Applying Functional Analytic Techniques to Evolution Equations

Mathematical models arising in the natural sciences often involve equations which describe how the phenomena under investigation evolve in time. In these notes, some mathematical techniques will be presented for analysing a range of evolution equations that can arise in a number of applied disciplines such as biomathematics and population dynamics. Different types of equations will be examined, but a unifying theme will be provided by developing methods from a dynamical systems point of view and using some elegant results from functional analysis. To fix ideas, we will begin with some simple finite-dimensional models from population dynamics which are expressed in terms of ordinary differential equations. We then go on to consider dynamical systems in an infinite-dimensional setting, and provide a gentle introduction to the theory of strongly continuous semigroups of operators. This theory is applied to an infinite system of nonlinear ordinary differential equations that models the time-evolution of the size distribution of a collection of particles that can coagulate to form larger particles or fragment into smaller particles.