Rotating spirals in segregated reaction-diffusion systems

Abstract

We give a complete characterization of the boundary traces ϕi(i = 1, . . . , K) supporting spiraling waves, rotating with a given angular speed ω, which appear as singular limits ofcompetition-diffusion systems of the type∂tui − ∆ui = µui − βui ∑j6=iaijujin Ω × R+ui = ϕi on ∂Ω × R+ui(x, 0) = ui,0(x) for x ∈ Ωas β → +∞. Here Ω is a rotationally invariant planar set and aij > 0 for every i and j. We tacklealso the homogeneous Dirichlet and Neumann boundary conditions, as well as entire solutionsin the plane. As a byproduct of our analysis we detect explicit families of eternal, entire solutionsof the pure heat equation, parameterized by ω ∈ R, which reduce to homogeneous harmonicpolynomials for ω = 0.