Non-dissipative system as limit of a dissipative one: comparison of the asymptotic regimes

Abstract

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} - Δ_p u & = & λu + g, & \text{ on } \quad (0,\infty)\times Ω, \\ u & = & 0, & \text{ in } \quad (0,\infty)\times \partial Ω, \\ u(0, \cdot) & = & u_0, & \text{ on } \quadΩ, \end{array} \right. \end{equation*} as $p⟶2^{+}$, where ${\Delta }_{p}u:=\mathrm{div}(|\nabla u{|}^{\mathrm{p}-2}\nabla u)$ denotes the $p$-laplacian operator.