On stable nonconstant stationary solutions and mesoscopic patterns for FitzHugh-Nagumo equations in higher dimensions

FitzHugh-Nagumo equation has been studied extensively in the field of mathematical biology. It has the mechanism of "lateral inhibition" which seems to play a big role in the pattern formation of plankton distribution. We consider FitzHugh-Nagumo equation in high dimension and show the existence of stable nonconstant stationary solutions which have fine structures on a mesoscopic scale. We construct spatially periodic stationary solutions. Moreover, we compute the singular limit energy, which suggests that the transition from planar structure to droplet pattern can occur when parameters change.