New generic mathematical theory for a class of partial differential equations

Seven years ago, Dr John Meyer was an undergraduate student, attending lectures given by Professor David Needham. Today, he is a lecturer who has co-authored a monograph with his former tutor that establishes a generic mathematical theory researchers around the world have spent decades trying to clarify.

The monograph, published by Cambridge University Press as part of the London Mathematical Society Lecture Note Series, is an early career triumph for John and the satisfying culmination of 20 years’ work for David.

The Cauchy Problem for Non-Lipschitz Semi-Linear Parabolic Partial Differential Equations is a theory for a class of partial differential equations (PDEs).

‘Specific questions in this area have been studied over the last 30 years. This monograph has unified these specific results into a generic theory,’ explains David, who is Director of Research and Deputy Head of the School of Mathematics.

Reaction-diffusion theory – of which PDEs are part – is a topic that has developed rapidly over the past 30 years, particularly in relation to applications in chemistry and life sciences. Of particular importance is the analysis of singular semi-linear parabolic PDEs.

‘This class of PDEs has had very little treatment to date,’ says David. ‘The generic theory John and I have developed enables us to understand the processes that are modelled by these PDEs.’

Although diverse, all these processes pose evolution problems – in other words, how structures change with time. There are applications in physical chemistry – particularly in combustion problems – and in life sciences, such as the spatial and temporal evolution of populations, the spread of infectious diseases, the propagation of large-scale forest fires and propagation of genetic structures.

‘All of these applications lead to PDEs, which are of semi-linear parabolic type. The interest is in obtaining solutions to this class of PDEs and in particular the qualitative structure of solutions.’

The class of regular semi-linear parabolic PDEs has been well understood for the past 60 years; the class of singular such PDEs far less so, says David.

‘What we have done is to provide a general theory for the particular class of what are called non-Lipschitz PDEs, which enables all of the models of this class of PDEs to be fully explored.’

Of particular interest is the emergent behaviour after ‘large times’. The theory reveals that there is significant bifurcation (change in structure) in ‘large times’ from that associated with the regular class of semi-linear parabolic PDEs.

‘The development of the theory therefore enables a much broader class of models to be studied and analysed in detail in relation to the application areas we’ve outlined,’ says David. ‘In addition to real-world applications, from a mathematical standpoint the theory provides a significant extension of the classical theory of the semi-linear parabolic PDEs.’

Various attempts at clarification have been made over recent decades, but although these have been successful in particular cases, what David and John have done is to take it a step further and produce a generic theory.

Both men, understandably, are proud to have developed this theory. And their collaboration is ongoing: they are working together in other areas of the same field.

David explains how the collaborative relationship came about. ‘I have been working on specific problems in this area of mathematics for 20 years. John was first introduced to it because he did a course of mine as an undergraduate, and then did a summer internship with me in this area – which highlights that summer internships can have a lot of value. That was seven years ago, and he was very interested and very able. We’ve worked together ever since.’

John, who says the undergraduate course he took of David’s was the ‘most illuminating’ of his first degree, subsequently did a PhD at Birmingham in the same area of mathematics; then became a Research Fellow and is now a Lecturer in Applied Mathematics.

‘Not only has our work together resulted in this monograph and additional papers; we’re still working together in that same broad area, because there’s still a lot more to develop,’ comments David. ‘The work that is the focus of the monograph is a nice general development of a theory almost to completion for a particular class of PDEs, but it’s not a closed book. It has set up a framework and a methodology, providing us with a good chance of answering questions in applied sciences, biomedical sciences and environmental sciences. A good mathematical theory should always have a relevance to serious models that arise in applied sciences.’