Topological Methods in Nonlinear Analysis

Conley index continuation and thin domain problems

Maria C. Carbinatto and Krzysztof P. Rybakowski

Full-text: Open access


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 $\varepsilon$-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.

Article information

Topol. Methods Nonlinear Anal., Volume 16, Number 2 (2000), 201-251.

First available in Project Euclid: 22 August 2016

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Carbinatto, Maria C.; Rybakowski, Krzysztof P. Conley index continuation and thin domain problems. Topol. Methods Nonlinear Anal. 16 (2000), no. 2, 201--251.

Export citation


  • J. Arrieta, Neumann eigenvalue problems on exterior perturbations of the domain , J. Differential Equations, 118 (1995), 54–103 \ref\key 2 ––––, Spectral behavior and upper semicontinuity of attractors , preprint \ref\key 3
  • J. Arrieta, J. Hale and Q. Han, Eigenvalue problems for nonsmoothly perturbed domains , J. Differential Equations, 91 (1991), 24–52 \ref\key 4
  • V. Benci, A new approach to the Morse–Conley theory and some applications , Ann. Mat. Pura Appl. (4), 158 (1991), 231–305 \ref\key 5
  • M. C. Carbinatto, J. Kwapisz and K. Mischaikow, Horseshoes and the Conley index spectrum , Ergodic Theory Dynam. Systems, to appear \ref\key 6
  • M. C. Carbinatto and K. Mischaikow, Horseshoes and the Conley index spectrum, Part II: the theorem is sharp, , Discrete Contin. Dynam. Systems, 5 (1999), 599–616 \ref\key 7
  • M. C. Carbinatto and K. P. Rybakowski, On a general Conley index continuation principle for singular perturbation problems . preprint \ref\key 8 ––––, On convergence, admissibility and attractors for damped wave equations on squeezed domains , preprint \ref\key 9 ––––, Continuation of the connection matrix for singular perturbation problems , in preparation \ref\key 10
  • I. S. Ciuperca, Spectral properties of Schrödinger operators on domains with varying order of thinness , J. Dynam. Differential Equations, 10(1998), 73–108 \ref\key 11
  • C. C. Conley, Isolated Invariant Sets and the Morse Index, CBMS, 38 , Amer. Math. Soc., Providence (1978) \ref\key 12
  • C. C. Conley and E. Zehnder, Morse type index theory for flows and periodic solutions for Hamiltonian Systems , Comm. Pure Appl. Math., 37(1984), 207–253 \ref\key 13
  • M. Degiovanni and M. Mrozek, The Conley index for maps in the absence of compactness , Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 75–94 \ref\key 14
  • B. Fiedler and C. Rocha, Heteroclinic orbits of semilinear parabolic equations , J. Differential Equations, 125(1996), 239–281 \ref\key 15
  • A. Floer, Morse theory for Lagrangian intersections , J. Differential Geometry, 28(1988), 513–547 \ref\key 16 ––––, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys., 120(1989), 575–611 \ref\key 17
  • R. Franzosa The connection matrix theory for Morse decompositions, Trans. Amer. Math. Soc., 311(1989), 561–592 \ref\key 18
  • R. Franzosa and K. Mischaikow, The connection matrix theory for semiflows on \rom(not necessarily locally compact\rom) metric spaces, J. Differential Equations, 71(1988), 270–287 \ref\key 19
  • J. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation , J. Differential Equations, 73(1988), 197–214 \ref\key 21 ––––, Reaction-diffusion equations on thin domains , J. Math. Pures Appl. (9), 71(1992), 33–95 \ref\key 22 ––––, A damped hyperbolic equation on thin domains , Trans. Amer. Math. Soc., 329(1992), 185–219 \ref\key 23 ––––, A reaction-diffusion equation on a thin $L$-shaped domain , Proc. Roy. Soc. Edinburgh Sect. A, 125(1995), 283–327 \ref\key 24
  • D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin (1981) \ref\key 25
  • Q. Huang, The continuation of Conley index for singular perturbations and the Conley indices in gradient-like systems I: Theory , preprint \ref\key 26
  • R. Johnson, P. Nistri and M. Kamenskiĭ, On periodic solutions of a damped wave equation in a thin domain using degree theoretic methods , J. Differential Equations, 140(1997), 186–208 \ref\key 27 ––––, On the existence of periodic solutions of the Navier-Stokes equations in a thin domain using the topological degree , preprint \ref\key 28
  • K. Mischaikow and M. Mrozek, Isolating neighbourhoods and chaos , Japan J. Indust. Appl. Math., 12(1995), 205–236 \ref\key 29
  • M. Mischaikow, M. Mrozek and J. F. Reineck, Singular index pairs , J. Dynam. Differential Equations, 11(1999), 399–425 \ref\key 30
  • M. Mrozek, Leray functor and cohomological Conley index for discrete dynamical systems , Trans. Amer. Math. Soc., 318(1990), 149–178 \ref\key 31
  • M. Mrozek and K. P. Rybakowski, A cohomological Conley index for maps on metric spaces , J. Differential Equations, 90(1991), 143–171 \ref\key 32
  • M. Mrozek, J. F. Reineck and R. Srzednicki, The Conley index over a base , Trans. Amer. Math. Soc., to appear \ref\key 33
  • M. Prizzi and K. P. Rybakowski, The effect of domain squeezing upon the dynamics of reaction-diffusion equations , J. Differential Equations, to appear \ref\key 34 ––––, Inertial manifolds on squeezed domains , preprint \ref\key 35
  • M. Prizzi, M. Rinaldi and K. P. Rybakowski, Curved thin domains and parabolic equations . preprint \ref\key 36
  • G. Raugel, Dynamics of partial differential equations on thin domains (R. Johnson, ed.), Dynamical systems. Lectures given at the 2nd session of the Centro Internazionale Matematico Estivo (CIME) held in Montecatini Terme, Italy, June 13–22, 1994, Lecture Notes in Math., 1609 , Springer-Verlag, Berlin (1995), 208–315 \ref\key 37
  • K. P. Rybakowski, On the homotopy index for infinite-dimensional semiflows , Trans. Amer. Math. Soc., 269(1982), 351–382 \ref\key 38 ––––, An index product formula for the study of elliptic resonance problems , J. Differential Equations, 56(1985), 408–425 \ref\key 39 ––––, The Homotopy Index and Partial Differential Equations, Universitext, Springer-Verlag, Berlin (1987) \ref\key 40
  • A. Szymczak, The Conley index for discrete semidynamical systems , Topology Appl., 66(1995), 215–240