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

Stability of the blow-up time and the blow-up set under perturbations

Pages: 43 - 61, Volume 26, Issue 1, January 2010      doi:10.3934/dcds.2010.26.43

 
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José M. Arrieta - Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain (email)
Raúl Ferreira - Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain (email)
Arturo de Pablo - Departamento de Matemática Aplicada, Universidad Carlos III de Madrid, 28911 Leganés, Spain (email)
Julio D. Rossi - IMDEA Matematicas, C-IX, Campus UAM, 28049 Madrid, Spain (email)

Abstract: In this paper we prove a general result concerning continuity of the blow-up time and the blow-up set for an evolution problem under perturbations. This result is based on some convergence of the perturbations for times smaller than the blow-up time of the unperturbed problem together with uniform bounds on the blow-up rates of the perturbed problems.
   We also present several examples. Among them we consider changing the spacial domain in which the heat equation with a power source takes place. We consider rather general perturbations of the domain and show the continuity of the blow-up time. Moreover, we deal with perturbations on the initial condition and on parameters in the equation. Finally, we also present some continuity results for the blow-up set.

Keywords:  Stability, blow-up, perturbations.
Mathematics Subject Classification:  35B20, 35B30, 35B35.

Received: November 2008;      Revised: July 2009;      Available Online: October 2009.