Spikes for the Gierer-Meinhardt system with discontinuous diffusion coefficients

Abstract

We rigorously prove results on spiky patterns for the Gierer-Meinhardt system with a jump discontinuity in the diffusion coefficient of the inhibitor. Firstly, we show the existence of an interior spike located away from the jump discontinuity, deriving a necessary condition for the position of the spike. In particular we show that the spike is located in one-and-only-one of the two subintervals created by the jump discontinuity of the inhibitor diffusivity. This localisation principle for a spike is a new effect which does not occur for homogeneous diffusion coefficients. Further, we show that this interior spike is stable.

Secondly, we establish the existence of a spike whose distance from the jump discontinuity is of the same order as its spatial extent. The existence of such a spike near the jump discontinuity is the second new effect presented in this paper.

To derive these new effects in a mathematically rigorous way, we use analytical tools like Liapunov-Schmidt reduction and nonlocal eigenvalue problems which have been developed in our previous work.

Finally, we confirm our results by numerical computations for the dynamical behavior of the system. We observe a moving spike which converges to a stationary spike located in the interior of one of the subintervals or near the jump discontinuity.