Asymptotic Solutions of Viscous Hamilton-Jacobi Equations with Ornstein-Uhlenbeck Operators

We study the long time behavior of solutions of the Cauchy problem for semilinear parabolic equations with the Ornstein–Uhlenbeck operator in R^N . The long time behavior in the main results is stated with help of the corresponding to ergodic problem, which complements, in the case of unbounded domains, the recent developments on long time behaviors of solutions of (viscous) Hamilton–Jacobi equations due to Namah (1996), Namah and Roquejoffre (1999), Roquejoffre (1998), Fathi (1998), Barles and Souganidis (2000, 2001). We also establish existence and uniqueness results for solutions of the Cauchy problem and ergodic problem for semilinear parabolic equations with the Ornstein–Uhlenbeck operator.