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

We describe the large time behavior of solutions of the scalar convection-diffusion 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 than 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 the rate of convergence of solutions towards the fundamental solution of the heat equation. We also prove pointwise global gaussian bounds for the gradient 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 of solutions with initial data in L-1(R-N) is given by a uniparametric family of self-similar solutions of the convection-diffusion equation with constant diffusion a a 1. When q>1+1/N, ne prove that the large time behavior of solutions is given by the heat kernel.