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November  2008, 7(6): 1375-1392. doi: 10.3934/cpaa.2008.7.1375

Stabilisation of differential inclusions and PDEs without uniqueness by noise

Tomás Caraballo 1, , José A. Langa 1, and  José Valero 2,

1. 

Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain
2. 

Centro de Investigación Operativa, Universidad Miguel Hernández, Avda Universidad s/n, 03202 Elche, Alicante

Received  October 2007 Revised  April 2008 Published  August 2008

We prove that the asymptotic behaviour of partial differential inclusions and partial differential equations without uniqueness of solutions can be stabilised by adding some suitable Itô noise as an external perturbation. We show how the theory previously developed for the single-valued cases can be successfully applied to handle these set-valued cases. The theory of random dynamical systems is used as an appropriate tool to solve the problem.
Keywords: Multivalued semiflows, stabilization by Ito's noise, random dynamical systems..
Mathematics Subject Classification: Primary: 37L55, 93D20; Secondary: 28B10, 34D2.
Citation: Tomás Caraballo, José A. Langa, José Valero. Stabilisation of differential inclusions and PDEs without uniqueness by noise. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1375-1392. doi: 10.3934/cpaa.2008.7.1375
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