Dynamics of an interior spike in the Gierer-Meinhardt system (Q2719186)

The authors deal with the Gierer-Meinhardt system NEWLINE\[NEWLINEA_t = D_A \Delta A - k_A A + l_A A^2 / H,\quad H_t = D_H \Delta H - k_H H + l_H A^2NEWLINE\]NEWLINE on a bounded domain \(\Omega \subset\mathbb{R}^N,\) \(N=2,3,\) with zero Neumann boundary conditions and positive initial densities \(A(x,0)> 0\) and \(H(x,0)>0\) of the slow-diffusing and/or fast-decaying activator \(A\) and fast-diffusing and/or slow-decaying inhibitor \(H\). Under certain assumptions on the system parameters and domain size, they show that a single interior spike of high activator density stays in the interior of the domain. More precisely, they show that the velocity of the center of the spike is proportional to the negative gradient of \(R(\xi, \xi)\), where \(R(x,\xi)\) is the regular part of the Green's function of the Laplacian with Neumann BC. These results contrast with prediction of the shadow Gierer-Meinhardt system, in which the inhibitor diffuses with infinite speed and/or decays infinitely slow, and single interior spikes move towards the nearest point of the boundary \(\partial \Omega\).