Homogenized model of reaction–diffusion in a porous medium
Leonid Pankratov, Andrey Piatnitskii, Volodymyr Rybalko · Comptes Rendus Mécanique · 2003
We study the initial boundary value problem for the reaction–diffusion equation, ∂ t u ϵ - ∇ · ( a ϵ ∇ u ϵ ) + g ( u ϵ ) = h ϵ in a bounded domain Ω with periodic microstructure ℱ ( ϵ ) ∪ ℳ ¯ ( ϵ ) , where a ε ( x ) is of order 1 in ℱ ( ϵ ) and κ ( ε ) in ℳ ( ϵ ) with κ ( ε )→0 as ε →0. Combining the method of two-scale convergence and the variational homogenization we obtain effective models which depend on the parameter θ =lim ε →0 κ ( ε )/ ε 2 . In the case of strictly positive finite θ the effective problem is nonlocal in time that corresponds to the memory effect.