LARGE-TIME ASYMPTOTIC OF THE SOLUTION TO THE DIFFUSION EQUATION AND ITS APPLICATION TO HOMOGENIZATION ESTIMATES

The Cauchy problem for a linear second order parabolic equation with 1-periodic measurable coefficients is considered Rd, d ≥ 2. The problem models diffusion in a nonhomogeneous periodic medium. The appropriate diffusion operator A is self-adjoint in L2 (Rd). The large-time behaviour of the solution of the Cauchy problem is of main interest for us. To this end, we study first of all the fundamental solution, in other words, the kernel of the exponential exp(-tA), more exactly, its large-time asymptotics. Its approximation is found with integral error estimate of order O(t-1), as time t tends to +∞. To construct this approximation and carry out its justification we use (i) the known fundamental solution to the homogenized diffusion equation (having constant coefficients); (ii) solutions of so-called auxiliary problems on a periodicity cell, which are formulated in a recurrent way. We substantiate this approximation under additional regularity condition on the diffusion matrix a(x): it should be Lipschitz continuous. The results of asymptotic behaviour of the fundamental solution are applied to obtain an approximation of order O(t-1) for the exponential exp(-tA) in operator Lp-norms, on the section t=const as t tends to +∞. There are also some corollaries of these results to operator estimates for a similar exponential exp(-tAε), Aε being a diffusion operator with quickly oscillating ε-periodic coefficients, as tends to zero. This exponential corresponds to the Cauchy problem: modelling diffusion in a strongly nonhomogeneous ε-periodic medium. Here ε is a small parameter. We construct approximations of order O(ε2) for the exponential exp(-tAε) in operator Lp - norm on the section t=const for arbitrary finite fixed t (say, t=1). This approximation is a sum of the exponential exp(-tA0) with a homogenized operator A0 and some correcting operator. The results have a broad range of applications, e.g., for computing heat flow in a periodic composite medium with a small periodicity cell or for bacterial density estimation in a periodic culture medium.

diffusion equation, effective diffusion, homogenized operator, fundamental solution, cell problems, large-time asymptotic, homogenization estimates in Lebesgue norms

1. Zhikov V.V. Asymptotic behavior and stabilization of solutions of a second-order parabolic equation with lowest terms // Trudy Moskovskogo Matematicheskogo Obshchestva (Proceedings of Moscow Mathematical Society). 1983. V. 46. P. 69–98. (in Russ.).

2. Zhikov V.V. Spectral approach to asymptotic diffusion problems // Differ. Uravn. (Partial Differential Equations). 1989. V. 25. № 1. P. 44–50. (in Russ.).

3. Suslina T.A. On Homogenization of Periodic Parabolic Systems // Funktsional. Anal. i Prilozhen. (Functional Analysis and Its Applications). 2004. V. 38. Iss.4. P. 86–90. (in Russ.).

4. Zhikov V.V., Pastukhova S.E. Estimates of homogenization of parabolic equation with periodic coefficients // Russian Journal of Math. Physics. 2006. V. 13. Iss. 2. P. 224–237.

5. Zhikov V.V., Pastukhova S.E. Operator estimates in homogenization theory // Uspekhi Mat. Nauk (Russian Mathematical Surveys). 2016. V. 71. Iss. 3. P. 27–122. (in Russ.).

6. Zhikov V.V., Kozlov S.M., Oleinik О.А. Averaging of differential operators. Moscow: Fizmatlit. 1993. 464 p. (in Russ.).

7. Korotkov V.B. Integral operators. Novosibirsk: Nauka, 1983. 224 p. (in Russ.).

8. Bensoussan A., Lions J.L., Papanicolaou G. Asymptotic Analysis for Periodic Structure. Amsterdam: North-Holland, 1978. 699 p.

9. Kinderlehrer D.,Stampacchia J. An intraduction to variational inequalities and their applications. Мoscow: Mir, 1983. 256 p. (in Russ.).

10. Pastukhova S.E. Approximations of the exponential of an operator with periodic coefficients // Journal of Mathematical Sciences. 2012. V. 181. № 5. P. 668–700.