Spatial independence of invariant sets of parabolic systems (Q1604762)

The authors deal with the following autonomous parabolic system, that is \[ \begin{cases}\frac{\partial u}{\partial t}- d\Delta u=f(u,v)\quad &\text{in }\Omega\\ \frac{\partial v_j}{\partial t} -\Delta v_j=v_j\left(h_j(u)-\sum^n_{k=1} c_{jk}(u)v_k\right)\quad &\text{in }\Omega,\;j=1,\dots, k\\ \frac{\partial u}{\partial\nu}=0,\;\frac{\partial v}{\partial\nu}=0 \quad &\text{on }\partial\Omega,\end{cases}\tag{1} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with boundary \(\partial\Omega\) of class \(C^{2+\ell}\), \(\ell>0\), \(d>0\) is a constant. Under some suitable assumptions on the data of (1), the author's show that \(\omega\)-limit sets for (1) consist of space in dependent solutions. Moreover the author's also prove similar results for shadow systems and some systems with periodic time dependence and calculate the Conley indices of invariant sets.