Large time behavior for convection-diffusion equations in R-N with asymptotically constant diffusion

We describe the large time behavior of solutions of the scalar convection-d iffusion equation u(t)-div(a(x)del u)=d.del(/u/(q-1)u) in (O, infinity) x R-N where d is an element of R-,(N) q greater than or equal to 1+ 1/N greater t han or equal to 1 and a(x)=1+b(x) with b(x)is an element of L-1(R-N)boolean AND C-1,C-alpha(R-N) such that parallel to b(-)(x)parallel to(infinity)< 1 , satisfying /b(x)/+(1+/x/(2))(1/2)/del b(x)/less than or equal to C(1+/x/(2))(-6/2) For All x is an element of R-N for some positive constants C and delta. First, we consider the linear problem (d=0) and prove sharp estimates on th e rate of convergence of solutions towards the fundamental solution of the heat equation. We also prove pointwise global gaussian bounds for the gradi ent of solutions that are valid for all t>0. In the nonlinear case, when q=1+1/N we prole that the large time behavior o f solutions with initial data in L-1(R-N) is given by a uniparametric famil y of self-similar solutions of the convection-diffusion equation with const ant diffusion a a 1. When q>1+1/N, ne prove that the large time behavior of solutions is given b y the heat kernel.