Global attractor for impulsive reaction-diffusion equation
In this paper we consider a reaction-diffusion equation with nonsmooth nonlinearity, whose solutions have impulse effects at fixed moments of time. We show how this object generates a nonautonomous multivalued dynamical system and prove the existence of a compact semiinvariant global attractor in t...
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irk-123456789-1780132021-02-18T01:27:27Z Global attractor for impulsive reaction-diffusion equation Iovane, G. Kapustyan, O.V. In this paper we consider a reaction-diffusion equation with nonsmooth nonlinearity, whose solutions have impulse effects at fixed moments of time. We show how this object generates a nonautonomous multivalued dynamical system and prove the existence of a compact semiinvariant global attractor in the phase space. Розглядається автономне рiвняння реакцiї-дифузiї з негладкою правою частиною, розв’язки якого зазнають iмпульсного збурення в фiксованi моменти часу. Доведено, що такий об’єкт породжує неавтономну багатозначну динамiчну систему, для якої в фазовому просторi iснує компактний напiвiнварiантний глобальний атрактор. 2005 Article Global attractor for impulsive reaction-diffusion equation / G. Iovane, O.V. Kapustyan // Нелінійні коливання. — 2005. — Т. 8, № 3. — С. 319-328. — Бібліогр.: 10 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/178013 517.9 en Нелінійні коливання Інститут математики НАН України |
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In this paper we consider a reaction-diffusion equation with nonsmooth nonlinearity, whose solutions
have impulse effects at fixed moments of time. We show how this object generates a nonautonomous multivalued dynamical system and prove the existence of a compact semiinvariant global attractor in the phase
space. |
| format |
Article |
| author |
Iovane, G. Kapustyan, O.V. |
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Iovane, G. Kapustyan, O.V. Global attractor for impulsive reaction-diffusion equation Нелінійні коливання |
| author_facet |
Iovane, G. Kapustyan, O.V. |
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Iovane, G. |
| title |
Global attractor for impulsive reaction-diffusion equation |
| title_short |
Global attractor for impulsive reaction-diffusion equation |
| title_full |
Global attractor for impulsive reaction-diffusion equation |
| title_fullStr |
Global attractor for impulsive reaction-diffusion equation |
| title_full_unstemmed |
Global attractor for impulsive reaction-diffusion equation |
| title_sort |
global attractor for impulsive reaction-diffusion equation |
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Інститут математики НАН України |
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2005 |
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http://dspace.nbuv.gov.ua/handle/123456789/178013 |
| citation_txt |
Global attractor for impulsive reaction-diffusion equation / G. Iovane, O.V. Kapustyan // Нелінійні коливання. — 2005. — Т. 8, № 3. — С. 319-328. — Бібліогр.: 10 назв. — англ. |
| series |
Нелінійні коливання |
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AT iovaneg globalattractorforimpulsivereactiondiffusionequation AT kapustyanov globalattractorforimpulsivereactiondiffusionequation |
| first_indexed |
2025-07-15T16:21:20Z |
| last_indexed |
2025-07-15T16:21:20Z |
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1837730649076662272 |
| fulltext |
UDC 517.9
GLOBAL ATTRACTOR FOR IMPULSIVE
REACTION-DIFFUSION EQUATION
ГЛОБАЛЬНИЙ АТРАКТОР ДЛЯ IМПУЛЬСНО ЗБУРЕНОГО
РIВНЯННЯ РЕАКЦIЇ-ДИФУЗIЇ
G. Iovane
Univ. Salerno
Ponte don Melillo Str., 84084 Fisciano (Salerno), Italy
e-mail: iovane@diima.unisa.it
O. V. Kapustyan
Kyiv Nat. Taras Shevchenko Univ.
Volodymyrs’ka Str., 64, Kyiv, 01033, Ukraine
e-mail: alexkap@univ.kiev.ua
In this paper we consider a reaction-diffusion equation with nonsmooth nonlinearity, whose solutions
have impulse effects at fixed moments of time. We show how this object generates a nonautonomous multi-
valued dynamical system and prove the existence of a compact semiinvariant global attractor in the phase
space.
Розглядається автономне рiвняння реакцiї-дифузiї з негладкою правою частиною, розв’язки
якого зазнають iмпульсного збурення в фiксованi моменти часу. Доведено, що такий об’єкт
породжує неавтономну багатозначну динамiчну систему, для якої в фазовому просторi iснує
компактний напiвiнварiантний глобальний атрактор.
Introduction. In this paper we study the asymptotic behaviour of solutions of an impulsive
reaction-diffusion equation from the point of view of the theory of global attractors. In the li-
terature there is a great number of results concerning the abstract theory of global attractors and
its applications [1]. But in the considered case, in spite the fact that the equation is autonomous,
the whole problem is nonautonomous because the moments of impulse effects are fixed. The
key idea of this paper is to investigate such a problem by the methods of the theory of global
attractors for nonautonomous dynamical systems.
The abstract classical (single-valued) theory of global attractors for nonautonomous dynami-
cal systems and its applications to almost-periodic and cascad systems are developed in [2].
However, if we want to relax the restrictive conditions imposed on the nonlinearity in the classi-
cal approach, the uniqueness of the Cauchy problem is lost. In this case we deal with a set of
solutions, so we need a generalization of the classical theory to the multivalued situation. On the
other hand, an important reason for developing the theory of multivalued dynamical systems
is justified by the well known models coming from the mathematical physics (Navier – Stokes
equations, Ginzburg – Landau equation and others), for which the problem of uniqueness of the
Cauchy problem is still open. In the recent years several approaches of multivalued analogous
constructions of the global attractor theory both in the autonomous and nonautonomous cases
were developed in [3 – 10]. Some applications of this theory to differential inclusions [6, 7],
evolution equations without uniqueness [8 – 10] have been studied.
c© G. Iovane, O. V. Kapustyan, 2005
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3 319
320 G. IOVANE, O. V. KAPUSTYAN
In this paper we develop abstract results from [7 – 10] and using methods, similar to [7],
apply them for proving the existence of global attractor of reaction-diffusion equation with a
nonsmooth nonlinearity and impulse effects occuring at fixed moments of time.
Setting of the problem. We consider the problem
∂u(t, x)
∂t
= a∆u(t, x)− f(u(t, x)) + h(x), (t, x) ∈ (τ, T )× Ω, (1)
u(t, x)|x∈∂Ω = 0,
(2)
u(t, x)|t=τ = uτ (x),
where a > 0, Ω ⊂ Rn is a bounded open subset with smooth boundary, τ ≥ 0, h ∈ L2(Ω),
f ∈ C ([0,+∞)) and satisfies the following conditions:
∃C1 C2 > 0 ∃p ≥ 2 ∃α > 0 ∀u ∈ R :
|f(u)| ≤ C1(1 + |u|p−1), (3)
f(u)u ≥ α |u|p − C2.
Let H = L2(Ω). By ‖·‖ and (·, ·) we denote the norm and the scalar product in H. We say that
a function u = u(t, x) ∈ L2(τ, T ;H1
0 (Ω)) ∩ Lp
(τ, T ;L
p
(Ω)) ∩ C([τ, T ] ;H) is a solution of (1),
(2) on (τ, T ), if for each v ∈ H1
0 (Ω) ∩ Lp(Ω),
d
dt
(u, v) + a(u, v)H1
0
+ (f(t, u), v)− (h, v) = 0
in the sense of scalar distributions on (τ, T ) and u(τ, x) = uτ (x).
Under conditions (3) it is known [9, 10] that for each T > τ , uτ ∈ H there exists at
least one solution u = u(t, x) of problems (1), (2), constructed by the Galerkin approximation
method. So we can talk about globally resolvability of (1), (2), that is, ∀τ ≥ 0 ∀uτ ∈ H ∃u ∈
∈ Wτ = L2
loc(τ,+∞;H1
0 (Ω)) ∩ Lp
loc(τ,+∞;L
p
(Ω)) ∩ C([τ,+∞);H), a solution of (1) on each
(τ, T ), u(τ) = uτ .
Now, we can correctly set the following impulsive problem. Let at the fixed moments of time
{τi}∞i=1 every solution of (1), (2) in the phase space H have impulse effects of the form
u(τi + 0)− u(τi) ∈ ψi(u(τi)), i ≥ 1, (4)
where ψi : H → 2H is some multivalued map, τ1 > 0, τi+1 − τi ≥ γ > 0 ∀i ≥ 1.
Let us denote τ0 := 0, ψ0(u) ≡ 0. Then ∀τ ∈ [τi, τi+1] the problem (1) – (4) is globally
resolved in the class Wτ (σ0), that is, ∀uτ ∈ H ∃u ∈ L2
loc(τ,+∞;H1
0 (Ω))∩Lp
loc(τ,+∞;L
p
(Ω))
is a solution of (1) on (τ, τi+1) , (τi+1, τi+2) , . . . , u(τ) = uτ and at the points {τi, τi+1, . . .} the
function u(·) has impulse effect (4) and is left-continuous.
Our aim is to investigate the qualitative behaviour, for t → ∞, of the solutions of problems
(1) – (4) in the phase space H using methods of the theory of global attractors of infinite di-
mensional multivalued dynamical systems. Since the moments {τi}∞i=1 are fixed, the problem
(1) – (4) is nonautonomous, so we should use the theory of nonautonomous dynamical systems.
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3
GLOBAL ATTRACTOR FOR IMPULSIVE REACTION-DIFFUSION EQUATION 321
Elements of abstract theory of global attractors of nonautonomous multivalued dynami-
cal systems. For some complete metric space (X, ρ) we denote by P (X) (β(X)) the set of
all nonempty (nonempty bounded) subsets of X , ∀A,B ⊂ X dist (A,B) = sup
x∈A
inf
y∈B
ρ(x, y),
Oδ(A) = {x ∈ X|dist (x,A) < δ}, Br = {x ∈ X| ρ(x, 0) ≤ r}, R + = [0,+∞), R+d =
= {(t, τ) ∈ R 2
+|t ≥ τ}. Let Σ be some complete metric space.
Definition 1. The family of multivalued maps {Uσ : R +d × X → P (X)}σ∈Σ is called a
family of Multivalued SemiProcesses (MSP), if on Σ there acts a continuous semigroup {T (h) :
Σ → Σ}h≥0 and ∀σ ∈ Σ ∀x ∈ X :
1) Uσ(τ, τ, x) = x ∀τ ≥ 0;
2) Uσ(t, τ, x) ⊂ Uσ(t, s, Uσ(s, τ, x)) ∀t ≥ s ≥ τ ;
3) Uσ(t+ h, τ + h, x) ⊂ UT (h)σ(t, τ, x) ∀t ≥ τ ∀h ≥ 0.
The family of MSP is called strict if in conditions 2), 3) there is equality.
We denote UΣ(t, τ, x) =
⋃
σ∈Σ
Uσ(t, τ, x).
Definition 2. A set θΣ ⊂ X is called a global attractor of the family of MSP {Uσ}σ∈Σ, if
θΣ 6= X and
1) θΣ is a uniformly attracting set, that is, ∀B ∈ β(X) ∀τ ≥ 0
dist (UΣ(t, τ, B), θΣ) → 0, t → ∞;
2) θΣ is a minimal uniformly attracting set, that is, for an arbitrary uniformly attracting set Y ,
we have θΣ ⊂ Y .
It is known [8 – 10] that if a family of MSP {Uσ}σ∈Σ satisfies the following conditions:
∀B ∈ β(X) ∃T = T (B) :
⋃
t≥T
UΣ(t, 0, B) ∈ β(X),
∀B ∈ β(X) ∀{ tn| tn ↗ ∞} arbitrary, (5)
the sequence {ξn| ξn ∈ UΣ(tn, 0, B)} is precompact inX,
then the family of MSP {Uσ}σ∈Σ has a global attractor,
θΣ =
⋃
τ≥0
θΣ(τ) = θΣ(0), (6)
where θΣ(τ) =
⋃
B∈β(X)
ωΣ(τ,B), and ωΣ(τ,B) =
⋂
s≥τ
⋃
t≥s
UΣ(t, τ, B) is compact in X . Moreover,
y ∈ ωΣ(τ,B) ⇐⇒ y = lim
n→∞
yn, yn ∈ UΣ(tn + τ, τ, B), tn↗∞.
Theorem 1. Let a family of MSP {Uσ}σ∈Σ satisfy the conditions (5).
1. If ∃B0 ∈ β(X) ∀B ∈ β(X) dist : (UΣ(t, 0, B), B0) → 0, t→ ∞, then θΣ is compact
in X.
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3
322 G. IOVANE, O. V. KAPUSTYAN
2. If for some t > 0 and for each τ ≥ 0, we have the following property:
if ξn ∈ UT (tn)Σ(t+ τ, τ, ηn), tn ↗ ∞, ξn → ξ, ηn → η, (7)
then ξ ∈ UΣ(t+ τ, τ, η),
then θΣ ⊂ UΣ(t+ τ, τ, θΣ).
If, additionally, ∃γ > 0 such that (7) holds ∀t ∈ (0, γ), and the family of MSP {Uσ}σ∈Σ is
strict, then θΣ is semiinvariant, that is ∀(t, τ) ∈ R +d
θΣ ⊂ UΣ(t, τ, θΣ).
Proof. 1. We have that ∀δ > 0 ∀B ∈ β(X) ∃T = T (δ,B) ∀t ≥ T : UΣ(t, 0, B) ⊂
⊂ Oδ(B0). From (6) we obtain that ωΣ(0, B) ⊂ Oδ(B0), so θΣ ⊂ Oδ(B0). Further, ∀t ≥ T
∀p ≥ 0: UΣ(t+ p, t, UΣ(t, 0, B)) ⊂ UΣ(t+ p, t, Oδ(B0)),
UΣ(t+ p, 0, B) ⊂ UT (t)Σ(p, 0, Oδ(B0)) ⊂ UΣ(p, 0, Oδ(B0)).
So ∀s ≥ T :
⋃
t′≥s+p
UΣ(t′, 0, B) ⊂ UΣ(p, 0, Oδ(B0)). Therefore, ∀s′ ≥ 0:
⋃
p≥s′
⋃
t′≥s+p
UΣ(t′, 0, B) ⊂
⋃
p≥s′
UΣ(p, 0, Oδ(B0)),
⋃
t′≥s+s′
UΣ(t′, 0, B) ⊂
⋃
p≥s′
UΣ(p, 0, Oδ(B0)),
and finally we obtain ωΣ(0, B) ⊂ ωΣ(0, Oδ(B0)). From this embedding and (6) we obtain θΣ =
= ωΣ(0, Oδ(B0)) is compact in X.
2. According to (6), θΣ =
⋃
τ≥0
θΣ(τ). Let ξ ∈ θΣ(τ). So, from the structure of θΣ(τ) there
exists B ∈ β(X), {σn} ⊂ Σ, {tn↗∞}, ξn ∈ Uσn(tn + τ, τ, B) such that ξn → ξ in X.
So ξn ∈ Uσn(tn + t − t + τ, τ, B) ⊂ Uσn(tn + t − t + τ, tn − t + τ, Uσn(tn − t + τ, τ, B)) ⊂
⊂ UT (tn−t)σn
(t+ τ, τ, ηn)), where ηn ∈ Uσn(tn− t+ τ, τ, B) ⊂ UT (τ)σn
(tn− τ, τ, θΣ)). From (5),
(6) on some subsequence ηn → η ∈ θΣ.Therefore from (7) ξ ∈ UΣ(t+τ, τ, η) ⊂ UΣ(t+τ, τ, θΣ),
so θΣ ⊂ UΣ(t+ τ, τ, θΣ).
Now let {Uσ}σ∈Σ be strict and (7) take place ∀t ∈ (0, γ). Then ∀n ≥ 1: UΣ(nt+ τ, τ, θΣ) =
= UΣ((n− 1) t + τ + t, τ, θΣ) = UΣ(t + τ + (n− 1) t, τ + (n− 1) t, UΣ((n− 1) t + τ, τ, θΣ)).
From this we immediately obtain that θΣ ⊂ UΣ(nt + τ, τ, θΣ) ∀n ≥ 1 ∀t ∈ (0, γ). So ∀t ≥ 0:
θΣ ⊂ UΣ(t+ τ, τ, θΣ) and the theorem is proved.
The family of MSP, generated by the problem (1) – (4). For using the abstract theory of
MSP we need to embed the problem (1) – (4) into a family of specially constructed problems.
For any h ∈ (τi−1, τi], i ≥ 1, we denote by σh the problem (1), whose solutions have impulse
effects of the form
u(τj − h+ 0)− u(τj − h) ∈ ψj(u(τj − h)), j ≥ i. (4)h
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3
GLOBAL ATTRACTOR FOR IMPULSIVE REACTION-DIFFUSION EQUATION 323
By σ0 we denote the problem (1) – (4).
By σ∞ we denote the problem (1), (2) without impulse effects.
Note that ∀τ ≥ 0 ∀h ∈ (τi−1, τi], i ≥ 1, the problem (1) – (4)h is globally resolved in the
same sense as problem (1) – (4). More exactly, we shall say that the problem (1) – (4)h is globally
resolved in the class Wτ (σh), if ∀uτ ∈ H ∃u ∈ L2
loc(τ,+∞;H1
0 (Ω)) ∩ Lploc(τ,+∞;Lp(Ω)) is
a solution of (1) on (τ, τj − h), (τj − h, τj+1 − h), . . . , u(τ) = uτ , where τj − h is the nearest
moment to τ , and at the points {τj − h, τj+1 − h, . . .} the function u(·) has impulse effects and
is left-continuous.
On the space Σ := {σh}h≥0 ∪ {σ∞} we define the function ρ : Σ× Σ → R,
ρ(σh1 , σh2) =
∣∣∣∣ 1
h1 + 1
− 1
h2 + 1
∣∣∣∣ , ρ(σh, σ∞) =
1
h+ 1
,
and ∀s ≥ 0 we define the map
T (s) : Σ → Σ, T (s)σh = σh+s, T (s)σ∞ = σ∞.
It is easy to show that (Σ, ρ) is a compact metric space and {T (s) : Σ → Σ}s≥0 is a continuous
semigroup acting on Σ.
Now for any σh ∈ Σ, 0 ≤ h ≤ ∞ we define the map
Uσh
(t, τ, uτ ) = {u(t) | u(·) ∈ Wτ (σh) is a solution of (1) – (4)h, u(τ) = uτ} . (8)
From noted above we have that formula (8) ∀σh ∈ Σ correctly defines a multivalued map,
Uσh
: R+d ×H → P (H).
Lemma 1. The family of maps, defined by (8) is a strict family of MSP.
Proof. The problem (1) – (4)∞ is autonomous and the required result for Uσ∞ can be easily
obtain from [10]. Let 0 ≤ h < ∞. From (8) ∀τ ≥ 0: Uσh
(τ, τ, uT ) = u(τ) = uτ , and we have
condition 1) from Definition 1.
Let ξ ∈ Uσh
(t, τ, x). Then ξ = u(t), u(·) ∈ Wτ (σh) is a solution of (1) – (4)h, u(τ) = x.
From this, ∀s ∈ (τ, t): u(s) ∈ Uσh
(s, τ, x). We put ω(p) = u(p), if p ≥ s. Then ω(·) ∈
∈ Ws(σh) is a solution of (1) – (4)h, ω(s) = u(s), so ξ = u(t) = ω(t) ∈ Uσh
(t, s, u(s)) ⊂
⊂ Uσh
(t, s, Uσh
(s, τ, x)).
Let ξ ∈ Uσh
(t, s, Uσh
(s, τ, x)). So ξ = u(t), where u(·) ∈ Ws(σh) is a solution (1) – (4)h,
u(s) = η, η = v(s), where v(·) ∈ Ws(σh) is a solution of (1) – (4)h, v(τ) = x. We put
ω(p) =
{
v(p), p ∈ [τ, s],
u(p), p > s.
Then ω(·) ∈ Wτ (σh) is solution of (1) – (4)h, ω(τ) = x, so ω(t) = u(t) = ξ ∈ Uσh
(t, τ, x).
Let ξ ∈ Uσh
(t + s, τ + s, x). Then ξ = u(t + s), u(·) ∈ Wτ+s(σh) is a solution of (1) – (4)h,
u(τ + s) = x. We put v(p) = u(p+ s), p ≥ τ. If τ + s ∈ (τi−1−h, τi−h], then u(·) is a solution
of (1) on (τ + s, τi − h), (τi − h, τi+1 − h), . . . which has impulse effect,
u(τj − h+ 0)− u(τj − h) ∈ ψj(u(τj − h)), j ≥ i.
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3
324 G. IOVANE, O. V. KAPUSTYAN
So, v(·) is a solution of (1) on (τ, τi − h − s), (τi − h − s, τi+1 − h − s), . . ., which has impulse
effect
v(τj − h− s+ 0)− v(τj − h− s) ∈ ψj(v(τj − h− s)), j ≥ i,
and v(τ) = u(τ + s) = x. Therefore, ξ = u(t + s) = v(t) ∈ Uσh+s
(t, τ, x) = UT (s)σh
(t, τ, x).
Let ξ ∈ UT (s)σh
(t, τ, x) = Uσh+s
(t, τ, x). Then ξ = u(t), u(·) ∈ Wτ (σh+s) is a solution of
(1) – (4)h+s, u(τ) = x. We put v(p) := u(p − s), p ≥ τ + s. Then analogously to the above
arguments we obtain that v(·) ∈ Wτ+s(σh) is a solution of (1) – (4)h, v(τ + s) = u(τ) = x. So,
ξ = u(t) = v(t+ s) ∈ Uσh
(t+ s, τ + s, x).
The lemma is proved.
We will assume that the following dissipative property holds:
∃R1 > 0 ∀τ ≥ 0 ∀r ≥ 0 ∀uτ ∈ H : ‖uτ‖ ≤ r and, for an arbitrary solution,
u(·) ∈ Wτ (σ0) of the problem (1) – (4), u(τ) = uτ , (9)
∃T1 = T1(r) ∀t ≥ T1 : ‖u(t+ τ)‖ ≤ R1,
that is ∀t ≥ T1: Uσ0
(t+ τ, τ, Br) ⊂ BR1 .
We shall discuss sufficient condition for (9) in the terms of the initial data given at the end
of this paper in Lemma 4.
Now we note that from (9) and Lemma 1, ∀σh ∈ Σ, 0 ≤ h < ∞:
Uσh
(t+ τ, τ, Br) = UT (h)σ0
(t+ τ, τ, Br) = Uσ0(t+ τ + h, τ + h,Br) ⊂ BR1 ∀t ≥ T1,
where T1 does not depend on τ and h. Using results from [9] we can write the estimate: ∀u(·) ∈
∈ Wτ , a solution of (1), ∀t ≥ 0 ∀τ ≥ 0:
‖u(t+ τ)‖2 ≤ ‖u(τ)‖2 e−δt +K, (10)
where the constants δ > 0,K > 0 do not depend on u(·), t, τ. From (10) we immediately obtain
∀τ ≥ 0 ∀r > 0 ∃T2 = T2(r) ∀t ≥ T2 : Uσ∞(t+ τ, τ, Br) ⊂ B√K+1.
So we have the following consequence of (9): ∃R0 > 0 ∀τ ≥ 0 ∀r ≥ 0 ∃T = T (r) ∀t ≥ T (r):
UΣ(t+ τ, τ, Br) ⊂ BR0 . (11)
Also we need the following condition:
∀r > 0 : sup
u∈Br
‖ψi(u)‖ → 0, i → ∞, (12)
where ‖ψi(u)‖ = supa∈ψi(u) ‖a‖.
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GLOBAL ATTRACTOR FOR IMPULSIVE REACTION-DIFFUSION EQUATION 325
The main result.
Theorem 2. Let the conditions (2), (9), (12) hold. Then the family of MSP {Uσ}σ∈Σ defined
by (8), has a global attractor θΣ ⊂ H , which is seminvariant and compact in H .
Proof. Since embedding (11) takes place, for the existence of a global attractor we need
to prove that for an arbitrary r > 0, {tn ↗ ∞}, the sequence {ξn| ξn ∈ UΣ(tn, 0, Br)} is
precompact in H. To do this, we need the following result which is a particular case of the result
obtained in [10].
Lemma 2. Let {un(·)} ⊂ Wτ be a sequence of solutions of (1) and un(τ) → uτ weakly in
H. Let us assume that ∀T ≥ τ we have a sequence {tn} ⊂ (τ, T ) such, that tn → t0 ∈ (τ, T ).
Then there exists u(·) ∈ Wτ , a solution of (1), such that u(τ) = uτ and, for some subsequence,
un(tn) → u(t0) in H. If un(τ) → uτ in H , then, moreover, for tn → τ on some subsequence,
un(tn) → uτ in H .
Now let ξn ∈ Uσhn
(tn, 0, Br). Then ∀t∗ ∈ (0, γ) ∀n ≥ N(r) we have ξn ∈ Uσhn
(tn −
−t∗ + t∗, tn − t∗, BR0) = UT (tn−t∗)σhn
(t∗, 0, BR0). So there exists {ηn} ⊂ BR0 such that ξn ∈
∈ UT (tn−t∗)σhn
(t∗, 0, ηn) and ηn → η weakly inH (we always can choose a suitable subsequence).
Let hn ∈ (tin−1, tin ]. Since tn ↗∞, we always have T (tn − t∗)σhn → σ∞ in Σ. Note that
for every n ≥ 1 we know the elements {τi − hn − tn}i≥1. Then ∀n ≥ 1 ∃! m(n) ≥ 1 such that
λn := τm(n) − hn − tn < 0, θn := τm(n)+1 − hn − tn ≥ 0. Moreover, m(n) → ∞, n → ∞ and
∀n ≥ 1: θn − λn ≥ γ > 0. Since we can always consider a subsequence, we should investigate
only two situations:
1. ∃ε ∈ (0, γ) ∀n ≥ N(r): λn < −ε. In this case we choose t∗ ∈ (0, ε) and obtain that
λn + t∗ < 0, θn + t∗ ≥ t∗. Since ξn ∈ Uσhn+tn−t∗ (t
∗, 0, ηn), we have ξn = un(t∗), where
un(·) ∈ W0(σhn+tn−t∗) is a solution of (1) – (4)hn+tn−t∗ , un(0) = ηn. Therefore, un(·) has the
first impulse effect at the moment θn + t∗ ≥ t∗, so it has no impulse on [0, t∗). From this we can
use Lemma 2 and obtain that on some subsequence ξn = un(t∗) → u(t∗), where u(·) ∈ W0 is
a solution of (1), u(0) = η. So, in this case the sequence {ξn} is precompact in H.
2. ∀n ≥ N(r) : λn < 0, but λn↗ 0, n → ∞. In this case we take arbitrary t∗ ∈ (0, γ) and
for sufficiently large n ≥ N(r) we haveλn + t∗ ∈ (0, t∗), θn + t∗ > t∗. So for ξn = un(t∗) the
solution un(·) has a unique impulse effect at the moment of time sn = λn + t∗ = τm(n) − hn −
−tn + t∗, sn ↗ t∗, which is characterized by the inclusion
un(sn + 0)− un(sn) ∈ ψm(n)(un(sn)).
According to Lemma 2, un(sn) → u(t∗), where u(·) is a solution of (1), u(0) = η. In particular,
∃r > 0 : ‖un(sn)‖ ≤ r. So ‖un(sn + 0)− un(sn)‖ ≤
∥∥ψm(n)(un(sn))
∥∥ ≤ supu∈Br
∥∥ψm(n)(u)
∥∥→
→ 0, n → ∞. Therefore, un(sn + 0) → u(t∗), n → ∞. Because ξn = un(t∗) ∈ Uσ∞(t∗, sn,
un(sn + 0)) = UT (sn)σ∞(t∗ − sn, 0, un(sn + 0)), we have ξn = vn(t∗ − sn), where vn(·) ∈ W0
is a solution of (1), vn(0) = un(sn + 0) → u(t∗). So from Lemma 2 on some subsequence,
ξn = un(t∗) = vn(t∗ − sn) → u(t∗), and we obtain precompactness of {ξn} in H.
Further, from embedding (11) and Theorem 1 we immediately obtain that the family of MSP
{Uσ}σ∈Σ has a compact global attractor θΣ ⊂ H.
Now we shall prove semiinvariance of θΣ. We need the following result.
Lemma 3. If ξn ∈ UT (tn)Σ(t, 0, ηn), where t ∈ (0, γ), tn ↗ ∞, ηn → η, then on some
subsequence ξn → ξ ∈ Uσ∞(t, 0, η).
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326 G. IOVANE, O. V. KAPUSTYAN
Proof. Our arguments in this case will be similar to the one described above, but not
analogous, because in this lemma we cannot choose t ∈ (0, γ), but we have a strong convergence
ηn → η in H.
Let ξn ∈ UT (tn)σhn
(t, 0, ηn), σhn ∈ Σ. For arbitrary n > 1 we use λn < 0 and θn ≥ 0
which are introduced above. We have ξn = un(t), where un(·) ∈ W0(σtn+hn) is a solution of
(1) – (4)tn+hn , un(0) = ηn → η. As t ∈ (0, γ), we have that un(·) on [0, t] is not bigger than one
impulse effect at the moment θn. If ∀n ≥ 1 : θn ≥ t, then un(·) on [0, t) has no impulse effect
and, according to Lemma 2, ξn = un(t) → u(t) ∈ Uσ∞(t, 0, η).
Let θn ∈ [0, t). Note that from estimate (10) ∃r > 0 ∀n ≥ 1 : ‖un(θn)‖ ≤ r, so
‖un(θn + 0)− un(θn)‖ ≤
∥∥ψm(n) (un(θn))
∥∥ ≤ supu∈Br
∥∥ψm(n)(u)
∥∥ → 0, n → ∞.
Now we should consider all possible cases for θn. If θn ↘ 0 (or θn = 0), then from Lemma
2 we obtain un(θn) → η in H. Therefore un(θn + 0) → η and ξn = un(t) ∈ Uσ∞(t, θn, un(θn +
+0)) = Uσ∞(t− θn, 0, un(θn + 0)), where t− θn↗ t. Using again Lemma 2, we have ξn → ξ ∈
∈ Uσ∞(t, 0, η).
If θn ↗ t, then ξn = un(t) ∈ Uσ∞(t, θn, un(θn + 0)) = Uσ∞(t − θn, 0, un(θn + 0)), where
t − θn ↘ 0. As un(θn + 0) → u(t) ∈ Uσ∞(t, 0, η), we can use Lemma 2 and obtain that ξn →
→ ξ = u(t) ∈ Uσ∞(t, 0, η).
If, finally, θn → θ ∈ (0, t), then ξn = un(t) ∈ Uσ∞(t, θn, un(θn +
+0)) = Uσ∞(t− θn, 0, un(θn + 0)), where t− θn → t− θ. Since, by Lemma 2, un(θn) → u(θ) ∈
∈ Uσ∞(θ, 0, η), so un(θn + 0) → u(θ). However ξn = vn(t − θn), where vn(·) ∈ W0 is a solu-
tion of (1), vn(0) = un(θn + 0). Therefore it follows from Lemma 2 that ξn = vn(t − θn) →
→ ξ = v(t− θn) ∈ Uσ∞(t− θ, 0, u(θ)) ⊂ Uσ∞(t, θ, Uσ∞(θ, 0, η)) = Uσ∞(t, 0, η) and the lemma
is proved.
Now according to Theorem 1, let ξn ∈ UT (tn)Σ(t+ τ, τ, ηn), where t ∈ (0, γ), τ ≥ 0, tn↗∞,
ξn → ξ, ηn → η. Then ξn ∈ UT (tn+t)Σ(t, 0, ηn) and, from Lemma 3, we obtain
ξn → ξ ∈ Uσ∞(t, 0, η) = UT (τ)σ∞(t, 0, η) = Uσ∞(t+ τ, τ, η) ⊂ UΣ(t+ τ, τ, η).
So from Theorem 1, it follows that θΣ is seminvariant.
The theorem is proved.
Lemma 4. If ∀i ≥ 1 ∀u ∈ H : ‖ψi(u)‖ ≤ a ‖u‖+ b, where a, b ≥ 0 and
−δ +
1
γ
ln(1 + (a+ 1)2) < 0, (13)
where δ > 0 is the constant from estimate (10), then (9) holds.
Proof. From [10] we have, that if u(·) ∈ Wτ is a solution of (1), (2), then the scalar function
t → ‖u(t)‖2 is absolutely continuous and for almost all t > τ it satisfies the inequality
d
dt
‖u(t)‖2 + δ ‖u(t)‖2 ≤ C̃, (14)
when the constant C̃ > 0 depends only on the constants if problem (1). Moreover, from (4) we
have
∣∣∣‖u(τi + 0)‖2 − ‖u(τi)‖2
∣∣∣ = |‖u(τi + 0)‖ − ‖u(τi)‖| |‖u(τi + 0)‖+ ‖u(τi)‖| ≤ (a ‖u (τi)‖+
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GLOBAL ATTRACTOR FOR IMPULSIVE REACTION-DIFFUSION EQUATION 327
+b) ((2 + a) ‖u (τi)‖+ b) ≤ ((a+ 1) ‖u (τi)‖+ b)2 ≤
(
ε+ (a+ 1)2
)
‖u (τi)‖2 + Cεb
2 ∀ε > 0. So
for the function t → ‖u (t)‖2 we have the following impulse problem:
d
dt
‖u(t)‖2 + δ ‖u(t)‖2 ≤ C̃,
(15)
‖u(τi + 0)‖2 − ‖u(τi)‖2 ≤
(
ε+ (a+ 1)2
)
‖u (τi)‖2 + Cεb
2.
Obviously, every solution of (15) is bounded by the solution of following problem:
d
dt
x(t) + δx(t) = C̃,
(16)
x(τi + 0)− x(τi) =
(
ε+ (a+ 1)2
)
x (τi) + Cεb
2.
For each τ ≥ 0, xτ ∈ R the solution x(·) of (16), x(τ) = xτ , is given by the formula
x(t) = e−δ(t−τ)(1 + ε+ (a+ 1)2)i(t,τ)xτ + C̃
t∫
τ
e−δ(t−p)(1 + ε+ (a+ 1)2)i(t,p)dp+
+ Cεb
2
∑
τ≤τi<t
e−δ(t−τi)(1 + ε+ (a+ 1)2)i(t,τi),
where i(t, s) is a number of moments τi, which belong to [s, t).
From (13), ∃ε > 0 ∃µ > 0 such that
−δ +
1
γ
ln(1 + ε+ (a+ 1)2) ≤ −µ < 0.
Then for xτ = ‖u(τ)‖2 we can easily obtain the following estimate:
‖u(t+ τ)‖2 ≤ x(t+ τ) ≤ e−µt ‖u(τ)‖2 +M
from what (9) follows.
The lemma is proved.
1. Temam R. Infinite-dimensional dynamical systems in mechanics and physics. — New York: Springer, 1997. —
645 p.
2. Chepyzhov V. V., Vishik M. I. Attractors of non-autonomous dynamical systems and their dimension // J.
Math. Pures et Appl. — 1994. — 73, № 3. — P. 279 – 333.
3. Chepyzhov V. V., Vishik M. I. Evolution equations and their trajectory attractors // Ibid. — 1997. — 76, № 10.
— P. 913 – 964.
4. Ball J. M. Continuity properties and global attractors of generalized semiflows and the Navier – Stokes equati-
ons // Mechanics: from Theory to Computation. — New York: Springer, 2000. — P. 447 – 474.
5. Melnik V. S. Multivalued dynamics of nonlinear infinite dimensional systems. — Kiev, 1996. — 73 p. —
(Preprint/Nat. Acad. Sci. Ukraine. IPSA).
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3
328 G. IOVANE, O. V. KAPUSTYAN
6. Kapustyan O. V., Valero J. Attractors of multivalued semiflows generated by differential inclusions and their
approximations // Abstract and Appl. Analysis. — 2000. — 5, № 1. — P. 33 – 46.
7. Kapustyan O. V., Perestyuk M. O. Global attractors of evolution inclusions with impulse effects at the fixed
moments of time // Ukr. Math. J. — 2003. — 55, № 8. — P. 1283 – 1294.
8. Kapustyan O. V., Melnik V. S., and Valero J. Attractors of multivalued dynamical processes generated by
phase-field equations // Int. J. Bifurcation and Chaos. — 2003. — 13, № 7. — P. 1969 – 1983.
9. Kapustyan O. V. Global attractors for nonautonomous reaction-diffusion equations // Different. Equat. —
2002. — 38, № 10. — P. 1378 – 1382.
10. Valero J., Kapustyan O. V. On the connectedness and asymptotic behaviour of solutions of reaction-diffusion
systems // J. Math. Anal. and Appl. (to apper).
Received 21.06.2005
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