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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 ($\delta =\delta (t)\geq 0,t\geq 0$) such that
$\delta (t)\rightarrow 0$ as $t\rightarrow +\infty $. 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 A for the limit system. We prove
that global weak solutions of the original non-autonomous system
converge as $t\rightarrow +\infty $ to the set A in a
weak sense. Consequently, A is also as the trajectory
attractor of the original non-autonomous reaction-diffusions
system.