吊桥方程的动力学性态

第47卷2011年第6期西北师范大学学报(自然科学版)Vol.472011No.6Journal of Northwest Normal University (Natural Science)吊桥方程的动力学性态马巧珍,李志宇,汪璇(西北师范大学数学与信息科学学院,甘肃兰州730070)摘要:研究弱阻尼非自治吊桥方程解的衰退估计,证明了讓方程在强拓扑空间中存在一致吸引子关键词:吊桥方程;一致嗄引子;一致条件(C);袁退估计中图分类号:O175文献标识码:A文章编号:1001-988X(2011)06-0001-07Dynamics of suspension bridge equationsMa Qiao-zhen, LI Zhi-yu, WANG Xuan(College of Mathematics and Information Science, Northwest Normal University, Lanzhou 730070, Gansu, China)Abstract: The decay estimates for the weakly damped nonautonomous suspension bridge equations areinvestigated in this paper, and then the existence of the uniform attractors in a stronger Hilbert space is proved.Key words: suspension bridge equation; uniform attractor; uniform condition(C); decay estimate1 Introductionbehavior of solution of 1 ) In [7 ],theexistence of global attractors in H(n)XL(n)forLet n be an open bounded subset of R2 with the autonomous suspension bridge equations, thatsufficiently smooth boundary r:. We study the is, h(r, t)=h(r) in(1), was investigated.following initial-boundary value problemConsequently, we achieved existence of stronguetAutaurtku+gu)=h(x,t),solutions and strong global attractors for the single(x,t)∈×R(1) suspension bridge equations and the coupled one=Va=0,x∈r,(2) respectively, in line with thermr=u,(r),a uImr = P, (x),rE R, (3) (C) introduced in [8, 9] and combining withwhere u(t, t)is an unknown function which techniques of the energy estimates(see [10, 11])represents the deflection of the road bed in the In this paper, we are going to focus on existencevertical plane, k>0 denotes a spring constant and of the uniform compact attractors of(1)-(38 is a positive constant.For a good survey of the literatures dedicated toThe suspension bridge equations were existence of attractors for the dynamical systemspresented by Lazer and McKenna as a new problem we would like to mention some monographs andin fields of the nonlinear analysis in[ 1 ] For works, such as [8-15] and so on(1), there were many literatures to studyFor the nonlinear function gE C(R, R)weexistence of solutions, see [1-6] and references presume the following general conditions: Theretherein. While we first studied the longtime exists a constant C>0 such that收稿日期:2011-03-18;修改稿收到日期:201109-15基金项目:国家自然科学基金资助项目(11101334);甘肃省自然科学基金资助项目(1107RJA223);甘肃省高校基本科研业务费项目中国煤化工作者简介:马巧珍(1972-),女,甘肃秦安人,教授,博士.主要研究方向为E-mail:maqzh@nwnu.edu.cnCNMH函统西北师范大学学报(自然科学版)第47卷Journal of Northwest Normal University (Natural ScienceVol 47exists to=to(r, B)>r such that UU,(t,t)BCBmify≥0G()=gx)dfor all t>to. a set Y Ce is said to be uniformly(G2) limsup4=0,v0≤y<∞;(wrt ges)attracting for the family of processesl→∞(G, )liminf _g(s)-C1G(s)U(t,r)},a∈Σ, if for any fixed t∈ R and every≥0B∈B(E),With the usual notation, we introduce thelim supdist(U, (t,T)B,Y)=0(7)spaces H=L'(n), V=H(n), and endow these where B(E)is the set of all bounded subset of E.spaces with the usual scalar products and normsDefinition 2 A closed set A=CE is said to be(…,),|,((…,),l1, respectively, wherethe uniform(wrt gE2)attractor of the family of(u, v)= u(r)v(r)dr,processes(L(,r)},o∈ E if it is uniformly(wrt∝∈E)attracting( attracting property)and it is contained(a,v)=△a(x)△v(x)dxin any closed uniformly (wrt oE2)attracting setWe write D(A)=H(n)n:(n), and forA' of the family of processes ( U(t,r)},d∈Σ,AxCA(minimal propertyevery u∈D(A),Aa=Δ2u. Especially, denoteI Au] the norm of D(A).The following basic assumption comes fromExploiting the Poincare inequality, thereapplications and is natural.exists a proper constant Ci>0, such thatAssumption 1 Let (T(r)Ir>o) be a family ofl|2≥clu|2,va∈V(4) operators acting on,?i)T(r)Σ=∑,r∈R;2 Non-autonomous systems and theirii)translation identity:U (t +r,+r)= Ur,(t,),attractorsd∈Σ,t≥r,r∈R,r≥0,(8)n this section we iterate some notations andNow we recall the results in [14]theorems in [13-15], which are necessary to getDefinition3 A family of processes (U, (t, t)our main results.∈Σ is said to be uniformly(wrta∈z) or limitLet e be a Banach space, and let a two compact if for any tER and BEB(E)the set Bparameter family of mappingsAUUU,(s, t)B is bounded for every t and{U(t,r)}={U(t,r)|t≥r,r∈R} act on EU(t,r):E→E,t≥r,r∈Rlim a(B, )=0. Where a is the Kuratowski measureDefinition 1 Let 2 be a parameter set. noncompactness defined by a(B)=inf (8 B admits a(U,(t,t)lt>r,tER), 0Ee is said to be a family of finite cover by set of diameter less than 8)processes in Banach space E, if for each g∈ΣTheorem 1 A family of processes (U, (t,t)U(t,r)} Is a process., that is, the twoσ∈Σ possesses compact uniform(wrta∈Σ)parameter family of mappings ( U, (t, r)) from E to attractor Ax satisfyingE satisfyAx=wx(Bo)=wr.(B),VTER (9)U2(t,s)·U(s,)=U(t,r),if and only if ityt≥s≥r,r∈Rbounded uniformly wrt oE 2U(r,r)=I,r∈R,6) absorbing set Boiwhere I is the identity operator, 2 is called the) is uniformly(wrta∈Σ) ar limit compact,symbol space and aE$ is the symbol.where a(B)=∩B, is the uniform(wrto∈E)a set B, Ce is said to be uniformly( wrt oEE) wr limit set中国煤化工absorbing set for the family of processes tU,(tNexted to verify theCNMHGr)},G∈∑, if for any t∈ R and B∈B(E), there uniform(wtVastness2011年第6期马巧珍等:吊桥方程的动力学性态2011No.6Dynamics of suspension bridge equationsDefinition 4 A family of processes(U, (t, r)), since the time-dependent term makes no essentialaEE is said to be satisfying the uniform ( wrt oE complicatIonsdirectly give the folleE)condition (C) if for any fixed rER, BEB(E) results of existence and uniqueness of the solutionand e>0, there exists a to=to(r, B,e)>T and a without proving. In fact, the proof is based onfinite dimensional subspace E of E such thatthe Faedo-Galerkin approximation approaches, see)P(U UU(t,r)B)is bounded[l, 10] for details. Denote R,=[r, +oo].i)l(I-Pn)(∪UU(t,r)x)lk≤e,Theorem 3 If h,ur,P. are given satisfyingyx∈B,h∈L(R;H),n∈VP∈H,then(1)~(3)where dimEm=m and PE-E is a boundedhas a unique solutionprojector.∈C(R:;V),au∈C(R-;H)Theorem 2 A family of processes U (t,t)If, furthermore,a∈D(A),h∈L(R;V),thend∈Σ satisfying the uniform(wrta∈E) conditionu sa(C) implies the uniform wrt gE 2)limita∈(R-,D(A),a,u∈L(R;V)compactness. Moreover, if e is a uniformlMoreover,(u, a,u)are weakly continuous functionconvex Banach space, then the converse is truefrom [r, t] to D(A)XV, that is u(r)EC(RNow we introduce a new class of functionsD(A),au∈C(R,;V)which are translation bounded but not translationNow we will rewrite(1)a(3)as ancompact, the details see [14]. First we need the evolutionary system by introducing y(r)=(u((), p(t))following property which can be found in [13].andy=(ur,p) for brevity. Denote E。=V×HProperty 1 A function h(s) is translation and E1= D(A)XV be the space of vector functionscompact in Lioe (R; X)if and only ify(r)=(u(x),P(r))with the finite norms,respeci) for any r∈R, the seth()dst∈RlylE =(lu2+Ipl2)i,Is pre-X:lyle, =(IAu|2+Ipl2)1i)lim supl h(s)-h(s+D)lds=0.which is equivalent to respectively,Definition 5 Let X be a Hilbert space. AlyE =|ul2+Ip+eu |2,function hE L(R; X) is said to satisfy Conditionll E,=| 12+p+eu 2.(C)if for any e >0, there exists a finite Then the system( 1)-(3)is equivalent to thedimensional subspace X, of X such thatfollowing systemsup, I(I-Pa)h(x, s) xds0a,y= Ao, (y),yler =y,(12)andr∈R, we havewhere o(s)=h(r, s) is the symbol of (11). Thus,sup e arI(I-P)h(s)ds Ho such thattopological space Lio(R; X)()|2+|a(t)|2≤,t∈[0,(B)]Thus, for any h(r, t)EH(ho), the problem So(11)with h instead of ho possesses a correspondingla(t)|2+|a,a(t)2≤,Vt≥0.(16)processes YUA(t,r)) acting on Eoor E1)Thus, exploiting (u*), Is|u, I we obtainProposition 1 If X is a reflexive seperablek(ut, Au)=k(ut, Au)then1) for all A1∈H(y),‖h1B(n,x≤hnk((ut),, Au)+Eok(ut, Au)>2)the translation group iT(t)) is weaklydk(,)-k|(+)||A|+ak(+,A)≥continuous on H(h),3) T(t)H(h)=H(h)for all tER.d k(u, Au)-klu, l- Au|+Therefore, the family of processes (U.(t,r))eok(u+,Aa)≥h∈H(h),U(t,x):E(orE1)→Eo(orE1)aredefined, t>t,tER. Furthermore, the translationd k(u, Au)-ku1 lAu +Eok(ut, Au)>semigroup{T(r)|r∈R} satisfies that yr∈Rk(a+,An)--。|An2T(r)H(ho)= H(h,), and the following translationIdene+Eok(u*,Au)(17)UA(t +r,r+r)=UT(t,r)According to Gs), Theorem 4 again and theh∈H(ho),t≥r,r∈R,r≥0.Sobolev embedding theorem we know that g (u),Theorem 416) Assume that(G1)-(G3)holdg(u) are uniformly bounded in L. That isand ho(x,s)ELC(R; H). Then there exists a there exists a constant K, >0 such thatpositive constant Ao, such thatlg(a)|≤K3,|g'(a)|x-≤K3.(18)12+|au|2≤3(13)Then combining with the Holder, Cauchy3. 1 A priori estimatesequalities and (16) yieldsFirst we prove that u and a, u are uniformly(g(a),A)=d(g(),A)-bounded in El. Choose 0(g(u), Au)+Eo(g(u), Au)1 Au 12+lul2)+Eo lAu 2+lg'(u)·|x|·| Au dr≥(-c0)lvl2-0(-)(An,v)+dk(ut, Au)+(g(u), Av)(g(u),A)+e(g(u),A)(h(t), Av)(14)KThanks to the holder and young inequalities, wehaaveg(u), Au)+Eo (g(u), Au)Eo Au 2+(8-Eo) vI2-Eo(8-Eo)(Au, v)>中国煤化工(19)n(1-e)Aa1+(3-)l15)CNMHGTogether with (14)-(15),(17)and(19),it2011年第6期马巧珍等:吊桥方程的动力学性态2011No.6Dynamics of suspension bridge equationsfollows thatby the gronwall lemmad(|Aa|2+lv2+2k(a+,An)+Y(t)≤Y(r)e"+C(1+a2)+δ(1+a1)lh2(g(u),An))+εoEoTherefore y(t)≤, where ui=2(C+δ1h")(1+ao), that is-e)lv『2+Aa()+g((t)+kn+(t)|2+lv(t)‖2≤吗Eok(ut, Au)+Eo(g(u), Au)o, ho (x, s)eTV凵中国煤化工 en the family(2K3+2kK3+2∈0kK3)+εK.rocessesCNMHGorrespondingRecall that Ih l 0 such thatan orthonormal basis of v and H. The(g(a)lx)21<4,corresponding eigenvalues are denoted by0u3)for any t>0 with the same as estimatesu= Pu+(I-P u=ui+uz,of(16)where P:vIs an orthogonal projector.Combining with( 25)-( 30)and using theSince 8: D(A)-V is compact by Lemma 2 Poincare inequality again, we conclude thatfor any e>0, there exists some m, such thatd(|Aa2|2+lw2|2+2k((u+)2,Aa2)+(I-Pn)g()≤5,2((g(u)2,Aa2))+a|Aa2|2+。lv2|2+yu∈BDA(0,p3)(24)2kx((4+)2,A2)+2o(g(u))2,Aa2)≤Chose O 0, there exists an m largeapplication[J]. Indiana University Math J, 2002enough such that51(6):1541-1559[9] ZHONG Cheng-kui, YANG Mei-hua, SUN Chun-exp(-a(t-s))(I-P)(t)'ds syou. The existence of global attractors for the norm-Ⅴh∈H(h),Vt≥(34)to-weak continuous semigroup and application to thenonlinear reaction diffusion equations [j]. J DiffLet t2=r+-In(3u4EI), then we haveEqu,2006,223(2):367-399[10] ZHONG Cheng-kui, MA Qiao-zhen, SUN Chun-2e"≤,Vtyou, Existence of strong solutions and globalClearly we can choose g=e(e, such that ce