Regular solutions and global attractors for reaction-diffusion systems without uniqueness

Oleksiy V. Kapustyan; Pavlo O. Kasyanov; José Valero

doi:10.3934/cpaa.2014.13.1891

Article Contents

2014, Volume 13, Issue 5: 1891-1906. Doi: 10.3934/cpaa.2014.13.1891

This issue Previous Article Non-isothermal viscous Cahn-Hilliard equation with inertial term and dynamic boundary conditions Next Article Reaction-diffusion equations with a switched--off reaction zone

Regular solutions and global attractors for reaction-diffusion systems without uniqueness

1.
Taras Shevchenko National University of Kyiv, 60, Volodymyrska Street, 01601, Kyiv
2.
Institute for Applied System Analysis, National Technical University of Ukraine "KPI", Kyiv
3.
Universidad Miguel Hernández, Centro de Investigación Operativa, Avda. Universidad s/n, Elche (Alicante), 03202

Received: August 31, 2013

Revised: August 31, 2013

Published: August 2015

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Abstract

In this paper we study the structural properties of global attractors of multi-valued semiflows generated by regular solutions of reaction-diffusion system without uniqueness of the Cauchy problem. Under additional gradient-like condition on the nonlinear term we prove that the global attractor coincides with the unstable manifold of the set of stationary points, and with the stable one when we consider only bounded complete trajectories. As an example we consider a generalized Fitz-Hugh-Nagumo system. We also suggest additional conditions, which provide that the global attractor is a bounded set in $(L^\infty(\Omega))^N$ and compact in $(H_0^1 (\Omega))^N$.

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Mathematics Subject Classification: Primary: 35B40, 35B41, 35K55; Secondary: 37B25, 58C06.

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