Minimal regularity solutions of semilinear generalized Tricomi equations

Abstract

We prove the local existence and uniqueness of minimal regularity solutions $u$ of the semilinear generalized Tricomi equation ${\partial }_t^{2}u-t^m\Delta u=F(u)$ with initial data $(u(0,\cdot ),{\partial }_{t}u(0,\cdot ))\in \dot{H^{\gamma }}({\mathbb{R}}^n)\times {\dot{H}}^{\gamma -\frac{2}{m+2}}({\mathbb{R}}^n)$ under the assumption that $|F(u)|\lesssim |u{|}^{\kappa }$ and $|F^{\prime }(u)|\lesssim |u{|}^{\kappa -1}$ for some $\kappa >1$. Our results improve previous results of M. Beals [2] and of ourselves [15-17]. We establish Strichartz-type estimates for the linear generalized Tricomi operator ${\partial }_t^2-t^m\Delta$ from which the semilinear results are derived.

38 pages