Global existence and asymptotic behavior of solutions to a semilinear parabolic equation on Carnot groups

In this paper we consider the semilinear parabolic equation ∑i,j=1maijXiXju−∂tu+Vup=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sum_{i,j = 1}^{m} {a_{ij} X_{i} X_{j} u} - \partial_{t} u + Vu^{p} = 0$\end{document} with a general class of potentials V=V(ξ,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$V=V(\xi,t)$\end{document}, where A={aij}i,j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A = \{a_{ij} \}_{i,j} $\end{document} is a positive definite symmetric matrix and the Xi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$X_{i}$\end{document}’s denotes a system of left-invariant vector fields on a Carnot group G. Based on a fixed point argument and by establishing some new estimates involving the heat kernel, we study the existence and large-time behavior of global positive solutions to the preceding equation.