非线性偏微分方程吸引子分歧问题的研究
摘要
在这篇博士学位论文中,我们主要研究如下的三类方程：
非线性广义Burgers方程非线性Chaffee-Infante方程和非线性反应扩散方程其中Ω(?)Rn(n≥3)是适当光滑有界开区域.我们研究了它们的吸引子分歧问题及分歧出的吸引子的结构.
首先,在第三章中,我们用文献[3]新建立的吸引子分歧理论,用中心流形约化方法,证明了系统(ε1)具有奇函数解的条件下,当参数λ穿越过第一特征值λ0=1时,系统(ε1)分歧出一个吸引子,吸引子由系统(ε1)的稳态解构成,见定理3.3.1；对无奇函数解的一般条件,也得到了一个类似结论,见定理3.3.2.在[0,2π]上,两个类似的结论也被得到,见定理3.4.1和定理3.4.2.
紧接着,在第四章中,对系统(ε2)也运用吸引子分歧理论,用中心流形约化方法,证明了在系统(ε2)具有奇函数解的条件下,当参数λ穿越过第一特征值λ0=1时,系统(ε1)也分歧出一个吸引子,吸引子由系统(ε1)的稳态解构成,见定理4.2.1；对无奇函数解的一般条件,我们也得到了类似结论,见定理4.2.2.
最后,在第五章中,在给定的非线性项f满足(5.2)-(5.5)条件下,利用吸引子分歧理论证明了,当参数λ穿越过Laplacian算子-△的第一特征值λ1时,系统(ε3)分歧一个吸引子,我们得到了两个结论,见定理5.3.1和定理5.3.2.我们还给出了u=0是系统(ε3)全局渐近稳定平衡点的另一种证明,见定理5.5.3,我们还得到了系统(ε3)具有Lyapunov函数,见定理5.5.4.
In this doctoral thesis, we are concerned with the following three classes of equations:
The nonlinear generalized Burgers equation the nonlinear Chaffee-Infante equation and Reaction-Diffusion equation whereΩ(?) Rn(n≥3) is an open bounded subset with smooth boundary (?)Q. We have considered attractor bifurcation and the structure of the bifurcated attractors for them.
Firstly, in chapter 3, by using the new developed Attractor Bifurcation The-ory in [3] and the center manifold Reduction Method, and under the condition which the system(ε1) have odd solutions, it is proved that (ε1) bifurcates an attractor, which is consist of steady solutions of (ε1), when the control param-eter A crosses the principal eigenvalueλ0= 1, see Theorem 3.3.1; and under the generalized condition which the system (ε1) have not odd solutions, then a similar result is obtained, see Theorem 3.3.2. On the [0,2π], we have gained two similar results, see Theorems 3.4.1. and 3.4.2.
Furthermore, in Chapter 4, by using the Attractor Bifurcation Theory and the center manifold Reduction Method, and under the condition which the sys-tem (ε2) have odd solutions, it is proved that (ε2) bifurcates an attractor,too, which is also consist of steady solutions of (ε2), when the control parameterλcrosses the principal eigenvalueλ0= 1, see Theorem 4.2.1; and under the generalized condition which the system (ε1) have not odd solutions, then a similar result is obtained, see Theorem 4.2.2.
Finally, in Chapter 5, under the conditions which the nonlinear term f satisfy (5.2)-(5.5), we have prove that the system (ε2) bifurcates an attractor by the Attractor Bifurcation Theory, as the control parameterλcrosses the principal eigenvalueλ1 of Laplacian operator -Δ, two results have been gotten, see Theorems 5.3.1. and 5.3.2. We also have given another proof for u= 0 is global asymptotical steady point of (ε3), see see Theorem 5.5.3. Moreover, we have gained that (ε3) has a Lyapunov function, see Theorem 5.5.4.
非线性广义Burgers方程非线性Chaffee-Infante方程和非线性反应扩散方程其中Ω(?)Rn(n≥3)是适当光滑有界开区域.我们研究了它们的吸引子分歧问题及分歧出的吸引子的结构.
首先,在第三章中,我们用文献[3]新建立的吸引子分歧理论,用中心流形约化方法,证明了系统(ε1)具有奇函数解的条件下,当参数λ穿越过第一特征值λ0=1时,系统(ε1)分歧出一个吸引子,吸引子由系统(ε1)的稳态解构成,见定理3.3.1；对无奇函数解的一般条件,也得到了一个类似结论,见定理3.3.2.在[0,2π]上,两个类似的结论也被得到,见定理3.4.1和定理3.4.2.
紧接着,在第四章中,对系统(ε2)也运用吸引子分歧理论,用中心流形约化方法,证明了在系统(ε2)具有奇函数解的条件下,当参数λ穿越过第一特征值λ0=1时,系统(ε1)也分歧出一个吸引子,吸引子由系统(ε1)的稳态解构成,见定理4.2.1；对无奇函数解的一般条件,我们也得到了类似结论,见定理4.2.2.
最后,在第五章中,在给定的非线性项f满足(5.2)-(5.5)条件下,利用吸引子分歧理论证明了,当参数λ穿越过Laplacian算子-△的第一特征值λ1时,系统(ε3)分歧一个吸引子,我们得到了两个结论,见定理5.3.1和定理5.3.2.我们还给出了u=0是系统(ε3)全局渐近稳定平衡点的另一种证明,见定理5.5.3,我们还得到了系统(ε3)具有Lyapunov函数,见定理5.5.4.
In this doctoral thesis, we are concerned with the following three classes of equations:
The nonlinear generalized Burgers equation the nonlinear Chaffee-Infante equation and Reaction-Diffusion equation whereΩ(?) Rn(n≥3) is an open bounded subset with smooth boundary (?)Q. We have considered attractor bifurcation and the structure of the bifurcated attractors for them.
Firstly, in chapter 3, by using the new developed Attractor Bifurcation The-ory in [3] and the center manifold Reduction Method, and under the condition which the system(ε1) have odd solutions, it is proved that (ε1) bifurcates an attractor, which is consist of steady solutions of (ε1), when the control param-eter A crosses the principal eigenvalueλ0= 1, see Theorem 3.3.1; and under the generalized condition which the system (ε1) have not odd solutions, then a similar result is obtained, see Theorem 3.3.2. On the [0,2π], we have gained two similar results, see Theorems 3.4.1. and 3.4.2.
Furthermore, in Chapter 4, by using the Attractor Bifurcation Theory and the center manifold Reduction Method, and under the condition which the sys-tem (ε2) have odd solutions, it is proved that (ε2) bifurcates an attractor,too, which is also consist of steady solutions of (ε2), when the control parameterλcrosses the principal eigenvalueλ0= 1, see Theorem 4.2.1; and under the generalized condition which the system (ε1) have not odd solutions, then a similar result is obtained, see Theorem 4.2.2.
Finally, in Chapter 5, under the conditions which the nonlinear term f satisfy (5.2)-(5.5), we have prove that the system (ε2) bifurcates an attractor by the Attractor Bifurcation Theory, as the control parameterλcrosses the principal eigenvalueλ1 of Laplacian operator -Δ, two results have been gotten, see Theorems 5.3.1. and 5.3.2. We also have given another proof for u= 0 is global asymptotical steady point of (ε3), see see Theorem 5.5.3. Moreover, we have gained that (ε3) has a Lyapunov function, see Theorem 5.5.4.
引文
[1]T. Ma, S. Wang, Dynamic bifurcation and stability in the Rayleigh-Benard convection, Comm. Math. Sci.,2004,2(2):159-183.
[2]T. Ma, S. Wang, Bifurcation and stability of superconductivity, Journal of Mathematical Physics,2005,46:095112-1-31.
[3]T. Ma, S. Wang, Bifurcation Theory and Applications, Vol.53 of World Scientific Series on Nonlinear Science, Series A, World Scientific,2005.
[4]T. Ma. S. Wang, Dynamic bifurcation of nonlinear evolution equations, Chinese Annals of Mathematics,2005,26(2):185-206.
[5]T. Ma, J. Park and S. Wang, Dynamic Bifurcation of the Ginzburg-Landau Equation, SIAM J. Applied Dynamical Systems,2004,3(4):620-635.
[6]J. Park, Dynamic bifurcation theory of Rayleigh-Benard convection with infinite Prandtl number, Discrete and continuous Dynamic Systems-Series B,6(3)(2006),591-604.
[7]R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics (2nd edition), Springer, New York.1997.
[8]D. Henry, Geometric theory of semi-linear parabolic equations,840 of Lecture Notes in Mathematics, Springer-Verlag, Berlin.(1981).
[9]B. Wang, Attractors for reaction-diffusion equations in unbounded do-mains, Physica D,128(1999),41-52.
[10]马天,汪守宏,非线性演化方程的稳定性与分歧,科学教育出版社,北京,2007.
[11]H.J.Poincare, Les figures equilibrium, Acta Math.7(1885),259-302.
[12]陆启韶,分岔与奇异性,,上海科技教育出版社,1995.
[13]钟承奎,范先令,陈文源,非线性泛函分析引论,兰州大学出版社,1998.
[14]Krasnosel'skii, M. A., Topological Methods in the Theory of Nonlinear Integral Equations. Translated by A. H. Armstrong:translation edited by J.Burlak.Macmillan, New York,1964.
[15]P.H.Rabinowitz, Some global results for nonlinear eigenvalue problems, J.Funct. Anal.,7(1971),487-513.
[16]H.Kielhofer, Bifurcation Theory. An Introduction with Applications to PDEs, Applied Mathematical Sciences, vol.156, Springer, New York, Berlin, Heidelberg,2004.
[17]S.-N. Chow, R.Lauterbach, A bifurcation theorem for critical points of variational problems, Nonlinear Anal.:Theory Methods Applications,12 (1988) 51-61.
[18]J. Cholewa. T.Dlotko, Global Attractor in Abstract parabolic Problems, Cambridge University Press,2000.
[19]S. Antman. Nonlinear Problems of Elasticity, Springer-Verlag, New York. 1995.
[20]P. Chossat and R.Lauterbach, Methods in Eqivariant Bifurcations and Dynamical Systems, World Scientific Publishing Corporation, Singapore,2000.
[21]S.-N. Chow and J.K. Hale. Methods in Bifurcations Theory. Springer-Verlag, New York-Berlin-Heidelberg,1982.
[22]M. Demazure. Bifurcations and Catastophes. Geometry of Nonlinear Problems. Berlin-Heidelberg-New York,2000.
[23]A. Vanderbauwhede, Local bifurcations and symmetry. Research Notes in Mathematics,75. Pitman, Boston-London-Melbourne,1982.
[24]E. Zeidler, Nonlinear Functional Analysis and its Applications, I:Fixed-Point Theorems, Springer-Verlag, New York-Berlin-Heidelberg,1986.
[25]K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New York-Berlin-Heidelberg-Tokyo,1980.
[26]C. Hsia, T. Ma, S. Wang, Stratified rotating Boussinesq equations in geophysical fluid dynamics:Dynamic bifurcation and periodic solutions, Journal of Mathematical Physics,48(2007),065602.
[27]T. Ma, S. Wang, Rayleigh-Benard Convection:Dynamics and Structure in the Physical Space, Commun. Math. Sci. International Press Vol. 5(2007), No.3, pp.553-574.
[28]T. Ma, S. Wang, Stability and Bifurcation of the Taylor Problem, Arch. Rational Mech.Anal.181 (2006),149-176.
[29]A. V.Babin, M. I.Vishik, Attactors of evolution equations, Nauka, Moscow, 1989. English translation, North-Holland,1992.
[30]V. V.Chepyzhov. M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc., Providence, RI,2002.
[31]M. Marion, Attractors for reaction-diffusion equations:Existence and estimate of their dimension, Appl.Anal.25(1987),101-147.
[32]J. C. Robinson, Infinite-dimensional Dynamical Systems an Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge,2001.
[33]G. B.Whitham. Linear and Nonlinear Waves, New York:John Wiely and Sons, Inc,1974.7..
[34]J. M. Burgers. A mathematical model illust rating the theory of turbu-lence, Adv.Appl.Mech.1948, (1):171-199.
[35]E. Hopf. The partial differential equation ut+uux= uxx, Comm. Pure Appl Math.1950, (3):201-230.
[36]J. L. Lions, Quelques Methodes de Resolution des Problems aux Limites Non Lineaires, Dunrod Gauthier-Villars, Paris,1969.
[37]孙春友,兰州大学博士论文,2005.
[38]陈光霞,兰州大学博士论文,2008.
[39]Riidiger Seydel, Practical Bifurcation and Stability Analysis:From Equi-librium to chaos, Second Edition,1999. Springer-Verlag,
[40]Masoud Yari, Attractor Bifurcztion and Final Patterns of the N-dimensional and Generalized Swift-Hohenberg Equations, Discrete and continuous Dynamic Systems-Series B,7(2)(2007),441-456.
[41]X. Y. Wang, The existence of global attractors for the norm-to-weak semi-group for nonlinear infinite dimensional dynamical systems with dissipation. Doctoral thesis,2004.
[42]C. K. Zhong, M. H. Yang, C. Y. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, Journal Differential Equations,223(2006), 367-399.
[43]张恭庆, 《临界点理论及其应用》,上海科学技术出版社,1986.
[44]Michael Struwe, Variational methods:Applications to nonlinear Par-tial Differential Equations and Hamiltonian Systems, Second Edition, Springer-Verlag,2000.
[45]Chun-Hsiung Hsia and Xiaoming Wang, On A Burgers'Type Equation, Discrete and continuous Dynamic Systems-Series B,2006; 6(5):1121-1139.
[46]B. Wang, Attractors for reaction-diffusion equations in unbounded do-mains, Physica D,1999; 128:41-52.
[47]N. Chafee, A Stability Analysis for a Semilinear Parabolic Partial Differ-ential Equation, Journal of Differential Equations,15 (1974),552-540.
[48]N. Chafee and E. F. Infante, A Bifurcation Problem for a Nonlinear Partial Differential Equation of Parabolic Type, Applicable Analysis,4 (1974),17-37.
[49]G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, Vol.143, Springer-Verlag, New York,2002.
[50]Roger Temam and Xiaoming Wang,Boundary Layer for Chaffee-Infante Type Equation, ARCHIVUM MATHEMATICUM (BRNO) Tomus,34 (1998),217-226
[51]H. Matano, Asymptotical behavior of solutions of semi-linear heat equa-tions on.S1,in Nonlinear Diffusions Equations and Equilibrium States, Springer, Berlin(1988).
[2]T. Ma, S. Wang, Bifurcation and stability of superconductivity, Journal of Mathematical Physics,2005,46:095112-1-31.
[3]T. Ma, S. Wang, Bifurcation Theory and Applications, Vol.53 of World Scientific Series on Nonlinear Science, Series A, World Scientific,2005.
[4]T. Ma. S. Wang, Dynamic bifurcation of nonlinear evolution equations, Chinese Annals of Mathematics,2005,26(2):185-206.
[5]T. Ma, J. Park and S. Wang, Dynamic Bifurcation of the Ginzburg-Landau Equation, SIAM J. Applied Dynamical Systems,2004,3(4):620-635.
[6]J. Park, Dynamic bifurcation theory of Rayleigh-Benard convection with infinite Prandtl number, Discrete and continuous Dynamic Systems-Series B,6(3)(2006),591-604.
[7]R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics (2nd edition), Springer, New York.1997.
[8]D. Henry, Geometric theory of semi-linear parabolic equations,840 of Lecture Notes in Mathematics, Springer-Verlag, Berlin.(1981).
[9]B. Wang, Attractors for reaction-diffusion equations in unbounded do-mains, Physica D,128(1999),41-52.
[10]马天,汪守宏,非线性演化方程的稳定性与分歧,科学教育出版社,北京,2007.
[11]H.J.Poincare, Les figures equilibrium, Acta Math.7(1885),259-302.
[12]陆启韶,分岔与奇异性,,上海科技教育出版社,1995.
[13]钟承奎,范先令,陈文源,非线性泛函分析引论,兰州大学出版社,1998.
[14]Krasnosel'skii, M. A., Topological Methods in the Theory of Nonlinear Integral Equations. Translated by A. H. Armstrong:translation edited by J.Burlak.Macmillan, New York,1964.
[15]P.H.Rabinowitz, Some global results for nonlinear eigenvalue problems, J.Funct. Anal.,7(1971),487-513.
[16]H.Kielhofer, Bifurcation Theory. An Introduction with Applications to PDEs, Applied Mathematical Sciences, vol.156, Springer, New York, Berlin, Heidelberg,2004.
[17]S.-N. Chow, R.Lauterbach, A bifurcation theorem for critical points of variational problems, Nonlinear Anal.:Theory Methods Applications,12 (1988) 51-61.
[18]J. Cholewa. T.Dlotko, Global Attractor in Abstract parabolic Problems, Cambridge University Press,2000.
[19]S. Antman. Nonlinear Problems of Elasticity, Springer-Verlag, New York. 1995.
[20]P. Chossat and R.Lauterbach, Methods in Eqivariant Bifurcations and Dynamical Systems, World Scientific Publishing Corporation, Singapore,2000.
[21]S.-N. Chow and J.K. Hale. Methods in Bifurcations Theory. Springer-Verlag, New York-Berlin-Heidelberg,1982.
[22]M. Demazure. Bifurcations and Catastophes. Geometry of Nonlinear Problems. Berlin-Heidelberg-New York,2000.
[23]A. Vanderbauwhede, Local bifurcations and symmetry. Research Notes in Mathematics,75. Pitman, Boston-London-Melbourne,1982.
[24]E. Zeidler, Nonlinear Functional Analysis and its Applications, I:Fixed-Point Theorems, Springer-Verlag, New York-Berlin-Heidelberg,1986.
[25]K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New York-Berlin-Heidelberg-Tokyo,1980.
[26]C. Hsia, T. Ma, S. Wang, Stratified rotating Boussinesq equations in geophysical fluid dynamics:Dynamic bifurcation and periodic solutions, Journal of Mathematical Physics,48(2007),065602.
[27]T. Ma, S. Wang, Rayleigh-Benard Convection:Dynamics and Structure in the Physical Space, Commun. Math. Sci. International Press Vol. 5(2007), No.3, pp.553-574.
[28]T. Ma, S. Wang, Stability and Bifurcation of the Taylor Problem, Arch. Rational Mech.Anal.181 (2006),149-176.
[29]A. V.Babin, M. I.Vishik, Attactors of evolution equations, Nauka, Moscow, 1989. English translation, North-Holland,1992.
[30]V. V.Chepyzhov. M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc., Providence, RI,2002.
[31]M. Marion, Attractors for reaction-diffusion equations:Existence and estimate of their dimension, Appl.Anal.25(1987),101-147.
[32]J. C. Robinson, Infinite-dimensional Dynamical Systems an Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge,2001.
[33]G. B.Whitham. Linear and Nonlinear Waves, New York:John Wiely and Sons, Inc,1974.7..
[34]J. M. Burgers. A mathematical model illust rating the theory of turbu-lence, Adv.Appl.Mech.1948, (1):171-199.
[35]E. Hopf. The partial differential equation ut+uux= uxx, Comm. Pure Appl Math.1950, (3):201-230.
[36]J. L. Lions, Quelques Methodes de Resolution des Problems aux Limites Non Lineaires, Dunrod Gauthier-Villars, Paris,1969.
[37]孙春友,兰州大学博士论文,2005.
[38]陈光霞,兰州大学博士论文,2008.
[39]Riidiger Seydel, Practical Bifurcation and Stability Analysis:From Equi-librium to chaos, Second Edition,1999. Springer-Verlag,
[40]Masoud Yari, Attractor Bifurcztion and Final Patterns of the N-dimensional and Generalized Swift-Hohenberg Equations, Discrete and continuous Dynamic Systems-Series B,7(2)(2007),441-456.
[41]X. Y. Wang, The existence of global attractors for the norm-to-weak semi-group for nonlinear infinite dimensional dynamical systems with dissipation. Doctoral thesis,2004.
[42]C. K. Zhong, M. H. Yang, C. Y. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, Journal Differential Equations,223(2006), 367-399.
[43]张恭庆, 《临界点理论及其应用》,上海科学技术出版社,1986.
[44]Michael Struwe, Variational methods:Applications to nonlinear Par-tial Differential Equations and Hamiltonian Systems, Second Edition, Springer-Verlag,2000.
[45]Chun-Hsiung Hsia and Xiaoming Wang, On A Burgers'Type Equation, Discrete and continuous Dynamic Systems-Series B,2006; 6(5):1121-1139.
[46]B. Wang, Attractors for reaction-diffusion equations in unbounded do-mains, Physica D,1999; 128:41-52.
[47]N. Chafee, A Stability Analysis for a Semilinear Parabolic Partial Differ-ential Equation, Journal of Differential Equations,15 (1974),552-540.
[48]N. Chafee and E. F. Infante, A Bifurcation Problem for a Nonlinear Partial Differential Equation of Parabolic Type, Applicable Analysis,4 (1974),17-37.
[49]G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, Vol.143, Springer-Verlag, New York,2002.
[50]Roger Temam and Xiaoming Wang,Boundary Layer for Chaffee-Infante Type Equation, ARCHIVUM MATHEMATICUM (BRNO) Tomus,34 (1998),217-226
[51]H. Matano, Asymptotical behavior of solutions of semi-linear heat equa-tions on.S1,in Nonlinear Diffusions Equations and Equilibrium States, Springer, Berlin(1988).