On ring-like solutions for the Gray–Scott model: existence, instability and self-replicating rings

We consider the following Gray–Scott model in B(R)(0) ={x: |x| < R} (subset) R^N, N=2,3: { v(t) = (epsilon)^2 (delta)v – v + A(v ^2)u in B(R)(0), (tau)(u)t = (delta)u + 1 –u – v^2u in B(R)(0), u,v > 0 ; (partial differential)u/(partial differential)v = (partial differential)v/(partial differential)v on (partial differential)B(R)(0) where (epsilon) > 0 is a small parameter. We assume that A=A (epsilon) 1/2. For each A < +(infinity) and R<(infinity), we construct ring-like solutions which concentrate on an (N-1)-dimensional sphere for the stationary system for all sufficiently small (epsilon). More precisely, it is proved the above problem has a radially symmetric steady state solution (v(epsilon,R), u(epsilon,R)) with the property that v(epsilon,R)- 0 in R^N \{ r (not equal ) r0 } for some r0 (element of) (0, R). Then we show that for N=2 such solutions are unstable with respect to angular fluctuations of the type (Phi)(r)e(radical)(-1m (theta)) for some m. A relation between A and the minimal mode m is given. Similar results are also obtained when (Omega)=R^N or (Omega)= B(R2)(0) \B(R1)(0) or (Omega)= R^N \ B(R)(0).