Geodesic
Homogenization of the Parabolic Cauchy Problem in the Sobolev Class $H^1(\mathbb{R}^d)$
Funkcionalʹnyj analiz i ego priloženiâ, Tome 44 (2010) no. 4, pp. 91-96 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Homogenization in the small period limit for the solution $\mathbf{u}_\varepsilon$ of the Cauchy problem for a parabolic equation in $\mathbb{R}^d$ is studied. The coefficients are assumed to be periodic in $\mathbb{R}^d$ with respect to the lattice $\varepsilon\Gamma$. As $\varepsilon\to 0$, the solution $\mathbf{u}_\varepsilon$ converges in $L_2(\mathbb{R}^d)$ to the solution $\mathbf{u}_0$ of the effective problem with constant coefficients. The solution $\mathbf{u}_\varepsilon$ is approximated in the norm of the Sobolev space $H^1(\mathbb{R}^d)$ with error $O(\varepsilon)$; this approximation is uniform with respect to the $L_2$-norm of the initial data and contains a corrector term of order $\varepsilon$. The dependence of the constant in the error estimate on time $\tau$ is given. Also, an approximation in $H^1(\mathbb{R}^d)$ for the solution of the Cauchy problem for a nonhomogeneous parabolic equation is obtained.
Mots-clés : parabolic equation, effective matrix
Keywords: Cauchy problem, homogenization, corrector, threshold effect. 
@article{FAA_2010_44_4_a8,
     author = {T. A. Suslina},
     title = {Homogenization of the {Parabolic} {Cauchy} {Problem} in the {Sobolev} {Class~}$H^1(\mathbb{R}^d)$},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {91--96},
     year = {2010},
     volume = {44},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2010_44_4_a8/}
}
TY  - JOUR
AU  - T. A. Suslina
TI  - Homogenization of the Parabolic Cauchy Problem in the Sobolev Class $H^1(\mathbb{R}^d)$
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2010
SP  - 91
EP  - 96
VL  - 44
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/FAA_2010_44_4_a8/
LA  - ru
ID  - FAA_2010_44_4_a8
ER  - 
%0 Journal Article
%A T. A. Suslina
%T Homogenization of the Parabolic Cauchy Problem in the Sobolev Class $H^1(\mathbb{R}^d)$
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2010
%P 91-96
%V 44
%N 4
%U http://geodesic.mathdoc.fr/item/FAA_2010_44_4_a8/
%G ru
%F FAA_2010_44_4_a8
T. A. Suslina. Homogenization of the Parabolic Cauchy Problem in the Sobolev Class $H^1(\mathbb{R}^d)$. Funkcionalʹnyj analiz i ego priloženiâ, Tome 44 (2010) no. 4, pp. 91-96. http://geodesic.mathdoc.fr/item/FAA_2010_44_4_a8/

[1] A. Bensoussan, J.-L. Lions, G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Stud. Math. Appl., 5, North-Holland Publishing Co., Amsterdam–New York, 1978 | MR

[2] N. S. Bakhvalov, G. P. Panasenko, Osrednenie protsessov v periodicheskikh sredakh, Nauka, M., 1984 | MR | Zbl

[3] V. V. Zhikov, S. M. Kozlov, O. A. Oleinik, Usrednenie differentsialnykh operatorov, Nauka, M., 1993 | MR | Zbl

[4] E. V. Sevostyanova, Matem. sb., 115:2 (1981), 204–222 | MR | Zbl

[5] V. V. Zhikov, Differentsialnye uravneniya, 25:1 (1989), 44–50 | MR | Zbl

[6] M. Sh. Birman, T. A. Suslina, Algebra i analiz, 15:5 (2003), 1–108 | MR | Zbl

[7] M. Sh. Birman, T. A. Suslina, Algebra i analiz, 17:5 (2005), 69–90 | MR | Zbl

[8] M. Sh. Birman, T. A. Suslina, Algebra i analiz, 17:6 (2005), 1–104 | MR | Zbl

[9] M. Sh. Birman, T. A. Suslina, Algebra i analiz, 18:6 (2006), 1–130 | MR | Zbl

[10] T. A. Suslina, Funkts. analiz i ego pril., 38:4 (2004), 86–90 | DOI | MR | Zbl

[11] T. A. Suslina, Nonlinear Equations and Spectral Theory, Amer. Math. Soc. Transl. (2), 220, Amer. Math. Soc., Providence, RI, 2007, 201–233 | MR | Zbl

[12] E. S. Vasilevskaya, Algebra i analiz, 21:1 (2009), 3–60 | MR | Zbl

[13] V. V. Zhikov, Dokl. RAN, 406:5 (2006), 597–601 | MR | Zbl

[14] V. V. Zhikov, S. E. Pastukhova, Russian J. Math. Phys., 12:4 (2005), 515–524 | MR | Zbl

[15] V. V. Zhikov, S. E. Pastukhova, Russian J. Math. Phys., 13:2 (2006), 224–237 | DOI | MR | Zbl

[16] T. A. Suslina, Math. Model. Nat. Phenom., 5:4 (2010), 390–447 | DOI | MR | Zbl