Minimal regularity solutions of semilinear generalized Tricomi equations

Zhuoping Ruan, Ingo Witt and Huicheng Yin

Vol. 296 (2018), No. 1, 181–226

DOI: 10.2140/pjm.2018.296.181

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 $\left(u\left(0,\cdot \right),{\partial }_{t}u\left(0,\cdot \right)\right)\in {Ḣ}^{\gamma }\left({ℝ}^n\right)\times {Ḣ}^{\gamma -2∕\left(m+2\right)}\left({ℝ}^n\right)$ under the assumptions that $\left|F\left(u\right)\right|\lesssim |u{|}^{\kappa }$ and $\left|F^{\prime }\left(u\right)\right|\lesssim |u{|}^{\kappa -1}$ 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

Mathematical Subject Classification 2010

Primary: 35L70

Secondary: 35L65

Milestones

Received: 23 March 2017

Revised: 5 November 2017

Accepted: 9 January 2018

Published: 1 May 2018