Long-time behavior of solutions to Hamilton-Jacobi equations with quadratic gradient term

We study a rate of convergence appearing in the long-time behavior of viscosity solutions of the Cauchy problem for the Hamilton-Jacobi equation u(t)(x, t) + alpha x . Du(x, t) + beta vertical bar Du(x, t)vertical bar(2) = f(x) in R(n) x (0, infinity), where alpha, beta > 0 are constants and f is a Lipschitz and semiconvex function on R(n). Our goal of this paper is to show that the semiconvexity property of f is an important factor which determines this rate of convergence. We also establish existence, uniqueness and Lipschitz continuity of viscosity solutions of the Cauchy problem and the corresponding ergodic problem for Hamilton-Jacobi equations in R(n)