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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.