Conley index continuation for singularly perturbed hyperbolic equations

Krzysztof P. Rybakowski

DOI: http://dx.doi.org/10.12775/TMNA.2003.037

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
$$
\alignedat 3
\varepsilon \partial_{tt}u+\partial_t u&=\Delta u+\phi(u)+\gamma(x) &\quad& x\in\Omega,&\ &t> 0,\cr
u(x,t)&=0&\quad& x\in \partial \Omega,&\ &t> 0.
\endalignedat
$$
Moreover, let $\pi'$ be the semiflow on $H^1_0(\Omega)$ generated
by the parabolic
equation
$$
\alignedat 3
\partial_t u&=\Delta u+\phi(u)+\gamma(x) &\quad& x\in\Omega,&\ &t> 0,\cr
u(x,t)&=0&\quad& x\in \partial \Omega,&\ &t> 0.
\endalignedat
$$
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.

Keywords


Parabolic equations; damped hyperbolic equations; singular perturbations; Conley index continuation; Morse decompositions

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