Fourth order spectral theory and diffusion-driven instability

Bi-Laplacian;cascading;avoided crossings;Turing diffusion-driven instability;reaction-diffusion system;fourth order

Chung, Jooyeon

关键词: Bi-Laplacian;    cascading;    avoided crossings;    Turing diffusion-driven instability;    reaction-diffusion system;    fourth order;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/101465/CHUNG-DISSERTATION-2018.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

In Part I, we study the spectrum of the one-dimensional vibrating free rod equation u′′′′ − τ u′′ = μu under tension (τ > 0) or compression (τ < 0). The eigenvalues μ as functions of the tension/compression parameter τ exhibit three distinct types of behavior. In particular, eigenvalue branches in the lower half-plane exhibit a cascading pattern of barely-avoided crossings.We provide a complete description of the eigenfunctions and eigenvalues by implicitly parameterizing the eigenvalue curves. We also establish properties of the eigenvalue curves such as monotonicity, crossings, asymptotic growth, cascading and phantom spectral lines.In Part II, we analyze diffusion-driven (Turing) instability of a reaction-diffusion system. The innovation is that we replace the traditional Laplacian diffusion operator with a combination of the fourth order bi-Laplacian operator and the second order Laplacian. We find new phenomena when the fourth order and second order terms are competing, meaning one of them stabilizes the system whereas the other destabilizes it. We characterize Turing space in terms of parameter values in the system, and also find criteria for instability in terms of the domain size and tension parameter.