Non-dissipative System as Limit of a Dissipative One: Comparison of the Asymptotic Regimes

Let $\Omega \subset \mathbb{R}^n$ be a bounded smooth domain (open and connected) in $\mathbb{R}^n$. Given $u_0\in L^2(\Omega)$, $g\in L^\infty(\Omega)$ and $\lambda \in \mathbb{R}$, our purpose is to describe the asymptotic behavior of weak solutions of the family of problems \begin{equation*} \left\{ \begin{array}{rcll} \dfrac{\partial u}{\partial t} - \Delta_p u & = & \lambda u + g, & \text{ on } \quad (0,\infty)\times \Omega, \\ u & = & 0, & \text{ in } \quad (0,\infty)\times \partial \Omega, \\ u(0, \cdot) & = & u_0, & \text{ on } \quad\Omega, \end{array} \right. \end{equation*} as $p \longrightarrow 2^+$, where $\Delta_p u:=\rm{div}\big(|\nabla u|^{p-2}\nabla u\big)$ denotes the $p$-laplacian operator.