Asymptotic behavior of a class of reaction--diffusion equations with delays (Q1399332)

The authors study systems of equations of the form \[ \begin{gathered} {\partial u\over\partial t}= \nabla(G(x, u)\circ\nabla u)- Bu+ F(u_t),\quad x\in\Omega,\quad t\geq t_0,\\ {\partial u\over\partial n}\Biggl|_{\partial\Omega}= 0,\quad t\geq t_0,\\ u(t_0+ s,x)= \phi(s,x)\quad -r\leq s\leq 0,\quad x\in\Omega,\end{gathered} \] where \(u_t(x)= u(t+ s,x)\), \(\phi(s, x)\in C([- r,0]\times \Omega;\mathbb{R}^m)\). By \(T(t)\phi= u_t(x)\) the semigroup of operators is defined for \(\phi\in C^L= C([- r,0], L^2(\Omega))\). The authors prove that the semigroup \(T(t)\) possesses a compact global and connected attractor in \(C^L\).