Portal de Periódicos da CAPES

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

2010; American Institute of Mathematical Sciences; Volume: 27; Issue: 4 Linguagem: Inglês

10.3934/dcds.2010.27.1493

1553-5231

Vladimir V. Chepyzhov, Mark Vishik,

Nonlinear Dynamics and Pattern Formation

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.