Convergence to steady states for a one-dimensional viscous Hamilton-Jacobi equation with Dirichlet boundary conditions
Résumé
The convergence to steady states of solutions to the one-dimensional viscous Hamilton-Jacobi equation ∂ t u−∂ 2 x u = |∂ x u| p , (t, x) ∈ (0, ∞)× (−1, 1) with homogeneous Dirichlet boundary conditions is investigated. For that purpose, a Liapunov functional is constructed by the approach of Zelenyak (1968). Instantaneous extinction of ∂ x u on a subinterval of (−1, 1) is also shown for suitable initial data.
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