TRAJECTORY ATTRACTOR FOR REACTION-DIFFUSION SYSTEM WITH DIFFUSION COEFFICIENT VANISHING IN TIME

We consider a non-autonomous reaction-diffusion system of two equations having in one equation a diffusion coefficient depending on time (delta = delta(t) >= 0, t >= 0) such that delta(t) -> 0 as t -> +infinity. The corresponding Cauchy problem has global weak solutions, however these solutions are not necessarily unique. We also study the corresponding "limit" autonomous system for delta = 0. This reaction-diffusion system is partly dissipative. We construct the trajectory attractor u for the limit system. We prove that global weak solutions of the originalnon-autonomous system converge as t -> +infinity to the set u in a weak sense. Consequently, u is also as the trajectory attractor of the original non-autonomous reaction-diffusions system.