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Singular Neumann problems and large-time behavior of solutions of noncoercive Hamilton-Jacobi equations. (English) Zbl 1286.35074
Summary: 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 et al. [Physica D 237, No. 22, 2845–2855 (2008; Zbl 1375.82118)]. 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.

MSC:
35F21 Hamilton-Jacobi equations
35B40 Asymptotic behavior of solutions to PDEs
35F25 Initial value problems for nonlinear first-order PDEs
35F30 Boundary value problems for nonlinear first-order PDEs
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
35D40 Viscosity solutions to PDEs
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