LAUSR

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