Advanced Studies in Pure Mathematics

Large-time asymptotics for Hamilton–Jacobi equations with noncoercive Hamiltonians appearing in crystal growth

Yoshikazu Giga, Qing Liu, and Hiroyoshi Mitake

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We investigate the large-time behavior of viscosity solutions of Hamilton–Jacobi equations with noncoercive Hamiltonian in a multi-dimensional Euclidean space. Our motivation comes from a model describing growing faceted crystals recently discussed by E. Yokoyama, Y. Giga and P. Rybka (Phys. D, 237 (2008), no. 22, 2845–2855). We prove that the average growth rate of a solution is constant only in a subset, which will be called effective domain, of the whole domain and give the asymptotic profile in the subset. This means that the large-time behavior for noncoercive problems may depend on the space variable in general, which is different from the usual results under the coercivity condition. 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 and comparison results for singular Neumann problems and apply the results for a large-time asymptotic profile on the effective domain.

Article information

Source
Nonlinear Dynamics in Partial Differential Equations, S. Ei, S. Kawashima, M. Kimura and T. Mizumachi, eds. (Tokyo: Mathematical Society of Japan, 2015), 235-242

Dates
Received: 2 December 2011
Revised: 10 February 2013
First available in Project Euclid: 30 October 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1540934219

Digital Object Identifier
doi:10.2969/aspm/06410235

Mathematical Reviews number (MathSciNet)
MR3381208

Zentralblatt MATH identifier
1339.35089

Subjects
Primary: 35B40: Asymptotic behavior of solutions 35F25: Initial value problems for nonlinear first-order equations 35F30: Boundary value problems for nonlinear first-order equations

Keywords
Large-time behavior noncoercive Hamilton–Jacobi equation singular Neumann problem gradient grow-up facet instability

Citation

Giga, Yoshikazu; Liu, Qing; Mitake, Hiroyoshi. Large-time asymptotics for Hamilton–Jacobi equations with noncoercive Hamiltonians appearing in crystal growth. Nonlinear Dynamics in Partial Differential Equations, 235--242, Mathematical Society of Japan, Tokyo, Japan, 2015. doi:10.2969/aspm/06410235. https://projecteuclid.org/euclid.aspm/1540934219


Export citation