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

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.