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

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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

Abstract
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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 & Continuous Dynamical Systems - A, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493

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