几何方法在变系数偏微分方程解稳定性上的应用
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摘要
波动方程是最重要,最早,和研究最多的一类偏微分方程,主要是应用泛函分析的知识来研究波动方程解的稳定性的问题.但由于原先线性波动方程的解法无法应用到变系数波动方程,其已经成为偏微分方程的一个重要课题,很多数学家在这方面进行研究并获得了很多成果.本文的主要工作是应用黎曼几何的方法分析记忆型边界反馈下变系数波动方程解的指数稳定性.
本文的组织结构是：首先在引言中介绍了Riemannian几何的一些基本概念及其与波动方程有关的一些等式关系为下面证明中的应用做好准备,其次讨论了有记忆型边界的耦合半线性系统：解的指数稳定性,即应用Riemannian几何的方法证明系统弱解的能量是指数衰减的.最后又应用相同的方法讨论了有记忆型边界的变系数波动方程：解的指数稳定性.
Wave equations are one type of the most important, the earliest and most frequently researched partial differential equations.Most researchers study the stabilization of solutions for the wave equations.But since the variable coefficients wave equations can not be analyzed by this method.And the theories of fixed coefficient wave equations can not be used in variable coefficient wave equations.Many mathematicians are working in this area and have got many results. We use the Rimannian geometric method to analyze the variable coefficient wave equations with a boundary control of memory type in this paper.
The organizational structure of this article:first,we introduce the basic concepts of Rie-mannian Geometry and the equations about the wave equations in this area for the used in the after demonstration. Second, we study the boundary stabilization of solutions for the coupled semilinear system with variable coefficients In the end, we study the boundary stabilization of solutions for the wave equation with a boundary control of memory type and with variable coefficients
本文的组织结构是：首先在引言中介绍了Riemannian几何的一些基本概念及其与波动方程有关的一些等式关系为下面证明中的应用做好准备,其次讨论了有记忆型边界的耦合半线性系统：解的指数稳定性,即应用Riemannian几何的方法证明系统弱解的能量是指数衰减的.最后又应用相同的方法讨论了有记忆型边界的变系数波动方程：解的指数稳定性.
Wave equations are one type of the most important, the earliest and most frequently researched partial differential equations.Most researchers study the stabilization of solutions for the wave equations.But since the variable coefficients wave equations can not be analyzed by this method.And the theories of fixed coefficient wave equations can not be used in variable coefficient wave equations.Many mathematicians are working in this area and have got many results. We use the Rimannian geometric method to analyze the variable coefficient wave equations with a boundary control of memory type in this paper.
The organizational structure of this article:first,we introduce the basic concepts of Rie-mannian Geometry and the equations about the wave equations in this area for the used in the after demonstration. Second, we study the boundary stabilization of solutions for the coupled semilinear system with variable coefficients In the end, we study the boundary stabilization of solutions for the wave equation with a boundary control of memory type and with variable coefficients
引文
[1]Salim A.Messaouli,Abdelaziz Soufyane. General decay of solution of a wave equation with a boundary control of memory type [J], Transactions of the American Society of Civil Engineers,2010,70:4226-4244.
[2]Peng-fei Yao.On the observability inequations for exact controllability of wave equations with variable coefficient [J].Society for Industrial and Applied Mathematics,1999,37,1568-1599.
[3]J.Y.Park,J.J.Bae,Hyo Jung.Uniform decay for wave equation of Kirchhoff type with nonlinear boundary damping and mrmory term [J]. Nonlinear Anal,2002,50:871-884.
[4]M.LSantos,F.Junior.As boundary condition with memory for Kirchhoff plates equations [J],Appl.Math.Comput,2004,148(2),475-496.
[5]M.M.Cavalcanti,V.N.Domingos Cavalcanti,J.S.Prates Filho,J.A.Soriano,Existence and uniform decay of solutions of a degenerate equaiton with nonlinear boundary memory source term [J]. Nonlinear Anal.1999,38,281-294.
[6]J.E.Munoz Rivera,D.Andarade. Exponential decay of solutions to a nonlinear wave equation with a viscoelastic boundary [J].Math.Methods Appl.Sci,2000,Vol.23:41-61.
[7]M.M.Cavalcanti,A.Guesmia,General decay rates of solutions to a nonlinear wave equa-tion with boundary conditions of memory type [J].Differential Integral Equations,2005,Vol.18:583-600.
[8]S.A.Messaoudi,A.Soufyane,Boundary stabilization of solutions of a nonlinear system of Timoshenko type.Nonlinear Anal,2007,Vol.67:2107-2121.
[9]M.M.Cavalcanti,V.N.Domingos Cavalcanti,J.S.Prate Filho,J.A.Soriano.Existence and uni-form decay rates for viscoelastic problems with nonlinear boundary damping[J].Differential Integral Equations,2001,Vol.14,85-116.
[10]S.A.Messaoudi,M.L.Mustafa.On the control of solutions of viscoelastic equations with boundary feedback[J].Nonlinear Anal.2009,Vol.10,3132-3140.
[11]M.M.Cavalcanti,V..N.Domingos Cavacanti,P.Marinez.Gerenal decay rate estimates for viscoelastic dissipative systems[J].Nonlinear Anal,2008,Vol.68,177-193.
[12]F.Alabau-Boussouira,Convexity and weighted integral inequations for energy decay rates of nonlinear dissipative hyperbolic systems[J].Appl.Math Optim,2005,Vol.51,61-105.
[13]M.M.Cavalcanti,V.N.Domingos Cavalcanti,M.L.Santos.Existence and uniform decay rates of solutions t a degenerate system with memory conditions at the boundary [J]. Appl.Math.Com 465.
[14]M.M.Cavalcanti,V.N.Domingos Cavalcanti,I.Lasiecks.Well-posedness and optional de-cay rates of the wave equation with nonlinear boundary damping-source interaction.Differential Equations,2007,Vol(236),407-459.
[15]F.Conrad,B.Rao.Decay of solution of wave equations in a star-shaped domain with nonlinear boundary feedback[J].Asympt.Anal,1993,Vol.7:159-177.
[16]G.C.Gorain,B.Rao.Boundary stabilization of nonlinear vibrations of a flexible structure in a bounded domain in Rn [J],Math Anal.Appl,2006,Vol(319):635-650.
[17]A.Gusesmia,S.A.Messaoudi.On the boundary stabilization of a compactly coupled sys-tem of nonlinear wave equations[j],Evol.Equ,2005,Vol(1):211-224.
[18]V.Komornik,E.Auazua,A direct method for the boundary stabilization of the wave equation[j],Math.Pure Appl.1990,Vol(69):33-54.
[19]V.Komornik,On the nonlinear boundary stabilization of the wave equations.Ann.Math,1993:153-164
[20]V.Komornik,Bomndary stabilization of compactly coupled wave equations.Asympt. Anal,1997:339-359
[21]M.L. Santos and F. Junior. A boundary condition with memory for Kirchhoff plates equations. Appl. Math. Comput,2004,148,475-496.
[22]M.M. Cavalcanti, V.N. Domingos Cavalcanti, J.S. Prates Filho and J.A. Soriano. Exis-tence and uniform decay of solutions of a degenerate equation with nonlinear boundary memory source term. Nonlinear Anal,1999,38,281-294.
[23]J.L. Lions, E. Magenes. Problems aux Limites Non-homogenes et Applications. Dunod(Paris), 1968,1,17-18.
[24]A.Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equa-tions. Springer-Verlag, New York,1983,27,309-311.
[2]Peng-fei Yao.On the observability inequations for exact controllability of wave equations with variable coefficient [J].Society for Industrial and Applied Mathematics,1999,37,1568-1599.
[3]J.Y.Park,J.J.Bae,Hyo Jung.Uniform decay for wave equation of Kirchhoff type with nonlinear boundary damping and mrmory term [J]. Nonlinear Anal,2002,50:871-884.
[4]M.LSantos,F.Junior.As boundary condition with memory for Kirchhoff plates equations [J],Appl.Math.Comput,2004,148(2),475-496.
[5]M.M.Cavalcanti,V.N.Domingos Cavalcanti,J.S.Prates Filho,J.A.Soriano,Existence and uniform decay of solutions of a degenerate equaiton with nonlinear boundary memory source term [J]. Nonlinear Anal.1999,38,281-294.
[6]J.E.Munoz Rivera,D.Andarade. Exponential decay of solutions to a nonlinear wave equation with a viscoelastic boundary [J].Math.Methods Appl.Sci,2000,Vol.23:41-61.
[7]M.M.Cavalcanti,A.Guesmia,General decay rates of solutions to a nonlinear wave equa-tion with boundary conditions of memory type [J].Differential Integral Equations,2005,Vol.18:583-600.
[8]S.A.Messaoudi,A.Soufyane,Boundary stabilization of solutions of a nonlinear system of Timoshenko type.Nonlinear Anal,2007,Vol.67:2107-2121.
[9]M.M.Cavalcanti,V.N.Domingos Cavalcanti,J.S.Prate Filho,J.A.Soriano.Existence and uni-form decay rates for viscoelastic problems with nonlinear boundary damping[J].Differential Integral Equations,2001,Vol.14,85-116.
[10]S.A.Messaoudi,M.L.Mustafa.On the control of solutions of viscoelastic equations with boundary feedback[J].Nonlinear Anal.2009,Vol.10,3132-3140.
[11]M.M.Cavalcanti,V..N.Domingos Cavacanti,P.Marinez.Gerenal decay rate estimates for viscoelastic dissipative systems[J].Nonlinear Anal,2008,Vol.68,177-193.
[12]F.Alabau-Boussouira,Convexity and weighted integral inequations for energy decay rates of nonlinear dissipative hyperbolic systems[J].Appl.Math Optim,2005,Vol.51,61-105.
[13]M.M.Cavalcanti,V.N.Domingos Cavalcanti,M.L.Santos.Existence and uniform decay rates of solutions t a degenerate system with memory conditions at the boundary [J]. Appl.Math.Com 465.
[14]M.M.Cavalcanti,V.N.Domingos Cavalcanti,I.Lasiecks.Well-posedness and optional de-cay rates of the wave equation with nonlinear boundary damping-source interaction.Differential Equations,2007,Vol(236),407-459.
[15]F.Conrad,B.Rao.Decay of solution of wave equations in a star-shaped domain with nonlinear boundary feedback[J].Asympt.Anal,1993,Vol.7:159-177.
[16]G.C.Gorain,B.Rao.Boundary stabilization of nonlinear vibrations of a flexible structure in a bounded domain in Rn [J],Math Anal.Appl,2006,Vol(319):635-650.
[17]A.Gusesmia,S.A.Messaoudi.On the boundary stabilization of a compactly coupled sys-tem of nonlinear wave equations[j],Evol.Equ,2005,Vol(1):211-224.
[18]V.Komornik,E.Auazua,A direct method for the boundary stabilization of the wave equation[j],Math.Pure Appl.1990,Vol(69):33-54.
[19]V.Komornik,On the nonlinear boundary stabilization of the wave equations.Ann.Math,1993:153-164
[20]V.Komornik,Bomndary stabilization of compactly coupled wave equations.Asympt. Anal,1997:339-359
[21]M.L. Santos and F. Junior. A boundary condition with memory for Kirchhoff plates equations. Appl. Math. Comput,2004,148,475-496.
[22]M.M. Cavalcanti, V.N. Domingos Cavalcanti, J.S. Prates Filho and J.A. Soriano. Exis-tence and uniform decay of solutions of a degenerate equation with nonlinear boundary memory source term. Nonlinear Anal,1999,38,281-294.
[23]J.L. Lions, E. Magenes. Problems aux Limites Non-homogenes et Applications. Dunod(Paris), 1968,1,17-18.
[24]A.Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equa-tions. Springer-Verlag, New York,1983,27,309-311.