ON THE GIERER-MEINHARDT SYSTEM WITH PRECURSORS

We consider the following Gierer-Meinhardt system with a precursor mu(x) for the activator A in R-1: {A(t) = epsilon(2)A '' - mu(x)A + A(2)/H in (-1, 1), tau H-t = DH '' ' -H + A(2) in (-1, 1), A'(-1) = A'(1) = H'(-1) = H'(1) = 0. Such an equation exhibits a typical Turing bifurcation of the second kind, i.e., homogeneous uniform steady states do not exist in the system. We establish the existence and stability of N-peaked steady-states in terms of the precursor mu(x) and the diffusion coefficient D. It is shown that mu(x) plays an essential role for both existence and stability of spiky patterns. In particular, we show that precursors can give rise to instability. This is a new effect which is not present in the homogeneous case.