1+ 1
σ
.
To prove the final estimate, we need to observe
sup
t∈R
‖u(t)‖3 ≤
∞∑
n=0
sup
t∈R
‖wn+1(t)− wn(t)‖3 ≤ sup
t∈R
‖h(t)‖3 ≤ CR.
This concludes the proof.
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