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November  2010, 27(4): 1493-1509. doi: 10.3934/dcds.2010.27.1493

Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time

Vladimir V. Chepyzhov 1, and  Mark I. Vishik 1,

1. 

Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoy Karetniy 19, Moscow 127994, GSP-4, Russian Federation, Russian Federation

Received  September 2009 Revised  December 2009 Published  March 2010

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.
Keywords: reaction-diffusion systems, partly dissipative systems., Trajectory attractor, vanishing diffusion.
Mathematics Subject Classification: Primary: 35K57; Secondary: 35B4.
Citation: Vladimir V. Chepyzhov, Mark I. Vishik. Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493
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