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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.
Mathematics Subject Classification: Primary: 35K57; Secondary: 35B41.

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