# American Institute of Mathematical Sciences

September  2014, 13(5): 1891-1906. doi: 10.3934/cpaa.2014.13.1891

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

 1 Taras Shevchenko National University of Kyiv, 60, Volodymyrska Street, 01601, Kyiv, Ukraine 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  September 2013 Revised  September 2013 Published  June 2014

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 $(H_0^1 (\Omega))^N$.
Citation: Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Regular solutions and global attractors for reaction-diffusion systems without uniqueness. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1891-1906. doi: 10.3934/cpaa.2014.13.1891
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