Large Time Behavior for Convection-Diffusion Equations in [formula omitted]N with Periodic Coefficients

Abstract

We describe the large time behavior of solutions of the convection-diffusion equationut−div(a(x)∇u)=d·∇(|u|q−1u)in(0, ∞)×RNwhere d∈RN and a=a(x) is a symmetric periodic matrix satisfying suitable ellipticity assumptions. We also assume that a∈W1, ∞(RN). First, we consider the linear problem (d=0) and prove that the large time behavior of solutions is given by the fundamental solution of the diffusion equation with a≡ah where ah is the homogenized matrix. In the nonlinear case, when q=1+1N, we prove that the large time behavior of solutions with initial data in L1(RN) is given by a uniparametric family of self-similar solutions of the convection-diffusion equation with constant homogenized diffusion a≡ah. When q>1+1N, we prove that the large time behavior of solutions is given by the fundamental solution of the linear-diffusion equation with a≡ah.