#### Vol. 314, No. 1, 2021

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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
##### 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 $m\in ℕ$, 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; 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). 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.

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##### 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
##### Authors
 Daoyin He Southeast University Nanjing China Ingo Witt University of Göttingen Göttingen Germany Huicheng Yin Nanjing Normal University Nanjing China