We investigate the large-time behavior of solutions of the Cauchy problem for Hamilton-Jacobi equations on the real line $R$. We establish a result on convergence of the solutions to asymptotic solutions as time $t$ goes to infinity.
"The Large-time Behavior of Solutions of Hamilton-Jacobi Equations on the Real Line." Methods Appl. Anal. 15 (2) 223 - 242, June 2008.