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 in $$\mathbb {R}^n$$. Given $$u_0\in L^2(\Omega )$$, $$g\in L^\infty (\Omega )$$ and $$\lambda \in \mathbb {R}$$, consider the family of problems parametrised by $$p \searrow 2$$, $$\begin{aligned} \left\{ \begin{array}{llll} &{}&{} \dfrac{\partial u}{\partial t} - \Delta _p u = \lambda u + g, \, \text { on } \quad (0,\infty )\times \Omega , \\ &{}&{} u = 0, \qquad \qquad \qquad \quad \quad \;\, \text { in } \quad (0,\infty )\times \partial \Omega , \\ &{}&{} u(0, \cdot ) = u_0, \qquad \qquad \quad \,\text { on } \quad \Omega , \end{array} \right. \end{aligned}$$where $$\Delta _p u:=\mathrm {div}\big (|\nabla u|^{p-2}\nabla u\big )$$ denotes the p-laplacian operator. Our aim in this paper is to describe the asymptotic behavior of this family of problems comparing compact attractors in the dissipative case $$p>2$$, with non-compact attractors in the non-dissipative limiting case $$p=2$$ with respect to the Hausdorff semi-distance between then.