Continuation of the connection matrix for singularly perturbed hyperbolic equations

Volume 196 / 2007

Maria C. Carbinatto, Krzysztof P. Rybakowski Fundamenta Mathematicae 196 (2007), 253-273 MSC: Primary 37B30, 35B25; Secondary 37L05, 37L65. DOI: 10.4064/fm196-3-3


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.


  • Maria C. CarbinattoDepartamento de Matemática, ICMC-USP
    Caixa Postal 668
    13.560-970 São Carlos SP, Brazil
  • Krzysztof P. RybakowskiInstitut für Mathematik
    Universität Rostock
    Universitätsplatz 1
    18055 Rostock, Germany

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