Highly oscillatory behavior of the activator in the Gierer and Meinhardt system (Q2476175)

The authors study the Gierer-Meinhardt system, i.e. a nonlinear system in one dimension of the form \(-\varepsilon^2u_{xx}+u= u^2/v\), \(-v_{xx}+v= u^2\), in \((0,L)\), with the boundary conditions \(u_x(0)= u_x(L)= v_x(0)= v_x(L)= 0\). \(u\) and \(v\) represent an activator and inhibitor model and \(\varepsilon\) tends to zero. They construct a family of positive solutions such that the activator \(u\) oscillates \(\sim \varepsilon^{-1}\) times, whereas the inhibitor stays close to a strictly decreasing function.