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Finite Time Blowup in L2 Sense of Solutions to SPDEs with Bernstein Functions of the Laplacian

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Abstract

The blowup in finite time in L2 sense of solutions to SPDEs

is investigated, where \(\dot {\xi }\) could be either a white noise or a colored noise and \(\phi :(0,\infty )\to (0,\infty )\) is a Bernstein function. The sufficient conditions on σ, \(\dot {\xi }\) and the initial value that imply the non-existence of the global solution are discussed. The results in this paper generalize the existing works on cases of the Laplacian and the fractional Laplacian by Chow (J Differential Equations 250(5), 2567-2580, 2011) and Foondun, Liu, Nane (J Differential Equations 266(5), 2575-2596, 2019), respectively.

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Data Availability statement

This paper describes entirely theoretical research. Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

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Acknowledgements

We would like to thank the editor and the reviewer for the valuable comments and suggestions, which helped to improve the manuscript. The research of Chang-Song Deng is supported in part by the National Natural Science Foundation of China (Grant No. 11831015). Wei Liu would like to thank the National Natural Science Foundation of China (Grant No. 11701378, 11871343, 11971316), and Science and Technology Innovation Plan of Shanghai (Grant No. 20JC1414200) for their supports. The research of Erkan Nane was partially supported by the Simons Foundation Collaboration Grants for Mathematicians.

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Authors and Affiliations

  1. School of Mathematics Statistics, Wuhan University, Wuhan, 430072, China

    Chang-Song Deng

  2. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

    Wei Liu

  3. Department of Mathematics and Statistics, Auburn University, Alabama, 36849, USA

    Erkan Nane

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  1. Chang-Song Deng
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  2. Wei Liu
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  3. Erkan Nane
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Correspondence to Wei Liu.

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Deng, CS., Liu, W. & Nane, E. Finite Time Blowup in L2 Sense of Solutions to SPDEs with Bernstein Functions of the Laplacian. Potential Anal 59, 565–588 (2023). https://doi.org/10.1007/s11118-021-09978-1

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  • DOI: https://doi.org/10.1007/s11118-021-09978-1

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