羅李平,楊柳

(衡陽師范學(xué)院數(shù)學(xué)與計算科學(xué)系,湖南衡陽 421008)

具連續(xù)偏差變元的中立型向量拋物偏微分方程的H-振動性

羅李平,楊柳

(衡陽師范學(xué)院數(shù)學(xué)與計算科學(xué)系,湖南衡陽 421008)

討論一類具連續(xù)偏差變元的中立型向量拋物偏泛函微分方程的H-振動性,利用內(nèi)積降維的方法和Green公式,得到了該類方程在Robin邊值條件下所有解H-振動的若干充分判據(jù),這里H是Rm中的單位向量.

向量;拋物型偏泛函微分方程;H-振動性;中立型;連續(xù)偏差變元

近年來,偏泛函微分方程振動理論在物理學(xué)、工程學(xué)和生物學(xué)等學(xué)科領(lǐng)域中有廣泛的應(yīng)用,對其研究取得了很大的發(fā)展,已有許多很好的研究成果[19],這為進一步研究向量偏泛函微分方程作了準(zhǔn)備.1970年,Domslak[10]在研究向量常微分方程時首次引入了H-振動性的概念,這里H是Rm中的單位向量.關(guān)于這一概念及其應(yīng)用,文[11]中作了很好的闡述.最近,文[12]研究了一類具泛函變元的向量拋物型偏泛函微分方程解的H-振動性;文[13]研究了一類具偏差變元的向量雙曲型偏泛函微分方程解的H-振動性;文[14]研究了一類具分布式中立項系數(shù)的向量雙曲型偏泛函微分方程解的H-振動性.但總的來說,目前這方面相關(guān)的研究工作還不夠豐富、完善,尚有待于深入展開研究.本文將著手考慮如下的一類具連續(xù)偏差變元的中立型向量拋物偏泛函微分方程

解的H-振動性,其中u(x,t)∈C2(?×[t0,∞),Rm)是向量函數(shù),?是Rn中具有逐片光滑邊界的有界域,?是Rn中的n維Laplacian算子.

考慮Robin邊界條件

其中N是??的單位外法向量,μ(x,t)∈C(??×R+,R+),R+=[0,∞).

參考文獻

[1]羅李平.具連續(xù)分布滯量的高階非線性偏泛函微分方程的強迫振動性[J].工程數(shù)學(xué)學(xué)報,2007,24(1):133-137.

[2]林文賢.偶數(shù)階中立型偏微分方程系統(tǒng)的振動性定理[J].純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué),2007,23(4):467-470.

[3]羅李平,高正暉,歐陽自根.具連續(xù)分布滯量的非線性中立型拋物方程的振動性[J].應(yīng)用數(shù)學(xué),2006,19(3):651-655.

[4]Lin S Z,Zhou Z X,Yu Y H.Oscillation criteria for a class of hyperbolic equations with continuous distributed deviating arguments[J].J.of Math.,2005,25(5):521-526.

[5]劉安平,何猛省.非線性中立雙曲型偏微分方程解的振動性質(zhì)[J].應(yīng)用數(shù)學(xué)與力學(xué),2002,23(6):604-610.

[6]王培光,葛渭高.一類非線性偏泛函微分方程的強迫振動性[J].系統(tǒng)科學(xué)與數(shù)學(xué),2000,20(4):454-461.

[7]王培光,傅希林,俞元洪.一類時滯雙曲方程的振動準(zhǔn)則[J].數(shù)學(xué)研究與評論,1998,18(1):105-111.

[8]Yu Y H,Liu B,Liu Z R.Oscillation of solutions of nonlinear partial differential equations of neutral type[J]. Acta.Math.Sini.,1997,13(4):563-570.

[9]崔寶同,俞元洪,林詩仲.具有時滯的雙曲型微分方程解的振動性質(zhì)[J].應(yīng)用數(shù)學(xué)學(xué)報,1996,19(1):80-88.

[10]Domslak Ju I.On the oscillation of solutions of vector differential equations[J].Soviet Math.Dokl., 1970,11:839-841.

[11]Courant R,Hilbert D.Methods of Mathematical Physics,Vol.I[M].New York:Interscience,1996.

[12]Minchev E,Yoshida N.Oscillation of solutions of vector differential equations of parabolic type with functional arguments[J].J.Comput.Appl.Math.,2003,151(1):107-117.

[13]Li W N,Han M A,Meng F W.H-oscillation of solutions of certain vector hyperbolic differential equations with deviating arguments[J].Appl.Math.Comput.,2004,158(3):637-653.

[14]王寧,王培光,孫曉玲.一類具分布式偏差變元的雙曲型向量泛函微分方程解的H-振動性[J].應(yīng)用泛函分析學(xué)報,2007,9(1):63-69.

[15]Ladde G S,Lakshmikantham V,Zhang B G.Oscillation Theory of Differential Equations with Deviating Arguments[M].New York:Marcel Dekker Inc.,1987.

H-oscillation of neutral vector parabolic partial differential equations with continuous deviating arguments

LUO Li-ping,YANG Liu

(Department of Mathematics and Computational Science,Hengyang Normal University, Hengyang421008,China)

The H-oscillation of a class of neutral vector parabolic partial functional differential equations with continuous deviating arguments is discussed.By using the method of reducing dimension with scalar product and Green formula,some sufficient criteria for the H-oscillation of all solutions of the equations are obtained under Robin boundary value condition,where H is a unit vector of Rm.

vector,parabolic partial functional differential equation,H-oscillation,neutral type,continuous deviating arguments

O175.26

A

1008-5513(2009)04-0801-06

2007-09-08.

湖南省教育廳科研計劃項目(07C164),湖南省自然科學(xué)基金(06JJ5001).

羅李平(1964-),教授,研究方向：偏泛函微分方程振動理論.

2000MSC:35B05,35K50,35R10