Large-Time Behavior of Unbounded Solutions of Viscous Hamilton-Jacobi Equations in

We study the large-time behavior of bounded from below solutions of parabolic viscous Hamilton-Jacobi Equations in the whole space (Formula presented.) in the case of superquadratic Hamiltonians. Existence and uniqueness of such solutions are shown in a very general framework, namely when the source term and the initial data are only bounded from below with an arbitrary growth at infinity. Our main result is that these solutions have an ergodic behavior when (Formula presented.) i.e., they behave like (Formula presented.) where (Formula presented.) is the maximal ergodic constant and (Formula presented.) is a solution of the associated ergodic problem. The main originality of this result comes from the generality of the data: in particular, the initial data may have a completely different growth at infinity from those of the solution of the ergodic problem. © 2020 Taylor & Francis Group, LLC.