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December  2010, 28(4): 1655-1667. doi: 10.3934/dcds.2010.28.1655

Blow-up in a subdiffusive medium with advection

W. Edward Olmstead 1, , Colleen M. Kirk 2, and  Catherine A. Roberts 3,

1. 

Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208, United States
2. 

Mathematics, California Polytechnic State University, San Luis Obispo, CA 93407, United States
3. 

Mathematics and Computer Science, College of the Holy Cross, Worcester, MA 01610, United States

Received  October 2009 Revised  February 2010 Published  June 2010

A mathematical model is presented for a localized energy source in a subdiffusive medium with advection. It is shown that blow-up cannot be prevented, regardless of the advection speed. This result holds for media associated with an unbounded spatial domain in one, two, or three dimensions. Results also suggest that increasing the advection speed will delay the time to blow-up, even though it does not prevent a blow-up. It is interesting to note that these results are in distinct contrast with the analogous classical diffusion problem, in which blow-up can be prevented by increasing sufficiently the advection speed. The asymptotic behavior of the temperature near the blow-up time is also presented.
Keywords: subdiffusion, moving source., heat equation, blow-up, advection.
Mathematics Subject Classification: Primary: 35K60, 45D05, 80A2.
Citation: W. Edward Olmstead, Colleen M. Kirk, Catherine A. Roberts. Blow-up in a subdiffusive medium with advection. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1655-1667. doi: 10.3934/dcds.2010.28.1655
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