Electron. Commun. Probab. 18 (2013), no. 45, 1–12.
DOI: 10.1214/ECP.v18-2858
ISSN: 1083-589X
ELECTRONIC
COMMUNICATIONS
in PROBABILITY
Asymptotic behavior for neutral stochastic partial
differential equations with infinite delays∗
Jing Cui†
Litan Yan‡
Abstract
This paper is concerned with the existence and asymptotic behavior of mild solutions
to a class of non-linear neutral stochastic partial differential equations with infinite
delays. By applying fixed point principle, we present sufficient conditions to ensure
that the mild solutions are exponentially stable in pth-moment (p ≥ 2) and almost
surely exponentially stable. An example is provided to illustrate the effectiveness of
the proposed result.
Keywords: Neutral stochastic partial differential equations; exponential stability; infinite delay.
AMS MSC 2010: 93E15; 34K50; 60H15.
Submitted to ECP on November 18, 2011, final version accepted on March 13, 2013.
1
Introduction
In recent years, there has been considerable interest in studying the quantitative
and qualitative properties of solutions to stochastic partial differential equations (SPDEs)
in a separable Hilbert space such as existence, uniqueness, stability, controllability and
asymptotic behavior (see, e.g., [1, 6, 17, 18, 21, 22, 23, 24, 25] and references therein).
Moreover, SPDEs with delays have also drawn much attention from many scholars.
For example, Liu and Truman [18] introduced some approximating systems with strong
solutions, and they presented several criteria about the asymptotic exponential stability for a class of non-autonomous stochastic evolution equations with variable delays;
Caraballo and Real [7] studied the stability of the strong solutions of semilinear stochastic evolution equations with delays; Caraballo and Liu [8] considered the exponential
stability of mild solutions of SPDEs with delays by utilizing the well-known Gronwall inequality, but the requirement of the monotone decreasing behavior of the delays should
be imposed; Liu and Truman [13] established the exponential stability of mild solutions
for SPDEs by establishing the corresponding Razuminkhin-type theorem.
However, in some cases, many stochastic dynamical systems depend not only on
present and past states, but also contain the derivatives with delays (see, e.g., [11]
and [12]). Neutral stochastic differential equations with delays are often used to describe such systems. To the best of our knowledge, there is only a little systematic
∗ Supported by the NSFC (11171062), Innovation Program of Shanghai Municipal Education Commission
(12ZZ063 ), Natural Science Foundation of Anhui Province (1308085QA14, 1208085MA11) and Key Natural
Science Foundation of Anhui Educational Committee (KJ2013A133, KJ2011A139).
† Department of Mathematics, Anhui Normal University, P.R. China.
E-mail: jcui123@126.com
‡ Department of Mathematics, College of Science, Donghua University, P.R. China.
E-mail: litanyan@hotmail.com (corresponding author)
Asymptotic behavior for NSPDEs with infinite delays
investigation on the stability of mild solutions to neutral SPDEs with delays. It is important to note that a number of difficulties exist in the study of the stability of mild
solutions to neutral SPDEs with delays due to the presence of the neutral items. For instance, the mild solutions do not have stochastic differentials, and many methods used
frequently fail to deal with the stability of mild solutions for neutral SPDEs with delays.
Recently, in [2, 3, 4, 5] Burton and his co-authors investigated the stability of deterministic differential equations by fixed point theory, and they pointed out that some of
these difficulties can be rectified. Luo [14] applied this valuable method to deal with the
stability for stochastic ordinary differential equations. We would also like to mention
that some topics similar to the above for stochastic partial functional differential equations have been studied by many authors (see, e.g., [9], [15], [16], [23] and references
therein).
Almost all the results on stability in the aforementioned papers are suited to stochastic differential equations with finite delay. However, as mathematical models of phenomena, neutral stochastic differential equations with infinite delays have become an
important issue in both the physical and social sciences. For this, see, for example,
[10, 19] with the theory development for the description of heat conduction in materials with fading memory. In this paper, motivated by the previous references, we are
concerned with the existence and asymptotic behavior of mild solutions to a class of
neutral SPDEs with infinite delays. Our approach is based on fixed point theorem and
operator theory.
The rest of this paper is organized as follows. In Section 2, we briefly present some
basic notations and preliminaries. Section 3 is devoted to the proof of the existence and
asymptotic behavior of mild solutions to a class of neutral SPDEs with infinite delays.
An example is provided to illustrate the effectiveness of the proposed result.
2
Preliminaries
Let (H, h·, ·iH , | · |H ) and (K, h·, ·iK , | · |K ) be two real, separable Hilbert spaces. Let
L(K, H) be the space of all linear bounded operators from K to H , equipped with the
usual operator norm k · k.
Let (Ω, F, {Ft }t≥0 , P ) be a filtered complete probability space satisfying the usual
conditions. We denote by W = (Wt )t≥0 a K -valued Wiener process defined on the
probability space (Ω, F, {Ft }t≥0 , P ) with covariance operator Q. That is,
Ehw(t), xiK hw(s), yiK = (t ∧ s)hQx, yiK ,
∀x, y ∈ K,
where Q is a positive, self-adjoint trace class operator on K . Furthermore, L02 (K, H)
denotes the space of all Q-Hilbert-Schmidt operators from K to H with the norm
|ξ|2L0 := tr(ξQξ ∗ ) < ∞,
2
ξ ∈ L(K, H).
For the construction of stochastic integral in Hilbert spaces, we refer to Da. Prato [21].
Let A, generally unbounded, be a linear operator from H to H . Assume that {S(t), t ≥
0} is an analytic semigroup with its infinitesimal generator A, then it is possible ([20])
under some circumstances to define the fractional power (−A)α for any α ∈ (0, 1] which
is a closed linear operator with domain D((−A)α ); moreover, the subspace D((−A)α ) is
dense in H and the expression
kxkα = |(−A)α x|H ,
x ∈ D((−A)α ),
defines a norm on D((−A)α ).
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Asymptotic behavior for NSPDEs with infinite delays
In this work, we consider the following neutral SPDEs with infinite delays of the
form:

Z 0



d[x(t) + G(t, x(t − ρ(t)))] = [Ax(t) + b(t,



Z 0

+
h(t,
σ(θ, x(t + θ))dθ)dW (t),



−∞


x(t) = φ(t),
g(θ, x(t + θ))dθ)]dt
−∞
(2.1)
t ≥ 0,
t ≤ 0,
where the mappings
h : [0, +∞) × H → L02 (K, H),
G, b : [0, +∞) × H → H,
g, σ : (−∞, 0] × H → H
are all Borel measurable, ρ(t) : [0, +∞) → [0, r] is continuous, the initial value φ :
(−∞, 0] → H is an F0 -measurable, continuous function with E[sups≤0 |φ(s)|pH ] < ∞.
Definition 2.1. An H -valued stochastic process {x(t), t ∈ (−∞, T ]}, 0 ≤ T < ∞, is said
to be a mild solution to (2.1) if
RT
(a) x(t) is an Ft -adapted, continuous process with 0 |x(t)|pH dt < ∞ almost surely;
(b) for t ≥ 0, x(t) satisfies the following integral equation:
x(t) =S(t)[φ(0) + G(0, φ(−ρ(0)))] − G(t, x(t − ρ(t)))
Z t
−
AS(t − s)G(s, x(s − ρ(s)))ds
0
Z
t
0
Z
g(θ, x(s + θ))dθ ds+
S(t − s)b s,
+
−∞
0
0
Z
t
Z
S(t − s)h s,
+
σ(θ, x(s + θ))dθ dW (s),
−∞
0
and x(t) = φ(t) for t ≤ 0.
Definition 2.2. Let p ≥ 2 be an integer. The mild solution x(t) of (2.1) with an initial
value φ(t) is said to be exponentially stable in pth-moment if there exist some constants
Mα ≥ 1, η > 0 such that
E|x(t)|pH < Mα E sup |φ(θ)|pH e−ηt ,
t ≥ 0.
θ≤0
In particular, when p = 2, the mild solution is said to be exponentially stable in mean
square.
Definition 2.3. The mild solution x(t) of (2.1) is said to be almost surely exponentially
stable if there exists a positive constant α > 0 such that
lim sup
t→∞
ln |x(t)|H
≤ −α
t
a·s·.
In order to obtain our main result, we shall impose the following assumptions:
(A1 ) A : D(A) → H is the infinitesimal generator of a strong continuous semigroup of
bounded linear operators {S(t), t ≥ 0} in H satisfying
kS(t)k ≤ e−γt ,
f or some constant γ > 0.
(A2 ) For p ≥ 2, there exist some constants α ∈ ( p−1
p , 1] and KG > 0 such that for any
x, y ∈ H , t ≥ 0, G(t, x), G(t, y) ∈ D((−A)α ) and
|(−A)α G(t, x) − (−A)α G(t, y)|pH ≤ KG |x − y|pH .
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Asymptotic behavior for NSPDEs with infinite delays
(A3 ) b : [0, +∞) × H → H , h : [0, +∞) × H → L02 (K, H) satisfy Lipschitz conditions, i.e.,
there exist some positive constants Kb , Kh such that for any x, y ∈ H , t ∈ R,
|b(t, x) − b(t, y)|H ≤ Kb |x − y|H ,
|h(t, x) − h(t, y)|L02 ≤ Kh |x − y|H .
Moreover, we assume that |b(t, 0)|H = |h(t, 0)|L02 = 0.
(A4 ) There exist some positive constants Lg , Lσ such that for all t ∈ R, x, y ∈ H ,
|g(t, x) − g(t, y)|H ≤ Lg e−ξ|t| |x − y|H ,
0 < ξ < γ;
|σ(t, x) − σ(t, y)|H ≤ Lσ e−ξ|t| |x − y|H ,
0 < ξ < γ,
we further assume that |G(t, 0)|H = |σ(t, 0)|H = |g(t, 0)|H = 0.
Lemma 2.1 (Pazy [20]). Let the assumption (A1 ) hold. Then for any 0 < β ≤ 1, the
following equality holds:
S(t)(−A)β x = (−A)β S(t)x,
x ∈ D (−A)β ,
and there exists a positive constant Mβ such that for any t > 0,
k(−A)β S(t)k ≤ Mβ t−β e−γt .
3
The main theorem
In this section, we establish the existence and asymptotic behavior of the mild solution to (2.1). Our main object is to explain and prove the following theorem.
Theorem 3.1. Let p ≥ 2 be an integer. Suppose that (A1 ) − (A4 ) are satisfied. Suppose
also that
p
4p−1 [KG k(−A)−α kp + M1−α
KG γ −αp Γp (α)
p
+ Kbp Lpg (ξγ)−p + (2γ)− 2 Khp Lpσ ξ −p cp ] < 1,
(3.1)
where Γ(·) is the Gamma function, M1−α is the corresponding constant in Lemma 2.1
p
and cp =
p(p−1) 2
.
2
If an initial value φ(t) to (2.1) satisfies
E|φ(t)|pH ≤ M0 E|φ(0)|pH e−µt ,
t ≤ 0,
for some M0 ≥ 1 and 0 < µ < ξ , then the mild solution to (2.1) exists uniquely which is
exponentially stable in pth-moment.
In order to prove the theorem, we first recall a useful lemma.
Theorem A (Theorem 6.13 in Da Prato and Zabczyk [21]). Let r ≥ 1. Then for an
arbitrary L02 -valued predictable process Φ(t),
Z
sup E s∈[0,t]
0
s
2r
Z t
r
r
2r r1
Φ(u)dW (u) ≤ (r(2r − 1))
(E|Φ(s)|L0 ) ds .
2
0
H
Proof of Theorem 3.1. Denote by S the Banach space of all Ft -adapted continuous prop
cesses x(t) endowed with a norm kxkS := supt≥0 E|x(t)|H such that there exist some
?
constants M ≥ 1, η > 0 satisfying
E|x(t)|pH < M ? E sup |φ(θ)|pH e−ηt ,
t≥0
θ≤0
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Asymptotic behavior for NSPDEs with infinite delays
and x(t) = φ(t) for t ≤ 0, where φ(t) is the initial value of (2.1). Define a mapping
π : S → S by π(x)(t) = φ(t) for t ≤ 0 and
π(x)(t) =S(t) φ(0) + G(0, φ(−ρ(0))) − G(t, x(t − ρ(t)))
Z t
−
AS(t − s)G(s, x(s − ρ(s)))ds
0
t
Z
0
Z
−∞
0
0
t
Z
Z
σ(θ, x(s + θ))dθ dW (s)
S(t − s)h s,
+
(3.2)
g(θ, x(s + θ))dθ ds
S(t − s)b s,
+
−∞
0
for t ≥ 0.
We begin by verifying the continuity of (πx)(t) on t ≥ 0. To this end, let x ∈ S , t1 ≥ 0
and |r| > 0 be sufficiently small. Notice that
(πx)(t1 )|pH
E|(πx)(t1 + r) −
p−1
≤5
5
X
E|Ii (t1 + r) − Ii (t1 )|pH .
i=1
Applying Theorem A together with assumption (A1 ), it follows that
Z t1 +r
Z 0
E|I5 (t1 + r) − I5 (t1 )| = E S(t1 + r − s)h s,
σ(θ, x(s + θ))dθ dW (s)
0
−∞
p
Z t1
Z 0
−
S(t1 − s)h s,
σ(θ, x(s + θ))dθ dW (s)
p
−∞
0
≤2
p−1
(Z
t1
cp
H
Z
E (S(t1 + r − s) − S(t1 − s))h(s,
t1 +r
+
−∞
(
≤2
0
Z
E S(t1 + r − s)h(s,
t1
p−1
p
t1
Z
kS(r) − Ik
cp
Z
t1 +r
+
L2
2
0
σ(θ, x(s +
−∞
0
Z
E|h(s,
σ(θ, x(s +
−∞
t1
p2 )
p p2
σ(θ, x(s + θ))dθ) L0 ds
E|h(s,
0
Z
p2
p p2
σ(θ, x(s + θ))dθ) 0 ds
−∞
0
Z
0
θ))dθ)|pL0
2
p2
θ))dθ)|pL0
2
p2
p2
ds
p2 )
ds
.
Noting that for any s ∈ [0, T ], 0 ≤ T < ∞, we have
Z
E h s,
0
−∞
p
Z
σ(θ, x(s + θ))dθ ≤ Khp E
0
≤ Khp Lpσ
s
eξ(τ −s) dτ
p−1 Z
−∞
s
−∞
Z
|σ(θ, x(s + θ))|H dθ
−∞
L2
Z
p
0
eξ(τ −s) E|x(τ )|pH dτ
0
≤ Khp Lpσ ξ 1−p
eξ(τ −s) M0 E|φ(0)|pH e−µτ dτ
−∞
Z s
+
eξ(τ −s) M ? E sup |φ(θ)|pH e−ητ dτ
θ≤0
0
≤
Khp Lpσ ξ 1−p
M E supθ≤0 |φ(θ)|pH −ηs M0 E supθ≤0 |φ(θ)|pH −ξs
e
+
e
ξ−η
ξ−µ
?
≤ L,
ECP 18 (2013), paper 45.
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Asymptotic behavior for NSPDEs with infinite delays
where L is a positive constant. By the strong continuity of S(t), we get
E|I5 (t1 + r) − I5 (t1 )|pH → 0
as
r → 0.
p
Similarly, we can verify that E|Ii (t1 + r) − Ii (t1 )|H → 0 as r → 0, i = 1, · · ·, 4.
Next, we show that π(S) ⊂ S . Let x ∈ S . Without loss of generality, we assume that
0 < η < ξ . From the definition of π , we have
p
E|(πx)(t)|pH ≤ 5p−1 E S(t) φ(0) + G(0, φ) + 5p−1 E|G(t, x(t − ρ(t)))|pH
H
p
Z t
AS(t − s)G(s, x(s − ρ(s)))ds
+ 5p−1 E H
0
Z
Z t
S(t − s)
+ 5p−1 E 0
−∞
0
0
Z
Z t
S(t − s)
+ 5p−1 E −∞
0
5
X
≡ 5p−1
p
g(θ, x(s + θ))dθds
(3.3)
H
p
σ(θ, x(s + θ))dθdW (s)
H
Ii .
i=1
By assumption (A2 ), we obtain
I2 = E|G(t, x(t − ρ(t)))|pH
≤ k(−A)−α kp E|(−A)α [G(t, x(t − ρ(t))) − G(t, 0)]|pH
(3.4)
≤ KG k(−A)−α kp E|x(t − ρ(t))|pH
≤ KG k(−A)−α kp (M ? eηr E sup |φ(θ)|pH e−ηt + M0 eµr E|φ(0)|pH e−µt ).
θ≤0
Employing Lemma 2.1 and Hölder’s inequality, we get
Z
t
I3 = E|
AS(t − s)G(s, x(s − ρ(s)))ds|pH
0
Z t
≤ E[
|(−A)1−α S(t − s)(−A)α G(s, x(s − ρ(s)))|H ds]p
0
Z t
p
p
≤ M1−α
E
(t − s)α−1 e−γ(t−s) |(−A)α G(s, x(s − ρ(s)))|H ds
0
≤
p
M1−α
Z
t
α−1 −γ(t−s)
(t − s)
Z
e
p−1
ds
0
t
·
0
(t − s)α−1 e−γ(t−s) E|(−A)α G(s, x(s − ρ(s)))|pH ds,
combining this with the assumption (A2 ), we further deduce that
p
KG γ −α(p−1) Γp−1 (α)
I3 ≤ M1−α
t
Z
(t − s)α−1 e−γ(t−s) E|x(s − ρ(s))|pH ds
0
≤
p
M1−α
KG γ −α(p−1) Γp−1
t
Z
(α)
0
+ M0 e
µr
(t − s)α−1 e−γ(t−s) [M ? eηr E sup |φ(θ)|pH e−ηs
θ≤0
E|φ(0)|pH e−µs ]ds
(3.5)
? ηr
M e
p
KG γ −α(p−1) Γp (α)
≤ M1−α
M0 eµr E|φ(0)|pH −µt
+
e
.
(γ − µ)α
E supθ≤0 |φ(θ)|pH −ηt
e
(γ − η)α
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Asymptotic behavior for NSPDEs with infinite delays
As for I4 , an application of Hölder’s inequality and assumption (A3 ), we have
t
Z
S(t − s)b s,
I4 ≤ E
Z
0
≤ Kbp E
≤ Kbp
≤
≤
t
Z
t
−∞
0
−∞
0
Z
Z
e−γ(t−s) 0
p
g(θ, x(s + θ))dθ H ds
p
g(θ, x(s + θ))dθ ds
H
p−1 Z
e−γ(t−s) ds
E
0
t
Kbp γ 1−p E
s
γ(t−s)
− p
Lg e
p
g(θ, x(s + θ))dθ ds
H
eξ(τ −s) |x(τ )|H dτ
p
ds
−∞
0
Z t "Z
Kbp γ 1−p Lpg
0
−∞
0
Z t Z
Z
− γ(t−s)
e
p
p−1 Z
s
e
ξ(τ −s)
e
dτ
−γ(t−s) ξ(τ −s)
e
−∞
−∞
0
#
s
E|x(τ )|pH dτ
ds
Z t Z 0
e−γ(t−s) eξ(τ −s) M0 E|φ(0)|pH e−µτ dτ
≤ Kbp γ 1−p Lpg ξ 1−p
−∞
0
Z s
p −ητ
−γ(t−s) ξ(τ −s)
?
)dτ ds
+
e
e
M E sup |φ(θ)|H e
(3.6)
θ≤0
0
≤ Kbp γ 1−p Lpg ξ 1−p
M ? E supθ≤0 |φ(θ)|pH −ηt
M0 E|φ(0)|pH −ξt
e
+
e
.
(γ − η)(ξ − η)
(γ − ξ)(ξ − µ)
Taking into account Theorem A and assumption (A3 ), we obtain that
p
Z t
Z 0
S(t − s)h s,
σ(θ, x(s + θ))dθ dW (s)
I5 = E 0
−∞
H

p
"
p # p2  2
Z 0
Z t
σ(θ, x(s + θ))dθ ≤ cp
E S(t − s)h s,
ds
 0

−∞
L0
2
p
" Z
p # p2  2
t
0
≤ Khp cp
ds
,
e−2γ(t−s) E σ(θ, x(s + θ))dθ
 0

−∞
H

Z
where cp = (
I5 ≤
p(p−1) p
)2 ,
2
Khp cp Lpσ
≤ Khp cp Lpσ
(Z
by assumption (A3 ) and Hölder’s inequality, we get
t
e
−2γ(t−s)
t
e−2γ(t−s)
Z
|x(τ )|H dτ )
s
eξ(τ −s) dτ
p−1 Z
−∞
0
≤ Khp cp Lpσ ξ 1−p
e
ξ(τ −s)
p
) p2
p2
ds
−∞
0
(Z
s
Z
E(
(Z
0
t
e−2γ(t−s)
Z
s
−∞
eξ(τ −s) E|x(τ )|pH dτ
s
−∞
eξ(τ −s) E|x(τ )|pH dτ
ECP 18 (2013), paper 45.
p2
p2
) p2
ds
) p2
ds
.
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Asymptotic behavior for NSPDEs with infinite delays
Noting that p ≥ 2, we then have
I5 ≤
Khp cp Lpσ ξ 1−p
e
−2γ(t−s)
s
Z
0
eξ(τ −s) M ? E sup |φ(θ)|pH e−ητ )dτ
θ≤0
0
0
Z
+
−∞
≤
t
Z
eξ(τ −s) M0 E|φ(0)|pH e−µτ dτ
Khp cp Lpσ ξ 1−p
(Z
t
e
−2γ(t−s)
) p2
p2
ds
"
0
M ? E supθ≤0 |φ(θ)|pH −ηs
e
ξ−η
) p2
p2
(3.7)
2 #
M0 E|φ(0)|pH −ξs p
ds
e
+
ξ−µ
"
p2
M ? E supθ≤0 |φ(θ)|pH
p
p
p 1−p p−2
≤ Kh cp Lσ ξ
2 2
e−ηt
ξ−η
2pγ − 2η
#
p2
M0 E|φ(0)|pH
p
−ξt
+
e
.
ξ−µ
2pγ − 2ξ
Recalling (3.3), from (3.4) to (3.7), we can deduce that there exist M1 ≥ 1 such that
E|(πx)(t)|pH ≤ M1 E sup |φ(θ)|pH e−ηt .
θ≤0
Since the Ft -measurability of (πx)(t) is easily verified, it follows that π is well defined.
Thus, we conclude that πS ⊂ S .
It remains to show that π has a unique fixed point. For any x, y ∈ S , we have
E|(πx)(t) −
(πy)(t)|pH
p−1
≤4
4
X
Ji .
(3.8)
i=1
We now estimate each Ji in (3.8). Noting that x(s) = y(s) = φ(s) for s ≤ 0, by assumption
(A2 ), we have
J1 = E|G(t, x(t − ρ(t))) − G(t, y(t − ρ(t)))|pH
≤ KG k(−A)−α kp E|x(t − ρ(t)) − y(t − ρ(t))|pH
≤ KG k(−A)−α kp sup E|x(s) − y(s)|pH .
s≥0
By standard computations involving Lemma 2.1 and Hölder’s inequality, we get
p
Z t
J2 = E AS(t − s)[G(s, x(s − ρ(s))) − G(s, y(s − ρ(s)))]ds
0
H
p
≤E
AS(t − s) G(s, x(s − ρ(s))) − G(s, y(s − ρ(s))) H ds
0
Z t
p
≤ M1−α E
(t − s)α−1 e−γ(t−s) |(−A)−α G(s, x(s − ρ(s)))
0
p
− G(s, y(s − ρ(s))) |H ds
Z
≤
t
p
M1−α
KG
Z
t
α−1 −γ(t−s)
(t − s)
e
p−1
ds
×
0
Z
≤
t
(t − s)α−1 e−γ(t−s) E|x(s − ρ(s)) − y(s − ρ(s))|pH ds
0
p
M1−α KG γ −αp Γp (α) sup E|x(s)
s≥0
− y(s)|pH .
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Asymptotic behavior for NSPDEs with infinite delays
A similar argument as before yields
p
g(θ, x(s + θ))dθ) − b(s, g(θ, y(s + θ))dθ ds
−∞
H
p
Z 0
0
g(θ, y(s + θ))dθ ds
g(θ, x(s + θ))dθ −
Z t
Z
S(t − s) b(s,
J3 = E 0
t
0
Z
e
≤
−∞
0
p−1
Z t
e−γ(t−s) ds
·
≤ Kbp
Kbp E
Z
−γ(t−s)
−∞
0
0
Z t Z
|e
Z t Z
Kbp γ 1−p Lpg
ds
p−1
s
e
ξ(τ −s)
·
dτ
−∞
0
s
Z
e
≤
p
g(θ, x(s + θ)) − g(θ, y(s + θ))|H dθ
−∞
0
≤
−γ(t−s)
p
−γ(t−s) ξ(τ −s)
e
−∞
p −p p −p
Kb γ Lg ξ
E|x(τ ) −
y(τ )|pH dτ
ds
sup E|x(s) − y(s)|pH ,
s≥0
and
p
Z t
Z 0
S(t − s) h(s,
σ(θ, x(s + θ))dθ − h(s, σ(θ, y(s + θ))dθ dW (s)
J4 = E 0
−∞
H
Z t
p
−2γ(t−s)
≤ cp Kh
e
·
0
p
p # p2  2
(σ(θ, x(s + θ)) − σ(θ, y(s + θ)))dθ
ds

−∞
H
" Z
E ≤ cp Khp Lpσ
≤
cp Khp Lpσ
"Z
0
(Z
t
Z
−2γ(t−s)
e
E
≤
eξ(τ −s) |x(τ ) − y(τ )|H dτ
) p2
p p2
ds
−∞
0
t
Z
e−2γ(t−s) ·
0
s
eξ(τ −s) dτ
p−1 Z
−∞
≤
s
cp Khp Lpσ ξ −p
# p2
s
eξ(τ −s) E|x(τ ) − y(τ )|H dτ
−∞
t
Z
e
−2γ(t−s)
p2
ds sup E|x(s) − y(s)|pH
s≥0
0
p
cp Khp Lpσ ξ −p (2γ)− 2
 p2

ds

sup E|x(s) − y(s)|pH .
s≥0
Consequently, we have
p
sup E|(πx)(t) − (πy)(t)|pH ≤ 4p−1 [KG k(−A)−α kp + M1−α
KG γ −αp Γp (α)
s≥0
p
+ Kbp Lpg (ξγ)−p + (2γ)− 2 Khp Lpσ ξ −p cp ] sup E|x(s) − y(s)|pH ,
s≥0
by (3.1) it follows that π is contractive. Thus, Banach fixed point principle implies that
there exists a unique x(t) ∈ S solves (2.1) with x(s) = φ(s) on (−∞, 0], moreover, x(t) is
exponentially stable in pth-moment. The proof is completed.
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Asymptotic behavior for NSPDEs with infinite delays
Corollary 3.2. Under the conditions of Theorem 3.1 with p = 2, the mild solution
of (2.1) exists uniquely which is exponentially stable in mean square.
Theorem 3.3. Suppose that all the conditions of Theorem 3.1 hold. Then the mild
solution of (2.1) is almost surely exponentially stable.
Proof. The proof is similar to [8], we omit it here.
At the end of this paper, we give an example to illustrate our results.
Example 3.4. Consider the following neutral stochastic partial differential equation
with infinite delays:
 x(t − ρ(t), u)
∂2
α1


x(t, u)dt
·
=
d
x(t,
u)
+

α

M1−α k(−A) k 1 + |x(t − ρ(t), u)|
∂x2


Z 0




+ f t,
α eξθ x(t + θ, u)dθ dt
1










2
−∞
0
Z
+ f2 t,
(3.9)
α3 eξθ x(t + θ, u)dθ dw(t),
−∞
2
t≥ 0;
u ∈ [0, π], t ≤ 0,
x(s, u) = φ(s, u) ∈ L [0, π], x(t, 0) = x(t, π) = 0,
where ξ, αi > 0, i = 1, 2, 3, M1−α > 0, α ∈ ( 12 , 1], w(t) denotes the standard cylindrical
Wiener process.
We now rewrite (3.9) into the abstract form of (2.1). Let H = L2 (0, π). Define
2
∂
A : H → H by −A = ∂x
2 with domain
n
o
0
00
D(−A) = x, x are absolutely continuous, x ∈ H and x(0) = x(π) = 0 .
Then (−A)x =
∞
P
n2 hx, en ien , x ∈ D(−A), where en (ξ) =
n=1
q
2
n
sin nξ , n = 1, 2, · · · is the
set of eigenvector of −A. It is well known that A is the infinitesimal generator of an
analytic semigroup S(t), t ≥ 0, in H and
S(t)x =
∞
X
2
e−n t hx, en ien ,
x ∈ H,
n=1
moreover, kS(t)k ≤ e−t , t ≥ 0.
Let
α1
x(t − ρ(t), u)
·
,
α
M1−α k(−A) k 1 + |x(t − ρ(t), u)|
Z 0
ξθ
g(θ, x(t + θ))dθ) = f1 t,
α2 e x(t + θ, u)dθ ,
G(t, x(t − ρ(t))) =
0
Z
b(t,
−∞
Z
0
h(t,
−∞
Z
σ(θ, x(t + θ))dθ) = f2 t,
−∞
0
ξθ
α3 e x(t + θ, u)dθ .
−∞
We assume that there exist some positive constants Kfi , i = 1, 2 such that for any
x, y ∈ H , t ≥ 0,
|f1 (t, x) − f1 (t, y)|H ≤ Kf1 |x − y|H , |f2 (t, x) − f2 (t, y)|L20 ≤ Kf2 |x − y|H ,
It is obvious that the assumptions (A1 ) − (A4 ) are satisfied with
γ = 1, KG =
α12
, Kb = Kf1 , Kh = Kf2 , Lg = α2 , Lσ = α3 .
2
M1−α
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Asymptotic behavior for NSPDEs with infinite delays
From the definition of (−A)−α (see page 70 of [20]),
k(−A)−α k ≤
1
Γ(α)
Z
∞
uα−1 kS(u)kdu ≤
0
1
.
π 2α
Thus, by Theorem 3.1 and Theorem 3.3, if E|φ(s)|2 ≤ M0 E|φ(0)|2 e−µs for s ≤ 0, where
M0 ≥ 1, 0 < µ < ξ , the mild solution of (3.9) exists uniquely and it is exponentially
stable in mean square and almost surely exponentially stable provided that
1 2 2 −2
α12
2 2
2 2 −2
+ α1 Γ (α)Kf1 α2 ξ + Kf2 α3 ξ
4
< 1.
(π 2α M1−α )2
2
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Acknowledgments. The authors would like to express their sincere gratitude to the
associate editor and the anonymous referees for their valuable comments.
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