Long time behavior of the solutions for coupled parabolic systems without monotone nonlinearities (Q2761991)

The paper deals with the long time behaviour of the solutions of the following parabolic system \(\frac{\partial u_i}{\partial t} -L_iu_i = f_i(x,{\mathbf u})\), \(t > 0\), \(x \in \Omega,\) \( B_iu_i = h_i(x)\), \(t > 0\), \(x \in \partial \Omega,\) \(u_i(0,x) = u_{i,0}(x)\), \(x \in \Omega,\) where \(i = 1,\dots ,n\), \({\mathbf u} = (u_1,\dots ,u_n)^T\), \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(L_i\) are uniformly elliptic operators. The author establishes monotone iterative technique which provides the existence of a unique solution. Using the monotone method the author constructs an attractor of the time-dependent solutions of the system without monotone conditions on the nonlinearities.