Instabilities in Sixth Order Cahn-Hilliard Type Equations

Abstract

This thesis is mostly concerned with understanding the multiplicity and geometric structure of asymptotic patterns that arise from the Cauchy problem for the equations ut = 3u m 􀀀 jujp􀀀1u ; p > 1; m = 1; 2; posed on the whole of RN. Higher order PDEs have become a popular research area over the last two decades but sixth order equations are still less common and less well understood than related equations of fourth order. Sixth order parabolic models continue to arise in various physical contexts, so understanding their behaviours is a matter of increasing importance. We dedicate two chapters to the case m = 1. In Chapter 2, we study the equation with negative sign, which is unstable and allows for solutions that blow up in nite time. In Chapter 4, we study the positive sign case, which is stable. The bulk of our results here concern so-called self-similar solutions, and we describe and categorize them as extensively as we can over a variety of parameter ranges. Our approach comprises a mix of rigorous analysis, careful numerics and asymptotic approximations. Chapter 5 addresses both the positively and negatively signed m = 2 case. We again focus on self-similar solutions, which demonstrate a complete departure from the lower-order theory. Chapter 3 details an adaptive numerical scheme suitable for simulating a wide class of sixth order parabolic PDEs. Based on earlier work for second and fourth order equations, it uses a robust collocation scheme over a grid whose mesh points can be made to move according to features of the solution as they develop. This is used at various points throughout the thesis to better understand aspects of the main problem.

Date of Award	27 Jun 2017
Original language	English
Awarding Institution	University of Bath
Supervisor	Jonathan Evans (Supervisor) & Victor Galaktionov (Supervisor)