#### 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