Singular limits of nonlinear elliptic and parabolic PDE

Strong-competition limits of competing-population systems with two components
have proved an effective way of obtaining information about the competition
system for large values of the competition parameter. Understanding the
dynamics of such systems is very difficult in general, due to their lack of
variational structure. On the other hand, the strong-interaction limit is a scalar
equation which has a Lyapunov function and is much more easily studied.
Such singular limits are both mathematically useful, and correspond to the important
biological and physical phenomena of spatial segregation and phase separation.

For competition systems with more than two components, however, the techniques
that are effective in the two-component case fail and new ideas are needed.
Some success has been had been recently, using tools from free-boundary problems,
to show convergence of solutions of the competition system to a solution of a limit
problem.

The central goal of this proposal is to prove results on the so-called converse problem for
multi-component systems; that is, on whether or not a solution of the limit problem
does in fact arise as the limit of solutions of the original system. Such converse problems
are an essential and important part of the understanding of any singular limit, and have
yet to be addressed at all for strong-competition limits of multi-component systems.
We also aim to prove results on the converse problem for competition systems of
two components when the interaction terms are large only on a subdomain, and for
systems with a different, cubic type of competitive coupling, which occurs in modelling
certain quantum mechanical systems.

The main beneficiaries of this research will be mathematicians, both in the UK and internationally,
particularly experts in the area of nonlinear partial differential equations. Since the proposed
mathematical problems have direct application to the fields of population dynamics and quantum
mechanical systems, mathematical biologists and physicists with interests in these topics will also benefit. The results arising from this work will be published in academic journals and disseminated through the medium of invited seminar and conference talks. Care will be taken both in the writing of journal articles and in the preparation of presentations to ensure that the introductory part is accessible to those from disciplines other than mathematics.

There is growing interest in strong-interaction limits of nonlinear elliptic and parabolic PDE because such limits both yield information when the original problem is mathematical intractable and also correspond to the biologically and physically interesting phenomena of spatial segregation and phase separation. Most work to date has focussed on the case of two interacting components, whereas there are often more than two components in real applications, and the proposed research will make a significant contribution to the understanding of such multi-component systems by characterising when a solution of the limit problem arises as the limit of solutions of the original problem. This is important for applications because it can be viewed as a physical/biological admissibility question about solutions of the limit problem. New mathematical methods will need to be developed to address the technical issues arising in the multi-component case, and it is expected that such ideas will be of benefit to mathematicians working on related problems.

This project will further strengthen the collaborative links between the UK and Australia, and benefit mathematicians in both countries through dissemination of the mathematical ideas and application areas of the project.