On a system of the ``reaction-diffusion'' class (Q1175358)

The asymptotic behavior of solutions is considered for the reaction- diffusion system: \[ \partial_ tu=\Delta u-f(u,T),\quad \partial_ tT=\Delta T+I_ 0+g(u,T),\quad (t,x)\in(0,+\infty)\times\Omega, \] \[ \partial u/\partial\nu|_{x\in\partial\Omega}=0,\quad\partial T/\partial\nu|_{x\in\partial\Omega}=0, \] where \(u=u(t,x)\), \(T=T(t,x)\), \(I_ 0=\hbox{const}>0\); \(\Omega\) is a bounded domain in \(\mathbb{R}^ n\) with smooth boundary \(\partial\Omega\); \(f,g\in C^ 1(\mathbb{R}^ 2)\). Under some natural assumptions on \(f\) and \(g\) the existence of the maximal attractor for some equivalent systems is proven. Results of \textit{M. Vishik} and \textit{A. Babin} [e.g.: Usp. Mat. Nauk 38, No. 4(232), 133-187 (1983; Zbl 0541.35038); Sib. Math. J. 24, 659-671 (1983); transl. from Sib. Mat. Zh. 24, No. 5(141), 15-30 (1983; Zbl 0528.35055)] are used essentially.