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

25 pages

References in corpus (2)

• From $p_0(n)$ to $p_0(n+2)$
• Minimal regularity solutions of semilinear generalized Tricomi equations