A description of the global attractor for a class of reaction-diffusion systems with periodic solutions. (Q2711599)

The author deals with the long-term behaviour of the reaction-diffusion system NEWLINE\[NEWLINE\frac{\partial}{\partial t}{u\choose v}=A\Delta{u \choose v}+g\left(\left\| {u\choose v}\right\|^2_{L^2}\right) {u\choose v},\tag{1}NEWLINE\]NEWLINE where \(t<0\), \(x\in\Omega=(0,1)\), \(u,v\) on \(\partial\Omega\) are zero. Under some natural conditions on \(g\) and \((u_0,v_0) \in L^2(\Omega)\times L^2(\Omega)\) he shows that each solution of (1) tends either to the zero solution or to a periodic orbit. To this end he uses Fourier expansion of the solution. Moreover, he shows that the number of periodic orbits is finite. The author also constructs a global attractor for (1).