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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

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

Pages: 1493 - 1509, Volume 27, Issue 4, August 2010      doi:10.3934/dcds.2010.27.1493

 
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Vladimir V. Chepyzhov - Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoy Karetniy 19, Moscow 127994, GSP-4, Russian Federation (email)
Mark I. Vishik - Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoy Karetniy 19, Moscow 127994, GSP-4, Russian Federation (email)

Abstract: 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:  Trajectory attractor, reaction-diffusion systems, vanishing diffusion, partly dissipative systems.
Mathematics Subject Classification:  Primary: 35K57; Secondary: 35B41.

Received: September 2009;      Revised: December 2009;      Available Online: March 2010.