Abstract
We study the long-time asymptotic behaviour of viscosity solutions [see formula in PDF] of the Hamilton-Jacobi equation [see formula in PDF] in [see formula in PDF] with a PDE approach, where [see formula in PDF] is coercive in [see formula in PDF], non-decreasing in [see formula in PDF] and strictly convex in [see formula in PDF], and establish the uniform convergence of [see formula in PDF] to an asymptotic solution [see formula in PDF] as [see formula in PDF]. Moreover, [see formula in PDF] is a viscosity solution of Hamilton-Jacobi equation [see formula in PDF].