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January  2010, 26(1): 43-61. doi: 10.3934/dcds.2010.26.43

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

José M. Arrieta 1, , Raúl Ferreira 1, , Arturo de Pablo 2, and  Julio D. Rossi 3,

1. 

Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain, Spain
2. 

Departamento de Matemática Aplicada, Universidad Carlos III de Madrid, 28911 Leganés, Spain
3. 

IMDEA Matematicas, C-IX, Campus UAM, 28049 Madrid

Received  November 2008 Revised  July 2009 Published  October 2009

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: blow-up, Stability, perturbations..
Mathematics Subject Classification: 35B20, 35B30, 35B3.
Citation: José M. Arrieta, Raúl Ferreira, Arturo de Pablo, Julio D. Rossi. Stability of the blow-up time and the blow-up set under perturbations. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 43-61. doi: 10.3934/dcds.2010.26.43
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