A DIFFUSION-CONVECTION EQUATION IN SEVERAL SPACE DIMENSIONS

We study the large-time behaviour of nonnegative solutions u(x,t) of t he diffusion-convection equation u(t) = DELTAu- a . del(u(q)), defined in the whole space RN for t > 0, with initial data uo is-an-element-o f L1(R(N)). The direction a is assumed to be constant. We concentrate in the exponent range 1 < q < (N + 1)/N and show that for very large t imes the effect of diffusion is neligible as compared to convection in the direction a, while in the directions orthogonal to a the motion i s explained by diffusion. More precisely, the asymptotic behaviour of the solutions to (1) is given by the fundamental entropy solutions of the reduced equation (2) u(t) = DELTA'u + a . V (u(q)), where DELTA' d enotes the (N - 1)-dimensional Laplacian in the hyperplane orthogonal to a. Existence and uniqueness of such special solutions, which have a selfsimilar form, is proved here, previous to establishing the asympt otic convergence. Such a phenomenon does not occur for q = 1 or for q greater-than-or-equal-to (N + 1)/N.