Doctor of Philosophy (PhD), Ohio University, 2024, Mathematics (Arts and Sciences)

The Lyapunov stability of equilibria in dynamical systems is determined by the interplay between the linearization and the nonlinear terms. In this work, we study the case when the spectrum of the linearization is diffusively stable with high-order spectral degeneracy at the origin. In particular, spatially periodic solutions called roll solutions at the zigzag boundary of the Swift-Hohenberg equation (SHE), typically selected by patterns and defects in numerical simulations, are shown to be nonlinearly stable. This also serves as an example where linear decay weaker than classical diffusive decay, together with quadratic nonlinearity, still gives nonlinear stability of spatially periodic patterns. The study is conducted on two physical domains: the 2D plane, $\R^2$, and the cylinder, $T_{2\pi}\times \R$. Linear analysis reveals that instead of the classical $t^{-1}$ diffusive decay rate, small localized perturbation of roll solutions with zigzag wavenumbers decay with slower algebraic rates ($t^{-\frac{3}{4}}$ for the 2D plane; $t^{-\frac{1}{4}}$ for the cylindrical domain) due to the high order degeneracy of the translational mode at the origin of the Bloch-Fourier spaces. The nonlinear stability proofs are based on decompositions of the neutral translational mode and the faster decaying modes, and fixed-point arguments, demonstrating the irrelevancy of the nonlinear terms.

Committee: Qiliang Wu (Advisor); Alexander Neiman (Committee Member); Todd Young (Committee Member); Tatiana Savin (Committee Member) Subjects: Mathematics