Bifurcating solutions and stabilities for multigroup neutron fission systems with temperature feedback (Q1910833)

The following reaction-diffusion system describing neutron fission in a reactor is studied: \hskip17mm \(\Delta u_i(x)+ \sum^m_{j= 1} H_{ij} (x, V(x)) u_j(x)= 0\text{ for all } x\in k\Omega,\;i= 1,\dots, m\), \hskip17mm \(\Delta V(x)- c(x) V(x)+ \sum^m_{j= 1} g_j(x) u_j(x)= 0\text{ for all } x\in k \Omega\), \hskip17mm \(u_i(x)= V(x)= 0\text{ for all } x\in \partial k \Omega,\;i= 1,\dots, m\). Here \(\Omega\) is a fixed domain in \(\mathbb{R}^N\), \(k\) is a positive parameter, \(u_i\) describes the neutron flux of the \(i\)th energy group, \(V(x)\) denotes the temperature, \(H_{ij}\) describes the temperature dependent fission and scattering rates of the various energy groups, \(c(x)\) is the cooling function and \(g_j(x)\) describes the rate of temperature increase corresponding to neutrons in the \(j\)th group. It is shown that positive steady states bifurcate from the trivial solution for \(k\) crossing a certain critical \(k_0\). The characterization of this bifurcation point \(k_0\) (i.e. of the critical size of the reactor) is given and the stability of bifurcating solutions is proved.