Large Time Behavior for Convection-Diffusion Equations in N with Periodic Coefficients

We describe the large time behavior of solutions of the convection-diffusion equation where d N and a=a(x) is a symmetric periodic matrix satisfying suitable ellipticity assumptions. We also assume that a W1, ∞( N). 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+ , we prove that the large time behavior of solutions with initial data in L1( N) is given by a uniparametric family of self-similar solutions of the convection-diffusion equation with constant homogenized diffusion a≡ah. When q>1+ , we prove that the large time behavior of solutions is given by the fundamental solution of the linear-diffusion equation with a≡ah.