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Quantitative homogenization of global attractors for reaction–diffusion systems with rapidly oscillating terms

2003; IOS Press; Volume: 34; Issue: 2 Linguagem: Inglês

10.3233/asy-2003-551

1875-8576

Bernold Fiedler, Mark Vishik,

Nonlinear Dynamics and Pattern Formation

For rapidly spatially oscillating nonlinearities f and inhomogeneities g we compare solutions u ε of reaction–diffusion systems ∂ t u ε =aΔu ε −f(ε,x,x/ε,u)+g(ε,x,x/ε) with solutions u 0 of the formally homogenized, spatially averaged system ∂ t u 0 =aΔu 0 −f 0 (x,u 0 )+g 0 (x,u 0 ). We consider a smooth bounded domain x∈Ω⊆ $\mathbb{R}^{n}$ , n≤3, with Dirichlet boundary conditions. We also impose sufficient regularity and dissipation conditions, such that solutions exist globally in time and, in fact, converge to their compact global attractors 𝒜 ε and 𝒜 0 , respectively, in L 2 (Ω). Based on ε‐independent a priori estimates we prove ‖u ε (·,t)−u 0 (·,t)‖ L 2 (Ω) ≤Cε e ρt , uniformly for all t≥0 and 0<ε≤ε 0 . Here the solutions u ε and u 0 start at the same initial condition u=u 0 (x)∈H 1 (Ω) for t=0, and C=C(‖u 0 ‖ H 1 ). Based on an ε‐independent H 2 ‐bound on the global attractors 𝒜 ε as well as an exponential attraction rate ν of the homogenized attractor 𝒜 0 in L 2 (Ω), we also prove fractional order upper semicontinuity of the global attractors for $\varepsilon\searrow 0$ , dist L 2 (Ω) (𝒜 ε ,𝒜 0 )≤Cε γ′ for γ′=(1+ρ/ν) −1 . This result requires the homogenized nonlinearity f 0 (x,w) to be near a potential vector function f 1 (x,w)=∇ w F(x,w) with scalar potential F. Both quantitative homogenization estimates are proved only for quasiperiodic dependence of f,g on the spatially rapidly oscillating variable x/ε. Moreover, the finitely many frequencies describing this quasiperiodic dependence are assumed to satisfy Diophantine conditions, as are familiar from small divisor problems in Kolmogorov–Arnold–Moser theory. Alternatively, the results hold if f, g admit a sufficiently regular divergence representation which describes their explicit spatial dependence. All results apply to, and are illustrated for, the case of FitzHugh–Nagumo systems with spatially rapidly oscillating quasiperiodic coefficients and inhomogeneities. For an earlier preprint version of the present paper see [6]. In the companion paper [7], based on analytic semigroup methods, similar results are obtained for the quantitative homogenization of solutions and invariant manifolds. Examples include homogenization of the Navier–Stokes system under periodic boundary conditions and for spatially rapidly oscillating quasiperiodic forces. For a recent extension to strongly continuous semigroups with an application to damped hyperbolic wave equations see [8].