Homogenization of the parabolic equation with periodic coefficients at the edge of a spectral gap
A. R. Akhmatova, E. S. Aksenova, Vladimir Anatolevich Sloushch, Tatiana Aleksandrovna Suslina · Complex Variables and Elliptic Equations · 2021
In L2(R), consider a second-order elliptic differential operator Aϵ, ϵ>0, of the form Aϵ=−ddxg(x/ϵ)ddx+ϵ−2p(x/ϵ) with periodic coefficients. For small ε, we study the behavior of the semigroup e−Aϵt, t>0, cut by the spectral projection of the operator Aϵ for the interval [ϵ−2ν,+∞). Here ϵ−2ν is the right edge of a spectral gap for the operator Aϵ. We obtain approximation for the ‘cut semigroup’ in the operator norm in L2(R) with error O(ϵ), and also a more accurate approximation with error O(ϵ2) (after singling out the factor e−tν/ϵ2). The results are applied to homogenization of the Cauchy problem ∂tvϵ=−Aϵvϵ, vϵ|t=0=fϵ, with the initial data fϵ from a special class.