Source-type solutions of heat equations with convection in several variables space

In this paper we study the source-type solution for the heat equation with convection: u(t) = Delta u + (b) over right arrowb . del u(n) for (x, t) is an element of S-T = R-N x (0, T] and u(x, 0) = delta(x) for x is an element of R-N, where delta(x) denotes Dirac measure in R-N, N >= 2, n >= 0 and (b) over right arrow = (b(1), ..., b(N)) is an element of R-N is a vector. It is shown that there exists a critical number p(c) = N+2/N such that the source-type solution to the above problem exists and is unique if 0 <= n < p(c) and there exists a unique similarity source-type solution in the case n = N+1/N, while such a solution does not exist if n > p(c). Moreover, the asymptotic behavior of the solution near the origin is studied. It is shown that when 0 < n < N+1/N the convection is too weak and the short time behavior of the source-type solution near the origin is the same as that for the heat equation without convection.