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

Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term

Pages: 4155 - 4182, Volume 34, Issue 10, October 2014      doi:10.3934/dcds.2014.34.4155

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Oleksiy V. Kapustyan - Kyiv National Taras Shevchenko University, 01033-Kyiv, Ukraine (email)
Pavlo O. Kasyanov - Institute for Applied System Analysis, National Technical University of Ukraine "KPI", Kyiv, Ukraine (email)
José Valero - Centro de Investigación Operativa, Universidad Miguel Hernández de Elche, Avda. de la Universidad, s/n, 03202 Elche, Spain (email)

Abstract: In this paper we study the structure of the global attractor for a reaction-diffusion equation in which uniqueness of the Cauchy problem is not guarantied. We prove that the global attractor can be characterized using either the unstable manifold of the set of stationary points or the stable one but considering in this last case only solutions in the set of bounded complete trajectories.

Keywords:  Reaction-diffusion equations, set-valued dynamical system, global attractor, unstable manifolds, asymptotic behaviour.
Mathematics Subject Classification:  35B40, 35B41, 35K55, 37B25, 58C06.

Received: September 2012;      Revised: January 2013;      Available Online: April 2014.