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Contracting sets and dissipation

Published online by Cambridge University Press:  14 November 2011

Alexandre N. Carvalho
Alexandre N. Carvalho
Affiliation:
Instituto de Ciências Matemáticas de São Carlos, Universidade de São Paulo, Campus de São Carlos, 13560-970 São Carlos, SP, Brazil e-mail: andcarva@icmsc.sc.usp.br
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Abstract

In this work we study reaction–diffusion systems in fractional power spaces Xα which are embedded in L∞. We prove that the solution operators T(t) to these problems are globally defined, point dissipative, locally bounded and compact. That ensures the existence of global attractors. We also find a set containing the range of every function in the attractor, providing good estimates on asymptotic concentrations. This is done under very few hypotheses on the reaction term. These hypotheses are natural and easy to verify in many applications. The tools employed are the theory of invariant regions for systems of parabolic partial differential equations, the notion of contracting sets and the variation of constants formula. Several examples are considered to emphasise the applicability of these techniques.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 6 , 1995 , pp. 1305 - 1329
Copyright
Copyright © Royal Society of Edinburgh 1995

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