On the global solution problem for semilinear generalized Tricomi equations, I

In this paper, we are concerned with the global Cauchy problem for the semilinear generalized Tricomi equation $partial_t^2 u-t^m Delta u=|u|^p$ with initial data $(u(0,cdot), partial_t u(0,cdot))= (u_0, u_1)$, where $tgeq 0$, $xin{mathbb R}^n$ ($nge 3$), $minmathbb N$, $p>1$, and $u_iin C_0^{infty}({mathbb R}^n)$ ($i=0,1$). We show that there exists a critical exponent $p_{text{crit}}(m,n)>1$ such that the solution $u$, in general, blows up in finite time when $1<p<p_{text{crit}}(m,n)$. We further show that there exists a conformal exponent $p_{text{conf}}(m,n)> p_{text{crit}}(m,n)$ such that the solution $u$ exists globally when $p>p_{text{conf}}(m,n)$ provided that the initial data is small enough. In case $p_{text{crit}}(m,n)<pleq p_{text{conf}}(m,n)$, we will establish global existence of small data solutions $u$ in a subsequent paper.