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 $t\ge 0$, $x\in \mathbb {R}^n$ ($n\ge 2$), $m\in \mathbb {N}$, $p>1$, 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 $p\ge p_{\text {conf}}(m,n)$ provided that the initial data is small enough. In case $p_{\text {crit}}(m,n)<p< p_{\text {conf}}(m,n)$, we will establish global existence of small data solutions u in a subsequent paper (He et al. 2015).