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 one-dimensional 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 $\mathcal A_\lambda$ when the control parameter $\lambda$ crosses $\lambda _{c} $, the principal eigenvalue of $(I+\Delta)^2$. The local behavior of solutions and their bifurcation to an invariant set near higher eigenvalues are analyzed as well.