Network for Biological Invasions and Dispersal Research

MITACS Seminar at the University of New Brunswick, Fredericton

Friday, 23 November, 2007,

Chunhua Ou

of Memorial University

will speak on

Analysis of Non-local Reaction Diffusion Equations

2:30 PM, Tilley Hall 205

Abstract: This talk includes two parts. The first part is about the metastable dynamics for a nonlocal PDE modeling the upwards propagation of a flame-front interface in a vertical channel. We analyze the one-dimensional case where the channel cross-section is taken to be the slab -1 < x < 1. In a certain asymptotic limit, the interface assumes a roughly concave parabolic shape, and the tip of the parabola drifts asymptotically exponentially slowly towards the boundary of the domain. In contrast to previous analyses that studied this behavior by transforming the governing nonlocal PDE to a convection-diffusion equation, a novel nonlinear transformation is introduced that transforms the problem to a singularly perturbed quasilinear PDE. The steady-state problem for this transformed PDE, for which the parabolic interface shape maps onto a one-spike solution, is closely related to a class of two-point boundary value problems with seemingly spurious solutions studied initially by G. Carrier in 1968. Asymptotic results for a one-spike solution to this transformed PDE are obtained together with a formal metastability analysis of certain time-dependent solutions.

The second part is about a nonlocal PDE which arises in population biology when age-structure is incorporated. We investigate the solution in unbounded space domain. In particular, traveling wave patterns are studied. We classify the non-linearity reaction term into four types in terms of monostable, bistable, monotone and non-monotone natures. Existence of traveling wavefront(s) is given for this non-local PDE via perturbation techniques and fixed point theory.