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On non-existence of global solutions to a class of stochastic heat equations


Authors: Mohammud Foondun and Rana D. Parshad
Journal: Proc. Amer. Math. Soc. 143 (2015), 4085-4094
MSC (2010): Primary 60H15; Secondary 82B44
DOI: https://doi.org/10.1090/proc/12036
Published electronically: April 6, 2015
MathSciNet review: 3359596
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Abstract: We consider nonlinear parabolic SPDEs of the form $ \partial _t u=-$
$ (-\Delta )^{\alpha /2} u + b(u) +\sigma (u)\dot w$, where $ \dot w$ denotes space-time white noise. The functions $ b$ and $ \sigma $ are both locally Lipschitz continuous. Under some suitable conditions on the parameters of this SPDE, we show that the above equation has no random-field solution. This complements recent works of Khoshnevisan and his coauthors.


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Additional Information

Mohammud Foondun
Affiliation: School of Mathematics, Loughborough University, LE11 3TU, United Kingdom & Mathematics and Computer Science and Engineering division, KAUST, Saudi Arabia
Email: m.i.foondun@lboro.ac.uk

Rana D. Parshad
Affiliation: School of Mathematics, Loughborough University, LE11 3TU, United Kingdom & Mathematics and Computer Science and Engineering division, KAUST, Saudi Arabia
Address at time of publication: Department of Mathematics, Clarkson University, Potsdam, NY 13699
Email: Rana.Parshad@kaust.edu.sa

DOI: https://doi.org/10.1090/proc/12036
Keywords: Stochastic partial differential equations, stable processes, Liapounov exponents, weak intermittence
Received by editor(s): August 22, 2012
Received by editor(s) in revised form: September 21, 2012, and April 21, 2014
Published electronically: April 6, 2015
Communicated by: Mark M. Meerschaert
Article copyright: © Copyright 2015 American Mathematical Society