Necessary and sufficient condition for the stabilization of the solution of the cauchy problem for a special heat equation

In the space ℝn, we obtain the solution of the Cauchy problem for the equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u'_t = a^2 (\bar x|\bar x|^{ - 1} )u''_{rr} $\end{document} degenerating at the origin, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\bar x = (x_1 ,x_2 ,...,x_n )$\end{document} and urr″ is the second derivative in the direction of the position vector of the point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\bar x$\end{document}. We study the stabilization of this solution.