On the dynamics of solutions for autonomous reaction-diffusion equation in Rⁿ with multivalued nonlinearity
We consider the dynamics of solutions for autonomous reaction-diffusion equation in Rⁿ with multivalued nonlinearity. The a priory estimates for solutions are obtained. The existence of compact invariant global attractor for m-semiflow was justified.
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irk-123456789-1243592017-09-25T03:02:44Z On the dynamics of solutions for autonomous reaction-diffusion equation in Rⁿ with multivalued nonlinearity Stanzhitsky, A.N. Gorban, N.V. We consider the dynamics of solutions for autonomous reaction-diffusion equation in Rⁿ with multivalued nonlinearity. The a priory estimates for solutions are obtained. The existence of compact invariant global attractor for m-semiflow was justified. 2009 Article On the dynamics of solutions for autonomous reaction-diffusion equation in Rⁿ with multivalued nonlinearity / A.N. Stanzhitsky, N.V. Gorban // Український математичний вісник. — 2009. — Т. 6, № 2. — С. 235-251. — Бібліогр.: 19 назв. — англ. 1810-3200 2000 MSC. 35B40, 35K55, 37B25 http://dspace.nbuv.gov.ua/handle/123456789/124359 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
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We consider the dynamics of solutions for autonomous reaction-diffusion equation in Rⁿ with multivalued nonlinearity. The a priory estimates for solutions are obtained. The existence of compact invariant global attractor for m-semiflow was justified. |
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Stanzhitsky, A.N. Gorban, N.V. |
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Stanzhitsky, A.N. Gorban, N.V. On the dynamics of solutions for autonomous reaction-diffusion equation in Rⁿ with multivalued nonlinearity Український математичний вісник |
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Stanzhitsky, A.N. Gorban, N.V. |
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Stanzhitsky, A.N. |
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On the dynamics of solutions for autonomous reaction-diffusion equation in Rⁿ with multivalued nonlinearity |
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On the dynamics of solutions for autonomous reaction-diffusion equation in Rⁿ with multivalued nonlinearity |
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On the dynamics of solutions for autonomous reaction-diffusion equation in Rⁿ with multivalued nonlinearity |
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On the dynamics of solutions for autonomous reaction-diffusion equation in Rⁿ with multivalued nonlinearity |
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On the dynamics of solutions for autonomous reaction-diffusion equation in Rⁿ with multivalued nonlinearity |
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on the dynamics of solutions for autonomous reaction-diffusion equation in rⁿ with multivalued nonlinearity |
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Інститут прикладної математики і механіки НАН України |
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2009 |
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On the dynamics of solutions for autonomous reaction-diffusion equation in Rⁿ with multivalued nonlinearity / A.N. Stanzhitsky, N.V. Gorban // Український математичний вісник. — 2009. — Т. 6, № 2. — С. 235-251. — Бібліогр.: 19 назв. — англ. |
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Український математичний вісник |
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AT stanzhitskyan onthedynamicsofsolutionsforautonomousreactiondiffusionequationinrnwithmultivaluednonlinearity AT gorbannv onthedynamicsofsolutionsforautonomousreactiondiffusionequationinrnwithmultivaluednonlinearity |
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2025-07-09T01:18:59Z |
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Український математичний вiсник
Том 6 (2009), № 2, 235 – 251
On the dynamics of solutions for autonomous
reaction-diffusion equation in R
N with
multivalued nonlinearity
Aleksandr N. Stanzhitsky, Nataliya V. Gorban
(Presented by A. M. Samoilenko)
Abstract. We consider the dynamics of solutions for autonomous
reaction-diffusion equation in R
n with multivalued nonlinearity. The a
priory estimates for solutions are obtained. The existence of compact
invariant global attractor for m-semiflow was justified.
2000 MSC. 35B40, 35K55, 37B25.
Key words and phrases. Reaction-diffusion equation, global attrac-
tor, m-semiflow, unbounded domain.
Introduction
The theory of global attractors for infinite-dimensional dynamical sys-
tems was founded in 70th of the last century in O. A. Ladyzhenskaya’s
works on studying the dynamics of two-dimensional system of Navier-
Stokes’s equations and in J. K. Hale’s works, which concerned with in-
vestigation of high-quality behavior of functional-differential equations.
However, swift development of this theory, which proceeds till today,
came in the middle of 80th, when it turned out that at abstract level
of those characteristic features which allowed to investigate the Navier–
Stokes’s equation and equations with delay from the point of global at-
tractors theory, are peculiar to the wide class of evolution equations,
which describe real natural phenomena: flow of viscid incompressible
liquid, processes of chemical kinetics, various wave processes, physical
processes assuming phase transitions, vibrations of shells in ultrafast gas
streams, etc. An important contribution to foundation and development
Received 10.03.2009
Partially supported by SFFI-BRFFI, Grant no. Φ29.1/025 and by SFFI-RFFI, Grant
no. Φ28.1/031.
ISSN 1810 – 3200. c© Iнститут математики НАН України
236 On the dynamics of solutions...
of classic theory of global attractors of the infinite-dimensional dynamic
systems was made by M. I. Vishik, O. A. Ladyzhenskaya, V. S. Melnik,
I. D. Chueshov, J. M. Ball, J. K. Hale, R. Temam, B. Wang, S. V. Zelik
and their apprentices [1]– [18].
The results concerning with existence and properties of solutions of
reaction-diffusion equation in the case of smooth by the phase variable
nonlinear term as well as the results dealing with existence of global
attractor under these conditions are classic and contain in [1, 16], for
non-autonomous equations with almost periodic dependence on time vari-
able — in [8], for inclusions in bounded domain in — [5,10], for equations
in unbounded domain — in [14, 17]. The global attractor for inclusions
in unbounded domain with continuous multivalued nonlinearity was con-
sidered in [19].
1. Problem definition
In the present paper it is investigated the asymptotic behavior of
solutions of reaction-diffusion equation with multivalued nonlinearity of
the next view:
yt ∈ △y − f(x, y), x ∈ R
N , t > 0, (1.1)
y(0) = y0 ∈ L2(RN ), (1.2)
where y is unknown function, yt = ∂y/∂t.
For numbers a, b ∈ R let [a, b] = {αa + (1 − α)b |α ∈ [0, 1]}. Let us
specify the conditions for parameters of the problem:
α1) for almost each (a.e.) x ∈ R
N , ∀u ∈ R f(x, u) = [f(x, u), f(x, u)],
where f, f : R
N+1 → R are measurable functions such that for a.e. x ∈
R
N f(x, ·) is lower semi-continuous (l.s.c.), and f(x, ·) is upper semi-
continuous (u.s.c.).
α2) ∃C1 ∈ L1(RN ), α > 0: for a.e. x ∈ R
N , ∀u ∈ R
f(x, u)u ≥ α|u|2 − C1(x), u ≥ 0,
f(x, u)u ≥ α|u|2 − C1(x), u ≤ 0.
(1.3)
α3) ∃C2 ∈ L1(RN ), C2 ≥ 0, ∃β > 0: for a.e. x ∈ R
N , ∀u ∈ R
|f(x, u)|2 ≤ C2(x) + β|u|2,
|f(x, u)|2 ≤ C2(x) + γ|u|2,
f(x, u) ≤ f(x, u).
(1.4)
A. N. Stanzhitsky, N. V. Gorban 237
Let us consider real spaces H = L2(RN ), V = H1(RN ) and V ∗ =
H−1(RN ) with corresponding norms ‖ · ‖, ‖ · ‖V and ‖ · ‖V ∗ . The norm
in R
N , inner product in H and in R
N we will denote by | · |, (·, ·)H , (·, ·)
respectively [14, p. 112].
The goal of this work is to prove the existence of compact global
attractor for the solutions of the problem (1.1)–(1.2) in the phase spaceH.
2. Preliminaries
From (1.4) it follows, that for arbitrary u, g ∈ L2(0, T ;H): g(x, t) ∈
f(x, u(x, t)) for a.e. (x, t) ∈ R
N × (0, T )
T
∫
0
∫
RN
|g(x, t)|2 dx dt ≤ K1
(
T + ‖u‖2
L2(0,T ;H)
)
. (2.5)
Definition 2.1. A function y(x, t), x ∈ R
N , t ∈ [0, T ] is called the
weak solution of (1.1) on [0, T ], if y ∈ L2(0, T ;V ) ∩ L∞(0, T ;H) and
for some selector d ∈ L2(0, T ;H): d(x, t) ∈ f(x, y(x, t)) for a.e. (x, t) ∈
R
N × (0, T ), the relation
−
T
∫
0
(y, vt) dt−
T
∫
0
(y,△v) dt+
T
∫
0
∫
RN
d(x, t)v(x, t) dx dt = 0, (2.6)
holds true for every v ∈ C∞
0 (RN × [0, T ]).
From (2.5), [14, p. 114] and from the Definition 2.1 of the weak so-
lution y of the inclusion (1.1) it follows, that yt ∈ L2(0, T ;V ∗), y ∈
C(0, T ;V ∗) and y ∈ C(0, T ;Hw). Thus, for every v ∈ L2(0, T ;V ) the
weak solution y satisfies such relation:
T
∫
0
〈yt, v〉V dt+
T
∫
0
(∇y,∇v) dt+
T
∫
0
∫
RN
d(x, t)v(x, t) dx dt = 0. (2.7)
where 〈·, ·〉V is the pairing in the space V , that coincides on H × V with
the inner product in H [2, p. 29].
The conditions α1–α3 don’t provide the uniqueness of the solution of
the problem (1.1)–(1.2) [10, p. 68], so let us introduce the definition of
multivalued, in the general case, semiflow and its global attractor (see for
example [10, p. 14]), that describe the dynamics of the solutions of initial
problem as t→ +∞.
238 On the dynamics of solutions...
Definition 2.2. A map G : R+ ×H → P (H) is called the multivalued
semiflow (m-semiflow) on H, if
1) G(0, ·) = IH is identical motion H;
2) G(t+ s, x) ⊂ G(t, G(s, x)) ∀ t, s ∈ R+, ∀x ∈ H.
M -semiflow is called the strict, if G(t+ s, x) = G(t, G(s, x)) ∀ t, s ∈ R+,
∀x ∈ H.
Definition 2.3 ([14, p. 123]). M -semiflow G is assiptotically compact,
if for any bounded B ∈ P (H) such, that γ+
T (B) is bounded for some
T = T (B) ≥ 0, an arbitrary sequence {ξn}n≥1, ξn ∈ G(tn, B), tn → +∞,
is pre-compact in H.
Definition 2.4. A set A ⊂ H, that satisfies the next properties:
1) A is absorbing set, i.e.,
dist(G(t, B),A) → 0, as t→ +∞,
for any bounded set B, where dist(C,A) = sup
c∈C
inf
a∈A
‖c− a‖.
2) A is semi-invariant, i.e.,
A ⊂ G(t, A), for every t ≥ 0,
3) A is minimal closed absorbing set (i.e. for any closed absorbing set
C we have, that A ⊂ C)
is called the global attractor A for m-semiflow G.
The global attractor is called invariant, if A = G(t, A), for every
t ≥ 0.
Let now Ω ⊂ R
N is bounded domain, T > 0, Q = Ω × (0, T ), Y =
L2(Q). The next lemma is necessary for proof of the main theorem.
Lemma 2.1. Let f satisfies α1, and {yn, dn}n≥0 ⊂ Y such, that
1) for a.e. (x, t) ∈ Q yn(x, t) → y0(x, t) as n→ +∞,
2) dn → d0 weakly in Y as n→ +∞,
3) ∀n ≥ 1 for a.e. (x, t) ∈ Q dn(x, t) ∈ f(x, yn(x, t)).
Then for a.e. (x, t) ∈ Q d0(x, t) ∈ f(x, y0(x, t)).
A. N. Stanzhitsky, N. V. Gorban 239
Proof. Let {yn, dn}n≥0 ⊂ Y satisfy the lemma conditions. Let us select
the complete measure set Q1 ⊂ Q such, that
∀ (x, t) ∈ Q1 yn(x, t) → y0(x, t) as n→ ∞. (2.8)
As the space L2(Q) is a Hilbert space, then in view of [2, Remark I.6.2] it
is uniformly convex (see for example [2, Definition I.5.9]). From the proof
of [3, Theorem 1, p. 64–66] it follows that from weakly convergent to 0̄ in
Y sequence {dn − d0}n≥1 we can choose a subsequence {dnk
− d0}k≥1 ⊂
{dn − d0}n≥1, for which the arithmetical means converge by norm to 0̄
in L2(Q) (in mentioned theorem from [3] it is proved stronger statement
than the Banach–Saks property), i.e.
∥
∥
∥
∥
1
k
k
∑
j=1
(dnj
− d0)
∥
∥
∥
∥
Y
→ 0 as k → +∞.
It means, that
1
k
k
∑
j=1
dnj
→ d0 strongly in L2(Q) as k → +∞. (2.9)
Further, ∃Q2 ⊂ Q1 such, that Q2 is measurable, mes(Q1\Q2) = 0 and
∀ (x, t) ∈ Q2 ∀ k ≥ 1
f(x, ynk
(x, t)) ≤ dnk
(x, t) ≤ f(x, ynk
(x, t)).
So, ∀ k ≥ 1, ∀ (x, t) ∈ Q2
1
k
k
∑
j=1
f(x, ynj
(x, t)) ≤ 1
k
k
∑
j=1
dnj
(x, t) ≤ 1
k
k
∑
j=1
f(x, ynj
(x, t)). (2.10)
From (2.9) there exists a subsequence { 1
kl
∑kl
j=1 dnj
}l≥1⊂{ 1
k
∑k
j=1 dnj
}k≥1
and a complete measure set Q3 ⊂ Q2:
∀ (x, t) ∈ Q3
1
kl
kl
∑
j=1
dnj
(x, t) → d0(x, t) as l → +∞. (2.11)
For a.e. (x, t) ∈ Q3 let us set ak = f(x, ynk
(x, t)), k ≥ 1, a0 =
f(x, y0(x, t)). From α1 and (2.8) it follows, that limk→∞ ak ≤ a0. Thus,
lim
k→+∞
1
k
k
∑
j=1
f(x, ynj
(x, t)) ≤ f(x, y0(x, t)).
240 On the dynamics of solutions...
Similarly,
lim
k→+∞
1
k
k
∑
j=1
f(x, ynj
(x, t)) ≥ f(x, y0(x, t)).
Taking into account (2.10)–(2.11), we obtain that for a.e. (x, t) ∈ Q
d0(x, t) ∈ f(x, y0(x, t)).
3. Main results
Firstly we will obtain some a priory estimations of solutions.
Lemma 3.1. For some weak solution y of the problem (1.1)–(1.2) we
have:
‖y‖X ≤ K1(‖y0‖, T ), (3.12)
‖yt‖U ≤ K2(‖y0‖, T ), (3.13)
where Ki are nondecreasing by each variable functions, X = L2(0, T ;V )∩
C([0, T ], H) and U = L2(0, T ;V ∗).
Proof. From [14, Lemma 3] it follows, that
1
2
d
dt
‖y‖2 = −(∇y,∇y) −
∫
RN
d · y dx, (3.14)
where d ∈ L2(0, T ;H): d(x, t) ∈ f(x, y(x, t)) for a.e. (x, t) ∈ R
N × (0, T ).
From α1 and α3 we obtain
1
2
d
dt
‖y‖2 + ‖∇y‖2 ≤ −α‖y‖2 +
∫
RN
C1(x) dx. (3.15)
So,
∀ t ∈ [0, T ] ‖y(t)‖2 + 2
t
∫
0
‖∇y‖2ds+ 2α
t
∫
0
‖y‖2ds ≤ ‖y0‖2 + 2MT,
(3.16)
from here it follows (3.12).
On the other hand, as −△ : V → V ∗ is linear bounded operator,
then from (3.12) it follows, that
‖ −△y‖L2(0,T ;V ∗) ≤ K‖y‖L2(0,T ;V ) ≤ KK1.
Finally, from yt = △y − d it follows (3.13).
A. N. Stanzhitsky, N. V. Gorban 241
Theorem 3.1. Let the suppositions α1–α3 hold true. Then for any y0 ∈
L2(RN ), T > 0, the problem (1.1)–(1.2) has at least one weak solution.
Proof. In order to prove the solvability let us consider the homogeneous
Dirichlet’s problem in bounded domain
yt ∈ △y − f(x, y), x ∈ ΩR, t > 0, (3.17)
y|∂Ω
= 0, t > 0, (3.18)
y(0) = y0,R, x ∈ ΩR, (3.19)
where ΩR = B(0;R) is open ball of the radius R ≥ 1 with the center in
zero, u0,R(x) = u0(x)ψR(|x|) and ψR is smooth function, which satisfies:
ψR(ξ) =
1, if 0 ≤ ξ ≤ R− 1,
0 ≤ ψR(ξ) ≤ 1, if R− 1 ≤ ξ ≤ R,
0, if ξ > R.
Let us set R ≥ 1, QR,T = ΩR × (0, T ) HR = L2(ΩR), VR = H1(ΩR),
V ∗
R = H−1(ΩR), XR = L2(0, T ;VR), X ∗
R = L2(0, T ;V ∗
R), Y∗
R ≡ YR =
L2(QR), 〈·, ·〉R is pairing in XR, coinciding on YR × XR with the inner
product in YR [2, p. 29], WR = {y ∈ XR | y′ ∈ X ∗
R}, where the derivative
y′ of an element y ∈ XR is considered in the sense of distributions space
D∗(0, T ;V ∗
R) [2, p. 168]. Let A : XR → X ∗
R is energetic extension −△,
B : YR ⇉ Y∗
R is the Nemitsky operator for f , i.e.
B(u) = {v ∈ Y∗
R | v(t, x) ∈ f(x, u(x, t)) for a.e. (x, t) ∈ QR,T }, u ∈ YR.
Let us denote, that A is linear, strongly monotone, bounded [2, 87–88].
From α1–α3 and Lemma 2.1 it follows, that B is bounded, and its graph
is closed in (Y∗
R)w × YR [6, p. 576]. Thus, there exist such c1, c2, c3 > 0,
that
〈A(y), y〉R = ‖y‖2
XR
, ‖A(y)‖X ∗
R
= ‖y‖XR
, y ∈ XR;
∀ y ∈ YR, ∀ d ∈ B(y) 〈d, y〉R ≥ c1‖y‖YR
− c2, ‖d‖Y∗
R
≤ c3(‖y‖YR
+ 1).
As the embedding VR ⊂ HR is compact, then WR ⊂ YR compactly [7,
p. 70]. From [6, p. 557] the λ0-pseudomonotony of C = A + B : XR ⇉
X ∗
R on WR [6, p. 543] follows. So, from upper considered suppositions
and from [4, Theorem 6.1.1] it follows, that the problem (3.17)–(3.19)
has at least one weak (generalized) solution for any y0,R ∈ L2(ΩR) (the
definition of the weak solution is the same as in the definition 2.1, but
242 On the dynamics of solutions...
there we considered ΩR instead R
N ). So, there exists dR ∈ L2(0, T ;HR)
such, that ∀ v ∈ XR
T
∫
0
〈
∂yR
∂t
, v
〉
V
dt+
T
∫
0
(∇yR,∇v) dt+
T
∫
0
∫
RN
dR(x, t)v(x, t) dx dt = 0,
(3.20)
dR(x, t) ∈ f(x, yR(x, t)) for a.e. (x, t) ∈ ΩR × (0, T ). (3.21)
Let yrj
, rj → +∞ be the sequence of solutions of (3.17)–(3.19) in the
sense of (3.20)–(3.21). Let us denote, that
|y0 − y0,rj
|2 =
∫
RN
(1 − ψrj
(|x|))2|y0|2 dx
≤
∫
|x|≥rj−1
|y0|2 dx→ 0, rj → +∞. (3.22)
Repeating the same suppositions as in the proof of Lemma 3.1, we will
obtain
‖y(t)‖2
Hrj
+ 2
t
∫
0
‖∇y‖2
rj
ds+ 2α
t
∫
0
‖y‖2
rj
ds
≤ ‖y0,rj
‖2 + 2T
∫
RN
C1(x) dx, ∀ t ∈ [0, T ].
So, from (3.22) it follows
‖yrj
‖Xrj
≤ K1(‖y0,rj
‖, T ) ≤ K̃1(‖y0‖, T ), (3.23)
where Xrj
= L2(0, T ;Vrj
) ∩ C([0, T ], Hrj
).
Let us continue solutions of (3.17)–(3.19) onto R
N :
ŷrj
(x, t) =
{
yrj
(x, t)ψrj
(|x|) in B(0, rj) × (0, T ),
0, else,
d̂rj
(x, t) =
{
drj
(x, t) in B(0, rj − 1) × (0, T ),
f(x, ŷrj
(x, t)), else,
d̂rj
(x, t) ∈ f(x, ŷrj
(x, t)) for a.e. (x, t) ∈ R
N × (0, T ). As yrj
uniformly
bounded in Xrj
by rj , then {ŷrj
, d̂rj
} is uniformly bounded sequence in
A. N. Stanzhitsky, N. V. Gorban 243
X × L2(0, T ;V ∗). Thus, there exists a subsequence, which we denote
again by {yrj
, drj
} such, that
yrj
→ y∞ weakly in L2(0, T ;V ),
yrj
→ y∞ weakly star in L∞(0, T ;H),
drj
→ d∞ weakly in L2(0, T ;V ∗).
(3.24)
Let us further prove, that y∞ is the weak solution of (1.1)–(1.2).
Let us fix an arbitrary rk. As rj → +∞, then we can suppose, that
rk ≤ rj − 1. The projection yrj
on B(0, rk) we will denote by:
yk,j = Lkyrj
.
From (3.23) it is obviously follows, that {yk,j}j is bounded in Xrk
. So, up
to the subsequence (let us denote it again by yrj
), yk,j = Lkyrj
→ yk,∞
weakly in L2(0, T ;Vrk
) and weakly star in L∞(0, T ;Hrk
). Following by
[14, p. 118], we obtain, that Lky∞ = yk,∞. In order to show, that y∞ is
the weak solution of the problem (1.1)–(1.2), it is sufficiently to check,
that Lky∞ is the weak solution on Ωrk
× (0, T ) [14, p. 118].
Let v ∈ C∞
0 (Ωrk
× [0, T ]). As B(0, rk) ⊂ B(0, rj), then v ∈ C∞
0 (Ωrj
×
[0, T ]). So,
T
∫
0
∫
Ωrk
(
−Lkyrj
· vt − Lkyrj
· △v + Lkdrj
· v
)
dx dt
=
T
∫
0
∫
Ωrj
(
−Lkyrj
· vt − Lkyrj
· △v + Lkdrj
· v
)
dx dt = 0. (3.25)
Following by the proof of the Lemma 3.1 and [14, Theorem 5], we will
obtain:
Lk
∂yrj
∂t
=
∂Lkyrj
∂t
→ ∂yk,∞
∂t
weakly in L2(0, T ;V ∗
rk
). (3.26)
As the embedding H1
0 (Ωrk
) ⊂ L2(Ωrk
) is compact, and the embedding
L2(Ωrk
) ⊂ H−1(Ωrk
) is continuous, using the compactness lemma [7,
p. 70] and Lemma 2.1, from convergences (3.12), (3.26) and from con-
ditions α1, α3, similarly to [14, p. 119] we will obtain, that d∞(x, t) ∈
f(x, y∞(x, t)) for a.e. (x, t) ∈ Ωrk
× (0, T ). In order to complete the
proof it remains to pass to the limit in (3.25) as rj → +∞.
244 On the dynamics of solutions...
Since in the Theorem 3.1 T > 0 is arbitrary, as the concatenation of
solutions is the solution (see the proof of 1◦ of the Theorem 3.2), then by
analogical suppositions to [14, p. 119], each solution can be continued to
the global, that is defined on [0,+∞).
Let us denote the family of all global solutions of the problem (1.1)–
(1.2), corresponding to the initial condition y0 by D(y0). It is obviously,
that D(y0) ⊂ L2
loc(0,+∞;V )∩C([0,+∞), H). Let us show, that D(y0) ⊂
L∞(0,+∞;H) ∀ y0 ∈ L2(RN ).
Lemma 3.2. If y is the weak solution of the problem (1.1)–(1.2), then
∀ t ≥ 0 ‖y(t)‖2 + 2
t
∫
0
e−α(t−s)‖∇y‖2 ds ≤ ‖y(0)‖2e−2αt +D, (3.27)
where D = ‖C1‖L1(RN )/α.
Proof. The proof follows from (3.15) and from the Gronwall–Bellman
lemma.
Let y0 ∈ H, P (H) = 2H \{∅}. We define (in general the multivalued)
map G : R+ ×H → P (H):
G(t, y0) = {z ∈ H | ∃u ∈ D(y0) : y(0) = y0, y(t) = z}.
Theorem 3.2. Under the conditions α1–α3, the problem (1.1)–(1.2) de-
fines m-semiflow in the phase space H, that possesses the invariant global
attractor.
Proof. 1◦ Firstly we will show, that G is the strict m-semiflow. The
proof of G(t+ s, x) ⊂ G(t, G(s, x)) repeats the proof of similar inclusion
from [14, Lemma 7]. Let us check G(t, G(s, x)) ⊂ G(t + s, x). Let
u ∈ G(t, G(s, x)). Then there exist z1, y1(·) ∈ D(x), y2(·) ∈ D(z1),
d1, d2 such, that
y1(0) = x, y1(s) = z1,
y2(0) = z1, y2(t) = u,
d1 = △y1 −
∂y1
∂t
, d1(ξ, ζ) ∈ f(ξ, y1(ξ, ζ)) for a.e. (ξ, ζ) ∈ R
N × R+,
d2 = △y2 −
∂y2
∂t
, d2(ξ, ζ) ∈ f(ξ, y2(ξ, ζ)) for a.e. (ξ, ζ) ∈ R
N × R+.
Let us show, that there exists y(·) ∈ D(y0): y(0) = x, y(t+ s) = u. Let
us define y by:
y(r) =
{
y1(r), 0 ≤ r ≤ s,
y2(r − s), s ≤ r.
A. N. Stanzhitsky, N. V. Gorban 245
For a.e. (ξ, ζ) ∈ R
N × R+ let us set
d(ξ, ζ) =
{
d1(ξ, ζ), 0 ≤ ζ ≤ s,
d2(ξ, ζ − s), s ≤ ζ.
We remark, that
d(ξ, ζ) ∈ f(ξ, y(ξ, ζ)) for a.e. (ξ, ζ) ∈ R
N × R+.
The next suppositions complete the proof
T
∫
0
〈
∂y
∂r
, v
〉
V
dr +
T
∫
0
[
(∇y,∇v) dr +
∫
RN
d(x, r)v(x, r) dx
]
dr
=
s
∫
0
〈
∂y1
∂r
, v
〉
V
dr +
s
∫
0
[
(∇y1,∇v) dr +
∫
RN
d1(x, r)v(x, r) dx
]
dr
+
T
∫
s
〈
∂y2(r − s)
∂r
, v
〉
V
dr +
T
∫
s
[
(∇y2(r − s),∇v) dr
+
∫
RN
d2(x, r − s)v(x, r − s) dx
]
dr = 0 +
T−s
∫
0
〈
∂y2
∂r
, v
〉
V
dr
+
T−s
∫
0
[
(∇y2,∇v) dr +
∫
RN
d2(x, r)v(x, r) dx
]
dr = 0
∀T > s+ t, ∀ v ∈ C∞
0 ([0, T ] × R
N ).
For fixed k > 0 the ball of the radius k with the center in 0 we denote
by Ωk.
2◦ Let us prove, that for an arbitrary nonempty bounded set B ⊂ H,
an arbitrary y0 ∈ B, an arbitrary weak solution y ∈ D(y0), an arbitrary
ε > 0 there exist T (ε,B), K(ε,B) such, that
∀ t ≥ T, k ≥ K
∫
|x|≥
√
2k
|y(x, t)|2 dx ≤ ε.
Indeed, let s ∈ R+. Let us define the smooth function
θ(s) =
0, 0 ≤ s ≤ 1,
0 ≤ θ(s) ≤ 1, 1 ≤ s ≤ 2,
1, s ≥ 2
246 On the dynamics of solutions...
such, that |θ′(s)| ≤ C ∀ s ∈ R+. Moreover, let us suppose, that
√
θ is
smooth too.
Let us apply [14, Lemma 3] to ρ(x) =
√
θ( |x|
2
k2 ). From the definition
of the weak solution of the equation (1.1) it follows, that
for a.e. t ≥ 0
1
2
d
dt
∫
RN
θ
( |x|2
k2
)
|y|2 dx = 〈yt, ρ
2y〉V
= 〈△y, ρ2y〉V −
∫
RN
θ
( |x|2
k2
)
d(x, t)y(x, t) dx, (3.28)
where
d = △y − ∂y
∂t
, d(ξ, ζ) ∈ f(ξ, y(ξ, ζ)) for a.e. (ξ, ζ) ∈ R
N × R+,
Similarly to [14, p. 122–123], the first term in the right part of the last
relation is estimated by the next way:
〈△y, ρ2y〉V ≤ ε′(1 + ‖∇y‖2) (3.29)
for an arbitrary k ≥ K1(ε
′), where ε′ > 0 is an arbitrary and rather
small.
For the second term from (3.28) with the help of α2 and α3 we will
obtain:
−
∫
RN
θ
( |x|2
k2
)
d(x, t)y(x, t) dx
≤ −α
∫
RN
θ
( |x|2
k2
)
|y(x, t)|2 dx+
∫
RN
θ
( |x|2
k2
)
C1(x) dx
≤ −α
∫
RN
θ
( |x|2
k2
)
|y(x, t)|2 dx+ 2ε′, (3.30)
as soon as k ≥ K2(ε
′). Let us set
Y (t) =
∫
RN
θ
( |x|2
k2
)
|y(x, t)|2 dx.
Then from (3.28)–(3.30) it follows, that
1
2
d
dt
Y (t) + αY (t) ≤ 3ε′ + ε‖∇y‖2,
A. N. Stanzhitsky, N. V. Gorban 247
as soon as k ≥ max{K1,K2}. By using the Gronwall–Bellman lemma
and Lemma 3.2, we obtain
Y (t) ≤ Y (0)e−2αt +
3
α
ε′ +
ε′
2
(‖y0‖2 +D).
Choosing ε′, T (ε,B) such, that
3
α
ε′ +
ε′
2
(‖y0‖2 +D) ≤ ε
2
, Y (0)e−2αt ≤ ε
2
, ∀ y0 ∈ B, t ≥ T,
we will obtain Y (t) ≤ ε and
∫
|x|≥
√
2k
|y(x, t)|2 dx ≤
∫
RN
θ
( |x|2
k2
)
|y(x, t)|2 dx ≤ ε.
3◦ For bounded set B ⊂ H and T ∈ R+ let us consider
γ+
T (B) =
⋃
t≥T
G(t, B).
Following by the proof of [14, Lemma 8] and the proof of the Theo-
rem 3.1 we obtain the next result, that is necessity for the proof of asymp-
totic compactness of m-semiflow G, namely, the graph of G(t, ·) is weakly
closed, i.e. if ξn → ξ∞, βn → β∞ weakly in H, where ξn ∈ G(t, βn)
∀n ≥ 1, then ξ∞ ∈ G(t, β∞).
4◦ Let us show, that m-semiflow G is assiptotically compact. Let
ξn ∈ G(tn, vn) vn ∈ B, B be bounded set inH. Since γ+
T (B)(B) is bounded
and ξn ∈ G(tn, vn) ⊂ γ+
T (B)(B) for n ≥ n0, then there exists the weakly
convergent in H subsequence (let us denote it by ξn again) to some ξ. Let
T0 > 0 be an arbitrary number. Using 1◦, we have, that ξn ∈ G(tn, vn) =
G(T0, G(tn − T0, vn)). Then there exists such βn ∈ G(tn − T0, vn), that
ξn ∈ G(T0, βn). Let us choose N(B, T0) such, that ∀n ≥ N(B, T0) tn −
T0 ≥ T (B) and such, that G(tn − T0, vn) ⊂ γ+
T (B)(B) is bounded, βn →
ξT0
weakly in H. From 3◦ it follows, that the graph G(T0, ·) is weakly
closed. So, ξ ∈ G(T0, ξT0
) and limn→∞ ‖ξn‖ ≥ ‖ξ‖. If we show, that up
to subsequence, limn→∞ ‖ξn‖ ≤ ‖ξ‖, then, up to subsequence, ξn → ξ
strongly in H, that we need to show.
From (3.14) it follows, that any weak solution y satisfies
1
2
d
dt
‖y‖2 +
1
2
‖y‖2 + ‖∇y‖2 = −
∫
RN
d · y dx+
1
2
‖y‖2, a.e. on [0, T ],
248 On the dynamics of solutions...
where d ∈ L2(0, T ;H): d(x, t) ∈ f(x, y(x, t)) for a.e. (x, t) ∈ R
N × (0, T ).
Let yn(·) is the sequence of weak solutions, for which yn(T0) = ξn and
yn(0) = βn. In view of the Gronwall–Bellman lemma,
‖ξn‖2 = e−T0‖βn‖2 − 2
T0
∫
0
e−(T0−s)‖∇yn‖2 ds
− 2
T0
∫
0
∫
RN
e−(T0−s)dn · yn dx ds+
T0
∫
0
e−(T0−s)‖yn‖2 ds, (3.31)
where dn ∈ L2(0, T ;H): dn(x, t) ∈ f(x, yn(x, t)) for a.e. (x, t) ∈ R
N ×
(0, T ). From the Lemma 3.1 and the Banach–Alaoglu theorem, up to
subsequence (we denote it again by {yn, dn}), yn converges to some weak
solution y by the next way:
yn → y weakly in L2(0, T ;V ),
yn → y weakly star in L∞(0, T ;H),
dn → d weakly in L2(0, T ;V ∗),
∂yn
∂t
→ ∂y
∂t
weakly in L2(0, T ;V ∗).
(3.32)
From 3◦, y(0) = ξT0
, y(T0) = ξ.
Since the sequence {βn} is bounded, then
∀n e−T0‖βn‖2 ≤ e−T0M. (3.33)
Further,
lim
n→∞
(
− 2
T0
∫
0
e−(T0−s)‖∇yn‖2 ds
)
≤ −2
T0
∫
0
e−(T0−s)‖∇y‖2 ds. (3.34)
On the other hand,
T0
∫
0
e−(T0−s)‖yn‖2 ds =
T0
∫
0
∫
Ωk
e−(T0−s)|yn|2 dx ds
+
T0
∫
0
e−(T0−s)
∫
|x|≥k
|yn|2 dx ds.
A. N. Stanzhitsky, N. V. Gorban 249
From 1◦ it follows, that yn(s) ∈ G(s,G(tn−T0, vn)) = G(s+ tn −T0, vn).
From 2◦, for any ε > 0 there exist such T (ε,B), K1(ε,B) > 0, that
∫
|x|≥k
|yn(s)|2 dx ≤ ε,
as soon as k ≥ K1, tn − T0 ≥ T. Repeating corresponding suppositions
from the proof of the Theorem 3.1, we have, that (up to subsequence)
Lkyn → Lky strongly in L2(0, T ;Hk). So,
lim
n→∞
T0
∫
0
e−(T0−s)‖yn‖2 ds ≤
T0
∫
0
e−(T0−s)‖y‖2 dx ds+ ε. (3.35)
Let us consider the “nonlinear term” from (3.25). At first we remark,
that from α3 it follows that
−2
T0
∫
0
∫
|x|≥k
e−(T0−s)dn · yn dx ds ≤ 4ε
T0
∫
0
e−(T0−s) ds ≤ 4ε,
as soon as k ≥ K2(ε). Since yn → y strongly in L2(0, T ;Hk), then up to
the subsequence, yn(t, x) → y(t, x) for a.e. (t, x) ∈ (0, T0) × Ωk. From
the Lemma 2.1 and (3.32) we have
lim
n→∞
(
− 2
T0
∫
0
∫
Ωk
e−(T0−s)dn · yn dx ds
)
= −2
T0
∫
0
∫
Ωk
e−(T0−s)d · y dx ds.
Thus,
lim
n→∞
(
− 2
T0
∫
0
∫
Ωk
e−(T0−s)dn · yn dx ds
)
≤ −2
T0
∫
0
∫
Ωk
e−(T0−s)d · y dx ds+4ε.
(3.36)
Passing to the limit as k → ∞ in (3.36) and using (3.31) and (3.33)–(3.36)
we find, that
lim
n→∞
‖ξn‖2 ≤ e−T0M − 2
T0
∫
0
∫
RN
e−(T0−s)|∇y|2 dx ds
+
T0
∫
0
∫
RN
e−(T0−s)|y|2 dx ds− 2
T0
∫
0
∫
RN
e−(T0−s)d · y dx ds+ 5ε
250 On the dynamics of solutions...
= ‖ξ‖2 + e−T0M − e−T0‖ξT0
‖2 + 5ε. (3.37)
Passing to the limit as T0 → +∞, and then directing ε → 0, we will
obtain the next inequality
lim
n→∞
‖ξn‖2 ≤ ‖ξ‖2.
5◦ Let us prove the semi-continuity of m-semiflow G [14, p. 126],
namely, let us prove, that the map G(t, ·) is upper semi-continuous and
has compact values for any t ≥ 0. Indeed, let ξn ∈ G(t, xn) and xn →
x0. Let us prove, that the sequence ξn is pre-compact in H. From the
Lemma 3.2, the sequence ξn is bounded, so, up to the subsequence it
is weakly convergent to some ξ. Supposing analogically to the proof of
4◦, there exist weak solutions yn(·), y(·) such, that yn(t) = ξn, yn(0) =
xn, y(t) = ξ, y(0) = x0 and yn converges to y in the sense of (3.32).
Repeating the suppositions from 4◦ we will obtain, that limn→∞ ‖ξn‖2 ≤
‖ξ‖2. Thus, ξn → ξ strongly in H. So, taking into account 3◦, G(t, x0) is
compact.
Now, if G(t, ·) is not upper semi-continuous, then there exists the
point x0, the neighborhood O of the set G(t, x0) and the sequence ξn ∈
G(t, xn) such, that ‖xn − x0‖ → 0 as n → +∞ and ξn /∈ O ∀n. Passing
to the subsequences we have, that ξnk
→ ξ, xnk
→ x0 strongly in H.
From 3◦ it follows, that ξ ∈ G(t, x0). We obtained the contradiction.
From the properties 1◦–5◦, there exists the global compact invariant
attractor for G (see [13, Theorem 3, Remark 8]), that is minimal closed
absorbing set. Thus, the Theorem 3.2 is proved.
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Contact information
Aleksandr
N. Stanzhitsky,
Nataliya V. Gorban
Taras Shevchenko National
University of Kyiv
64, Volodymyrs’ka St.,
01033 Kyiv,
Ukraine
E-Mail: gorbannv@i.ua
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