In this article, we use the cone Sobolev inequality and the 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 H 1, n 2 2,0 (B). Finally, we obtain existence result of global solutions with exponential decay and show the blow-up in finite time of solutions.

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In this paper, we apply the family of potential wells to the initial boundary value problem of semilinear hyperbolic equations on the cone Sobolev spaces. We not only give some results of global existence and nonexistence of solutions but also obtain the vacuum isolating of solutions. Finally, we show blow‐up in finite time of solutions on a manifold with conical singularities. Copyright © 2017 John Wiley & Sons, Ltd.

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