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

Abstract

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 $t\geq 0$, $x\in{\mathbb R}^n$ ($n\ge 3$), $m\in\mathbb N$, $p>1$, and $u_i\in C_0^{\infty}({\mathbb R}^n)$ ($i=0,1$). 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 $p_{\text{crit}}(m,n)<p\leq p_{\text{conf}}(m,n)$, we will establish global existence of small data solutions $u$ in a subsequent paper.