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Reaction-diffusion systems for multigroup neutron fission with temperature feedback: positive steady-state and stability. (English) Zbl 0892.35023

Reaction-diffusion systems describing the dynamics of fission reactors with temperature feedback are considered. First, the elliptic system \[ \Delta u_i(x) + \sum _{j=1}^{m+1} H_{i,j}(x,u_{m+1})u_j(x) = 0 \quad \text{in } k\Omega,\qquad u_i(x) = 0 \quad \text{on } \partial k\Omega,\qquad i=1,\dots,m+1 \] is studied where \(\Omega \) is a bounded domain in \(\mathbb{R}^N\) with \(C^{2+\mu }\) boundary, \(k\Omega = \{x=ky;\) \(y \in \Omega \}\), \(k\) is a positive parameter, \(m \geq 2\). The domain \(k\Omega \) represents the reactor core, \(u_j(x)\), \(j=1,\dots,m\), is the neutron flux of the \(j\)-th energy group, \(u_{m+1}(x)\) is the temperature, \(H_{i,j}\), \(i,j=1,\dots,m\) describe the temperature dependent fission and scattering rates of various energy groups, \(M_{m+1,m+1}\) denotes the cooling coefficient, \(H_{m+1,j},\^^Mj=1,\dots,m\) denotes the rate of temperature increase due to neutrons in the group \(j\). We have \(H_{m+1,m+1} \leq 0\), \(H_{i,j} \geq 0\) for all \([i,j] \neq [m+1,m+1]\). It is proved that positive steady-states bifurcate from the trivial solution at some parameter \(k\), i.e. for some critical size of the reactor core. Further, the linearized stability of these steady-states as solutions of the corresponding parabolic system is shown and the asymptotic stability is proved by using the abstract theorem for sectorial operators. The main advantage with respect to previous results is in treating all the \(m+1\) equations simultaneously, without eliminating the equation for the temperature.
Reviewer: M.Kučera (Praha)

MSC:

35B32 Bifurcations in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35K57 Reaction-diffusion equations
35B35 Stability in context of PDEs
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