Studies On The Attractor Bifurcation Of Nonlinear Partial Differential Equations

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.