On the strauss index of semilinear tricomi equation

Daoyin He; Ingo Witt; Huicheng Yin

doi:10.3934/cpaa.2020213

Article Contents

2020, Volume 19, Issue 10: 4817-4838. Doi: 10.3934/cpaa.2020213

This issue Previous Article Global bifurcation for the Hénon problem Next Article Boundedness in prey-taxis system with rotational flux terms

On the strauss index of semilinear tricomi equation

1.
School of Mathematics, Southeast University, Nanjing 211189, China
2.
Mathematical Institute, University of Göttingen, Bunsenstr. 3-5, 37073 Göttingen, Germany
3.
School of Mathematical Sciences and Mathematical Institute, Nanjing Normal University, Nanjing 210023, China

^* Corresponding author
^* Corresponding author

Received: September 2019

Revised: May 2020

Published: July 2020

Authors are supported by National Natural Science Foundation of China (grant No. 11901103, No. 11571177, No. 11731007)

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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{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*} $

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 $.

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Mathematics Subject Classification: Primary: 35L70, 35L71; Secondary: 35L80, 35L81.

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