Large-Time Asymptotics of Fundamental Solutions for Diffusion Equations in Periodic Media and its Application to Averagi
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LARGE-TIME ASYMPTOTICS OF FUNDAMENTAL SOLUTIONS FOR DIFFUSION EQUATIONS IN PERIODIC MEDIA AND ITS APPLICATION TO AVERAGING-THEORY ESTIMATES UDC 517.956.8
V. V. Zhikov* and S. E. Pastukhova
Abstract. The diffusion equation is considered in an infinite 1-periodic medium. We find large-time approximations for its fundamental solution. The approximation precision has pointwise and integral d+j+1 j+1 estimates of orders O(t− 2 ) and O(t− 2 ), j = 0, 1, . . . , respectively. The approximations are constructed on the base of the known fundamental solution of the averaged equation with constant coefficients, its derivatives, and solutions of a family of auxiliary problems on the periodicity cell. The family of problems on the cell is generated recurrently. These results are used to construct approximations of the operator exponential of the diffusion equation with precision estimates in operator norms in Lp -spaces, 1 ≤ p ≤ ∞. For the analogous equation in an ε-periodic medium, where ε is a small parameter, we obtain approximations of the operator exponential in Lp -operator norms for a fixed time with precision of order O(εn ), n = 1, 2, . . .
CONTENTS 1. 2. 3. 4. 5. 6. 7. 8. 9.
Introduction . . . . . . . . . . . . . . . . . . . . . . The Bloch Representation of the Exponential e−tA On the Spectrum of the Operator A(k) . . . . . . The Principal Term of the Asymptotics . . . . . . Auxiliary Cell Problems . . . . . . . . . . . . . . . Complete Analytic Expansions . . . . . . . . . . . High-Order Approximations . . . . . . . . . . . . . Equations with ε-Periodic Coefficients . . . . . . . Proof of Lemma 7.1 . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .
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Introduction
1.1. Consider the following Cauchy problem for a function u = u(x, t), x ∈ Rd , t ≥ 0: ⎧ ∂u ⎪ ⎨ − div(a(x)∇u) = 0, t > 0, ∂t ⎪ ⎩ u|t=0 = f ∈ C0∞ (Rd ).
(1.1)
The matrix a(x) is assumed to be real, symmetric, and uniformly elliptic, i.e., νξ 2 ≤ a(x)ξ · ξ ≤ ν −1 ξ 2
∀ξ ∈ Rd , ν > 0.
(1.2)
We have the diffusion equation in an inhomogeneous medium, where a(x) is the diffusion matrix. Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 63, No. 2, Proceedings of the Crimean Autumn Mathematical School-Symposium, 2017. ∗ Deceased. c 2020 Springer Science+Business Media, LLC 1072–3374/20/2504–0569
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Represent problem (1.1) in the form ⎧ ∂u ⎪ ⎨ + Au = 0, t > 0, ∂t ⎪ ⎩ u|t=0 = f, where A is the operator in L2 (Rd ), defined by the quadratic form a∇u ·
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