| Abstract: |
| In this talk, we consider the following initial-boundary value problem for a reaction diffusion system:
\begin{equation}
\tag{NR}
\left\{
\begin{aligned}
& \partial_t u_1-\Delta u_1 = u_1u_2 - bu_1, && x\in\Omega,~t>0,\
& \partial_t u_2-\Delta u_2 = au_1, && x\in\Omega,~t>0,\
& \partial_{\nu}u_1 + \alpha u_1 = \partial_{\nu}u_2 + \beta|u_2|^{\gamma-2}u_2 = 0, && x\in\partial\Omega,~t>0,\
& u_1(x,0)=u_{10}(x)\ge 0,~u_2(x,0)=u_{20}(x)\ge 0, &&x\in\Omega,
\end{aligned}
\right.
\end{equation}
where \(\Omega\subset\mathbb{R}^N\) is a bounded domain with smooth boundary \(\partial\Omega\),
\(\nu\) denotes the unit outward normal vector on \(\partial\Omega\) and
\(\partial_\nu u_i = \nabla u_i\cdot\nu \) ~(\(i=1,2\)).
Moreover, \(u_1\), \(u_2\) are real-valued unknown functions, \(a\), \(b>0\), \(\alpha\ge0\), \(\beta>0\), \(\gamma\ge2\) are parameters and \(u_{10}\), \(u_{20}\in L^{\infty}(\Omega)\) are
given initial data. This system describes diffusion phenomena of neutrons and heat in nuclear reactors, introduced by Kastenberg and Chambr\`{e}. In this model, the unknwon functions \(u_1\) and \(u_2\) represent the neutron density and the temperature in nuclear reactors respectively. From physical point of view it would be natural to consider the nonlinear boundary condition for the temperature. Indeed, it is well known that the power type nonlinearity on the boundary condition for \(u_2\) is justified by Stefan-Boltzmann`s law. We show the existence of positive stationary solutions and investigate some threshold property of them to determine blow-up or global existence of solutions to (NR). |
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