Czechoslovak Mathematical Journal, first online, pp. 1-18
Homogenization of linear parabolic problems with a certain matching between the microscopic scales
Pernilla JOHNSEN
Received January 8, 2024. Published online September 8, 2026. OPEN ACCESS
Abstract: The focus of this paper is the homogenization of, and development of suitable compactness results to, linear parabolic problems exhibiting an arbitrary finite number of both spatial and temporal microscopic periodic oscillations and a vanishing coefficient in front of the temporal derivative. We establish compactness results for evolution multiscale convergence and very weak evolution multiscale convergence, which are key in obtaining the homogenization result. In the homogenization, we note that we have an elliptic homogenized problem although the original problem is parabolic. We also note that we obtain a parabolic local problem for a different matching between the spatial and temporal microscopic scales than the standard matching, where by standard matching we mean that the temporal microscopic scale is the square of the spatial one.
References: [1] G. Allaire: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992), 1482-1518. DOI 10.1137/0523084 | MR 1185639 | Zbl 0770.35005
[2] G. Allaire, M. Briane: Multiscale convergence and reiterated homogenization. Proc. R. Soc. Edinb., Sect. A, Math. 126 (1996), 297-342. DOI 10.1017/S0308210500022757 | MR 1386865 | Zbl 0866.35017
[3] G. Allaire, A. Piatnitski: Homogenization of nonlinear reaction-diffusion equation with a large reaction term. Ann. Univ. Ferrara, Sez. VII, Sci. Mat. 56 (2010), 141-161. DOI 10.1007/s11565-010-0095-z | MR 2646529 | Zbl 1205.35019
[4] A. Bensoussan, J.-L. Lions, G. Papanicolaou: Asymptotic Analysis for Periodic Structures. Studies in Mathematics and its Applications 5. North-Holland, Amsterdam (1978). DOI 10.1090/chel/374 | MR 0503330 | Zbl 0404.35001
[5] T. Danielsson, P. Johnsen: Homogenization of the heat equation with a vanishing volumetric heat capacity. Available at https://arxiv.org/abs/1809.11019 (2018), 14 pages. DOI 10.48550/arXiv.1809.11019
[6] T. Danielsson, P. Johnsen: Homogenization of the heat equation with a vanishing volumetric heat capacity. Progress in Industrial Mathematics at ECMI 2018. Mathematics in Industry 30. Springer, Cham (2019), 343-349. DOI 10.1007/978-3-030-27550-1_43 | MR 4180925 | Zbl 1469.80006
[7] T. Danielsson, P. Johnsen: Homogenization of linear parabolic equations with three spatial and three temporal scales for certain matchings between the microscopic scales. Math. Bohem. 146 (2021), 483-511. DOI 10.21136/MB.2021.0087-19 | MR 4336552 | Zbl 1499.35049
[8] H. Douanla, J. L. Woukeng: Homogenization of reaction-diffusion equations in fractured porous media. Electron. J. Differ. Equ. 2015 (2015), Article ID 253, 23 pages. MR 3414107 | Zbl 1336.35046
[9] L. Flodén, A. Holmbom, M. Olsson Lindberg: A strange term in the homogenization of parabolic equations with two spatial and two temporal scales. J. Funct. Spaces Appl. 2012 (2012), Article ID 643458, 9 pages. DOI 10.1155/2012/643458 | MR 2875184 | Zbl 1242.35030
[10] L. Flodén, A. Holmbom, M. Olsson Lindberg, J. Persson: A note on parabolic homogenization with a mismatch between the spatial scales. Abstr. Appl. Anal. 2013 (2013), Article ID 329704, 6 pages. DOI 10.1155/2013/329704 | MR 3111807 | Zbl 1293.35027
[11] L. Flodén, A. Holmbom, M. Olsson Lindberg, J. Persson: Homogenization of parabolic equations with an arbitrary number of scales in both space and time. J. Appl. Math. 2014 (2014), Article ID 101685, 16 pages. DOI 10.1155/2014/101685 | MR 3176810 | Zbl 1406.35140
[12] L. Flodén, A. Holmbom, M. Olsson, J. Persson: Very weak multiscale convergence. Appl. Math. Lett. 23 (2010), 1170-1173. DOI 10.1016/j.aml.2010.05.005 | MR 2665589 | Zbl 1198.35023
[13] A. Holmbom: Homogenization of parabolic equations: An alternative approach and some corrector-type results. Appl. Math., Praha 42 (1997), 321-343. DOI 10.1023/A:1023049608047 | MR 1467553 | Zbl 0898.35008
[14] P. Johnsen, T. Lobkova: Homogenization of a linear parabolic problem with a certain type of matching between the microscopic scales. Appl. Math., Praha 63 (2018), 503-521. DOI 10.21136/AM.2018.0350-17 | MR 3870146 | Zbl 1499.35053
[15] T. Lobkova: Homogenization of linear parabolic equations with a certain resonant matching between rapid spatial and temporal oscillations in periodically perforated domains. Acta Math. Appl. Sin., Engl. Ser. 35 (2019), 340-358. DOI 10.1007/s10255-019-0810-1 | MR 3950176 | Zbl 1416.35032
[16] D. Lukkassen, G. Nguetseng, P. Wall: Two-scale convergence. Int. J. Pure Appl. Math. 2 (2002), 35-86. MR 1912819 | Zbl 1061.35015
[17] G. Nguetseng: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989), 608-623. DOI 10.1137/0520043 | MR 0990867 | Zbl 0688.35007
[18] G. Nguetseng, J. L. Woukeng: $\Sigma$-convergence of nonlinear parabolic operators. Nonlinear Anal., Theory Methods Appl., Ser. A 66 (2007), 968-1004. DOI 10.1016/j.na.2005.12.035 | MR 2288445 | Zbl 1116.35011
[19] F. Paronetto: Homogenization of degenerate elliptic-parabolic equations. Asymptotic Anal. 37 (2004), 21-56. DOI 10.3233/ASY-2004-599 | MR 2035361 | Zbl 1052.35025
[20] J. Persson: Selected Topics in Homogenization: Doctoral Thesis. Mid Sweden University, Östersund (2012).
[21] N. Svanstedt, J. L. Woukeng: Periodic homogenization of strongly nonlinear reaction- diffusion equations with large reaction terms. Appl. Anal. 92 (2013), 1357-1378. DOI 10.1080/00036811.2012.678334 | MR 3169106 | Zbl 1271.35006
[22] E. Zeidler: Nonlinear Functional Analysis and Its Applications. II/A. Linear Monotone Operators. Springer, New York (1990). DOI 10.1007/978-1-4612-0985-0 | MR 1033497 | Zbl 0684.47028
Affiliations: Pernilla Johnsen, Department of Engineering, Mathematics and Science Education, Mid Sweden University, Kunskapens väg 8, S-83125 Östersund, Sweden, e-mail: pernilla.johnsen@miun.se
