Some problems in nonlinear diffusion
In this thesis we investigate mathematical models for a number of topics in the field of nonlinear diffusion, using similarity, asymptotic and numerical methods and focussing on the time-asymptotic behaviour in most cases. Firstly, we consider `fast' diffusion in the vicinity of a mask-edge, with application to dopant diffusion into a semiconductor. A variety of approaches are used to determine concentration contours and aspect ratios. Next we consider flow by curvature. Using group analysis, we determine a number of new symmetries for the governing equations in two and three dimensions. By tracking a moving front numerically, we also construct single and double spiral patterns (reminiscent of those observed in the Belousov-Zhabotinskii chemical reaction), and classify the types of behaviour that can occur. Finally, we analyse travelling wave solutions and the behaviour near to extinction for closed loops. We next consider relaxation waves in a system that can be used to model target patterns, also observed in the Belousov-Zhabotinskii reaction. Numerical and asymptotic results are presented, and a number of new cases of front behaviour are obtained. Finally, we investigate a number of systems using an approach based on the WKB method, analysing the motion of invasive fronts and also the form of the pattern left behind. For Fisher's equation, we demonstrate how modulated travelling waves can be obtained by prescribing an oscillatory initial profile. The method is then extended, firstly to Turing systems and then to oscillatory systems, for which we use an additional periodic plane wave argument to determine the unequal front and pattern speeds, as well as the periodicity. Finally, we illustrate how these methods apply to a recently-used `chaotic' model from ecology.