Continuation of the connection matrix for singularly perturbed hyperbolic equations

Maria C. Carbinatto ¹   ; Krzysztof P. Rybakowski ²

¹ Departamento de Matemática, ICMC-USP Caixa Postal 668 13.560-970 São Carlos SP, Brazil
² Institut für Mathematik Universität Rostock Universitätsplatz 1 18055 Rostock, Germany

Fundamenta Mathematicae, Tome 196 (2007) no. 3, pp. 253-273

Résumé

Let ${\mit\Omega}\subset \mathbb R^N$, $N\le 3$, be a bounded domain with smooth boundary, $\gamma\in L^2({\mit\Omega})$ 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 $H^1_0({\mit\Omega})\times L^2({\mit\Omega})$ generated by the damped wave equation $$ \eqalign{ \varepsilon \partial_{tt}u+\partial_t u={\mit\Delta} u+\phi(u)+\gamma(x), \quad\ x\in{\mit\Omega},\,t>0,\cr u(x,t)=0,\quad\ x\in \partial {\mit\Omega},\,t>0. } $$ Moreover, let $\pi'$ be the semiflow on $H^1_0({\mit\Omega})$ generated by the parabolic equation $$ \eqalign{ \partial_t u={\mit\Delta} u+\phi(u)+\gamma(x), \quad\ x\in{\mit\Omega}, \,t>0,\cr u(x,t)=0,\quad\ x\in \partial {\mit\Omega},\,t>0. } $$ Let ${\mit\Gamma}\colon H^2({\mit\Omega})\to H^1_0({\mit\Omega})\times L^2({\mit\Omega})$ be the imbedding $u\mapsto (u,{\mit\Delta} u+\phi(u)+\gamma)$. We prove that whenever $K'$ is a compact isolated $\pi'$-invariant set and $(M_p')_{p\in P}$ is a partially ordered Morse decomposition of $K'$ then the imbedded sets $K={\mit\Gamma}(K')$ and $M_{p,{0}}={\mit\Gamma}(M_p')$, $p\in P$, continue, for $\varepsilon>0$ small, to an isolated $\pi_\varepsilon$-invariant set $K_\varepsilon$ a Morse decomposition $(M_{p,\varepsilon})_{p\in P}$ of $K_\varepsilon$, relative to $\pi_\varepsilon$, such that the homology index braid of $(\pi_\varepsilon,K_\varepsilon, (M_{p,\varepsilon})_{p\in P})$ is isomorphic to the homology index braid of $(\pi',K',(M_p')_{p\in P})$. This, in particular, implies a connection matrix continuation principle.

Détail

DOI : 10.4064/fm196-3-3

Keywords: mit omega subset mathbb bounded domain smooth boundary gamma mit omega arbitrary phi colon mathbb mathbb function satisfying subcritical growth condition every varepsilon infty consider semiflow varepsilon mit omega times mit omega generated damped wave equation eqalign varepsilon partial partial mit delta phi gamma quad mit omega quad partial mit omega moreover semiflow mit omega generated parabolic equation eqalign partial mit delta phi gamma quad mit omega quad partial mit omega mit gamma colon mit omega mit omega times mit omega imbedding mapsto mit delta phi gamma prove whenever compact isolated pi invariant set partially ordered morse decomposition imbedded sets mit gamma mit gamma continue varepsilon small isolated varepsilon invariant set varepsilon morse decomposition varepsilon varepsilon relative varepsilon homology index braid varepsilon varepsilon varepsilon isomorphic homology index braid particular implies connection matrix continuation principle