TY - JOUR

T1 - Space-time homogenization for nonlinear diffusion

AU - Akagi, Goro

AU - Oka, Tomoyuki

N1 - Funding Information:
We thank the anonymous referees for their useful comments. The first author is supported by JSPS KAKENHI Grant Number JP21KK0044 , JP21K18581 , JP20H01812 , JP18K18715 , JP16H03946 , JP20H00117 and JP17H01095 . The second author is partially supported by Division for Interdisciplinary Advanced Research and Education, Tohoku University , Grant-in-Aid for JSPS Fellows (No. JP20J10143 ) and JSPS KAKENHI Grant Number JP22K20331 . This work was also supported by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University.
Publisher Copyright:
© 2023 The Author(s)

PY - 2023/6/15

Y1 - 2023/6/15

N2 - The present paper is concerned with a space-time homogenization problem for nonlinear diffusion equations with periodically oscillating (in space and time) coefficients. Main results consist of a homogenization theorem (i.e., convergence of solutions as the period of oscillation goes to zero) as well as a characterization of homogenized equations. In particular, homogenized matrices are described in terms of solutions to cell-problems, which have different forms depending on the log-ratio of the spatial and temporal periods of the coefficients. At a critical ratio, the cell problem turns out to be a parabolic equation in microscopic variables (as in linear diffusion) and also involves the limit of solutions, which is a function of macroscopic variables. The latter feature stems from the nonlinearity of the equation, and moreover, some strong interplay between microscopic and macroscopic structures can explicitly be seen for the nonlinear diffusion. As for the other ratios, the cell problems are always elliptic (in micro-variable only) and do not involve any macroscopic variables, and hence, micro- and macrostructures are weakly interacting each other. Proofs of the main results are based on the two-scale convergence theory (for space-time homogenization). Furthermore, finer asymptotics of gradients, diffusion fluxes and time-derivatives with certain corrector terms are provided, and a qualitative analysis on homogenized matrices is also performed.

AB - The present paper is concerned with a space-time homogenization problem for nonlinear diffusion equations with periodically oscillating (in space and time) coefficients. Main results consist of a homogenization theorem (i.e., convergence of solutions as the period of oscillation goes to zero) as well as a characterization of homogenized equations. In particular, homogenized matrices are described in terms of solutions to cell-problems, which have different forms depending on the log-ratio of the spatial and temporal periods of the coefficients. At a critical ratio, the cell problem turns out to be a parabolic equation in microscopic variables (as in linear diffusion) and also involves the limit of solutions, which is a function of macroscopic variables. The latter feature stems from the nonlinearity of the equation, and moreover, some strong interplay between microscopic and macroscopic structures can explicitly be seen for the nonlinear diffusion. As for the other ratios, the cell problems are always elliptic (in micro-variable only) and do not involve any macroscopic variables, and hence, micro- and macrostructures are weakly interacting each other. Proofs of the main results are based on the two-scale convergence theory (for space-time homogenization). Furthermore, finer asymptotics of gradients, diffusion fluxes and time-derivatives with certain corrector terms are provided, and a qualitative analysis on homogenized matrices is also performed.

KW - Fast diffusion equation

KW - Nonlinear diffusion

KW - Periodic space-time homogenization

KW - Porous media equation

KW - Two-scale convergence

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U2 - 10.1016/j.jde.2023.01.044

DO - 10.1016/j.jde.2023.01.044

M3 - Article

AN - SCOPUS:85148667062

SN - 0022-0396

VL - 358

SP - 386

EP - 456

JO - Journal of Differential Equations

JF - Journal of Differential Equations

ER -