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On inertial manifolds for reaction-diffusion equations on genuinely high-dimensional thin domains

100%

Prizzi M., Rybakowski K.

Studia Mathematica

2003

tom 154

nr 3

253-275

We study a family of semilinear reaction-diffusion equations on spatial domains $Ω_{ε}$, ε > 0, in $ℝ^{l}$ lying close to a k-dimensional submanifold ℳ of $ℝ^{l}$. As ε → 0⁺, the domains collapse onto (a subset of) ℳ. As proved in [15], the above family has a limit equation, which is an abstract semilinear parabolic equation defined on a certain limit phase space denoted by $H¹_{s}(Ω)$. The definition of $H¹_{s}(Ω)$, given in the above paper, is very abstract. One of the objectives of this paper is to give more manageable characterizations of the limit phase space. Under additional hypotheses on the domains $Ω_{ε}$ we also give a simple description of the limit equation. If, in addition, ℳ is a k-sphere and the nonlinearity of the above equations is dissipative, then for every ε > 0 small enough the corresponding equation on $Ω_{ε}$ has an inertial manifold, i.e. an invariant manifold containing the attractor of the equation. We thus obtain the existence of inertial manifolds for reaction-diffusion equations on certain classes of thin domains of genuinely high dimension.

Curved thin domains and parabolic equations

81%

Prizzi M., Rinaldi M., Rybakowski K.

Studia Mathematica

2002

tom 151

nr 2

109-140

Consider the family uₜ = Δu + G(u), t > 0, $x ∈ Ω_{ε}$, $∂_{ν_{ε}}u = 0$, t > 0, $x ∈ ∂Ω_{ε}$, $(E_{ε})$ of semilinear Neumann boundary value problems, where, for ε > 0 small, the set $Ω_{ε}$ is a thin domain in $ℝ^{l}$, possibly with holes, which collapses, as ε → 0⁺, onto a (curved) k-dimensional submanifold of $ℝ^{l}$. If G is dissipative, then equation $(E_{ε})$ has a global attractor $𝒜_{ε}$. We identify a "limit" equation for the family $(E_{ε})$, prove convergence of trajectories and establish an upper semicontinuity result for the family $𝒜_{ε}$ as ε → 0⁺.

Ograniczanie wyników

2 Studia Mathematica

2 Prizzi M.

2 Rybakowski K.

1 Rinaldi M.

1 2003

1 2002

Wyniki wyszukiwania

On inertial manifolds for reaction-diffusion equations on genuinely high-dimensional thin domains

Curved thin domains and parabolic equations