Asymptotic Homogenization in A Parabolic Semilinear Problem With Periodic Coefficients and Integrable Initial Data

In this paper, we study the large time behavior of solutions of the parabolic semilinear equation ∂tu-div(a(x)∇u)=-α|u|u in (0,∞)×ℝN, where α > 0 is constant and a ∈ is a symmetric periodic matrix satisfying some ellipticity assumptions. Considering an integrable initial data u0 and α ∈ (2/N,3/N), we prove that the large time behavior of solutions is given by the solution U(t,x) of the homogenized linear problem ∂tU – div(ah∇ U) = 0, U(0) = Cδ, where ah is the homogenized matrix of a(x), C is a positive constant and δ is the Dirac measure at 0.