### Conley index continuation and thin domain problems

DOI: http://dx.doi.org/10.12775/TMNA.2000.039

#### Abstract

Given $\varepsilon> 0$ and

a bounded Lipschitz domain $\Omega$ in

$\mathbb R^M\times \mathbb R^N$ let $\Omega_\varepsilon:=\{(x,\varepsilon y)\mid (x,y)\in

\Omega\}$ be the $\varepsilon$-{\it squeezed domain\/}.

Consider the reaction-diffusion equation

$$

u_t = \Delta u + f(u) \leqno(\widetilde E_\varepsilon)

$$

on $\Omega_\varepsilon$ with Neumann boundary

condition.

Here $f$ is an appropriate nonlinearity

such that $(\widetilde E_\varepsilon)$ generates a (local) semiflow $\widetilde\pi_ \varepsilon$ on

$H^1(\Omega_\varepsilon)$. It was proved by Prizzi and Rybakowski (J. Differential Equations, to appear),

generalizing some previous results of Hale and Raugel, that there are

a closed subspace $H^1_s(\Omega)$ of $H^1(\Omega)$,

a closed subspace $L^2_s(\Omega)$ of $L^2(\Omega)$ and a sectorial operator

$A_0$ on $L^2_s(\Omega)$ such that the semiflow $\pi_0$ defined on $H^1_s(\Omega)$

by the abstract equation

$$\dot u+A_0u=\widehat f(u)$$ is the limit of the semiflows $\widetilde\pi_\varepsilon$

as $\varepsilon\to 0^+$.

In this paper we prove a singular Conley index continuation principle stating

that every isolated invariant set $K_0$ of

$\pi_0$ can be continued to a nearby family $\widetilde K_\varepsilon$ of

isolated invariant sets of $\widetilde \pi_\varepsilon$ with the same Conley

index. We present various applications of this result

to problems like connection lifting or resonance

bifurcation.

a bounded Lipschitz domain $\Omega$ in

$\mathbb R^M\times \mathbb R^N$ let $\Omega_\varepsilon:=\{(x,\varepsilon y)\mid (x,y)\in

\Omega\}$ be the $\varepsilon$-{\it squeezed domain\/}.

Consider the reaction-diffusion equation

$$

u_t = \Delta u + f(u) \leqno(\widetilde E_\varepsilon)

$$

on $\Omega_\varepsilon$ with Neumann boundary

condition.

Here $f$ is an appropriate nonlinearity

such that $(\widetilde E_\varepsilon)$ generates a (local) semiflow $\widetilde\pi_ \varepsilon$ on

$H^1(\Omega_\varepsilon)$. It was proved by Prizzi and Rybakowski (J. Differential Equations, to appear),

generalizing some previous results of Hale and Raugel, that there are

a closed subspace $H^1_s(\Omega)$ of $H^1(\Omega)$,

a closed subspace $L^2_s(\Omega)$ of $L^2(\Omega)$ and a sectorial operator

$A_0$ on $L^2_s(\Omega)$ such that the semiflow $\pi_0$ defined on $H^1_s(\Omega)$

by the abstract equation

$$\dot u+A_0u=\widehat f(u)$$ is the limit of the semiflows $\widetilde\pi_\varepsilon$

as $\varepsilon\to 0^+$.

In this paper we prove a singular Conley index continuation principle stating

that every isolated invariant set $K_0$ of

$\pi_0$ can be continued to a nearby family $\widetilde K_\varepsilon$ of

isolated invariant sets of $\widetilde \pi_\varepsilon$ with the same Conley

index. We present various applications of this result

to problems like connection lifting or resonance

bifurcation.

#### Keywords

Thin domains; singular perturbations; reaction-diffusion equations; Conley index

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