Exact periods and exact critical values for Hopf bifurcations from multi-peak solutions of the shadow Gierer-Meinhardt model in one spatial dimension

We consider the shadow Gierer-Meinhardt model{∂tU=ε2Uxx−U+Upξqfor0<x<1,0<t<T,τ∂tξ=∫01(−ξ+Urξs)dxfor0<t<T,Ux(0,t)=Ux(1,t)=0, where ε>0, τ>0, p>1, q>0, r>0, s≥0 and qr/(p−1)(s+1)>1. Wei and Winter showed that if (p,r)=(3,2) or (2,2), then as τ increases, a stable stationary monotone solution is destabilized by Hopf bifurcation, and hence periodic solutions appear. In this paper we consider two cases (p,r)=(3,1) and (3,3). We show that a Hopf bifurcation occurs for n-mode stationary solutions, n≥1, in a rigorous way, studying eigenvalues in detail. Exact periods and exact critical values of τ can be written by using complete elliptic integrals. A relationship between a period and a shape of a stationary solution is also studied. In particular, a maximum point of the period is studied.