楊 歡, 陳光淦, 何 興

(四川師范大學(xué) 數(shù)學(xué)與軟件科學(xué)學(xué)院, 四川 成都 610066)

有界區(qū)間上隨機(jī)分?jǐn)?shù)階反應(yīng)擴(kuò)散方程鞅解的存在性

楊 歡, 陳光淦*, 何 興

(四川師范大學(xué) 數(shù)學(xué)與軟件科學(xué)學(xué)院, 四川 成都 610066)

有界區(qū)間上隨機(jī)分?jǐn)?shù)階反應(yīng)擴(kuò)散方程在分?jǐn)?shù)階非相對(duì)量子力學(xué)中起到很重要的作用.由于噪聲和有界區(qū)間上分?jǐn)?shù)階Laplace算子的擾動(dòng)和影響,使隨機(jī)分?jǐn)?shù)階反應(yīng)擴(kuò)散方程的研究變得復(fù)雜.通過引入一個(gè)適當(dāng)?shù)募訖?quán)函數(shù)來構(gòu)造加權(quán)空間,運(yùn)用算子理論來克服有界區(qū)間上的分?jǐn)?shù)階Laplace算子帶來的困難.運(yùn)用Prokhorov定理和Skorokhod嵌入定理來解決噪聲帶給系統(tǒng)的通常緊性不成立的收斂問題.利用It公式和一系列精致的不等式技巧,以及Galerkin方法,最終獲得系統(tǒng)鞅解的存在性.

有界區(qū)間上分?jǐn)?shù)階Laplace算子; 白噪聲; 反應(yīng)擴(kuò)散方程; 鞅解

有界區(qū)域上的分?jǐn)?shù)階反應(yīng)擴(kuò)散方程是常擴(kuò)散方程的一種拓展,它源于反常擴(kuò)散模型,用于描述具有分形結(jié)構(gòu)的多孔介質(zhì)中的反常擴(kuò)散現(xiàn)象,在物理、財(cái)經(jīng)、水文學(xué)、以及工程學(xué)、材料科學(xué)等方面,都有廣泛應(yīng)用[1-6].

本文考慮如下隨機(jī)分?jǐn)?shù)階反應(yīng)擴(kuò)散方程

(1)

這里,有界區(qū)間D=(-1,1)?R1,Dc=R1D,W(t)是一個(gè)Wiener過程.分?jǐn)?shù)階Laplace算子(-Δ)s定義如下

0

(2)

其中Cs是依賴于s的常數(shù).有界區(qū)域上的分?jǐn)?shù)階Laplace算子(-Δ)s與正常的Laplace算子有很大的不同[7].因此,對(duì)于有界區(qū)域上的分?jǐn)?shù)階Laplace算子驅(qū)動(dòng)的發(fā)展方程,受到了許多數(shù)學(xué)家的關(guān)注[7-9].

本文關(guān)心有界區(qū)域上分?jǐn)?shù)階Laplace算子驅(qū)動(dòng)的隨機(jī)分?jǐn)?shù)階反應(yīng)擴(kuò)散方程.分析了分?jǐn)?shù)階Laplace算子和白噪聲的特征.為了克服有界區(qū)域上分?jǐn)?shù)階Laplace算子帶來的困難,引入一個(gè)新的加權(quán)函數(shù),來構(gòu)造加權(quán)Sobolev空間,進(jìn)而再在這個(gè)空間上對(duì)方程進(jìn)行研究.由于噪聲的擾動(dòng),系統(tǒng)(1)通常意義下的緊性不再成立,運(yùn)用胎緊來代替.運(yùn)用Prokhorov定理和Skorokhod嵌入定理來解決序列的收斂問題,最終獲得系統(tǒng)鞅解的存在性.

1 預(yù)備引理

設(shè)s∈(0,1),D?R1且為有界區(qū)域,定義如下:

Ws,2(D):={u∈L2(D):

其中,‖u‖Ws,2(D)和[u]Ws,2(D)分別為經(jīng)典的分?jǐn)?shù)階Sobolev空間Ws,2(D)的范數(shù)和半范數(shù)[10].

引理 1.1[10]設(shè)0

‖u‖Ws,2(D)≤C‖u‖W1,2(D),

W1,2(D)?Ws,2(D).

引理 1.3[11]設(shè)B0?B?B1,均為Banach空間,B0與B1是自反的,B0緊嵌入到B.設(shè)γ∈(0,1),X=L2(0,T;B0)∩Wγ,2(0,T;B1),那么X緊嵌入到L2(0,T;B).

根據(jù)文獻(xiàn)[8-9]有

x∈R1,

J*(u)(x,y)=-(u(y)-u(x))β(x,y),

x,y∈R1,

β(x,y)(Θ(x,y)β(x,y))dy,

其中,J是非局部散度算子,J*為J的伴隨算子,V(x,y),β(x,y):R1×R1→Rk,β是反對(duì)稱的,u(x):R1→R1,Θ(x,y)=Θ(y,x)為一個(gè)二階張量滿足Θ=ΘT.

(3)

設(shè)D?R1是一個(gè)開的有界區(qū)域,由文獻(xiàn)[10]及(2)式得

(-Δ)su(x)=J(Θ·J*u)(x)=J(J*u)(x).

〈(-Δ)su,u〉L2(D)=〈J*u(x),J*u(x)〉=

(5)

由x∈D=(-1,1),知ρ(x)有嚴(yán)格的正下界和上界,因此,加權(quán)分?jǐn)?shù)階Sobolev空間范數(shù)定義為

從而

(6)

2 鞅解的存在性

本文記

假設(shè)(H1):f(u)滿足

k2|u|p-α2|u|2≤f(u)u≤

k1|u|p+α1|u|2,

2

其中k1、k2、α1、α2均為正常數(shù).

假設(shè)(H2):g:H→L2(U,H)是連續(xù)的,且滿足

?u,v∈H,

其中C和λ為正常數(shù).在本文中,假定C均為正常數(shù),但是出現(xiàn)的C有所不同.

u∈L∞(0,T;H)∩L2(0,T;V)∩C([0,T];V1),

且使得對(duì)任意的t∈[0,T],v∈V有

(7)

則稱(Ω,F,,{Ft}t≥0,W,u)是方程(1)的鞅解.

定理 2.1 假定s∈(1/2,1),初值u0滿足F0可測(cè),且u0∈L2(Ω,H).設(shè)(H1)和(H2)成立,則方程(1)存在一個(gè)鞅解.

任給n∈N,在有限維空間Hn上考慮以下隨機(jī)方程

(8)

由于在有限維空間上的隨機(jī)微分方程(8)滿足局部Lipschitz條件和線性增長,方程(8)有唯一強(qiáng)解un(t)∈L2(Ω;C([0,T];Hn))[12],滿足

(9)

由假設(shè)(H1),于是得

再利用Gronwall不等式,可得

(10)

由(10)式,知方程(8)的解{un}n∈N在空間L2(Ω,L2(0,T;V))上一致有界.

-Pn(-Δ)sun(r)-Pnf(un(r))〉dr+

于是,由Burkholder-Davis-Gundy不等式和Young不等式得

由假設(shè)(H1),從而

又由Gronwall不等式得

(11)

以上所有的C不依賴于n.

根據(jù)(9)式,令

(-Δ)sφ(x)=

C‖un‖H.

(12)

故由(10)、(11)和(12)式,以及內(nèi)插不等式和假設(shè)(H1)得

2k1α1‖

(13)

再由Burkholder-Davis-Gundy不等式及假設(shè)(H2)得

(14)

定義

Mn(t)=un(t)-Pnu0+

由(9)式,于是Mn(·)是一個(gè)鞅,其二階變差為

E([Mn(t)-Mn(r)]φ(un(·)))=0,

進(jìn)一步

E([〈Mn(t),a〉〈Mn(t),b〉-

φ(un(·)))=0,

(15)

(16)

r.

(17)

從而對(duì)任意的t∈[0,T],v∈V有

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2010 MSC：35K05; 60H15; 60G46

(編輯 周 俊)

The Existence of Martingale Solution of the Stochastic FractionalReaction-diffusion Equation on Bounded Intervals

YANG Huan, CHEN Guanggan, HE Xing

(College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, Sichuan)

This paper studies the stochastic fractional reaction-diffusion equation on bounded intervals, which plays a very important role in fractional nonrelativistic quantum mechanics. Due to interacting and disturbing of the fractional Laplacian operator on a bounded interval with white noise, the stochastic fractional reaction-diffusion equation is too complicated to be understood. By introducing a suitable weight function to construct the weighted space, we apply the operator theory to overcome the difficulties caused by the fractional Laplacian operator on bounded intervals. We apply Prokhorov theorem and Skorokhod embedding theorem to solve the convergence problem instead of losing the common compactness of the system caused by the white noise. On the basis of Itformula, a series of exquisite inequalities and Galerkin approximation, we finally establish the existence of the martingale solution of the stochastic fractional reaction-diffusion equation on bounded intervals.

fractional Laplacian operator on bounded intervals; white noise; reaction-diffusion equation; martingale solution

2015-11-08

國家自然科學(xué)基金(11571245和11401409)和四川省教育廳重點(diǎn)科研項(xiàng)目(15ZA0031)

O175.2

A

1001-8395(2016)06-0794-07

10.3969/j.issn.1001-8395.2016.06.003

*通信作者簡(jiǎn)介：陳光淦(1978—),男,教授,主要從事隨機(jī)偏微分方程的研究,E-mail:chenguanggan@hotmail.com