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Large-time asymptotics for one-dimensional Dirichlet problems for Hamilton-Jacobi equations with noncoercive Hamiltonians. (English) Zbl 1230.35038
Summary: We study large-time asymptotics for a class of noncoercive Hamilton-Jacobi equations with Dirichlet boundary conditions in one space dimension. We prove that the average growth rate of a solution is constant only in a subset of the whole domain and give the asymptotic profile in the subset. We show 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. This work is an extension with more rigorous analysis of a recent paper by E. Yokoyama, Y. Giga and P. Rybka [Physica D 237, No. 22, 2845–2855 (2008; Zbl 1375.82118)], in which a growing crystal model is established and the asymptotic behavior described above is first discovered.

MSC:
35F31 Initial-boundary value problems for nonlinear first-order PDEs
35B40 Asymptotic behavior of solutions to PDEs
35F21 Hamilton-Jacobi equations
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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