LARGE TIME BEHAVIOR FOR CONVECTION-DIFFUS ION EQUATIONS IN R(N) WITH ASYMPTOTICALLY CONSTANT DIFFUSION

We describe the large time behavior of solutions of the convection-dif fusion equation u(t) - div (a (x) del u) = (d) over right arrow . del (/u/(q-1) u) in (0, infinity) x R(N) where (d) over right arrow is an element of R(N), q greater than or equal to 1 + (1/N), 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)) and //b(-) (x)//infinity < 1. When q = 1 + (1/N), we prove 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 = 1. When q > 1(1/N), we prove that the large time behavi or of solutions is given by the heat kernel. In this case we also find the second term in the asymptotic development of solutions as t --> i nfinity.