  In the paper, we consider the large time behavior of solutions to the
convection-diffusion equation u_t - Delta u + nabla cdot f(u) = 0 in R^n times
[0,infinity), where f(u) ~ u^q as u --> 0. Under the assumption that q >=
1+1/(n+beta) and the initial condition u_0 satisfies: u_0 in L^1(R^n),
integral_{R^n} u_0(x) dx = 0, and |e^{t Delta}u_0|_{L^1(R^n)} <= Ct^{-beta/2}
for fixed beta in (0,1), all t>0, and a constant C, we show that the L^1-norm
of the solution to the convection-diffusion equation decays with the rate
t^{-beta/2} as t --> infinity. Moreover, we prove that, for small initial
conditions, the exponent q^* = 1+1/(n+beta) is critical in the following sense.
For q > q^* the large time behavior in L^p(R^n), 1 <= p <= infinity, of
solutions is described by self-similar solutions to the linear heat equation.
For q = q^*, we prove that the convection-diffusion equation with f(u) =
u|u|^{q^*-1} has a family of self-similar solutions which play an important
role in the large time asymptotics of general solutions.
