2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/224838We study the long-time behavior of the unique viscosity solution $u$ of the viscous Hamilton-Jacobi Equation $u_t-Δu + |Du|^m = f\hbox{in }Ω\times (0,+\infty)$ with inhomogeneous Dirichlet boundary conditions, where $Ω$ is a bounded domain of $\mathbb{R}^N$. We mainly focus on the superquadratic case ($m>2$) and consider the Dirichlet conditions in the generalized viscosity sense. Under rather natural assumptions on $f,$ the initial and boundary data, we connect the problem studied to its associated stationary generalized Dirichlet problem on one hand and to a stationary problem with a state constraint boundary condition on the other hand.Analysis of PDEstext