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

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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.

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R.P.S. is partially supported by FAPDF \(\#193.001.372/2016\).

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Correspondence to Ricardo Parreira da Silva.

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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

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

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da Silva, R.P. Non-dissipative System as Limit of a Dissipative One: Comparison of the Asymptotic Regimes. Bull Braz Math Soc, New Series 51, 125–137 (2020). https://doi.org/10.1007/s00574-019-00146-z

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  • Dissipative systems
  • Non-dissipative systems
  • Global attractors
  • Non-compact attractors
  • Upper-semicontinuity
  • Lower-semicontinuity

Mathematics Subject Classification

  • 35B40
  • 35B41