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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Singular Neumann problems and large-time behavior of solutions of noncoercive Hamilton-Jacobi equations


Authors: Yoshikazu Giga, Qing Liu and Hiroyoshi Mitake
Journal: Trans. Amer. Math. Soc. 366 (2014), 1905-1941
MSC (2010): Primary 35B40, 35F25, 35F30, 49L25
Published electronically: September 4, 2013
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Abstract: We investigate the large-time behavior of viscosity solutions of Hamilton-Jacobi equations with noncoercive Hamiltonian in a multidimensional Euclidean space. Our motivation comes from a model describing growing faceted crystals recently discussed by E. Yokoyama, Y. Giga and P. Rybka. Surprisingly, growth rates of viscosity solutions of these equations depend on the $ x$-variable. In a part of the space called the effective domain, growth rates are constant, but outside of this domain, they seem to be unstable. Moreover, on the boundary of the effective domain, the gradient with respect to the $ x$-variable of solutions blows up as time goes to infinity. Therefore, we are naturally led to study singular Neumann problems for stationary Hamilton-Jacobi equations. We establish the existence, stability and comparison results for singular Neumann problems and apply the results for a large-time asymptotic profile on the effective domain of viscosity solutions of Hamilton-Jacobi equations with noncoercive Hamiltonian.


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Additional Information

Yoshikazu Giga
Affiliation: Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan — and — Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Email: labgiga@ms.u-tokyo.ac.jp

Qing Liu
Affiliation: Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan
Address at time of publication: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Email: qingliu@pitt.edu

Hiroyoshi Mitake
Affiliation: Department of Applied Mathematics, Graduate School of Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi-Hiroshima-shi 739-8527, Japan
Address at time of publication: Department of Applied Mathematics, Faculty of Science, Fukuoka University, Fukuoka 814-0180, Japan
Email: mitake@math.sci.fukuoka-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05905-3
PII: S 0002-9947(2013)05905-3
Keywords: Large-time behavior, noncoercive Hamilton-Jacobi equation, effective domain, singular Neumann problems, gradient grow-up, faceted crystals, facet instability, viscosity solution
Received by editor(s): October 22, 2010
Received by editor(s) in revised form: June 7, 2012
Published electronically: September 4, 2013
Additional Notes: The work of the first author was partly supported by Grant-in-Aid for Scientific Research, No. 21224001 (Kiban S) and No 23244015 (Kiban A), the Japan Society for the Promotion of Science (JSPS)
The work of the second author was partly supported by Research Fellowship for Young Researcher from JSPS, No. 21-7428
The work of the third author was partly supported by Research Fellowship for Young Researcher from JSPS, No. 22-1725
Article copyright: © Copyright 2013 American Mathematical Society