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Global well-posedness of coupled parabolic systems

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Abstract

The initial boundary value problem of a class of reaction-diffusion systems (coupled parabolic systems) with nonlinear coupled source terms is considered in order to classify the initial data for the global existence, finite time blowup and long time decay of the solution. The whole study is conducted by considering three cases according to initial energy: the low initial energy case, critical initial energy case and high initial energy case. For the low initial energy case and critical initial energy case the suffcient initial conditions of global existence, long time decay and finite time blowup are given to show a sharp-like condition. In addition, for the high initial energy case the possibility of both global existence and finite time blowup is proved first, and then some suffcient initial conditions of finite time blowup and global existence are obtained, respectively.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11471087), the China Postdoctoral Science Foundation International Postdoctoral Exchange Fellowship Program, the Heilongjiang Postdoctoral Foundation (Grant No. LBH-Z13056) and the Fundamental Research Funds for the Central Universities. Part of this work was finished when the first author visited Professor Zhouping Xin in the Institute of Mathematical Sciences, the Chinese University of Hong Kong. The authors appreciate the referees’ valuable suggestions, which help so much with the improving and modification of this paper. Thanks are also to Dr. Xingchang Wang and Dr. Yuxuan Chen for helping revise the paper.

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Authors and Affiliations

  1. College of Automation, Harbin Engineering University, Harbin, 150001, China

    Runzhang Xu, Wei Lian & Yi Niu

  2. College of Science, Harbin Engineering University, Harbin, 150001, China

    Runzhang Xu

  3. The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Hong Kong, China

    Runzhang Xu

  4. School of Information Science and Engineering, Shandong Normal University, Jinan, 250014, China

    Yi Niu

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  1. Runzhang Xu
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  2. Wei Lian
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  3. Yi Niu
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Correspondence to Runzhang Xu.

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Xu, R., Lian, W. & Niu, Y. Global well-posedness of coupled parabolic systems. Sci. China Math. 63, 321–356 (2020). https://doi.org/10.1007/s11425-017-9280-x

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  • DOI: https://doi.org/10.1007/s11425-017-9280-x

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