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.
Citation
Krzysztof P. Rybakowski. "Conley index continuation for singularly perturbed hyperbolic equations." Topol. Methods Nonlinear Anal. 22 (2) 203 - 244, 2003.
Information