Topological Methods in Nonlinear Analysis

Conley index continuation for singularly perturbed hyperbolic equations

Krzysztof P. Rybakowski

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Abstract

Let $\Omega\subset \mathbb R^N$, $N\le 3$, be a bounded domain with smooth boundary, $\gamma\in L^2(\Omega)$ be arbitrary and $\phi\colon \mathbb R\to \mathbb R$ be a $C^1$-function satisfying a subcritical growth condition. For every $\varepsilon\in]0,\infty[$ consider the semiflow $\pi_\varepsilon$ on $H^1_0(\Omega)\times L^2(\Omega)$ generated by the damped wave equation $$ \begin{alignedat}{3} \varepsilon \partial_{tt}u+\partial_t u&=\Delta u+\phi(u)+\gamma(x) &\quad& x\in\Omega,&\ &t> 0,\\ u(x,t)&=0&\quad& x\in \partial \Omega,&\ &t> 0. \end{alignedat} $$ Moreover, let $\pi'$ be the semiflow on $H^1_0(\Omega)$ generated by the parabolic equation $$ \begin{alignedat}{3} \partial_t u&=\Delta u+\phi(u)+\gamma(x) &\quad& x\in\Omega,&\ &t> 0,\\ u(x,t)&=0&\quad& x\in \partial \Omega,&\ &t> 0. \end{alignedat} $$ Let $\Gamma\colon H^2(\Omega)\to H^1_0(\Omega)\times L^2(\Omega)$ be the imbedding $u\mapsto (u,\Delta u+\phi(u)+\gamma)$. We prove in this paper that every compact isolated $\pi'$-invariant set $K'$ lies in $H^2(\Omega)$ and the imbedded set $K_0=\Gamma(K')$ continues to a family $K_\varepsilon$, $\varepsilon\ge0$ small, of isolated $\pi_\varepsilon$-invariant sets having the same Conley index as $K'$. This family is upper-semicontinuous at $\varepsilon=0$. Moreover, any (partially ordered) Morse-decomposition of $K'$, imbedded into $H^1_0(\Omega)\times L^2(\Omega)$ via $\Gamma$, continues to a family of Morse decompositions of $K_\varepsilon$, for $\varepsilon\ge 0$ small. This family is again upper-semicontinuous at $\varepsilon=0$.

These results extend and refine some upper semicontinuity results for attractors obtained previously by Hale and Raugel.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 22, Number 2 (2003), 203-244.

Dates
First available in Project Euclid: 30 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1475266334

Mathematical Reviews number (MathSciNet)
MR2036374

Zentralblatt MATH identifier
1083.37011

Citation

Rybakowski, Krzysztof P. Conley index continuation for singularly perturbed hyperbolic equations. Topol. Methods Nonlinear Anal. 22 (2003), no. 2, 203--244. https://projecteuclid.org/euclid.tmna/1475266334


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