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.

Appendix

For reader’s convenience we collect some of the no so standard inequalities that we had use in the paper.

Lemma A.1

(Tartar’s Inequality, Sakaguchi 1985, Lemma 3.1) For all $\xi , \eta \in \mathbb {R}^n$ and $p\ge 2$ the following inequality hold

$$\begin{aligned} 2^{2-p} |\xi - \eta |^p \le (|\xi |^{p-2}\xi - |\eta |^{p-2}\eta ) \cdot (\xi - \eta ). \end{aligned}$$

Lemma A.2

(Ghidaglia’s Inequality, Temam 1988, III-Lemma 5.1) Let $y:(0,\infty ) \rightarrow \mathbb {R}$ be a positive absolutely continuous function which satisfies

$$\begin{aligned} \dfrac{dy}{dt} + \gamma y^{p/2} \le \delta , \end{aligned}$$

with $p>2$, $\gamma >0$ and $\delta \ge 0$. Then

$$\begin{aligned} y(t) \le \left( \dfrac{\delta }{\gamma } \right) ^{2/p} + \left( \dfrac{\gamma (p-2)}{2}t \right) ^{-2/(p-2)}, \quad \; \forall \, t>0. \end{aligned}$$