Stability analysis of Turing patterns generated by the Schnakenberg model

We consider the following Schnakenberg model on the interval (- 1, 1): u(t) = D(1)u" - u + vu(2) in (-1, 1), v(t) = D(2)nu" + B - vu(2) in (-1, 1) u (- 1) = u' (1) v' (- 1) = v'(1) = 0, where D-1 > 0, D-2 > 0, B > 0. We rigorously show that the stability of symmetric N-peaked steady-states can be reduced to computing two matrices in terms of the diffusion coefficients D-1, D-2 and the number N of peaks. These matrices and their spectra are calculated explicitly and sharp conditions for linear stability are derived. The results are verified by some numerical simulations.