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

Giga, Yoshikazu; Liu, Qing; Mitake, Hiroyoshi

doi:10.1090/S0002-9947-2013-05905-3

Singular Neumann problems and large-time behavior of solutions of noncoercive Hamilton-Jacobi equations
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by Yoshikazu Giga, Qing Liu and Hiroyoshi Mitake PDF

Trans. Amer. Math. Soc. 366 (2014), 1905-1941 Request permission

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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