Asymptotic profiles of solutions to convection-diffusion equations

The large time behavior of zero-mass solutions to the Cauchy problem for the convection-diffusion equation u(t) - u(xx) + (\u\(q))(x) = 0, u(x, 0) = u(0)(x) is studied when q > 1 and the initial datum u(0) belongs to L-1(R, (1 + \x\) dx) and satisfies integral(R) u(0) (x) dx = 0. We provide conditions on the size and shape of the initial datum u(0) as well as on the exponent q > 1 such that the large time asymptotics of solutions is given either by the derivative of the Gauss-Weierstrass kernel, or by a self-similar solution of the equation, or by hyperbolic N-waves. (C) 2004 Academie des sciences. Published by Elsevier SAS. All rights reserved.