Ladder methods, length scales and positivity of solutions in dissipative partial differential equations.

In this thesis we have investigated some physically interesting dissipative partial differential equations through Ladder Methods. The technique originally used in [1] enables us to obtain estimates on some of the most important features of these equations including the dissipative length scale. We illustrate this method by applying it to the Kuramoto-Sivashinsky equation and to a Generalised Diffusion Model, that models the population density of a single animal species under a more general diffusive mechanism than Fickian Diffusion. We have also looked at some of the problems arising from Ladder Methods. Specifically we have studied the rate of decay of solutions in non-linear dissipative Partial Differential equations (PDEs). This is important when finding upper bounds on the time average of the dissipative length scales which arise naturally from the Ladder. We show that using a method first employed in [13,14,15] it is possible to bound several dissipative PDEs below by an exponential. Furthermore we have addressed the following important problem; in the study of the behaviour of solutions of dissipative partial differential equations, an important question is whether solutions with positive initial data remain positive for all time. We find the necessary conditions for positivity of solutions of a class of dissipative PDEs possessing linear diffusion. Moreover we extend our analysis of positivity to a PDE with a non-linear diffusion term. Proving positivity of solutions is also important in obtaining an upper bound for the bottom rung of the ladder for the Generalised Diffusion Model Equation. Finally, following the work in reference [23], we also study the application of weighted norms in finding tighter bounds on the bottom rung of ladders