Global-in-time behavior of the solution to a Gierer-Meinhardt system

Gierer-Meinhardt system is a mathematical model to describe biological pattern formation due to activator and inhibitor. Turing pattern is expected in the presence of local self-enhancement and long-range inhibition. The long-time behavior of the solution, however, has not yet been clarified mathematically. In this paper, we study the case when its ODE part takes periodic-in-time solutions, that is, $\tau=s+1$. Under some additional assumptions on parameters, we show that the solution exists global-in-time and absorbed into one of these ODE orbits. Thus spatial patterns eventually dis- appear if those parameters are in a region without local self-enhancement or long-range inhibition

Exponential suppression of radiatively induced mass in the truncated overlap

A certain truncation of the overlap (domain wall fermions) contains $k$ flavors of Wilson-Dirac fermions. We show that for sufficiently weak lattice gauge fields the effective mass of the lightest Dirac particle is exponentially suppressed in $k$. This suppression is seen to disappear when lattice topology is non-trivial. We check explicitly that the suppression holds to one loop in perturbation theory. We also provide a new expression for the free fermion propagator with an arbitrary additional mass term.Comment: 26 pages, plain Te