MURPHYS-HSFS-2014 - 7th International Workshop on Multi-Rate Processes & Hysteresis, 2nd International Workshop on Hysteresis and Slow-Fast Systems, April 7-11, 2014 - Abstract
Gurevich, Pavel
(joined paper with Sergey Tikhomirov) In biological and population models, one often has the situation where diffusive and nondiffusive substances interact with each other according to hysteresis law. The simplest prototype model then involves a reaction-diffusion equation of the form $$ u_t=Delta u + mathcal R(u) $$ with $u=u(x,t)$, where $mathcal R(u(x,cdot))(t)$ ($= h_1$ or $-h_2$) is the non-ideal relay operator defined for each fixed $xinmathbb R^N$. Such a model has been studied since 1980s, mostly in the setting where the non-ideal relay $mathcal R(u)$ is replaced by its multi-valued analog, which allows one to prove existence results but says little about the qualitative behavior of solutions. In the talk, we will consider the discrete counterpart $$ dot u_n=dfracu_n-1-2u_n+u_n+1varepsilon^2+mathcal R(u_n),quad ninmathbb Z, $$ with $u_n=u_n(t)$ and $varepsilon>0$. This equation becomes well posed, and one can concentrate on the qualitative and quantative behavior of solutions. As the next step, this should allow one to properly pass to the continuous limit $varepsilonto 0$ and determine $u(x,t)$. Under appropriate initial conditions, we will see that the switching nodes $u_n$ form a certain pattern depending on $h_1$ and $h_2$, but surprisingly not on $varepsilon$. In the examplary case $h_2=0$, we will present the main ingredients that allow us to prove pattern formation.