Existence theorem and global solution for semilinear edge-degenerate hypoelliptic equations

In this article, we use the edge-type of Sobolev inequality,Hardy inequlity and Poincaré inequality to prove the existence theorem for a class of semilinear degenerate hypoelliptic equation on manifolds with conical singularities. In this paper we shall find the existence theorem for the problem 1.1 in cone Sobolev space H2,01,N2(E).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {H}}^{1,\frac{N}{2}}_{2,0}({\mathbb {E}}).$$\end{document} Finally, we obtain existence result of global solutions with exponential decay and show the blow-up in finite time of solutions.