Conley index continuation for singularly perturbed hyperbolic equations

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.