Infinitely many fine modes bifurcating from radially symmetric internal layers

Abstract

For a propagator–controller type system of reaction–diffusion equations on an N‐dimensional ball, it is shown that there exists a family of radially symmetric stationary transition layer solutions when the propagator diffuses slowly compared with the controller. A spectral analysis is also carried out for the linearization around the layered solutions, which reveals that these layer‐solutions are unstable. More importantly, it is found that for activator(propagator)–inhibitor(controller) systems there exist infinitely many static bifurcation points from the radially symmetric solutions as the diffusion rate of the activator goes to zero. This confirms the validity of a conjecture raised in Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), 359–401, in the special situation where the domain is a ball.