Target pattern solutions to reaction-diffusion equations in the presence of impurities

AbstractWe consider reaction-diffusion equations of the special type ct = F(c,x) +▽ 2c, with F(c,x) = (λ−ΩΩ λ)c, λ = λ(¦c¦2), Ω = \̄gw(¦c¦2) + ε2Ω(x), λ′ < 0,λ(1) = 0, \̄gw′ < 0, and Ω(x) having compact support in x. Assumptions about the relevant space scales and size of the catalytic effect exactly parallel those of Hagan (Advances in Appl. Math., 2 (1981), 400–416). The results are also parallel: For x of dimension one or two, if Ω(x) ≥ 0, Ω ≭ 0, then ∃ a unique target pattern solution which stays locally close to the homogeneous limit cycle solution. If x has dimension three, there is such a solution provided that Ω(x) is sufficiently large. Thus this paper shows that the phenomena uncovered formally by Hagan for a much larger class of kinetic equations can be rigorously substantiated for λ — ω systems