On a jump-type stochastic fractional partial differential equation with fractional noises

Nonlinear Analysis 75 (2012) 6060–6070

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On a jump-type stochastic fractional partial differential equation with fractional noises✩ Junfeng Liu a,∗ , Litan Yan b , Yuquan Cang a a

School of Mathematics and Statistics, Nanjing Audit University, 86 West Yushan Rd., Pukou, Nanjing 211815, PR China

b

Department of Mathematics, Donghua University, 2999 North Renmin Rd., Songjiang, Shanghai 201620, PR China

article

info

Article history: Received 15 March 2012 Accepted 11 June 2012 Communicated by Enzo Mitidieri Keywords: Stochastic fractional partial differential equation Fractional derivative operator Poisson measure Fractional noise

abstract We study a class of stochastic fractional partial differential equations of order α > 1 driven by a (pure jump) Lévy space–time white noise and a fractional noise. We prove the existence and uniqueness of the global mild solution by the fixed point principle under some suitable assumptions. © 2012 Elsevier Ltd. All rights reserved.

1. Introduction In the past two decades, fractional calculus (see, for example, [1] and references therein) has attracted many physicists, mathematicians and engineers, and notable contributions have been made to both theory and applications of fractional (partial) differential equations. Fractional (partial) differential equations draw a great application in many physical phenomena such as seepage flow in porous media and in fluid dynamic traffic models. The most important advantage of using fractional (partial) differential equations in these and other applications is their nonlocal property. This means that the next state of a system depends not only upon its current state but also upon all of its historical states. This is probably the most relevant feature for making this fractional tool useful from an applied standpoint and interesting from a mathematical standpoint and in turn led to the sustained study of the theory of fractional (partial) differential equations. Besides, noise or stochastic perturbation is unavoidable and omnipresent in nature as well as in man-made systems. Therefore it is of great significance to import the stochastic effects into the investigation of fractional (partial) differential systems. However few publications treat stochastic partial differential equations involving fractional derivatives. In fact, these kinds of equations can be widely used in physics, fractal medium, quantum fields, risk management, among other areas (cf. [2,3] for a survey of applications). Most of them investigate evolution type equations driven by a fractional power of the Laplacian. These operators generate symmetric stable semigroups when the order of derivative is less than 2. Mueller [4] and Wu [5] proved the existence of a solution of the stochastic fractional heat, respectively Burgers, equation perturbed by a stable noise. The work of Bonaccorsi and Tubaro [6] can be applied to stochastic evolution equations with fractional time derivatives. Cui and Yan [7] studied the existence of mild solutions for a class of fractional neutral stochastic integrodifferential equations with infinite delay in Hilbert spaces. In [8], Dalang and Mueller have studied the stochastic equation

✩ The Project—sponsored by NSFC (11171062), NSFC (81001288), NSRC (10023), Innovation Program of Shanghai Municipal Education Commission (12ZZ063) and Major Program of Key Research Center in Financial Risk Management of Jiangsu Universities Philosophy Social Sciences (No: 2012JDXM009). ∗ Corresponding author. E-mail address: [email protected] (J. Liu).

0362-546X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2012.06.012

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which is of second order in time, driven by a power of the Laplacian and perturbed by colored noises (white in time and homogeneous in space). In probabilistic terms, replacing the Laplacian by its fractional power (which is an integrodifferential operator) leads to interesting and largely open questions of extensions of results for Brownian motion driven stochastic equations to those driven by Lévy stable processes. In the physical literature, such fractal anomalous diffusions have been recently enthusiastically embraced by a slew of investigators in the context of hydrodynamics, acoustics, trapping effects in surface diffusion, statistical mechanics, relaxation phenomena, and biology. On the other hand, there has been some recent interest in studying stochastic partial differential equations driven by a fractional noise. Linear stochastic evolution equations in a Hilbert space driven by an additive cylindrical fractional Brownian motion with Hurst parameter H were studied by Duncan et al. [9] in the case H ∈ (1/2, 1) and by Tindel et al. [10] in the general case, where they provided necessary and sufficient conditions for the existence and uniqueness of an evolution solution. Hu and Nualart [11] studied the d-dimensional stochastic heat equation with a multiplicative Gaussian noise which is white in space and has the covariance of a fractional Brownian motion with Hurst parameter H ∈ (0, 1) in time. First they considered the equation in the Itô–Skorohod sense, and later in the Stratonovich sense. An explicit chaos expansion for the solution was obtained. Moreover, the moments of the solution are expressed in terms of the exponential moments of some weighted intersection local time of the Brownian motion. In the meanwhile, there also have been some works on stochastic partial differential equations involving Lévy space–time white noises (e.g. [12,13,4,14,5,15,16]). In [12,15,16], the authors established the existence and uniqueness of the solutions for a class of stochastic heat equations driven by compensated Poisson random measures and Lévy space–time white noises in the L2 -sense, respectively. Løkka et al. [13] studied the stochastic partial differential equations driven by a d-parameter (pure jump) Lévy white noise. As an example they used this theory to solve the stochastic Poisson equation with respect to the Lévy white noise for any dimension d. Shi and Wang [14] studied the existence and uniqueness of the global mild solution for a stochastic fractional partial differential equation driven by the Lévy space–time white noise. Moreover the flow property for the solution was also studied. We notice that the fixed point principle and the Picard iteration scheme work in [12,15,16], since the Burkhölder–Davis–Gundy (abbr. B–D–G) inequality can be applied to estimate the stochastic integral with respect to the compensated Poisson random measure in the L2 -sense. Unfortunately, the usual B–D–G inequality cannot work in estimating the stochastic integral with respect to the compensated Poisson random measure in the Lp -sense (p > 2). Hence a new version of the B–D–G inequality will be adopted for Lp (p ≥ 2)-estimates on the solution to Eq. (1.1) (see [17]). Motivated by the above results, in this paper, we will study the following jump type stochastic fractional partial differential equation with fractional noises:

  ∂u = Dαδ u + f (t , x, u) + σ (t , x, u)L˙ + B˙ H , ∂t  u(0, ·) = u0 (·),

in [0, T ] × R,

(1.1)

where Dαδ is the fractional differential operator with respect to the spatial variable, to be defined in the Appendix which was recently introduced by Debbi [18] and Debbi and Dozzi [19], B˙ H denotes the fractional noise on [0, T ] × R with Hurst index H > 1/2 defined on a complete probability space (Ω , F , P ), and L˙ is a (pure jump) Lévy space–time white noise on [0, T ] × R defined on a complete probability space (Ω , F , P ) (see Section 2 for precise definitions). Actually, we understand this equation as Walsh [20] sense, and so we can rewrite Eq. (1.1) as follows: u( t , x ) =



Gα (t , x − y)u0 (y)dy +

 t 0

R

 t

Gα (t − s, x − y)f (s, y, u(s, y))dyds

+ 0

R

 t + 0

Gα (t − s, x − y)BH (ds, dy)

R

Gα (t − s, x − y)σ (s, y, u(s, y))L˙ (y, s)dyds,

(1.2)

R

for all t ∈ [0, T ] and x ∈ R, where Gα (·, ∗) denotes the Green function associated to Eq. (1.1). The main subject of this paper is to establish the existence and uniqueness of the solution of Eq. (1.2) via the fixed point principle. The rest of the paper is organized as follows. In Section 2, we give the definitions of the fractional noises, Lévy space–time white noises and a generalized B–D–G inequality. Section 3 is devoted to proving the existence and uniqueness of the mild solution to Eq. (1.2) in the Lp (p ≥ 2) sense under some appropriate conditions. Most of the estimates in this paper contain unspecified constants. An unspecified positive and finite constant will be denoted by C , which may not be the same in each occurrence. Sometimes we shall emphasize the dependence of these constants upon parameters. 2. Preliminaries In this section, we will present the definitions of the fractional noises, Lévy space–time white noises and a generalized B–D–G inequality.

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2.1. Fractional noises Let (Ω , F , (Ft )t ≥0 , P ) be a complete probability space equipped (Ft )t ≥0 satisfying the usual  with the filtration  conditions. Let Bb (R) denote the class of bounded Borel sets in R. BH ([0, t ] × A) (t ,A)∈[0,T ]×B (R) is a centered Gaussian b

family of random variables with the covariance, for H ∈ (0, 1)

E BH ([0, t ] × A)BH ([0, s] × B) = |A ∩ B|RH (t , s),





(2.1)

with s, t ∈ [0, T ], A, B ∈ Bb (R) and covariance kernel RH (t , s) = 12 (t 2H + s2H − |t − s|2H ). Here |A| denotes the Lebesgue measure of the set A ∈ Bb (R). We denote by E the set of step functions on [0, T ] × R. Let H be the Hilbert space defined as the closure of E with respect to the scalar product



1[0,t ]×A , 1[0,s]×B



= |A ∩ B|RH (t , s).

H

Thus the mapping 1[0,t ]×A → BH ([0, t ] × A) is an isometry between E and the linear space span of BH ([0, t ] × A), A ∈  Bb (R), t ∈ [0, T ] . Moreover, the mapping can be extended to an isometry from H to the Gaussian space associated with BH . This isometry will be denoted by ϕ → BH (ϕ) for ϕ ∈ H . Therefore, we can regard BH (ϕ) as the stochastic integral with respect to BH . In general, we use the notation



BH (ϕ) =

T

 0



ϕ(s, y)BH (ds, dy),

ϕ ∈H.

R

On the other hand, it is known that the covariance kernel RH (t , s) satisfies R H ( t , s) =

t ∧s



KH (t , r )KH (s, r )dr , 0

where KH (t , s) is the square kernel defined, for 0 < s < t, by KH (t , s) = cH

  1 H− 2 t

s

( t − s)



H − 12

− H−

1 2

 s

1 −H 2



t



u

H − 32

( u − s)

H − 12

du ,

(2.2)

s

where cH2 = (1−2H )β(12H (β(·, ·) denotes the Beta function). In particular, for H > 1/2, −2H ,H +1/2) RH (t , s) = H (2H − 1)

 t 0

s

|u − v|2H −2 dudv. 0

Define a linear operator KH∗ : E → L2 ([0, T ] × R) by

(KH∗ ψ)(s, x) = KH (T , s)ψ(s, x) +



T

(ψ(t , x) − ψ(s, x))

s

∂ KH (t , s)dt . ∂t

Then the operator KH ψ gives an isometry from H to L2 ([0, T ] × R). Consequently, ∗

W (t , A) = BH (KH∗ )−1 (1[0,t ]×A ) ,





(t , A) ∈ [0, T ] × Bb (R),

defines a space–time white noise. Moreover one can regard BH as B ([0, t ] × A) = H

 t 0

KH (t , s)W (ds, dy). A

2.2. Lévy space–time white noises Let (Ei , Ei , µi ), i = 1, 2 be two σ -finite measurable spaces. We call N : (E1 , E1 , µ1 ) × (E2 , E2 , µ2 ) × (Ω , F , P ) → N ∪ {0} ∪ {∞} a Poisson noise on (E1 , E1 , µ1 ), if for all A ∈ E1 , B ∈ E2 and n ∈ N ∪ {0} ∪ {∞}, e−µ1 (A)µ2 (B)[µ1 (A)µ2 (B)]

n

P (N (A, B) = n) =

n!

.

(2.3)

In particular, if (E1 , E1 , µ1 ) = ([0, ∞) × R, B ([0, ∞) × R), dt × dx), then define a compensated random martingale measure by M (B, A, t ) = N ([0, t ] × A, B) − µ1 ([0, t ] × A)µ2 (B),

(2.4)

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by assuming that µ1 ([0, t ] × A)µ2 (B) < ∞ for all (t , A, B) ∈ [0, ∞) × B (R) × E2 . Further, let φ : E1 × E2 × Ω → R be an (Ft )t ≥0 -predictable function satisfying

 t  

 |φ(t , x, y)|2 µ2 (dy)dxds < ∞,

E 0

A

(2.5)

B

for all t > 0 and (A, B) ∈ E1 × E2 . We can define a stochastic integral process



t+



 

φ(s, x, y)M (dy, dx, ds)

Rt = 0

A

 (2.6)

B

which is a square integrable (P , (Ft )t ≥0 )-martingale. It is well-known that, a (pure jump) Lévy space–time white noise possesses the following structure: L˙ (x, t ) =



˙ (dy, x, t ) + h1 (t , x, y)M



h2 (t , x, y)N˙ (dy, x, t )

(2.7)

E2 \U0

U0

for some U0 ∈ E2 such that µ2 (E2 \ U0 ) < ∞ and

 U0

z 2 µ2 (dz ) < +∞. Here h1 , h2 : [0, ∞) × R × E2 → R are some

˙ and N˙ are the Radon–Nikodym derivatives given by measurable functions; M M (dy, dx, dt )

˙ (dy, x, t ) = M

dt × dx

,

N˙ (dy, x, t ) =

N (dt × dx, dy) dt × dx

,

(2.8)

with (t , x, y) ∈ [0, ∞) × R × E2 . 2.3. A generalized B–D–G inequality In order to estimate the higher order moments of the mild solution to (1.2), we need the following B–D–G inequality (see e.g. [17]). Proposition 2.1. Let φ : [0, ∞) × R × E2 × Ω → R be (Ft )t ≥0 -predictable and satisfy (2.5). Define the integral process by



t+



 φ(s, y, z )M (dz , dy, ds), t ≥ 0 .

 

Xt = 0

E2

R

Then for any T > 0 and p > 1, there exists a constant Cp,T > 0 such that sup E |Xt |p ≤ Cp,T





T



t ∈[0,T ]

0

 2p 2 (E[|φ(s, y, z )|p ]) p µ2 (dz )dyds .

  R

(2.9)

E2

3. Existence and uniqueness In this section, we shall prove the existence and uniqueness of the global mild solution to (1.1). Recall (1.2) and (2.7). Then for all (t , x) ∈ [0, T ] × R, u( t , x ) =



Gα (t , x − y)u0 (y)dy +

 t 0

R

 t

Gα (t − s, x − y)f (s, y, u(s, y))dyds

+ 0

R

 t

Gα (t − s, x − y)σ (s, y, u(s, y))ψ(s, y)dyds

+ 0

R

t+



 

Gα (t − s, x − y)σ (s, y, u(s, y))h(s, y, z )M (dz , dy, ds),

+ 0

Gα (t − s, x − y)BH (ds, dy)

R

R

(3.1)

E2

with the mappings

ψ(t , y) =



h2 (t , y, z )µ2 (dz ),

(3.2)

h(t , y, z ) = h1 (t , y, z )1U0 (z ) + h2 (t , y, z )1E2 \U0 (z ),

(3.3)

E2 \U0

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J. Liu et al. / Nonlinear Analysis 75 (2012) 6060–6070

with indicator 1A (·) of the set A ∈ E2 . On the other hand, as in Section 2, the fractional integral term in (3.1) can be represented as

 t 0

Gα (t − s, x − y)BH (ds, dy) =

 t 0

R

(KH∗ Gα )(t − ·, x − ·)W (ds, dy),

(3.4)

R

with the space–time white noise (W (t , x))(t ,x)∈[0,T ]×R mentioned in Section 2. In what follows, we will show that such a mild solution indeed exists and is unique. First, we state the main result of this section. Theorem 3.1. Assume that the following conditions are satisfied. (1) f , σ are uniformly Lipschitzian, i.e. there exists a constant C > 0 such that for (t , x) ∈ [0, T ] × R and u, v ∈ R

|f (t , x, u) − f (t , x, v)| + |σ (t , x, u) − σ (t , x, v)| ≤ C |u − v|.

(3.5)

(2) f has linear  growth onR, i.e. there exists a constant C > 0 such that |f (t , x, u)| ≤ C (1 + |u|), for all u ∈ R. 2(α+1) α−1

(3) For p ∈

, +∞ with α > 1,

sup ∥ψ(t , ·)∥pp < ∞,

(3.6)

t ∈[0,T ]

 p  2 2  sup  |h(t , ·, z )| µ2 (dz )  p < ∞. t ∈[0,T ] E2

(3.7)

2

p

Then, for all F0 -measurable u0 : R × Ω → R satisfying E ∥u0 (·)∥p



(u(t , x))(t ,x)∈[0,T ]×R to Eq. (1.1) and for all p ∈   sup E ∥u(t , ·)∥pp < ∞.



2(α+1) α−1



< ∞, there exists a unique mild solution



, +∞ ,

t ∈[0,T ]

To prove the theorem, we need the following lemmas. Lemma 3.1. Let p ∈ [1, ∞), ρ ∈ [1, p] and r ∈ [1, ∞) such that 1 r

=

1 p

−

1

ρ

+ 1 ∈ [0, 1].

Let Gα = Gα (t , x − y) be the Green kernel, H = Gα , or G2α with (t , x, y) ∈ [0, T ] × R × R. Define an operator J by J (v)(t , x) =

 t 0

H (t − s, x − y)v(s, y)dyds,

(3.8)

R

with v ∈ L1 ([0, T ]; Lρ ()). Then J : L1 ([0, T ]; Lρ ()) → L∞ ([0, T ]; Lρ ()) is a bounded linear operator and satisfies the following. (1) If H = Gα , then there exists a constant C > 0 such that for all r ∈ [1, 1 + α)

∥J (v)(t , ·)∥p ≤ C

t



( t − s) −

1−r

α

∥v(x, ·)∥ρ ds,

∀t ∈ [0, T ].

(3.9)

0

(2) If H = G2α , then there exists a constant C > 0 such that for all r ∈ [1, 1+α ) 2

∥J (v)(t , ·)∥p ≤ C

t



(t − s)−

2−r

α

∥v(x, ·)∥ρ ds,

∀t ∈ [0, T ].

(3.10)

0

Proof. We only prove case (1), since the proof of (2) is similar. From Minkowski’s inequality, (6) of Lemma A.1 and the Young inequality, it follows that

 t      ∥J (v)(t , ·)∥p =  G ( t − s , · − y )v( s , y ) dyds α   0 R p   t     Gα (t − s, · − y)v(s, y)dy ds ≤   0 R p    t    1  −α  − α1 ≤C (t − s)  Gα 1, (t − s) (· − y) |v(s, y)|dy  ds 0

R

p

J. Liu et al. / Nonlinear Analysis 75 (2012) 6060–6070 t



1

     1    Gα 1, (t − s)− α · ∗ |v(s, ·)| (·) ds

1

   1   Gα 1, (t − s)− α ·  · ∥v(s, ·)∥ρ ds

(t − s)− α

≤C 0 t



(t − s)− α

≤C 0 t



(t − s)−

≤C

6065

p

r

1−r

α

∥v(s, ·)∥ρ ds

(3.11)

0

where we have used the fact that, for r ∈ [1, 1 + α)

      1r 1 1   Grα 1, (t − s)− α y dy Gα (1, (t − s)− α ·) = r

R

r

≤ (t − s) α



Grα (1, z )dz

 1r

r

≤ C (t − s) α .

R

So we complete the proof of this lemma.



The following embedding proposition (see [21]) is useful for our derivations below. Lemma 3.2. If H >

1 , 2

then

1

L H ([0, T ] × R) ⊂ H .

(3.12)

We mainly adopt the fixed point principle to prove Theorem 3.1. Let B be the space of all Lp (R)-valued Ft -adapted processes u(t , ·)0≤t ≤T : [0, T ] × R × Ω → R with the norm

 ∥u∥B :=

sup e−ηt E ∥u(t , ·)∥pp



1  p

,

η > 0,

(3.13)

0≤t ≤T

with ∥ · ∥p the usual norm of Lp (R). Then (B, ∥ · ∥B ) forms a Banach space. Further, for u ∈ B, let us define an operator Aα by (as follows)

Aα (t , x) =

5 

Aiα (u)(t , x),

(3.14)

i=1

where

Aα (u)(t , x) = 1



Gα (t , x − y)u0 (y)dy, R

A2α (u)(t , x) = A3α (u)(t , x) = A4α (u)(t , x) = A5α (u)(t , x) =

 t 0

Gα (t − s, x − y)f (s, y, u(s, y))dyds, R

 t 0

Gα (t − s, x − y)BH (dy, ds), R

 t 0

Gα (t − s, x − y)σ (s, y, u(s, y))ψ(s, y)dyds, R

 t  0

R

Gα (t − s, x − y)σ (s, y, u(s, y))h(s, y, z )M (dz , dy, ds). E2

According to (3.14), we have the following. 2(α+1)

Proposition 3.1. Under the assumptions of Theorem 3.1, for each p > α+1 and u ∈ B, it holds that Aα (u) ∈ B. Proof. Applying (6) of Lemma A.1, Corollary A.2 and the Young inequality, we conclude that

     ∥Aα (u)(t , x)∥p =  Gα (t , x − y)u0 (y)dy  R p      1  1 −α  −α ≤ t  Gα 1, t (· − y) u0 (y)dy  1

R

p

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J. Liu et al. / Nonlinear Analysis 75 (2012) 6060–6070 1

≤ t− α 1

≤ t− α

     1    Gα 1, t − α · ∗ u0 (·) (·) p     − α1  Gα 1, t ·  · ∥u0 (·)∥p 1

≤ C ∥u0 (·)∥p < ∞,   which is due to the fact that E ∥u0 (·)∥pp < ∞. Now we turn to A2α (u). From (1) of Lemma 3.1 with

(3.15) 1 r

=

1 p

− 1p + 1 = 1 and

condition (2) of Theorem 3.1, it follows that

E ∥Aα (u)(t , x)∥



2

p p



t



( t − s)

≤ CE

r − 1− α

p

∥f (s, ·, u(s, ·))∥p ds

0

p

t





≤ CE

1 + ∥u(s, ·)∥p ds



0



≤ Cp,T 1 + sup E ∥u(s, ·)∥pp



0≤t ≤T

 ≤ Cp,T 1 + ∥u(s, ·)∥pB < ∞, 

(3.16)

since u ∈ B. In what follows, let us consider A3α (u)(t , x). We deduce that

p 

 t  



E∥A3α (u)(t , x)∥pp =

R

E 

0

R

Gα (t − s, x − y)BH (dy, ds) dx

 t  p   = E  (KH∗ Gα (t − ·, x − ·))(s, y)W (dy, ds) dx R 0 R   ∗ p ≤ Cp KH Gα (t − ·, x − ·), KH∗ Gα (t − ·, x − ·) L22 ([0,T ]×R) dx 

R



p

2 ⟨Gα (t − ·, x − ·), Gα (t − ·, x − ·)⟩H dx

= Cp R



∥Gα (t − ·, x − ·)∥p 1

≤ Cp

L H ([0,T ]×R)

R

dx

(3.17)

1

where we have used the fact that L H ([0, T ] × R) ⊂ H when H > 1/2. Note that

∥Gα (t − ·, x − ·)∥

T



p 1



|Gα (t − s, x − y)| dyds

=

L H ([0,T ]×R)

0

R

 t 

1 H

|Gα (t − s, x − y)| dyds

= 0

− α1H

( t − s)

= 0

pH     H1   − α1H (x − y)  dyds Gα 1, (t − s) R

t



pH

R t



pH

1 H

1

1

( t − s) − α H + α

= 0



1

|Gα (1, z )| H dzds

pH

R

   t 1 1 ≤ Cα,H  (t − s)− αH + α  0

 ≤ Cα,H

t

R

pH 1

1 + |z |1+α

  H1 dzds

pH 1 1 (t − s)− αH + α ds < ∞,

0

under the assumption 1 − α1H + α1 > 0 (i.e. α >

1 H

− 1). So we have A3α (u)(t , x) ∈ B for p ≥ 2.

(3.18)

J. Liu et al. / Nonlinear Analysis 75 (2012) 6060–6070

6067

∈ (0, 1] and assumption (3.6), it follows that for p ∈ [2, ∞),  t p   − 1 1− 1r ( t − s) α E∥A4α (u)(t , x)∥pp ≤ C E ∥1 + |u(s, ·)|∥p · ∥ψ∥p ds

From (1) of Lemma 3.1 with

1 r

=

1 p

−

2 p

+1=1−

1 p

0

  ≤ Cp 1 + ∥u(·)∥pB sup ∥ψ(t , ·)∥pp · 0≤t ≤T

≤ Cp,T sup ∥ψ(t , ·)∥p 1 + ∥u(·)∥  p

0≤t ≤T

p B

t



  p − α1 1− 1r p− 1

( t − s)

p−1 ds

0



(3.19)

with 1 − α(p1−1) > 0, i.e. p > 1 + α1 . Finally, let us estimate A5α (u)(t , x). This is a key step in the proof of Proposition 3.1. From the condition (3.7), (2) of Lemma 3.1 with 1r = 1p − 4p + 1 = 1 − 2p ∈ (0, 1] and Proposition 2.1, it follows that, for 2(α+1)

p > α−1 ,

E∥A5α (u)(t , ·)∥pp

  =E  

t+

 

0

E2

R

p   Gα (t − s, · − y)σ (s, y, u(s, y))h(s, y, z )M (dy, dz , ds) 

≤ Cp 0

R

p

 2p  2 |Gα (t − s, x − y)h(s, y, z )|2 E[1 + |u(s, y)|p ] p µ2 (dz )dyds dx

  t   E2

R

 t   2p      2 2 2 p p  = Cp  |Gα (t − s, x − y)| |h(s, y, z )| µ2 (dz ) 1 + E|u(s, ·)| dyds  0



E2

R

t

(t − s)

≤ Cp

− pα+p2

E2

0



t

(t − s)

≤ Cp

 2p       2p  2 p  |h(s, y, z )| µ2 (dz ) 1 + E|u(s, ·)|    p ds

− pα+p2

4

     |h(s, y, z )|2 µ2 (dz )  p E2

0

2

 2p     2p  p  1 + E|u(s, ·)|  ds  p 2

 p   p   t  p−2 2  2  2 2 − α(pp+−22) 2 p p    sup  |h(s, y, z )| µ2 (dz ) · sup  1 + E|u(s, ·)|  · (t − s) p p 0≤t ≤T 0≤t ≤T

 ≤ Cp

E2

2

2

0

p   2    p |h(s, y, z )|2 µ2 (dz ) sup  1 + ∥u(·)∥B < ∞.   p 0 ≤t ≤T

 ≤ Cp,T

E2

(3.20)

2

Thus we have proved the operator Aα defined by (3.14) is an operator from B to itself. On the other hand, from the similar argument as in (3.15)–(3.20), let η > 0 sufficiently large, then Aα ∈ B. Thus we complete the proof of the proposition.  In what follows, we will prove that the operator Aα : B → B is a contract operator. 2(α+1)

Proposition 3.2. For p > α−1 , the operator Aα is an contraction on B under the conditions of Theorem 3.1. In other words, there exists a constant ϱ ∈ (0, 1) such that

∥Aα (u) − Aα (v)∥B ≤ ϱ∥u − v∥B ,

for u, v ∈ B.

(3.21)

Proof. Suppose u0 and v0 are initials of (Ft )t ≥0 -adapted random fields u, v ∈ B such that u0 = v0 . Let us begin by p considering A1α (u). Note that, for ρ = 3 , by (1) of Lemma 3.1 with 1r = 1p − ρ1 + 1 = 1 − 2p and condition (1) of Theorem 3.1, we have

 t p  p  2 2 − α1 (1−r )   E Aα (u)(t , ·) − Aα (v)(t , ·) p ≤ C E ( t − s) ∥f (s, y, u(s, y)) − f (s, y, v(s, y))∥p ds 0

t

 ≤ Cp 0

1

(t − s)− α (1−r ) E∥u(s, ·) − v(s, ·)∥p ds

p

.

(3.22)

6068

J. Liu et al. / Nonlinear Analysis 75 (2012) 6060–6070

So

  2   A (u)(t , ·) − A2 (v)(t , ·)p = sup e−ηt E A2 (u)(t , ·) − A2 (v)(t , ·)p α α α α B p 0≤t ≤T

t



− ηp (t −s)

≤ C sup E

e

0≤t ≤T

η − α1 (1−r ) − p s

(t − s)

e

p

∥u(s, ·) − v(s, ·)∥p ds

0

0 ≤t ≤T

  t 

t



e−ηs ∥u(s, ·) − v(s, ·)∥pp ds

≤ Cp sup E

−ηs

≤ Cp sup

e

0 ≤t ≤T

(t − s)− α (1−r )

− ηp (t −s)

− α1 (1−r )

1

 p−p 1 p−1

0

0 t



− ηp (t −s)

e

E∥u(s, ·) − v(s, ·)∥

  t 

p p ds

e

(t − s)

 p−p 1 p−1

0

0

= Cp χ (p, T )∥u − v∥pB ,

(3.23)

where

χ (p, T ) =

 t 

− ηp (t −s)

e

1

(t − s)− α (1−r )

 p−p 1 p−1

.

0

Let τ = p−1 α1 (1 − r ), then p

χ (p, T ) ≤

+∞





e

− ηp (t −s)

(t − s)

− α1 (1−r )

 p−p 1 p−1

0



(p − 1)τ +1 η τ +1



τ +1

= =

(p − 1) η τ +1

+∞



p−1

e−x xτ dx

0

Γ (τ + 1)

p−1

with p ≥ 3. Therefore

 2  1 A (u)(t , ·) − A2 (v)(t , ·) ≤ Cp T p α α B



(p − 1)τ +1 Γ (τ + 1) η τ +1

 p−p 1

· ∥u − v∥B ≤ ϱ∥u − v∥B ,

(3.24)

with ϱ ∈ (0, 1) by choosing η > 0 large enough. Next we shall consider the term A5α (u)(t , x). From a similar argument as in (3.20), thanks to the generalized B–D–G inequality and (A.5), we derive from the conditions (1), (2) and (3) of Theorem 3.1 that

  5   A (u)(t , ·) − A5 (v)(t , ·)p = sup e−ηt E A5 (u)(t , ·) − A5 (v)(t , ·)p α α α α B p 0≤t ≤T   t +    Gα (t − s, x − y)h(s, y, z ) = sup e−ηt E  0≤t ≤T

0

R

R

E2

p  × [σ (s, y, u(s, y)) − σ (s, y, v(s, y))] M (dy, dz , ds) dx   t   ≤ Cp sup e−ηt (E |Gα (t − s, x − y)h(s, y, z ) 0≤t ≤T

R

0

R

E2

 2p

2

µ2 (dz )dyds dx × [σ (s, y, u(s, y)) − σ (s, y, v(s, y))]|   t   |Gα (t − s, x − y)h(s, y, z )|2 ≤ Cp sup e−ηt p p

0 ≤t ≤T

R

0

R

E2

 2p 2 × E|u(s, y) − v(s, y)|p p µ2 (dz )dyds dx   t   2η s |Gα (t − s, x − y)h(s, y, z )|2 e p ≤ Cp sup e−ηt 

0 ≤t ≤T



−ηs

× e

R

0

R

E2

E|u(s, y) − v(s, y)|

2

p p

µ2 (dz )dyds

 2p dx

J. Liu et al. / Nonlinear Analysis 75 (2012) 6060–6070

  t  

e−ηs E|u(s, y) − v(s, y)|p µ2 (dz )dyds

≤ Cp sup 0≤t ≤T

0

R

 t   R

E2 2p

|Gα (t − s, x − y)h(s, y, z )| p−2 e−

× 0

R

6069



2η(t −s) p−2

µ2 (dz )dyds

 p−2 2 dx

E2

≤ Cp,T µ2 (E2 )∥u − v∥pB   t  |Gα (t − s, x − y)h(s, y, z )|p e−η(t −s) µ2 (dz )dydsdx × sup 0≤t ≤T

≤ ϱ∥u − v∥

R

0

R

E2

p B

(3.25)

with ϱ ∈ (0, 1) by choosing η > 0 large enough. Then Aα (u) is a contraction on B. A similar procedure as the above arguments yields that A4α (u) is a contraction on B by letting η > 0 large enough. Therefore, it follows from (3.14) that Aα (·) is a contraction on B if η > 0 large enough. Thus the proof of Proposition 3.2 is complete.  5

Based on Propositions 3.1 and 3.2 and the fixed point principle on the set {u ∈ B : u(0) = u0 }, we conclude that (1.1) admits a unique solution u ∈ B. Thus the conclusion of Theorem 3.1 follows. Acknowledgments We would like to thank the Editor and an anonymous referee whose comments and suggestions greatly improved the presentation of this paper. Appendix. The Green function The fractional differential operator Dαδ is an extension of the inverse of the generalized Riesz–Feller potential when α > 2. It is given for α > 0 by Definition A.1. The fractional differential operator Dαδ is given by Dαδ ϕ = F −1 {ψα (λ)F (ϕ(x), λ)} α −iδ π2 sgn(λ)

ψα (λ) = −|λ| e

(A.1)

,

(A.2)

δ ≤ min{α − [α]2 , 2 + [α]2 − α}, [α]2 is the largest even integer less than or equal to α (even part of α ), and for α = 0 when α ∈ 2N + 1, and F (respectively F −1 ) is the Fourier (respectively Fourier inverse) transform. The operator Dαδ is a closed, densely defined operator on L2 (R) and it is the infinitesimal generator of a semigroup which is not symmetric and not a contraction. This operator is a generalization of various well-known operators, such as the Laplacian operator (when α = 2), the inverse of the generalized Riesz–Feller potential (when α > 2), the Riemann–Liouville differential operator (when δ = 2 + [α]2 − α or δ = α − [α]2 , see [18,19,22] for more details). It is self-adjoint only when δ = 0 and in this case, it coincides with the fractional power of the Laplacian. We refer the readers to Debbi [18] and Debbi and Dozzi [19] for more details about this operator. Let the Green function Gα (t , x) associated with Eq. (1.1) be the fundamental solution of the Cauchy problem

 ∂ Gα (t , x) = Dαδ Gα (t , x), ∂t  Gα (0, x) = δ0 (x),

t > 0, x ∈ R,

(A.3)

where δ0 is the Dirac distribution at the point zero. Using Fourier’s calculus we obtain Gα (t , x) = F −1 {eψα (λ)t } =

1 2π





π



exp −iλx − t |λ|α e−iδ 2 sgn(λ) dλ. R

The function Gα (t , ·) has the following properties (see [18,19]). Lemma A.1. For α ∈ (0, ∞) \ N (1) R Gα (t , x)dx = 1. (2) Gα (t , x) is real but in general it is not symmetric relatively to x and it is not everywhere positive.



(A.4)

6070

J. Liu et al. / Nonlinear Analysis 75 (2012) 6060–6070

(3) Gα (t , x) satisfies the semigroup property, or the Chapman–Kolmogorov equation, i.e. for 0 < s < t Gα (t + s, x) =



Gα (t , ξ )Gα (s, x − ξ )dξ . R

(4) For 0 < α ≤ 2, the function Gα (t , ·) is the density of a Lévy stable process in time t. β

(5) For fixed t , Gα (t , ·) ∈ C ∞ and ∂∂xβ Gα (t , x) is bounded and tends to zero when |x| tends to ∞ for β ∈ R+ . n n+1 ∂ n (6) ∂∂xn Gα (t , x) = t − α ∂ζ n Gα (1, ζ )| − 1 , for all n ≥ 0 (when n = 0, it is called the scaling property). ζ =t α x

Corollary A.1. Let α ∈ (1, +∞). Then there exists a constant Kα such that

|Gα (1, x)| ≤ Kα (1 + |x|1+α )−1 , |G(αn) (1, x)| ≤ Kα

(A.5)

α+n−1

1 + | x| . (1 + |x|α+n )2

(A.6)

1 Corollary A.2. Let α ∈ (1, +∞), for n ≥ 1, and T ≥ 0, for γ such that α+1n+1 < γ < α+ , n+1 T

 0

γ   n  ∂    ∂ xn Gα (t , x) dxdt < ∞.

(A.7)

R

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