In this paper, we are concerned with the global Cauchy problem for the semilinear generalized Tricomi equation ∂t2u-tmΔu=|u|p\documentclass[12pt]{minimal}
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\begin{document}$$\partial _t^2 u-t^m \Delta u=|u|^p$$\end{document} with initial data (u(0,·),∂tu(0,·))=(u0,u1)\documentclass[12pt]{minimal}
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\begin{document}$$(u(0,\cdot ), \partial _t u(0,\cdot ))=(u_0, u_1)$$\end{document}, where t≥0\documentclass[12pt]{minimal}
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\begin{document}$$t\ge 0$$\end{document}, x∈Rn\documentclass[12pt]{minimal}
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\begin{document}$$x\in \mathbb {R}^n$$\end{document} (n≥2\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 2$$\end{document}), m∈N\documentclass[12pt]{minimal}
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\begin{document}$$m\in \mathbb {N}$$\end{document}, p>1\documentclass[12pt]{minimal}
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\begin{document}$$p>1$$\end{document}, and ui∈C0∞(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$u_i\in C_0^{\infty }(\mathbb {R}^n)$$\end{document} (i=0,1\documentclass[12pt]{minimal}
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\begin{document}$$i=0,1$$\end{document}). We show that there exists a critical exponent pcrit(m,n)>1\documentclass[12pt]{minimal}
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\begin{document}$$p_{\text {crit}}(m,n)>1$$\end{document} such that the solution u, in general, blows up in finite time when 1<p<pcrit(m,n)\documentclass[12pt]{minimal}
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\begin{document}$$1<p<p_{\text {crit}}(m,n)$$\end{document}. We further show that there exists a conformal exponent pconf(m,n)>pcrit(m,n)\documentclass[12pt]{minimal}
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\begin{document}$$p_{\text {conf}}(m,n)> p_{\text {crit}}(m,n)$$\end{document} such that the solution u exists globally when p≥pconf(m,n)\documentclass[12pt]{minimal}
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\begin{document}$$p\ge p_{\text {conf}}(m,n)$$\end{document} provided that the initial data is small enough. In case pcrit(m,n)<p<pconf(m,n)\documentclass[12pt]{minimal}
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\begin{document}$$p_{\text {crit}}(m,n)<p< p_{\text {conf}}(m,n)$$\end{document}, we will establish global existence of small data solutions u in a subsequent paper (He et al. 2015).
