張媛媛, 王宏偉

(1.開封大學(xué) 數(shù)學(xué)教研部，河南 開封 475000；2.安陽師范學(xué)院 數(shù)學(xué)與統(tǒng)計(jì)學(xué)院，河南 安陽 455000)

一類四階非線性發(fā)展方程整體解的存在唯一性

張媛媛1, 王宏偉2

(1.開封大學(xué) 數(shù)學(xué)教研部，河南 開封 475000；2.安陽師范學(xué)院 數(shù)學(xué)與統(tǒng)計(jì)學(xué)院，河南 安陽 455000)

研究一類四階非線性發(fā)展方程整體解的存在唯一性,借助偏微分方程的一些標(biāo)準(zhǔn)技巧對非線性項(xiàng)進(jìn)行估計(jì),利用嵌入定理和算子半群的方法證明了在相對較弱的條件下上述問題整體解的存在唯一性.

整體解；存在性；唯一性

1 預(yù)備知識

具耗散項(xiàng)非線性發(fā)展方程的整體解是近年來偏微分方程研究的熱點(diǎn),目前關(guān)于它的研究主要集中在其存在性和惟一性方面.本文的目的是研究下列一類四階具耗散項(xiàng)方程

(1)

u|?Ω=0,t>0，

(2)

u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω.

(3)

(Ω是RN中具有光滑邊界的有界區(qū)域)整體解的存在唯一性.方程(1)有一定的物理模型,它主要描述的是一類具黏彈性材料的振動[1-4],此類方程也頻繁出現(xiàn)在具黏彈性構(gòu)形且服從Voight非線性模型的縱向運(yùn)動中[5].考慮到偏微分方程和無窮維動力系統(tǒng)的密切關(guān)系,因此筆者嘗試用算子半群的方法探討該問題的整體解,最終得到在相對較弱的條件下整體解的存在性和唯一性.

(4)

u(x,0)=u0(x),ut(x,0)=u1(x).

(5)

引理1[6]設(shè)X，Y是Banach空間,并且X?Y.若φ∈L∞(0,T;X)ICw([0,T];Y),則

φ∈Cw([0,T];X).

引理2[7](Aubin-Lions) 設(shè)B0,B和B1是Banach空間,且B0??B?B1,B0,B1是自反的.W={v|v∈Lp0(0,T;B0),vt∈Lp1(0,T;B1)},這里0

引理3[7]設(shè)z(t)非負(fù),是[0,∞)上的絕對連續(xù)函數(shù),滿足下列不等式

2 主要結(jié)論

定理假定下列條件成立

(ii)βi∈C1(R),βi(s)s≥B3|s|q+2,|βi(s)|≤B4(1+|s|q+1)；

(iii)(u0,u1)∈X2+2δ.

證明:(4)式與ut作內(nèi)積,得:

(6)

(7)

(8)

(8)式積分,并結(jié)合(7)式,得:

≤C(‖(u0,u1)‖X2,T),t∈[0，T].

(9)

設(shè)v=u1+εu，則v滿足

(10)

(10)式與v作內(nèi)積,得:

(11)

(12)

將上式代入(11)式,得:

(13)

通過計(jì)算,得:

K(u,v)-ερH(u,v)

將上式代入(13)式,得:

(14)

由(12),(14)式并利用引理3,得:

(15)

設(shè)R>0,對于給定的(u0,u1)∈X2+2δ且‖(u0,u1)‖X2+2δ≤R,則‖(u0,u1)‖X2≤R.

(16)

(17)

(18)

(19)

(20)

(21)

將(19)-(21)式代入(18)式,得:

(22)

(23)

由(4),(22),(23)式和假定,得：

‖utt‖L2(0,T;V2δ-1)≤‖u‖L2(0,T;V1+2δ)+‖ut‖L2(0,T;V1+2δ)+‖u‖L2(0,T;V3+2δ)

≤C(R)(‖u‖L2(0,T;V3+2δ)+‖ut‖L2(0,T;V1+2δ)+1)≤C(R,T), t∈[0,T].

(24)

t>0,j=1,2,…,n，

(25)

顯然,估計(jì)(22)-(23)式對un仍成立.因此,可從中抽取子序列仍記作un,使得

un→u 在L∞(0,T;V2+2δ) weak*;

(25)式在(0,t)上積分,得到等價方程

(26)

在(26)式中n→∞,對t求導(dǎo),可得u是問題(4)-(5)的解,且(u,ut)∈L∞(0,T;V2+2δ×V2δ),

ut∈L2(0,T;V2+2δ),utt∈L2(0,T;V2δ-1).

所以,u∈H1(0,T;V2+2δ)?C(0,T;V2+2δ),ut∈H1(0,T;V2δ-1)?Cw(0,T;V2δ-1).

由引理1,ut∈Cw(0,T;V1+2δ).(16)式在(t0,t)上積分,得:

所以,方程在(t0,t)上是可積的.因此,(u,ut)∈C([0,T];V2+2δ×V2δ).

下證(u,ut)連續(xù)依賴X2+2δ上的初值.

設(shè)u,v是問題(4)-(5)分別對應(yīng)于初值u0,u1和v0,v1的兩個解,則ω=u-v滿足方程

(27)

ω(0)=u0-v0≡ω0, ωt(0)=u1-v1≡ω1.

(27)式與Aδωt作內(nèi)積,由假定(i),(ii),得:

(28)

(29)

(30)

將(29),(30)式代入(28)式,得:

所以,解連續(xù)依賴X2+2δ上的初值,即解的唯一性得證.

[1] ANDREWS G. On the existence of solutions to the equationutt-uxxt=σ(ux)x[J]. Journal of Differential Equations, 1980,35(1):200-231.

[2] KOBAYASHI T, PECHER H, SHIBATA Y. On a global in time existence theorem of smooth solutions to a nonlinear wave equation with viscosity[J]. Mathematische Annalen， 1993,296(1):215-234.

[3] KAVASHIMA S, SHIBATA Y. Global existence and exponential stability of small solutions to nonlinear viscoelasticity[J]. Communications in Mathematical Physics, 1992,148(1):189-208.

[4] 姜禮尚,陳亞浙，等.數(shù)學(xué)物理方程講義.第二版[M].北京：高等教育出版社，38-40.

[5] ANG DD, DINH APN. Strong solutions of quasilinear wave equation with nonlinear damping[J]. SIAM Journal on Mathematical Analysis,1988,19(2):337-347.

[6] NAKAO M. Existence of global smooth solution to the initial boundary value problem for the quasi-linearhyperbolic equation with a degenerate dissipative term[J]. Differ Eqs, 1992,98(1):299-327.

[7] YANG Zhijian. Global attrator for the Kirchhoff type equation with a strong dissipation[J]. Differ Eqs, 2010,249(6):3258-3278.

TheExistenceandUniquenessoftheGlobalSolutionstoaClassofNonlinearEvolutionEquationsofFourth-Order

ZHANG Yuan-yuan1, WANG Hong-wei2

(1.Teaching and Research Department of Mathematics, Kaifeng University,Kaifeng 475000, China; 2.Department of Mathematics and Statistics, Anyang Normal University, Anyang 455000, China)

The existence and uniqueness of the global solutions to a class of nonlinear evolution equations of fourth order were studied. By some standard methods the non-linear terms were estimated. By use of embedding theorem and the method of semigroup,under rather mild conditions the existence and uniqueness of the global solutions for the above-mentioned problem were obtained.

global solutions; existence; uniqueness

2013-09-19

國家自然科學(xué)基金(10971199)；河南省教育廳科學(xué)技術(shù)研究重點(diǎn)項(xiàng)目(13B110137)；開封市科技發(fā)展計(jì)劃(1401012).

張媛媛(1979-),女,河南省商丘市人,講師,碩士,主要從事偏微分方程與無窮維動力系統(tǒng)的研究.

張媛媛,王宏偉.一類四階非線性發(fā)展方程整體解的存在唯一性[J].安徽師范大學(xué)學(xué)報(bào):自然科學(xué)版,2014，37(4)：330-334.

O175.29

A

1001-2443(2014)04-0330-05