The regular attractor for the reaction-diffusion system with a nonlinearity rapidly oscillating in time and its averaging

Abstract

The longtime behaviour of solutions of a reaction-diffusion system with the nonlinearity rapidly oscillating in time ($f=f(t/{\varepsilon},u)$) is studied. It is proved that (under the natural assumptions) this behaviour can be described in terms of global (uniform) attractors $\mathcal A^{\varepsilon}$ of the corresponding dynamical process and that these attractors tend as ${\varepsilon}\to 0$ to the global attractor $\mathcal A^0$ of the averaged autonomous system. Moreover, we give a detailed description of the attractors $\mathcal A^{\varepsilon}$, ${\varepsilon}\ll1$, in the case where the averaged system possesses a global Liapunov function.