Attractor Bifurcation and Final Patterns of the N-dimensional and Generalized Swift-hohenberg Equations

In this paper I will investigate the bifurcation and asymptotic behavior of solutions of the Swift-Hohenberg equation and the generalized Swift-Hohenberg equation with the Dirichlet boundary condition on a onedimensional domain (0, L). I will also study the bifurcation and stability of patterns in the n-dimensional Swift-Hohenberg equation with the odd-periodic and periodic boundary conditions. It is shown that each equation bifurcates from the trivial solution to an attractor Aλ when the control parameter λ crosses λc, the principal eigenvalue of (I+∆). The local behavior of solutions and their bifurcation to an invariant set near higher eigenvalues are analyzed as well.