Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. M. Abramowitz and I. A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications Inc., New York, 1992, reprint of the 1972 edition. MR 1225604 (94b:00012)
  • 2. S. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math. 390 (1988), 79-96. MR 953678 (89j:35015)
  • 3. -, Solutions of the one-dimensional porous medium equation are determined by their free boundary, J. London Math. Soc. (2) 42 (1990), no. 2, 339-353. MR 1083450 (92i:35130)
  • 4. G. I. Barenblatt, Scaling, self-similarity, and intermediate asymptotics, Cambridge Texts in Applied Mathematics, vol. 14, Cambridge University Press, Cambridge, 1996, with a foreword by Ya. B. Zeldovich. MR 1426127 (98a:00005)
  • 5. P. Brunovský and B. Fiedler, Simplicity of zeros in scalar parabolic equations, J. Differential Equations 62 (1986), no. 2, 237-241. MR 833419 (88c:35079)
  • 6. X.-Y. Chen, A strong unique continuation theorem for parabolic equations, Math. Ann. 311 (1998), no. 4, 603-630. MR 1637972 (99h:35078)
  • 7. J. Chung, E. Kim, and Y.-J. Kim, Asymptotic agreement of moments and higher order contraction in the Burgers equation, J. Differential Equations 248 (2010), no. 10, 2417-2434. MR 2600962
  • 8. J. Denzler and R. J. McCann, Phase transitions and symmetry breaking in singular diffusion, Proc. Natl. Acad. Sci. USA 100 (2003), no. 12, 6922-6925 (electronic). MR 1982656 (2004c:35210)
  • 9. -, Fast diffusion to self-similarity: complete spectrum, long-time asymptotics, and numerology, Arch. Ration. Mech. Anal. 175 (2005), no. 3, 301-342. MR 2126633 (2005k:35214)
  • 10. J. Duoandikoetxea and E. Zuazua, Moments, masses de Dirac et décomposition de fonctions, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 6, 693-698. MR 1183805 (94a:42046)
  • 11. A. Friedman, Partial differential equations of parabolic type, Prentice-Hall Inc., Englewood Cliffs, N.J., 1964. MR 0181836 (31:6062)
  • 12. V. A. Galaktionov, Geometric Sturmian theory of nonlinear parabolic equations and applications, Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series, 3, Chapman & Hall/CRC, Boca Raton, FL, 2004. MR 2059317 (2005h:35002)
  • 13. V. A. Galaktionov and P. J. Harwin, Sturm's theorems on zero sets in nonlinear parabolic equations, Sturm-Liouville theory, Birkhäuser, Basel, 2005, pp. 173-199. MR 2145082
  • 14. Y.-J. Kim and W.-M. Ni, Higher order approximations in the heat equation and the truncated moment problem, SIAM J. Math. Anal. 40 (2009), no. 6, 2241-2261. MR 2481293 (2010h:35170)
  • 15. K.-A. Lee and J. L. Vázquez, Geometrical properties of solutions of the porous medium equation for large times, Indiana Univ. Math. J. 52 (2003), no. 4, 991-1016. MR 2001942 (2004h:35113)
  • 16. H. Matano, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), no. 2, 401-441. MR 672070 (84m:35060)
  • 17. N. Mizoguchi, Asymptotic behavior of zeros of solutions for parabolic equations, J. Differential Equations 170 (2001), no. 1, 51-67. MR 1813099 (2002d:35092)
  • 18. S. Osher and R. Fedkiw, Level set methods and dynamic implicit surfaces, Applied Mathematical Sciences, vol. 153, Springer-Verlag, New York, 2003. MR 1939127 (2003j:65002)
  • 19. S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys. 79 (1988), no. 1, 12-49. MR 965860 (89h:80012)
  • 20. S. Sakaguchi, The number of peaks of nonnegative solutions to some nonlinear degenerate parabolic equations, J. Math. Anal. Appl. 203 (1996), no. 1, 78-103. MR 1412482 (97i:35096)
  • 21. T. P. Witelski and A. J. Bernoff, Self-similar asymptotics for linear and nonlinear diffusion equations, Stud. Appl. Math. 100 (1998), no. 2, 153-193. MR 1491842 (99d:35081)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2010): 35B05, 35B40, 35C11, 35C20

Retrieve articles in all journals with MSC (2010): 35B05, 35B40, 35C11, 35C20

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.

American Mathematical Society