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Continuation of the connection matrix for singularly perturbed hyperbolic equations

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Carbinatto M., Rybakowski K.

Fundamenta Mathematicae

2007

tom 196

nr 3

253-273

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.

Ograniczanie wyników

1 Fundamenta Mathematicae

1 Carbinatto M.

1 Rybakowski K.

1 2007

Wyniki wyszukiwania

Continuation of the connection matrix for singularly perturbed hyperbolic equations