Asymptotic behavior of viscosity solutions of first order Hamilton-Jacobi equations (Q1098399)

The author gives results about the asymptotic behavior when \(t\to \infty\) of viscosity solutions of first order Hamilton-Jacobi equations of the following type \(\partial u/\partial t + H(x,t,u,Du) = 0\) in \(\Omega\times (0,\infty)\), \(u(x,0)=u_ 0(x)\) in \(\Omega\), \(u(x,t)=\phi(x,t)\) in \(\partial \Omega \times (0,\infty)\). He studies the following cases: I. \(\Omega ={\mathcal R}^ n.\) He obtains properties of convergence for non- negative Hamiltonians and for Lipschitz initial conditions. Also, some estimates on the rate of convergence are offered. II. General domains \(\Omega\), bounded or unbounded. It is analyzed the convex domain case in a first step, and after that, a result for general domains is given.