您好,欢迎访问三七文档
当前位置:首页 > 商业/管理/HR > 管理学资料 > 一类广义耦合的非线性波动方程组时间周期解的存在性Ξ
:1000-0887(2003)06-0595-10X1,2,3(1.,512005;2.,100088;3.,100088)():Galerkin,Laray-Schauder,,:;;:O175.25;O175.29:A[1]ut=uxxx+buux+2vvx,(1)vt=2(uv)x(2)ItoM.[2],(1)(2)P.F.He[3]KdV[4]ut=a(uxxx+buux)+2bvvx,(3)vt=-vxxx-3uvx,(4)abM.E.Schonbek[5][6]ut=uxxx-uux-vx,(5)vt=-(uv)x(6)L1Dunford,ut+f(u)x-uxx+uxxx+2vvx=G1(u,v)+h1(x),(7)595,246(20036)AppliedMathematicsandMechanicsX:2001-05-28;:2003-03-03:(1964),,,,,:(E-mail:dz90@163.net)©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.vt-vxx+(2uv)x+g(v)x=G2(u,v)+h2(x),(8)u(x,t+)=u(x,t),v(x,t+)=v(x,t)xR,tR,(9)u(x+D,t)=u(x-D,t),v(x+D,t)=v(x-D,t)xR,tR,(10)0D0000u(x,t)v(x,t)t,;x,u(x,t)v(x,t),f(u)g(v)G1(u,v)G2(u,v)(7)(10),L2,pLp,mHm,=(-D,D),t0,0,L2()(u,v)=uvdxHilbertXBanach,CK(,X)X1K(),:uCK(,X)=sup0t6Ki=1DituXLp(,X)(1p)XuLp(,X)=0upX1/p(1p),uL(,X)=sup0tuX1GalerkinLaray-Schauder(7)(10)j(x)(j=1,2,)j+jj=0(9)(10)j(j=1,2,)L2j(x)(7)(10)uN(x,t),vN(x,t)uN(x,t)=6Nj=1jN(t)j(x),vN(x,t)=6Nj=1jN(t)j(x),(11)jN(t),jN(t)(j=1,2,,Nj;N=1,2,)tR+Galerkin,jN(t)jN(t)(12)(13):(uNt+f(uN)x-uNxx+uNxxx+2vNvNx-G1(uN,vN)-h1(x),j(x))=0,(12)(vNt+g(vN)x-vNxx+2(uNvN)x-G2(uN,vN)-h2(x),j(x))=0(13),Laray-Schauder(12)(13)N,HN1,2,,N,HN=span1,2,,N695©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.N(x,t)=6Nj=1jN(t)j(x),N(x,t)=6Nj=1jN(t)j(x)(14)T:(N,N)(uN,vN)(01)(N,N)C1(,HN),(uN,vN)C1(,HN):(uNt-uNxx+uNxxx+M1(N,N)-h1(x),j(x))=0,(15)(vNt-vNxx+M2(N,N)-h2(x),j(x))=0,(16)M1(N,N)=f(N)x+2NNx-G1(N,N),M2(N,N)=g(N)x+2(NN)x-G2(N,N)T:(N,N)(uN,vN)C1(,HN)=0,(15)(16)(uN,vN)=(jN(t),jN(t))C1(,HN),T0(uN,vN)C1(,HN)Leray-SchauderT,M1(N,N)M2(N,N)M1(uN,vN)M2(uN,vN)(15)(16),uN2+vN21+1E1,(17)E1N,h1h21.11)Gi(0,0)=0(i=1,2),(,)-G1u-G2u-G1v-G2vb0(||2+||2),(,)R2,b002)hi(x)L2()(i=1,2),=(-D,D)3)uN(x,t),vN(x,t)C1(,HN)(17)(15)(16)(uNt+f(uN)x-uNxx+uNxxx+2vNvNx-G1(uN,vN)-h1(x),j(x))=0,(18)(vNt+g(vN)x-vNxx+2(uNvN)x-G2(uN,vN)-h2(x),j(x))=0(19)jN(t)jN(t)(18)(19),1Nj,(uNt+f(uN)x-uNxx+uNxxx+2vNvNx-G1(uN,vN)-h1(x),uN)=0,(20)(vNt+g(vN)x-vNxx+2(uNvN)x-G2(uN,vN)-h2(x),vN)=0(21)(uN,vN)=D-DuN(x,t)vN(x,t)dx,(uN,f(uN)x)=0,(uN,-uNxx)=uNx2,(vN,-vNxx)=vNx2,(uN,uNxxx)=0,(vN,g(vN)x)=0,(uN,2vNvNx)+(vN,2(uNvN)x)=2uNvNvNxdx-2uNvNvNxdx=0,795©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.(uN,G1(uN,vN))+(vN,G2(uN,vN))=((uN,G1uNuN+G1vNvN)+(vN,G2uNuN+G2vNvN))=(uN,vN)-G1uN-G2uN-G1vN-G2vNuNvN-b0(uN2+vN2),(uN,h1(x))b02u2+12b0h12,(vN,h2(x))b02v2+12b0h22(20)(21),12ddt(uN2+vN2)+uNx2+vNx2+b0(2-1)(uN2+vN2)12b0(h12+h22)(22)0,(22),0(uN(,t)2+vN(,t)2)dtb20(2-1)(h1(x)2+h2(x)2)=E1(23),t30,,uN(,t3)2+vN(,t3)21b20(2-1)(h1(x)2+h2(x)2)=E1(24)(22)t3t+(t0,),uN(,t)2+vN(,t)2(uN(,t3)2+vN(,t3)2)+b20(2-1)(h1(x)2+h2(x)2)=1+1E1,(25)uN(,t)2+vN(,t)21/+1E1,NE1,Laray-Schauder,(15)(16)C1(,HN),=1,C1(,HN)(15)(16)(uN,vN)1.1N,(15)(16)(uN,vN)C1(,HN)21,(7)(10)(uN,vN),(uN,vN),(7)(10),(7)(10)2.1(Sobolevs[8])uLq(),1r,q,RnC0,DjuLp()CDmuaLr()u1-aLq(),0jm,j/ma1,1p1/p=j/n+a1/r-m/n+(1-a)/q895©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.2.21.1,1)f(u)C1,g(v)C1,Gi(u,v)C1(i=1,2);2)|f(u)|A|u|5-,|g(v)|B|v|6-,A0,B0,0,|Gi|Ci(|u|5+|v|5),Ci0(7)(10),uNx2+vNx21/+1E2,(26)E2N(uNt+f(uN)x-uNxx+uNxxx+2vNvNx-G1(uN,vN)-h1(x),j)=0,(27)(vNt+g(vN)x-vNxx+2(uNvN)x-G2(uN,vN)-h2(x),j)=0(28)jN(27),j1N,(uNt+f(uN)x-uNxx+uNxxx+2vNvNx-G1(uN,vN)-h1(x),uNxx)=0,(29)(uNxx,uNt)=-12ddtuNx2,(uNxx,f(uN)x)=-(uNxxx,f(uN))=1(uNt+f(uN)x-uNxx+2vNvNx-G1(uN,vN)-h1(x),f(uN))=1ddtF(uN)dx-(uNxx,f(uN))+2(vNvNx,f(uN))-1(G1(uN,vN)+h1(x),f(uN))Sobolev,|(uNxx,f(uN))|uNxxf(uN)AuNxxuN5-10-212uNxx2+C(uN),2|(vNvNx,f(uN))|2||vN4vNxf(uN)212uNxx2+8vNxx2+C(uN,vN),1(G1(uN,vN),f(uN))4C1||(uN10-10-+vN5-10-2+vN510)12uNxx2+8vNxx2+C(uN,vN),|(uNxx,h1)|uNxxh112uNxx2+3h12,(uNxx,-uNxx)=-uNxx2,(uNxx,uNxxx)=0(29),12ddtuNx2-2F(uN)dx+712uNxx2-2(uNxx,vNvNx)-(uNxx,G1(uN,vN))+4uNxx2+3h12+C(30)jN(28),j1N,(vNt+g(vN)x-vNxx+2(uNvN)x-G2(uN,vN)-h2(x),vNxx)=0,(31)995©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.(vNxx,vNt)=-12ddtvNx2,(vNxx,-vNxx)=-vNxx2,|(vNxx,g(vN)x)|CvNxxg(vN)x16vNxx2+C,|(vNxx,h2)|vNxxh216vNxx2+4h22,|2(uNxx,vNvNx)+2(vNxx,(uNvN)x)|=3uNxv2Nxdx3uNx2vNx2424uNxx2+16vNxx2+C,(uNxx,G1(uN,vN))+(vNxx,G2(uN,vN))=-(uNx,G1x(uN,vN))-(vNx,G2x(uN,vN))=-(uNx,G1uNuNx+G1vNvNx)-(vNx,G2uNuNx+G2vNvNx)=-(uNx,vNx)G1uNG2uNG1vNG2vNuNxvNx-b0(uNx2+vNx2)(30)(31),ddt(t)+2b0(t)+uNxx2+vNxx23h12+4h22+C,(32)(t)=uNx2+vNx2-2F(uN)dx0,(32),2F(u)dxAu6-6-12uxx2+C,0(uNx2+vNx2)dt2b03h12+4h22=E2(33)t30,,uNx(,t3)2+vNx(,t3)212b03h12+4h22=E2(34)(32),t3t+(t0,),uNx(,t)2+vNx(,t)2uNx(,t3)2+vNx(,t3)2+E21/+1E2,(35)(26)2.32.2,1)f(u)C2,g(v)C2,Gi(u,v)C2(i=1,2);2)hi(x)H1()(i=1,2)(7)(10),uNxx2+vNxx21+1E3,(36)E3N006©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.(uNt+f(uN)x-uNxx+uNxxx+2vNvNx-G1(uN,vN)-h1(x),(4)j)=0,(37)(vNt+g(vN)x-vNxx+2(uNvN)x-G2(uN,vN)-h1(x),(4)j)=0(38)jN(37),j1N,(uNt+f(uN)x-uNxx+uNxxx+2vNvNx-G1(uN,vN)-h1(x),uNxxxx)=0,(39)|(uNxxxx,f(uN)x)|=|uNxxx,f(uN)u2N(x)+f(uN)uNxx|f(uN)uNxxxuNx24+f(uN)uNxxxuNxx10uNxxx2+C,|(uNxxxx,-uNxx)|=uNxxx2,(uNxxxx,uNxxx)=0,|(uNxxxx,2vNvNx)|=|(uNxxx,2(v2Nx+vNvNx))|8vNxxx2+10uNxxx2+C,|(uNxxxx,h1)|uNxx
本文标题:一类广义耦合的非线性波动方程组时间周期解的存在性Ξ
链接地址:https://www.777doc.com/doc-725547 .html