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Dynamics of a parabolic problem arising in nuclear engineering. (English) Zbl 1340.35111
The authors analyze the dynamics of a spatially heterogeneous parabolic system which is a refinement of a classical prototype model in nuclear engineering proposed by W. E. Kastenberg and P. L. Chambré [“On the stability of nonlinear space-dependent reactor kinectics”, Nuclear Science and Engineering 31, No. 1, 67–79 (1968)] and introduced to study the interactions between the density of fast neutrons and the temperature in the reactor $\begin{cases} \dfrac{\partial u}{\partial t}-\Delta u= \lambda u -b(x) uv, & x\in \Omega,\, t>0,\\ \dfrac{\partial v}{\partial t}- \Delta v =c(x)u-d(x)uv-e(x)v, & x\in \Omega,\, t>0,\\ (u,v)=(0,0), & x\in \partial \Omega,\, t>0,\\ u(x,0)=u_0>0, \;v(x,0)=v_0\geq 0, & x\in \Omega.\end{cases}$ The dynamics is completely characterized within the ranges of values of the parameters where the model does not admit a positive steady state, as well as in the region where it admits a positive steady state and the system can be transformed, through an appropriate change of variables, into an irreducible cooperative system.
##### MSC:
 35K40 Second-order parabolic systems 35K55 Nonlinear parabolic equations 35B09 Positive solutions to PDEs
##### Keywords:
parabolic system; steady state; positive solution