Abstract
In this paper, we study the dynamical properties inside the global attractor for multivalued semiflows. Given a disjoint finite family of isolated weakly invariant sets, we prove, extending a previous result from the single-valued case, that the existence of a Lyapunov function, the property of being a dynamically gradient semiflow and the existence of a Morse decomposition are equivalent properties. We apply this abstract theorem to a reaction–diffusion inclusion.
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Acknowledgments
The first author was partially supported by the Fundação de Amparo a Pesquisa do Estado de São Paulo under Grant Fapesp 2011/21456-7, and by the Fundação CAPES (Ministério da Educação do Brasil) under CAPES-DGU proyecto n\({{}^\circ }\) 238/11. The second author was partially supported by FEDER and Ministerio de Economía y Competitividad (Spain) under Grants MTM2011-22411 and MTM2012-31698, and by Junta de Andalucía under Proyecto de Excelencia P12-FQM-1492.
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da Costa, H.B., Valero, J. Morse decompositions and Lyapunov functions for dynamically gradient multivalued semiflows. Nonlinear Dyn 84, 19–34 (2016). https://doi.org/10.1007/s11071-015-2193-z
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DOI: https://doi.org/10.1007/s11071-015-2193-z