Target pattern and spiral solutions to reaction-diffusion equations with more than one space dimension

Abstract

We prove the existence of homogeneous target pattern and spiral solutions to equations of the form c t = F(c) + ▽ 2c, where F(c) = ( λ−Ω Ωλ )c, λ = λ(¦c¦ 2), ω = ω(¦c¦ 2), λ′ < 0, λ(1) = 0, ω′ <0 and ¦ω′¦⪡ 1 ; the spatial dimension is greater than one. As in the one-dimensional case, such solutions exist for discrete values of the asymptotic wave number (or equivalently, the frequency of oscillation of the entire solution). For target patterns, we construct solutions for a sequence of frequencies. For spirals, we construct only the “lowest mode” solution.