Singular limits of nonlinear elliptic and parabolic PDE

点击翻译按钮获取中文摘要

Strong-competition limits of competing-population systems with two componentshave 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 techniquesthat 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 limitproblem. 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 problemsare 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 oftwo components when the interaction terms are large only on a subdomain, and forsystems with a different, cubic type of competitive coupling, which occurs in modelling certain quantum mechanical systems.