An interface problem in an activator-inhibitor system depending on the spatial average of an inhibitor (Q2721579)

The author discusses an activator-inhibitor system of the form NEWLINE\[NEWLINE\begin{cases} \varepsilon\sigma u_t=\varepsilon^2 u_{xx}+f(u)-v\\ v_t=v_{xx}+ g(u,v)+\langle u\rangle \end{cases} \quad t>0,\;x\in (0,1).NEWLINE\]NEWLINE Here \(\langle u\rangle\) denotes the average, \(f(u)=-u+H(u-a_0)\), \( H=\)Heaviside step function, \(0<a_0< {1\over 2},g(u,v)= \mu u-v\). Neumann boundary conditions are imposed. He studies the dynamics of interfaces \(s(t)\) in \((0,1)\). The main methods used are abstract evolution equations and Hopf bifurcation.