Continuation of the connection matrix for singularly perturbed hyperbolic equations

• Autorzy: Maria C. Carbinatto , Krzysztof P. Rybakowski
• Języki publikacji: EN

Abstrakty

EN

Let $Ω ⊂ ℝ^{N}$, N ≤ 3, be a bounded domain with smooth boundary, γ ∈ L²(Ω) be arbitrary and ϕ: ℝ → ℝ be a C¹-function satisfying a subcritical growth condition. For every ε ∈ ]0,∞[ consider the semiflow $π_{ε}$ on H¹₀(Ω) × L²(Ω) generated by the damped wave equation
ε∂ₜₜu + ∂ₜu = Δu + ϕ(u) + γ(x), x ∈ Ω, t > 0,
u(x,t) = 0, x ∈ ∂Ω, t > 0
Moreover, let π' be the semiflow on H¹₀(Ω) generated by the parabolic equation
∂ₜu = Δu + ϕ(u) + γ(x), x ∈ Ω, t > 0,
u(x,t) = 0, x ∈ ∂Ω, t > 0
Let Γ: H²(Ω) → H¹₀(Ω) × L²(Ω) be the imbedding u ↦ (u,Δu+ϕ(u)+γ). We prove that whenever K' is a compact isolated π'-invariant set and $(Mp')_{p∈P}$ is a partially ordered Morse decomposition of K' then the imbedded sets K = Γ(K') and $M_{p,0} = Γ(Mp')$, p ∈ P, continue, for ε > 0 small, to an isolated $π_{ε}$-invariant set $K_{ε}$ a Morse decomposition $(M_{p,ε})_{p∈P}$ of $K_{ε}$, relative to $π_{ε}$, such that the homology index braid of $(π_{ε},K_{ε},(M_{p,ε})_{p∈P}) is isomorphic to the homology index braid of $(π',K',(M'_{p})_{p∈P})$. This, in particular, implies a connection matrix continuation principle.

Kategorie tematyczne

• 37L05: General theory, nonlinear semigroups, evolution equations
• 37B30: Index theory, Morse-Conley indices
• 37L65: Special approximation methods (nonlinear Galerkin, etc.)
• 35B25: Singular perturbations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2007
• Tom: 196
• Numer: 3
• Strony: 253-273

Daty

wydano

2007

Twórcy

autor

Maria C. Carbinatto

• Departamento de Matemática, ICMC-USP, Caixa Postal 668, 13.560-970 São Carlos SP, Brazil

autor

Krzysztof P. Rybakowski

• Institut für Mathematik, Universität Rostock, Universitätsplatz 1, 18055 Rostock, Germany

Identyfikatory

DOI

10.4064/fm196-3-3