On the strauss index of semilinear tricomi equation

Abstract

In our previous papers, we have given a systematic study on the global existence versus blowup problem for the small-data solution $ u $ of the multi-dimensional semilinear Tricomi equation \begin{document}$ \begin{equation*} \partial_t^2 u-t\, \Delta u = |u|^p, \quad \big(u(0, \cdot), \partial_t u(0, \cdot)\big) = (u_0, u_1), \end{equation*} $\end{document} where $ t>0 $, $ x\in \mathbb R^n $, $n\geq2$, $ p>1 $, and $ u_i\in C_0^{\infty}( \mathbb R^n) $ ($ i = 0, 1 $). In this article, we deal with the remaining 1-D problem, for which the stationary phase method for multi-dimensional case fails to work and the large time decay rate of $\|u(t, \cdot)\|_{L^\infty_x(\mathbb R)}$ is not enough. The main ingredient of the proof in this paper is to use the structure of the linear equation to get the suitable decay rate of $u$ in $t$, then the crucial weighted Strichartz estimates are established and the global existence of solution $ u $ is proved when $ p>5 $.