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: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u_t = \Delta u + \vec b \cdot \nabla u^n$\end{document} for (x, t) ∈ ST = ℝN × (0, T] and u(x, 0) = δ(x) for x ∈ ℝN, where δ(x) denotes Dirac measure in ℝN, N ⩾ 2, n ⩾ 0 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\vec b = \left( {b_1 , \ldots ,b_N } \right) \in \mathbb{R}^N$\end{document} is a vector. It is shown that there exists a critical number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p_c = \frac{{N + 2}} {N}$\end{document} such that the source-type solution to the above problem exists and is unique if 0 ⩽ n < pc and there exists a unique similarity source-type solution in the case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n = \frac{{N + 1}} {N}$\end{document} , while such a solution does not exist if n > pc. Moreover, the asymptotic behavior of the solution near the origin is studied. It is shown that when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0 < N < \frac{{N + 1}} {N}$\end{document} 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.