基于临界点理论的次二次四阶半线性常微分方程周期解的存在性研究
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摘要
微分方程已成为研究自然科学和社会科学的一个强有力工具,在科技和经济的发展过程中,越来越多的实际课题都可以建立关于四阶或者更高阶的常微分方程数学模型.
经典的Fisher-Kolmogorov(FK)'方程为
1988年,Dee and Van Saarloos在研究双稳态物理系统时建立了Extended Fisher-Kolmogorov(EFK)[2]方程
1977年,Swift and Hohenberg在研究流体的不稳定性时建立了Swift-Hohenberg(SH)[3]方程
人们感兴趣的是以上方程的驻波解,如果引入适当的变量变换上述方程可简化为下述四阶常微分方程
当p>0时,方程即为EFK方程；当p 本文主要是对下述更一般的四阶半线性常微分方程2T-周期解的存在性进行研究,其中A,B是常数,V(t,u)∈C~1(O,T]×R,R)具有以下性质：
(HO) V(t,O)=O,v(t+2T,u)=V(t,u),V(t,-u)=V(t,u),(?)t,∈[o,T],u∈R.
(H1)2V(t,u)-uV_u,(t,u)→-∞,|u|→∞,t∈[O,T],或2V(t,u)一uV_u(t,u)→∞,|u|→∞,t∈[O,T].假设ū＝ū(t)为边值问题的解,那么在区间[-T,T]上作奇扩充
根据条件(HO),ū=ū(t)在R上进行2T周期扩充即可得到方程(工)的2T-周期解.
为了研究边值问题(P)的解的存在性,我们将其转化为讨论泛函的非平凡临界点的存在性,研究空间为X(T)=H~2(O,T)∩H10(O,T).此泛函的临界点即为边值问题(P)的经典解.
本文内容安排如下：第一章是引言,介绍了本文的研究背景、研究内容和相关研究综述；第二单足预备知识,介绍了临界点理论的相关定理内容；第三章是针对A>O,B>O,A>O,BO,AIn recent years there has been a considerable interest in fourth-order model equations such as the extended Fisher-Kolmogorov equation proposed in1988by Dee and Van Saarloos as a model for bi-stable systems, and the Swift-Hohenberg equation by Swift and Hohenberg in studies of hydrodynamic instability. After an appropriate transformation, standing wave solutions of these equations lead to the equation in which p>0corresponds to EFK equation and p<0to the SH equation. Many profound results about many other types of fourth-order nonlinear differential equations have been obtained in the use of the methods of the critical point theory developed in recent decades.
To be more specific, we shall mainly discuss the existence of2T-periodic solutions for more general forth-order semi-linear differential equations Where A,B are constants, V(t,u)∈C1([O,T]×R,R) satisfies2V(t,u)-uVu(t,u)→∞,|u|→∞,t∈[O,T],or2V(t,u)→-∞,|u|→∞,t∈[O,T].
Firstly, we will consider the following boundary problem If u is a solution of the above problem, since is f an even2T-periodic function with respect to x and odd with respect to u, then the IT-periodic extensionu=u(t) of the odd extension of the solution u to the interval [-T, T] is a2T-periodic solution of (1) on R.
Problem (P) has a variational structure and its solutions can be found as critical points of the functional In the Sobolev space It is easy to prove that the critical point of I(u;T) is the classical solution of the boundary value problem (P). So, in this paper we mainly obtain nontrivial critical points of the functional/using different variational means.
The paper contains three chapters. Chapter I is an introduction of research background and cents; Chapter II is preliminary knowledge which introduces the basic theory of critical points; In Chapter III, we mainly obtain nontrivial critical points of the functional/using different variational means such as the Mountain-pass theorem, the minimizing methods, the local linking theorem due to Brezis and Nirenberg, the Silva-linking theorem as A,B respectively satisfies corresponding conditions.
经典的Fisher-Kolmogorov(FK)'方程为
1988年,Dee and Van Saarloos在研究双稳态物理系统时建立了Extended Fisher-Kolmogorov(EFK)[2]方程
1977年,Swift and Hohenberg在研究流体的不稳定性时建立了Swift-Hohenberg(SH)[3]方程
人们感兴趣的是以上方程的驻波解,如果引入适当的变量变换上述方程可简化为下述四阶常微分方程
当p>0时,方程即为EFK方程；当p
(HO) V(t,O)=O,v(t+2T,u)=V(t,u),V(t,-u)=V(t,u),(?)t,∈[o,T],u∈R.
(H1)2V(t,u)-uV_u,(t,u)→-∞,|u|→∞,t∈[O,T],或2V(t,u)一uV_u(t,u)→∞,|u|→∞,t∈[O,T].假设ū＝ū(t)为边值问题的解,那么在区间[-T,T]上作奇扩充
根据条件(HO),ū=ū(t)在R上进行2T周期扩充即可得到方程(工)的2T-周期解.
为了研究边值问题(P)的解的存在性,我们将其转化为讨论泛函的非平凡临界点的存在性,研究空间为X(T)=H~2(O,T)∩H10(O,T).此泛函的临界点即为边值问题(P)的经典解.
本文内容安排如下：第一章是引言,介绍了本文的研究背景、研究内容和相关研究综述；第二单足预备知识,介绍了临界点理论的相关定理内容；第三章是针对A>O,B>O,A>O,B
To be more specific, we shall mainly discuss the existence of2T-periodic solutions for more general forth-order semi-linear differential equations Where A,B are constants, V(t,u)∈C1([O,T]×R,R) satisfies2V(t,u)-uVu(t,u)→∞,|u|→∞,t∈[O,T],or2V(t,u)→-∞,|u|→∞,t∈[O,T].
Firstly, we will consider the following boundary problem If u is a solution of the above problem, since is f an even2T-periodic function with respect to x and odd with respect to u, then the IT-periodic extensionu=u(t) of the odd extension of the solution u to the interval [-T, T] is a2T-periodic solution of (1) on R.
Problem (P) has a variational structure and its solutions can be found as critical points of the functional In the Sobolev space It is easy to prove that the critical point of I(u;T) is the classical solution of the boundary value problem (P). So, in this paper we mainly obtain nontrivial critical points of the functional/using different variational means.
The paper contains three chapters. Chapter I is an introduction of research background and cents; Chapter II is preliminary knowledge which introduces the basic theory of critical points; In Chapter III, we mainly obtain nontrivial critical points of the functional/using different variational means such as the Mountain-pass theorem, the minimizing methods, the local linking theorem due to Brezis and Nirenberg, the Silva-linking theorem as A,B respectively satisfies corresponding conditions.
引文
[1]A. Kolmogorov, I. Petrovski, and N. Piscounov, Etude de l'equation de la diffusion avec croissance de laquantite de matiere at son application a un probleme biologique, Bull. Univ. Moskou, ser. Internat, Sec A,1 (1937),1-25.
[2]G.T. Dee and W. van Saarloos, Bistable systems with propagating fronts leading to pattern formation, Phys. Rev. Lett,1988,60:2641-2644.
[3]J.B. Swift, P.C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A,1977,15:319-328.
[4]李永祥,一类四阶边值问题正解的存在性,纯粹数数学与应用数学,2000年9月16卷3期.
[5]S.Tersian and J.Chaparova, Periodic and homoclinic solutions of extended Fisher-Kolmogorov equations, J.Math.Anal. Appl.260 (2001) 490-506.
[6]J. Chaparova, L. Peletier, S. Tersian, Existence and nonexistence of nontrivial solutions of semi linear fourth and sixth-order differential equations, Adv. Differential Equations 8 (2003) 1237-1258.
[7]Tihomir gyulov & Stepan terslan, Existence of trivial and nontrivial solutions of a fourth-order ordinary differential equation, Electronic Journal of Differential Equations, Vol.2004(2004), No. 41,1-14.
[8]李永祥,四阶边值问题正解的存在性与多解性,应用数学学报,2003年9月26卷1期.
[9]Lambertus A.Peletier and Vivi Roitschafer, Large time behavior of solutions of the Swift-Hohenberg equation Comportement des solutions de l'equation de Swift Hohenberg en grand temps, C. R. Acad. Sci. Paris, Ser. I 336 (2003) 225-230.
[10]Maria do Rosario Grossinho, Luis Sanchez, Stepan A. Tersian, On the solvability of a boundary value problem for a fourth-order ordinary differential equation, Applied Mathematics Letters 18 (2005) 439-444.
[11]P.C. Carriao, L.F.O. Faria, O.H. Miyagaki, Periodic solutions for extended Fisher-Kolmogorov and Swift-Hohenberg equations by truncature techniques, Nonlinear Analysis 67 (2007) 3076-3083.
[12]Yang Yang, Jinhui Zhang, Nontrivial solutions for some fourth order boundary value problems with parameters, Nonlinear analysis 70(2009) 3966-3977.
[13]ChengYue Li, ChangHua Shi,Linking method for periodic non-autonomous fourth-order differential equations with super quadratic potentials. Electronic Journal of Differential Equations, Vol.2009(2009), No.120,1-11.
[14]尹海霞,李成岳,满足Costa型非二次条件的四阶非线性常微分方群周期解的存在性研究,中央民族大学学报(自然科学版).2011年,第2期.
[15]A.Ambrosetti,P.H.Rabinowtiz. Dual variational methods in critical point theory and applications.J.Funct.Anal.14(1973),349-381.
[16]J.Mawhin,M.Willem,Critical Point Theory and Hamilatonian Systems. New York ect, Springer-Verlag,1989.
117] P.H.Rabinowtiz.Mimimax Methods in critical point Theory with applications to Differential Equations.CBMS Reg.Conf. Ser. In Math.65,1986.
[18]H.Brezis and L.Nirenberg, Remarks on finding critical points, Comm. Pure and Appl.Math.(1991),939-963.
[19]李成岳,变分方法与哈密顿系统同宿轨道引论,北京：科学技术文献出版社,2006.
[20]P. Bartolo, V.Benci, D.Fortunato, Abstract critical point theorems and applications to some nonlinear problems with strong resonance, Nonlinear Analysis:Theory, Methods and Applications, Vol.7, No.9 (1983),981-1012.
[21]M.Schechter,Periodic non-autonomous second-order dynamical systems, Journal of Differential Equations 223(2006),290-302.
[22]S.Tersian and J.Chaparova,Periodic and homoclinic solutions of extend Fisher-Kolmogorov equation,J.Math Anal.Appl.,266(2001),490-506.
[23]L.A.Peletier, W.C.Troy, Spatial Patterns: Higher Order Models in Physics and Mechanics,Boston:Birkhaser,2001.
[24]郭大钧,非线性泛函分析,山东科学技术出版社,2004.
[25]Benci V., Some critical point theorems and applications, Commons pure appl. Math.33(1980), 147-172.
[26]Li Cheng-yue, Remarks on homoclinic solutions for semilinear fourth order order ordinary differential equations without periodicity,2009.
[27]张恭庆,临界点理论及其应用,上海：上海科学技术出版社,1986.
[28]Benjin Xuan, Variational Methods-Theory and Applications. University of Science and Technology of China Press,2006.
[29]David G. Costa, An Invitation to Variational Methods in Differential Equation.
[2]G.T. Dee and W. van Saarloos, Bistable systems with propagating fronts leading to pattern formation, Phys. Rev. Lett,1988,60:2641-2644.
[3]J.B. Swift, P.C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A,1977,15:319-328.
[4]李永祥,一类四阶边值问题正解的存在性,纯粹数数学与应用数学,2000年9月16卷3期.
[5]S.Tersian and J.Chaparova, Periodic and homoclinic solutions of extended Fisher-Kolmogorov equations, J.Math.Anal. Appl.260 (2001) 490-506.
[6]J. Chaparova, L. Peletier, S. Tersian, Existence and nonexistence of nontrivial solutions of semi linear fourth and sixth-order differential equations, Adv. Differential Equations 8 (2003) 1237-1258.
[7]Tihomir gyulov & Stepan terslan, Existence of trivial and nontrivial solutions of a fourth-order ordinary differential equation, Electronic Journal of Differential Equations, Vol.2004(2004), No. 41,1-14.
[8]李永祥,四阶边值问题正解的存在性与多解性,应用数学学报,2003年9月26卷1期.
[9]Lambertus A.Peletier and Vivi Roitschafer, Large time behavior of solutions of the Swift-Hohenberg equation Comportement des solutions de l'equation de Swift Hohenberg en grand temps, C. R. Acad. Sci. Paris, Ser. I 336 (2003) 225-230.
[10]Maria do Rosario Grossinho, Luis Sanchez, Stepan A. Tersian, On the solvability of a boundary value problem for a fourth-order ordinary differential equation, Applied Mathematics Letters 18 (2005) 439-444.
[11]P.C. Carriao, L.F.O. Faria, O.H. Miyagaki, Periodic solutions for extended Fisher-Kolmogorov and Swift-Hohenberg equations by truncature techniques, Nonlinear Analysis 67 (2007) 3076-3083.
[12]Yang Yang, Jinhui Zhang, Nontrivial solutions for some fourth order boundary value problems with parameters, Nonlinear analysis 70(2009) 3966-3977.
[13]ChengYue Li, ChangHua Shi,Linking method for periodic non-autonomous fourth-order differential equations with super quadratic potentials. Electronic Journal of Differential Equations, Vol.2009(2009), No.120,1-11.
[14]尹海霞,李成岳,满足Costa型非二次条件的四阶非线性常微分方群周期解的存在性研究,中央民族大学学报(自然科学版).2011年,第2期.
[15]A.Ambrosetti,P.H.Rabinowtiz. Dual variational methods in critical point theory and applications.J.Funct.Anal.14(1973),349-381.
[16]J.Mawhin,M.Willem,Critical Point Theory and Hamilatonian Systems. New York ect, Springer-Verlag,1989.
117] P.H.Rabinowtiz.Mimimax Methods in critical point Theory with applications to Differential Equations.CBMS Reg.Conf. Ser. In Math.65,1986.
[18]H.Brezis and L.Nirenberg, Remarks on finding critical points, Comm. Pure and Appl.Math.(1991),939-963.
[19]李成岳,变分方法与哈密顿系统同宿轨道引论,北京：科学技术文献出版社,2006.
[20]P. Bartolo, V.Benci, D.Fortunato, Abstract critical point theorems and applications to some nonlinear problems with strong resonance, Nonlinear Analysis:Theory, Methods and Applications, Vol.7, No.9 (1983),981-1012.
[21]M.Schechter,Periodic non-autonomous second-order dynamical systems, Journal of Differential Equations 223(2006),290-302.
[22]S.Tersian and J.Chaparova,Periodic and homoclinic solutions of extend Fisher-Kolmogorov equation,J.Math Anal.Appl.,266(2001),490-506.
[23]L.A.Peletier, W.C.Troy, Spatial Patterns: Higher Order Models in Physics and Mechanics,Boston:Birkhaser,2001.
[24]郭大钧,非线性泛函分析,山东科学技术出版社,2004.
[25]Benci V., Some critical point theorems and applications, Commons pure appl. Math.33(1980), 147-172.
[26]Li Cheng-yue, Remarks on homoclinic solutions for semilinear fourth order order ordinary differential equations without periodicity,2009.
[27]张恭庆,临界点理论及其应用,上海：上海科学技术出版社,1986.
[28]Benjin Xuan, Variational Methods-Theory and Applications. University of Science and Technology of China Press,2006.
[29]David G. Costa, An Invitation to Variational Methods in Differential Equation.