Global existence and nonexistence for semilinear parabolic equations with conical degeneration

In this article we study the initial boundary value problem of semilinear parabolic equations \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${u_t-\triangle_\mathbb{B}u=|u|^{p-1}u}$$\end{document} on a manifold with conical singularity, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\triangle_\mathbb{B}}$$\end{document} is Fuchsian type Laplace operator investigated in Chen et al. (Calc Var 43:463–484, 2012) with totally characteristic degeneracy on the boundary x1 = 0. By using a family of potential wells, we obtain existence theorem of global solutions with exponential decay and show the blow-up in finite time of solutions. Especially, the relation between the above two phenomena is derived as a sharp condition.