## Topological Methods in Nonlinear Analysis

### Conley index continuation for singularly perturbed hyperbolic equations

Krzysztof P. Rybakowski

#### 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.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 22, Number 2 (2003), 203-244.

Dates
First available in Project Euclid: 30 September 2016

https://projecteuclid.org/euclid.tmna/1475266334

Mathematical Reviews number (MathSciNet)
MR2036374

Zentralblatt MATH identifier
1083.37011

#### Citation

Rybakowski, Krzysztof P. Conley index continuation for singularly perturbed hyperbolic equations. Topol. Methods Nonlinear Anal. 22 (2003), no. 2, 203--244. https://projecteuclid.org/euclid.tmna/1475266334

#### References

• H. Bauer, Measure and Integration Theory, Walter de Gruyter, Berlin (2001) \ref\no \dfaCR4
• M. C. Carbinatto and K. P. Rybakowski, Conley index continuation and thin domain problems , Topol. Methods Nonlinear Anal., 16 (2000), 201–252 \ref\no \dfaCR1––––, On a general Conley index continuation principle for singular perturbation problems , Ergodic Theory Dynam. Systems, 22 (2002), 729–755 \ref\no \dfaCR3 ––––, On convergence, admissibility and attractors for damped wave equations on squeezed domains , Proc. Roy. Soc. Edinburgh Sect. A, 132(2002), 765–791 \ref\no \dfaCR––––, Morse decompositions in the absence of uniqueness , Topol. Methods Nonlinear Anal., 18(2001), 205–242 \ref\no \dfaCR5––––, Morse decompositions in the absence of uniqueness, II , Topol. Methods Nonlinear Anal., to appear \ref\no \dfaFM
• R. D. Franzosa and K. Mischaikow, The connection matrix theory for semiflows on (not necessarily locally compact) metric spaces , J. Differential Equations, 71 (1988), 270–287 \ref\no \dfaHR
• J. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation , J. Differential Equations, 73 (1988), 197–214 \ref\no \dfaHr
• A. Haraux, Two remarks on hyperbolic dissipative problems , Nonlinear Partial Differential Equations and their Applications: Collège de France Seminar, Volume VII (H. Brezis and J. L. Lions, eds.), Pitman Advanced Publishing Program, Boston (1985), 161–179 \ref\no \dfaHE
• D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer–Verlag, Berlin (1981)
• \ref\no\dfaLA O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge (1991) \ref\no \dfaRy1
• K. P. Rybakowski, On the homotopy index for infinite-dimensional semiflows , Trans. Amer. Math. Soc., 269 (1982), 351–382 \ref\no \dfaKPR ––––, The Homotopy Index and Partial Differential Equations, Springer–Verlag, Berlin (1987) \ref\no \dfaSE
• I. Segal, Non-linear semigroups , Ann. of Math., 78 (1963), 339–364