一类耦合非线性梁方程组的整体解
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摘要
随着应用数学的发展以及相关学科的推动,偏微分方程成为数学理论与实际应用的一座重要桥梁。自然科学和工程技术领域很多重大问题都可归结为对非线性偏微分方程的研究,而梁方程是非线性偏微分方程的一个重要组成部分
目前关于梁方程的研究主要以局部解的存在性,整体解的存在性、正则性及能量衰减估计等为主。
本文建立如下梁方程组：并研究了在初始条件和边界条件下,该方程组的整体弱解、强解的存在性和唯一性。
具体研究内容如下：
首先,对与本文相关的非线性偏微分方程(组)的发展和研究现状进行了简单的总结和评述；
其次,给出了文章中需要的一些重要概念和引理,并对部分符号做了说明；
第三,利用Galerkin方法证明了方程组(1)—(6)的近似方程：整体弱解的存在性和唯一性；
第四,证明来了方程组(1)—(6)的强解的存在性和唯一性。
With the development of applied mathematics and promotion of relevant branch of learning, partial differential equation is an important bridge between mathematics theory and actual application. Many problems in natural science and engineering field can be approximately solved with nonlinear partial differential equation. And also beam equation is an important content of nonlinear partial differentia] equation.
In recent years, the study on nonlinear partial differential equation mainly focuses on the existence of partial solutions, the existence of global solutions, regularity and the estimation on the energy decrease.
In this paper, we present the following nonlinear system of partial differential equations And consider the problem of finding u and v solutions of the system (1)-(2), verifying the initial conditions And the boundary conditions
The particular content is following:
1. We made simple sum-up and comment on the developing and actuality of study on partial differential equations relevant with this paper.
2. We give some important definitions and lemmas, and the part of the text symbols is explained.
3. We proved the existence and uniqueness of weak solutions of nonlinear system of beam equations (1)-(6) using Galerkin method.
4. Proved the existence and uniqueness of the strong solution.
目前关于梁方程的研究主要以局部解的存在性,整体解的存在性、正则性及能量衰减估计等为主。
本文建立如下梁方程组：并研究了在初始条件和边界条件下,该方程组的整体弱解、强解的存在性和唯一性。
具体研究内容如下：
首先,对与本文相关的非线性偏微分方程(组)的发展和研究现状进行了简单的总结和评述；
其次,给出了文章中需要的一些重要概念和引理,并对部分符号做了说明；
第三,利用Galerkin方法证明了方程组(1)—(6)的近似方程：整体弱解的存在性和唯一性；
第四,证明来了方程组(1)—(6)的强解的存在性和唯一性。
With the development of applied mathematics and promotion of relevant branch of learning, partial differential equation is an important bridge between mathematics theory and actual application. Many problems in natural science and engineering field can be approximately solved with nonlinear partial differential equation. And also beam equation is an important content of nonlinear partial differentia] equation.
In recent years, the study on nonlinear partial differential equation mainly focuses on the existence of partial solutions, the existence of global solutions, regularity and the estimation on the energy decrease.
In this paper, we present the following nonlinear system of partial differential equations And consider the problem of finding u and v solutions of the system (1)-(2), verifying the initial conditions And the boundary conditions
The particular content is following:
1. We made simple sum-up and comment on the developing and actuality of study on partial differential equations relevant with this paper.
2. We give some important definitions and lemmas, and the part of the text symbols is explained.
3. We proved the existence and uniqueness of weak solutions of nonlinear system of beam equations (1)-(6) using Galerkin method.
4. Proved the existence and uniqueness of the strong solution.
引文
[1]J.M. BALL, Initial boundary value problems for an extensible beam, J, Math.Anal.42(1973),61-90
[2]Ball JM, Stability theory for an extensible beam, Journal of Differential Equation,1973,14(3),61-90Equation,1973,14(3),61-90
[3]Senkyrik M, Fourth order boundary value problems and nonlinear beams, Applicable Aualusis,59(1995),15-25
[4]Yamada Y, Some nonlinear degenerate wave equation, Nonlinear Anaalysis, TMA,1987,11(10),1155-1168
[5]Pereira DC, Existence uniqueness and asymptotic behavior for solution of the nonlinear beam equation, Nonlinear Analysis TMA,1990,14 (8), 613-623
[6]De Brito EH, Decay estimates for the generalized damped extensible String and beam equations, Nonlinear Analysis,8(1984),1489-1496
[7]张建文等,具强迫项非线性梁方程的渐进性,应用数学,2001,14(1),60-66
[8]张建文等,具强阻尼项非线性粘弹性梁方程的整体解,应用数学学报,2003,20(2),30-34
[9]Ono K, Mildly degenerate Kirchhoff equations with damping in unbounded domain, Applicable Analysis,67(1997),221-132
[10]Ono K, On global solutions and blow-up solutions of nonlinear Kirchhoff strings with nonlinear dissipation, J. Math. Appl.,216(1997) 321-342
[11]Arosio A&th. Garavaldi S, On the mildly degenerate Kirchhhoff string,Math.Meth.Appl. Sci.14(1991),177-195
[12]Manfrin R,Global solutions to Kirchhoff equations with non-local nonlinearities definedby potential Analysis,69(1998),1-13
[13]Webb GF, Existence and asymptotic behavior for a strongly damped nonlinear wave eqution, Can. J. Math., Vol.32,3(1980),631-643
[14]Kouemou-Patcheu S,Gloal esistence and exponential decay estimates for a damped quasilinearvequation,Commum, in Partial Diff. Eps.22 (11&12),1997,20072024
[15]牛丽芳,张建文Initial-boundary value problems for a kind of nonlinear system of partial differential equations, International Journal of Pure and Applied Mathematics,2008
[16]牛丽芳,张建文一类非线性梁方程强解的存在性,太原理工大学学报,2008
[17]Ebihara Y,Local solution for a nonlinear degenerate hyperbolic equation, Nonlinear Analysis TMA,1986,10(1),27-40.
[18]R.W. Dickey, Infinite systems of nonlinear oscillations equations related to the string, Proc. Amer. Math. Soc. (1969),459-468.
[19]De Brito E H. Decay eatimates for the generalized damped extensible Sting and beam equations[J]. Nonlinear Analysis,1984,8:1489-1496
[20]Ono K. On global existence, asymptotic stability and blowing up solutions for some degenerate nonlinear wave equations of Kirchhoff type with a strong dissipation[J]. Math. Appl. Sci.,1997,20:151-177.
[21]Solange Kouemou Patcheu. On a global solution and asymptotic behaviour for the generalized damped extensible beam equation[J]. J Differential Equa-tions,1997,135:299-314.
[22]Jong Yeoul Park and Jeong Ja Bae, Pusan. On solutions of quasilinear wave equations with nonlinear damping term[J]. Czechoslovak Mathematical Journal,2000,50(125):565-584.
[23]Lions JL. Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunond, Gauthier-Villars,Paris,1969.
[24]Le Xuan Ttuong,Le Thi Phuong Ngoc, Nguyen Thanh Long, High-order iterative schemes for a nonlinear Kirchhoff-Carrier wave eqution associated with the mixed homogeneous condtions, Nonlinear Analysis 71(2009)476-484
[25]Nguyen Thanh Long, Vo Giai, Existence and asymptotic expansion for a nonlinear wave equation associated with nonlinear boundary conditions, Nonlinear Analysis 67(2007)1791-1819.
[26]Xinfu Chen, Jong-Shenq Guo,Existence and uniqueness of entire solutions for a reaction-diffusion equation, J. Differential Equations 212(2005)62-84.
[27]J.-L Guermond, Finite-element-based Faedo-Galerkin weak solutions to the Navier-Stokes equations in three-dimensional torus are suitable, J. Math. Pures Appl.85(2006)451-464.
[28]吴奕飞.一类耦合非线性系统解的整体存在性和渐进性[D].广州：华南理工大学,2008.
[29]李桂莲.非线性弹性杆动力学方程解的存在性和唯一性研究[D].太原：太原理工大学,2005.
[30]呼青英.具结构阻尼的非线性梁方程的初边值问题[D].郑州：郑州大学,2003.
[31]张建文等.具非线性耗散粘弹性梁方程的整体解[J].工程数学学报,2001,18(2)：61-68.
[32]Takeshi Taniguchi. Existence and asymptotic behaviour of solutions to weakly damped wave equations of Kirchhoff type with nonlinear damping and source terms[J].J Math Anal Appl,2010,361:566-578.
[2]Ball JM, Stability theory for an extensible beam, Journal of Differential Equation,1973,14(3),61-90Equation,1973,14(3),61-90
[3]Senkyrik M, Fourth order boundary value problems and nonlinear beams, Applicable Aualusis,59(1995),15-25
[4]Yamada Y, Some nonlinear degenerate wave equation, Nonlinear Anaalysis, TMA,1987,11(10),1155-1168
[5]Pereira DC, Existence uniqueness and asymptotic behavior for solution of the nonlinear beam equation, Nonlinear Analysis TMA,1990,14 (8), 613-623
[6]De Brito EH, Decay estimates for the generalized damped extensible String and beam equations, Nonlinear Analysis,8(1984),1489-1496
[7]张建文等,具强迫项非线性梁方程的渐进性,应用数学,2001,14(1),60-66
[8]张建文等,具强阻尼项非线性粘弹性梁方程的整体解,应用数学学报,2003,20(2),30-34
[9]Ono K, Mildly degenerate Kirchhoff equations with damping in unbounded domain, Applicable Analysis,67(1997),221-132
[10]Ono K, On global solutions and blow-up solutions of nonlinear Kirchhoff strings with nonlinear dissipation, J. Math. Appl.,216(1997) 321-342
[11]Arosio A&th. Garavaldi S, On the mildly degenerate Kirchhhoff string,Math.Meth.Appl. Sci.14(1991),177-195
[12]Manfrin R,Global solutions to Kirchhoff equations with non-local nonlinearities definedby potential Analysis,69(1998),1-13
[13]Webb GF, Existence and asymptotic behavior for a strongly damped nonlinear wave eqution, Can. J. Math., Vol.32,3(1980),631-643
[14]Kouemou-Patcheu S,Gloal esistence and exponential decay estimates for a damped quasilinearvequation,Commum, in Partial Diff. Eps.22 (11&12),1997,20072024
[15]牛丽芳,张建文Initial-boundary value problems for a kind of nonlinear system of partial differential equations, International Journal of Pure and Applied Mathematics,2008
[16]牛丽芳,张建文一类非线性梁方程强解的存在性,太原理工大学学报,2008
[17]Ebihara Y,Local solution for a nonlinear degenerate hyperbolic equation, Nonlinear Analysis TMA,1986,10(1),27-40.
[18]R.W. Dickey, Infinite systems of nonlinear oscillations equations related to the string, Proc. Amer. Math. Soc. (1969),459-468.
[19]De Brito E H. Decay eatimates for the generalized damped extensible Sting and beam equations[J]. Nonlinear Analysis,1984,8:1489-1496
[20]Ono K. On global existence, asymptotic stability and blowing up solutions for some degenerate nonlinear wave equations of Kirchhoff type with a strong dissipation[J]. Math. Appl. Sci.,1997,20:151-177.
[21]Solange Kouemou Patcheu. On a global solution and asymptotic behaviour for the generalized damped extensible beam equation[J]. J Differential Equa-tions,1997,135:299-314.
[22]Jong Yeoul Park and Jeong Ja Bae, Pusan. On solutions of quasilinear wave equations with nonlinear damping term[J]. Czechoslovak Mathematical Journal,2000,50(125):565-584.
[23]Lions JL. Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunond, Gauthier-Villars,Paris,1969.
[24]Le Xuan Ttuong,Le Thi Phuong Ngoc, Nguyen Thanh Long, High-order iterative schemes for a nonlinear Kirchhoff-Carrier wave eqution associated with the mixed homogeneous condtions, Nonlinear Analysis 71(2009)476-484
[25]Nguyen Thanh Long, Vo Giai, Existence and asymptotic expansion for a nonlinear wave equation associated with nonlinear boundary conditions, Nonlinear Analysis 67(2007)1791-1819.
[26]Xinfu Chen, Jong-Shenq Guo,Existence and uniqueness of entire solutions for a reaction-diffusion equation, J. Differential Equations 212(2005)62-84.
[27]J.-L Guermond, Finite-element-based Faedo-Galerkin weak solutions to the Navier-Stokes equations in three-dimensional torus are suitable, J. Math. Pures Appl.85(2006)451-464.
[28]吴奕飞.一类耦合非线性系统解的整体存在性和渐进性[D].广州：华南理工大学,2008.
[29]李桂莲.非线性弹性杆动力学方程解的存在性和唯一性研究[D].太原：太原理工大学,2005.
[30]呼青英.具结构阻尼的非线性梁方程的初边值问题[D].郑州：郑州大学,2003.
[31]张建文等.具非线性耗散粘弹性梁方程的整体解[J].工程数学学报,2001,18(2)：61-68.
[32]Takeshi Taniguchi. Existence and asymptotic behaviour of solutions to weakly damped wave equations of Kirchhoff type with nonlinear damping and source terms[J].J Math Anal Appl,2010,361:566-578.