【作者】 王仲平；

【导师】 钟承奎；

【作者基本信息】 兰州大学 ， 基础数学， 2010， 博士

【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.更多还原

【Abstract】 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.更多还原