On the global weak solution problem of semilinear generalized Tricomi equations, II

Daoyin He, Ingo Witt and Huicheng Yin

Vol. 314 (2021), No. 1, 29–80

DOI: 10.2140/pjm.2021.314.29

Abstract

In Part I (Calc. Var. Partial Differential Equations 56:2, (2017), 1–24), for the semilinear generalized Tricomi equation ${\partial }_t^{2}u-t^m\Delta u=|u{|}^p$ with the initial data $\left(u\left(0,x\right),{\partial }_{t}u\left(0,x\right)\right)=\left(u_0\left(x\right),u_1\left(x\right)\right)$, $t\ge 0$, $x\in {ℝ}^n$ $\left(n\ge 3\right)$, $p>1$ and $m\in \mathbb{N}$, we have shown that there exists a critical exponent $p_{crit}\left(m,n\right)>1$ such that the weak solution $u$ generally blows up in finite time when $1<p<p_{crit}\left(m,n\right)$; and meanwhile there exists a conformal exponent $p_{conf}\left(m,n\right)$ $\left(>p_{crit}\left(m,n\right)\right)$ such that the weak solution $u$ exists globally when $p\ge p_{conf}\left(m,n\right)$ provided that $\left(u_0\left(x\right),u_1\left(x\right)\right)$ are small. In the present paper, we shall prove that the small data weak solution $u$ of ${\partial }_t^{2}u-t^m\Delta u=|u{|}^p$ exists globally when $p_{crit}\left(m,n\right)<p<p_{conf}\left(m,n\right)$. Hence, collecting the results in this paper and the previous paper, we have given a basically systematic study on the blowup or global existence of small data weak solution $u$ to the equation ${\partial }_t^{2}u-t^m\Delta u=|u{|}^p$ for $n\ge 3$. Here we point out that the study on the equation ${\partial }_t^{2}u-t^m\Delta u=|u{|}^p$ is closely related to those of the semilinear wave equation ${\partial }_t^{2}u-\Delta u+\frac{\mu }{1+t}{\partial }_{t}u=|u{|}^p$ for $0<\mu <1$ or other related physical problems.

Keywords

generalized Tricomi equation, Fourier integral operator, global existence, weighted Strichartz estimate, weak solution

Mathematical Subject Classification

Primary: 35L65, 35L70

Milestones

Received: 19 November 2018

Revised: 11 September 2020

Accepted: 22 May 2021

Published: 15 October 2021