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We study the large-time behavior of bounded from below solutions of parabolicviscous Hamilton-Jacobi Equations in the whole space $\mathbb{R}^N$ in the caseof superquadratic Hamiltonians. Existence and uniqueness of such solutions areshown in a very general framework, namely when the source term and the initialdata are only bounded from below with an arbitrary growth at infinity. Our mainresult is that these solutions have an ergodic behavior when $t\to +\infty$,i.e., they behave like $\lambda^*t + \phi(x)$ where $\lambda^*$ is the maximalergodic constant and $\phi$ is a solution of the associated ergodic problem.The main originality of this result comes from the generality of the data: inparticular, the initial data may have a completely different growth at infinityfrom those of the solution of the ergodic problem.

Large-time behavior of unbounded solutions of viscous Hamilton-Jacobi Equations in $\mathbb{R}^N$