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Small data blow-up for the weakly coupled system of the generalized Tricomi equations with multiple propagation speeds

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Abstract

In the present paper, we study the Cauchy problem for the weakly coupled system of the generalized Tricomi equations with multiple propagation speeds. Our aim of this paper is to prove a small data blow-up result and an upper estimate of lifespan of the problem for a suitable compactly supported initial data in the subcritical and critical cases of the Strauss type. The proof is based on the framework of the argument in the paper (Ikeda et al. in J Diff Equs 267:5165–5201, 2019). One of our new contributions is to construct two families of special solutions to the free equation (see (2.16) or (2.18)) as the test functions and prove their several properties. We emphasize that the system with two different propagation speeds is treated in this paper and the assumption on the initial data is improved from the point-wise positivity to the integral positivity.

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Acknowledgements

The first author is supported by JST CREST Grant Number JPMJCR1913, Japan and Grant-in-Aid for Young Scientists Research (B) No.15K17571 and Young Scientists Research (No.19K14581), Japan Society for the Promotion of Science. This work was done when the second author was visiting Riken and Keio University, with the support of Chinese Scholarship Council. The second author is supported by NSFC No. 11501511 and Zhejiang Provincial Nature Science Foundation of China under Grant No. LQ15A010012.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan

    Masahiro Ikeda

  2. Center for Advanced Intelligence Project, RIKEN, Japan

    Masahiro Ikeda

  3. Department of Mathematics and Science, School of Sciences, Zhejiang Sci-Tech University, 310018, Hangzhou, China

    Jiayun Lin

  4. School of Data Science, Zhejiang University of Finance and Economics, 310018, Hangzhou, China

    Ziheng Tu

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  1. Masahiro Ikeda
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  2. Jiayun Lin
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  3. Ziheng Tu
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Correspondence to Jiayun Lin.

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Ikeda, M., Lin, J. & Tu, Z. Small data blow-up for the weakly coupled system of the generalized Tricomi equations with multiple propagation speeds. J. Evol. Equ. 21, 3765–3796 (2021). https://doi.org/10.1007/s00028-021-00703-4

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