This thesis is mostly concerned with understanding the multiplicity and geometric
structure of asymptotic patterns that arise from the Cauchy problem for the
ut = 3u m 􀀀
; 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
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|
|Supervisor||Jonathan Evans (Supervisor) & Victor Galaktionov (Supervisor)|