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 $(u(0,\cdot), \partial_t u(0,\cdot)) \in \dot{H^{\gamma}}(\mathbb R^n) \times \dot{H}^{\gamma-\frac2{m+2}}(\mathbb R^n)$ under the assumption that $|F(u)|\lesssim |u|^\kappa$ and $|F'(u)| \lesssim |u|^{\kappa -1}$ 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.