On Transmission Problem for Berger Plates on an Elastic Base
A nonlinear transmission problem for a Berger plate on an elastic base is studied. The plate consists of thermoelastic and isothermal parts. The problem generates a dynamical system in a suitable Hilbert space. In the paper the existence of a compact global attractor is proved.
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Potomkin, M. 2016-10-01T18:45:25Z 2016-10-01T18:45:25Z 2011 On Transmission Problem for Berger Plates on an Elastic Base / M. Potomkin // Журнал математической физики, анализа, геометрии. — 2011. — Т. 7, № 1. — С. 96-102. — Бібліогр.: 12 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106666 A nonlinear transmission problem for a Berger plate on an elastic base is studied. The plate consists of thermoelastic and isothermal parts. The problem generates a dynamical system in a suitable Hilbert space. In the paper the existence of a compact global attractor is proved. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии On Transmission Problem for Berger Plates on an Elastic Base Article published earlier |
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A nonlinear transmission problem for a Berger plate on an elastic base is studied. The plate consists of thermoelastic and isothermal parts. The problem generates a dynamical system in a suitable Hilbert space. In the paper the existence of a compact global attractor is proved.
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On Transmission Problem for Berger Plates on an Elastic Base / M. Potomkin // Журнал математической физики, анализа, геометрии. — 2011. — Т. 7, № 1. — С. 96-102. — Бібліогр.: 12 назв. — англ. |
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2025-11-24T16:02:14Z |
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Journal of Mathematical Physics, Analysis, Geometry
2011, vol. 7, No. 1, pp. 96–102
On Transmission Problem
for Berger Plates on an Elastic Base
M. Potomkin
Mathematics Division, B. Verkin Institute for Low Temperature Physics and Engineering
National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkiv, 61103, Ukraine
E-mail:mika potemkin@mail.ru
Received November 5, 2010
A nonlinear transmission problem for a Berger plate on an elastic base
is studied. The plate consists of thermoelastic and isothermal parts. The
problem generates a dynamical system in a suitable Hilbert space. In the
paper the existence of a compact global attractor is proved.
Key words: transmission problem, thermoelasticity, dynamical systems,
attractors.
Mathematics Subject Classification 2000: 35B41, 35B35.
1. Introduction
Let Ω, Ω1 and Ω2 be bounded open sets in R2 with smooth boundaries Γ1,
Γ1∪Γ0 and Γ0, respectively, such that Ω = Ω1∪Ω2 and Ω1∩Ω2 = ∅. An example
is when Ω2 is completely surrounded by Ω1. In what follows below ν denotes the
outward vector on Γ1 and Γ0. Also we assume that Ω2 is a star-shaped domain,
i.e., the following condition holds
(x− x0) · ν(x) ≥ 0 on Γ0 for some x0 ∈ R2. (1.1)
We study an asymptotic behavior of the following system:
ρ1utt + β1∆2u + µ∆θ + F1(u, v) = 0 in Ω1 × R+, (1.2)
ρ0θt − β0∆θ − µ∆ut = 0 in Ω1 × R+, (1.3)
ρ2vtt + β2∆2v + F2(u, v) = 0 in Ω2 × R+. (1.4)
Boundary conditions imposed on u along Γ1 are clamped
u =
∂u
∂ν
= 0 on Γ1 × R+. (1.5)
c© M. Potomkin, 2011
On Transmission Problem for Berger Plates on an Elastic Base
We assume that θ satisfies Newton’s law of cooling (with the coefficient λ ≥ 0)
through the Γ1 and θ vanishes along Γ0
θ = 0 on Γ0 × R+,
∂θ
∂ν
+ λθ = 0 on Γ1 × R+. (1.6)
Also we impose the following boundary conditions along Γ0:
u = v,
∂u
∂ν
=
∂v
∂ν
, β1∆u = β2∆v, β1
∂∆u
∂ν
+µ
∂θ
∂ν
= β2
∂∆v
∂ν
on Γ0×R+. (1.7)
Real parameters ρi, βi and µ are strictly positive and the relations
ρ1 ≥ ρ2 and β1 ≤ β2 (1.8)
hold. Nonlinearities are given by
F1(u, v) = −M(||∇u||2Ω1
+ ||∇v||2Ω2
)∆u + a1(x)u|u|p−1 + g1(x, u),
F2(u, v) = −M(||∇u||2Ω1
+ ||∇v||2Ω2
)∆v + a2(x)v|v|p−1 + g2(x, v),
where M(s) = s1+α with α > 0, a1(x) ∈ L∞(Ω1) and a2(x) ∈ L∞(Ω2). We
assume that the following condition holds:
either a(x) ≥ c0 ∀x ∈ Ω or 2(α + 2) > p + 1, p ≥ 1.
Here a = {a1, a2} , and c0 > 0 is a small number. The functions g1(x, u) and
g2(x, v) are scalar and satisfy the growth condition for some ε0 > 0 and any
xi ∈ Ωi
∣∣∣∣
∂
∂u
g1(x1, u)
∣∣∣∣ +
∣∣∣∣
∂
∂v
g2(x2, v)
∣∣∣∣ ≤ C(1 + |u|max{0,p−1−ε0} + |v|max{0,p−1−ε0}),
and, for the sake of simplicity, we assume that g2(x, 0) = 0.
The plate equations with nonlocal nonlinearity were introduced in [2] and
their asymptotic behavior was deeply studied in [4] and [5]. Different models
with partial damping were considered in [3, 7] (see also the references therein).
Exponential stability of linear equations (1.2)–(1.7) (Fi = 0) was obtained in [12].
In [11] we proved the existence of a compact global attractor for the case when
α = 0 and ai = gi = 0.
Our main result is to prove the existence of a compact global attractor (Theo-
rem 3.1). To obtain the result we need to overcome two difficulties. The first is to
show that the corresponding energy of the system is a strict Lyapunov function,
here we use the observability estimate from [1]. The second is to prove asymptotic
smoothness. Here the idea of the stabilizability estimates from [5] (see also [6])
is used.
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 1 97
M. Potomkin
2. Preliminaries
Below the equality w = {u, v} denotes that w(x) = u(x) if x ∈ Ω1 and
w(x) = v(x) if x ∈ Ω2. We introduce a Hilbert space H1
D as a space of such
function φ ∈ H1(Ω1) that φ = 0 on Γ0. The space H1
D is equipped with the
following inner product:
(w, φ)H1
D
:=
∫
Ω1
β0∇w · ∇φdx +
∫
Γ1
β0λwφdx.
Denote H = H2
0 (Ω) × L2(Ω) × L2(Ω1). This space plays the role of a phase
space for the dynamical system to be introduced below. The following set, which
is densely embedded in H, is needed for the statement about strong solutions:
D0 =
w ∈ [
H2
0 (Ω) ∩ (
H4(Ω1)×H4(Ω2)
)]×H2
0 (Ω)× [
H2(Ω1) ∩H1
D
]
:
β1∆w1 = β2∆w2 and β1
∂∆w1
∂ν + µ ∂θ
∂ν = β2
∂∆w2
∂ν on Γ0,
∂w5
∂ν + λw5 = 0 on Γ1
.
We introduce the potential
Π(w) =
1
2(α + 2)
||∇w||2(α+2)
L2(Ω)
+
1
p + 1
∫
Ω
a(x)|w(x)|p+1dx+
∫
Ω
∫ w(x)
0
g(x, s)dsdx,
where a = {a1, a2} and g = {g1, g2}. We have that Π′(w) = {F1(w), F2(w)}.
Energy functional (or Lyapunov function) E : H −→ R is defined for an
argument w = (w1, w2, w3, w4, w5) (here {w1, w2} ∈ H2
0 (Ω), {w3, w4} ∈ L2(Ω)
and w5 ∈ L2(Ω)) as follows:
E(w) =
1
2
[ ∫
Ω1
β1|∆w1|2 + ρ1|w3|2 + ρ0|w5|2dx
+
∫
Ω2
β2|∆w2|2 + ρ2|w4|2dx + 2Π(w1, w2)
]
. (2.1)
Theorem 2.1. Next statements hold true:
(i) For any initial w0 ∈ H and T > 0 there exists a unique mild solution
w(t) ∈ C([0, T ];H). Moreover, it satisfies the energy equality
E(w(T ))− E(w(t)) = −
∫ T
t
∫
Ω1
β0|∇w5|2dxdτ −
∫ T
t
∫
Γ1
β0λ|w5|2dΓdτ
(2.2)
for all 0 ≤ t ≤ T . If one set S(t)w0 = w(t), then (H, S(t)) is a continuous
dynamical system.
(ii) If w0 ∈ D0, then the corresponding mild solution is strong.
98 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 1
On Transmission Problem for Berger Plates on an Elastic Base
We take the same definitions of mild and strong solutions as in [10, Ch. 4].
To prove this theorem we use the standard methods from the theory of semigroups
of linear operators and their perturbations, see [10]. For some details for the
similar model we refer to [11].
3. Main Result
Our main result is the following theorem:
Theorem 3.1. Let (1.1) and (1.8) hold. Then (H, S(t)) possesses a compact
global attractor.
To prove this theorem, we have to prove that the energy E is a strict Lyapunov
function for (H, S(t)) (see Sec. 4) and (H, S(t)) is asymptotically smooth (see
Sec. 5) For how to prove the existence of a compact global attractor, taking into
consideration the results of Secs. 4 and 5, we refer to [5, Cor. 2.29].
4. Strict Lyapunov Function
Proposition 4.1. If E(S(T )U) = E(U) for any T > 0, then S(t)U = U for
any t ≥ 0.
In compare with [11], our model is more complicated because of the presence
of the scalar nonlinearity and the assertion is stronger since, in contrast with the
proposition above, Proposition 4.13 in [11] requires E(S(T )U) = E(U) to hold
for any T ∈ R. To prove Proposition 4.1 we use the Carleman-type inequalities
formulated in the following auxiliary lemma (see [1, Th. 3.4]):
Lemma 4.2. Let w be a solution to wtt + ∆2w = f in Ω2 and
w|Γ0 =
∂w
∂ν
|Γ0 =
∂2w
∂ν2
|Γ0 =
∂3w
∂ν3
|Γ0 = 0.
Then there exists such τ0 > 0 that for all τ > τ0 there holds
||eτφw||22,τ̃ ≤ C||eτφτ̃−1/2f ||, (4.1)
where
||eτφw||22,τ̃ :=
∫ T
0
∫
Ω2
τ̃4|eτφw|2 + τ̃2|∇(eτφw)|2 + |∂t(eτφw)|2 + |∆(eτφw)|2dxdt
τ̃ = τgeψ, ψ(x) = |x− x|2 with x ∈ R2\Ω2, g(t) = 1
t(T−t) and
φ(t,x) = g(t)(eψ(x) − 2e||ψ||L∞(Ω2)).
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 1 99
M. Potomkin
P r o o f of Proposition 4.1. Let us consider such T > 0 and U0 ∈ H that
E(S(T )U0) = E(U0). Energy equality (2.2) implies that θ ≡ 0, then equation (1.3)
implies that ut = 0. Equation (1.2) implies that either u ≡ 0 for all t ∈ [0, T ]
(case 1) or
M(||∇u||2Ω1
+ ||∇v||2Ω2
) ≡ M (4.2)
does not depend on t (case 2). Both cases are considered below.
Case 1. Let us assume u ≡ 0. Assume also that ||∆v(t)||2Ω2
+ ||vt(t)||2Ω2
≤ r.
Then for any t ∈ [0, T ] and x ∈ Ω2 we have
|F2(0, v)|2≤
[
| ||∇w||1+α
Ω2
∆v|+ ||a2||L∞ |v|p−1|v|+ C(r)|v|
]2
≤ C(r)
[|∆v|2 + |v|2] .
Using the following inequality that holds for any t ∈ [0, T ] and x ∈ Ω2:
|eτφ∆w|2 ≤ |∆(eτφw)|2 + Cτ̃2|∇(eτφw)|2 + Cτ̃4|eτφw|2,
τ̃−1 < C/τ , 1 ≤ Cτ̃4 and (4.1) with f = F2(0, v), we finally get
||eτφw||22,τ̃ ≤
C(r)
τ
||eτφw||22,τ̃ .
Choosing τ large enough we get the conclusion that v ≡ 0.
Case 2. Assume that ||∇v||Ω2 does not depend on t and (4.2) takes place.
In this case we consider an application of (4.1) for wh(t) = v(t + h) − v(t) with
some h > 0, and
f = F2(u, v(t + h))− F2(u, v(t))
= M∆wh + a2
[|v(t + h)|p−1v(t + h)− |v(t)|p−1v(t)
+g2(v(t + h))− g2(v(t))] .
Using the arguments as in case 1, we obtain wh(t) ≡ 0 and, hence, v does not
depend on t.
5. Asymptotic Smoothness
The proof of the asymptotic smoothness is based on the method of compen-
sated compactness function suggested in [8] and developed in [5] (see also [6]).
Let (u1(t), v1(t), θ1(t)) and (u2(t), v2(t), θ2(t)) be solutions to the problem
(1.2)–(1.7) and assume that for any t > 0 there exists R > 0 such that
∫
Ω1
ρ1|ui
t|2 + β1|∆ui|2 + ρ0|θi|2dx +
∫
Ω2
ρ2|vi
t|2 + β2|∆vi|2dx ≤ R2.
100 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 1
On Transmission Problem for Berger Plates on an Elastic Base
Let u(t) = u1(t) − u2(t), v(t) = v1(t) − v2(t), θ(t) = θ1(t) − θ2(t). The
triple (u(t), v(t), θ(t)) satisfies boundary conditions (1.5)–(1.7) and the following
system:
ρ1utt + β1∆2u + µ∆θ = G1,
ρ0θt − β0∆θ − µ∆ut = 0,
ρ2vtt + β2∆2v = G2.
with G1(t) = F1(u2, v2)− F1(u1, v1) and G2(t) = F2(u2, v2)− F2(u1, v1).
Also we denote
E(t) =
1
2
∫
Ω1
ρ1|ut|2 + β1|∆u|2 + ρ0|θ|2dx +
1
2
∫
Ω2
ρ2|vt|2 + β2|∆v|2dx.
Proposition 5.1. Let (1.1) and (1.8) hold. There exists k, C > 0 and a
functional R(u, v, ut, vt, θ), continuous on H, such that if
R(t) := R(u(t), v(t), ut(t), vt(t), θ(t)),
then |R(t)| ≤ CE(t) and
d
dt
R(t) ≤ −kE(t) + C
[ ∫
Ω1
|∇θ|2dx +
∫
Ω
| {u, v} |2 + |∆−1
D {ρ1ut, ρ2vt} |2dx
]
.
Our proof of Proposition 5.1 mostly follows the line of arguments given in
[11]. We only give here the formula for R:
R = J1 +
η
β1,
J2 +
(µ
2
− ηC
)
J3 + η1/2J4
with sufficiently small η > 0 and Ji defined as follows:
J1 = −
∫
Ω1
ρ1utw1dx−
∫
Ω2
ρ2vtw2dx,
J2 =
∫
Ω1
ρ1uth · ∇udx +
∫
Ω2
ρ2vth · ∇vdx, J3(t) =
∫
Ω1
ρ1utφudx,
J4 =
∫
Ω1
ρ1utψm · ∇udx +
∫
Ω2
ρ2vtψm · ∇vdx.
Here {w1, w2} := ∆−1
D {ρ0φ1θ, 0}, where ∆−1
D is an inverse Laplace operator with
the Dirichlet boundary conditions on Γ1, a vector field h = (h1, h2) ∈ [C2(Ω)]2
satisfies h(x) = −ν(x) if x ∈ Γ1, m(x) = x − x0, where x0 is the same as in
(1.1). Functions φ and ψ are scalar from C2(Ω) and φ(x) = 1 if x ∈ Ω1 \U4δ(Γ0)
and φ(x) = 0 if x ∈ U2δ(Γ0) ∩ Ω1; ψ(x) = 1 if x ∈ U4δ(Ω2) and ψ(x) = 0 if
Ω1 \U8δ(Ω2). Number δ > 0 is chosen sufficiently small. The idea of such Ji was
used by many authors (see, e.g., [3, 6, 9, 11, 12] and the references therein).
Proposition 5.1 is a key step of the proof. We get the asymptotic smoothness
using the arguments from [5, Ch. 3].
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 1 101
M. Potomkin
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