#### Vol. 296, No. 1, 2018

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Minimal regularity solutions of semilinear generalized Tricomi equations

### Zhuoping Ruan, Ingo Witt and Huicheng Yin

Vol. 296 (2018), No. 1, 181–226
##### Abstract

We prove the local existence and uniqueness of minimal regularity solutions $u$ of the semilinear generalized Tricomi equation ${\partial }_{t}^{2}u-{t}^{m}\Delta u=F\left(u\right)$ with initial data $\left(u\left(0,\cdot \phantom{\rule{0.3em}{0ex}}\right),{\partial }_{t}u\left(0,\cdot \phantom{\rule{0.3em}{0ex}}\right)\right)\in {Ḣ}^{\gamma }\left({ℝ}^{n}\right)×{Ḣ}^{\gamma -2∕\left(m+2\right)}\left({ℝ}^{n}\right)$ under the assumptions that $|F\left(u\right)|\lesssim |u{|}^{\kappa }$ and $|{F}^{\prime }\left(u\right)|\lesssim |u{|}^{\kappa -1}$ for some $\kappa >1$. Our results improve previous results of M. Beals and ourselves. We establish Strichartz-type estimates for the linear generalized Tricomi operator ${\partial }_{t}^{2}-{t}^{m}\Delta$ from which the semilinear results are derived.

##### Keywords
generalized Tricomi equation, minimal regularity, Fourier integral operators, Strichartz estimates
Primary: 35L70
Secondary: 35L65
##### Milestones
Received: 23 March 2017
Revised: 5 November 2017
Accepted: 9 January 2018
Published: 1 May 2018
##### Authors
 Zhuoping Ruan Department of Mathematics Nanjing University Nanjing China Ingo Witt Mathematical Institute University of Göttingen Göttingen Germany Huicheng Yin School of Mathematical Sciences and Mathematical Institute Nanjing Normal University Nanjing China