Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Long-time asymptotics of the zero level set for the heat equation


Author: Jaywan Chung
Journal: Quart. Appl. Math. 70 (2012), 705-720
MSC (2010): Primary 35B05, 35B40; Secondary 35C11, 35C20
DOI: https://doi.org/10.1090/S0033-569X-2012-01262-4
Published electronically: June 21, 2012
MathSciNet review: 3052086
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Abstract: The zero level set $ Z(t) := \{ \textbf {x} \in \mathbf {R}^d : u(\textbf {x},t) = 0 \}$ of a solution $ u$ to the heat equation in $ \mathbf {R}^d$ is considered. Under vanishing conditions on moments of the initial data, we will prove that the set $ Z(t)$ in a ball of radius $ C\sqrt {t}$ for any $ C>0$ converges to a specific graph as $ t \rightarrow \infty $ when the set is divided by $ \sqrt {t}$. Solving a linear combination of the Hermite polynomials gives the graph, and coefficients of the linear combination depend on moments of the initial data. Also the graphs to which the zero level set $ Z(t)$ converges are classified in some cases.


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Additional Information

Jaywan Chung
Affiliation: Department of Mathematical Sciences, KAIST, 335 Gwahangno, Yuseong-gu, Daejeon 305-701, Republic of Korea
Email: jaywan.chung@gmail.com

DOI: https://doi.org/10.1090/S0033-569X-2012-01262-4
Received by editor(s): November 17, 2010
Published electronically: June 21, 2012
Additional Notes: This work was supported by the National Research Foundation of Korea (No. 2009-0077987).
Article copyright: © Copyright 2012 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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