Homogenization of the Heat Equation with a Vanishing Volumetric Heat Capacity

  • T. Danielsson
  • P. JohnsenEmail author
Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 30)


This paper is a study of the homogenization of the heat conduction equation, with a homogeneous Dirichlet boundary condition, having a periodically oscillating thermal conductivity and a vanishing volumetric heat capacity. In particular, the volumetric heat capacity equals εq and the thermal conductivity oscillates with period ε in space and εr in time, where 0 < q < r are real numbers. By using certain evolution settings of multiscale and very weak multiscale convergence we investigate, as ε tends to zero, how the relation between the volumetric heat capacity and the microscopic structure affects the homogenized problem and its associated local problem. It turns out that this relation gives rise to certain special effects in the homogenization result.


  1. 1.
    Danielsson, T., Johnsen, P.: Homogenization of the heat equation with a vanishing volumetric heat capacity (2018). arXiv: 1809.11019Google Scholar
  2. 2.
    Flodén, L., Holmbom, A., Olsson Lindberg, M., Persson, J.: Homogenization of parabolic equations with an arbitrary number of scales in both space and time. J. Appl. Math. 2014, 16 pp. (2014)Google Scholar
  3. 3.
    Holmbom, A.: Homogenization of parabolic equations: an alternative approach and some corrector-type results. Appl. Math. 42, 321–343 (1997)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Johnsen, P., Lobkova, T.: Homogenization of a linear parabolic problem with a certain type of matching between the microscopic scales. Appl. Math. 63, 503–521 (2018)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Persson, J.: Homogenisation of monotone parabolic problems with several temporal scales. Appl. Math. 57, 191–214 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Zeidler, E.: Nonlinear functional analysis and its applications II/A: linear monotone operators. Springer, New York (1990)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Science EducationMid Sweden UniversityÖstersundSweden

Personalised recommendations