Blow-up in asubdiffusive medium with advection

W. Edward Olmstead; Colleen M. Kirk; Catherine A. Roberts

doi:10.3934/dcds.2010.28.1655

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2010, Volume 28, Issue 4: 1655-1667. Doi: 10.3934/dcds.2010.28.1655

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Blow-up in a subdiffusive medium with advection

1.
Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208
2.
Mathematics, California Polytechnic State University, San Luis Obispo, CA 93407
3.
Mathematics and Computer Science, College of the Holy Cross, Worcester, MA 01610

Received: October 2009

Revised: February 2010

Published: December 2010

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Abstract

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.

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Mathematics Subject Classification: Primary: 35K60, 45D05, 80A20.

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