Chalmers Conferences, 9th European Conference on Mathematical and Theoretical Biology

How can we approach reaction-diffusion systems with concentrated terms on the rough border of a thin channel?
Marcone Correa Pereira

Last modified: 2014-03-27


In this work we propose a mathematical model based on a reaction-diffusion system in order to model concentrated interactions of organisms close to the border of a two-dimensional thin channel highly rough. Using methods from asymptotic analysis we introduce an one-dimensional limit system to approach the original one that captures the actual effect of the roughness, thickness and geometry of the channel as well as of the region of concentration of the involved agents.

We discuss how our singular boundary value problem can be approximated by this one-dimensional equation where the variable profile and the oscillatory behavior of the thin domain and the narrow strip where the concentrations take place are captured by a homogenized diffusion coefficient and a reaction limit term respectively. The homogenized limit equation obtained is not singular being an effective option to replace the original one in a accurate way. Also, it displays some features of the original system pointing out their emergent properties giving us conditions to get the qualitative behavior of the modeled problem.

Potential applications of our results include field of lubrication, nanotechnology, fluid-structure interaction mechanism in vascular dynamics, management and control of aquatic ecological systems where one can find localized concentrations in connection with boundary complexity in thin channels.


reaction-diffusion equations, asymptotic analysis, homogenization