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

Abstract

For rapidly spatially oscillating nonlinearities f and inhomogeneities g we compare solutions u^ε of reaction–diffusion systems

∂_tu^ε=aΔu^ε−f(ε,x,x/ε,u)+g(ε,x,x/ε)

with solutions u⁰ of the formally homogenized, spatially averaged system

∂_tu⁰=aΔu⁰−f⁰(x,u⁰)+g⁰(x,u⁰).

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 𝒜⁰, respectively, in L₂(Ω).

Based on ε‐independent a priori estimates we prove

‖u^ε(·,t)−u⁰(·,t)‖_L2(Ω)≤Cε e^ρt,

uniformly for all t≥0 and 0<ε≤ε₀. Here the solutions u^ε and u⁰ start at the same initial condition u=u₀(x)∈H¹(Ω) for t=0, and C=C(‖u₀‖_H¹).

Based on an ε‐independent H²‐bound on the global attractors 𝒜^ε as well as an exponential attraction rate ν of the homogenized attractor 𝒜⁰ in L₂(Ω), we also prove fractional order upper semicontinuity of the global attractors for $\varepsilon\searrow 0$ ,

dist_L2(Ω)(𝒜^ε,𝒜⁰)≤Cε^γ′

for γ′=(1+ρ/ν)⁻¹. This result requires the homogenized nonlinearity f⁰(x,w) to be near a potential vector function f¹(x,w)=∇_wF(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].