Long-time asymptotic solutions of convex Hamilton-Jacobi equations with Neumann type boundary conditions

We study the long-time asymptotic behavior of solutions u of the Hamilton-Jacobi equation u_t(x,t)+H(x,Du(x,t))=0 in \Omega \times (0,\infty), where \Omega is a bounded open subset of R^n, with Hamiltonian H=H(x,p) being convex and coercive in p, and establish the uniform convergence of u to an asymptotic solution as t goes to \infty.