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On global existence and blowup of solutions of Stochastic Keller–Segel type equation

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Abstract

In this paper we consider a stochastic Keller–Segel type equation, perturbed with random noise. We establish that for special types of random pertubations (i.e. in a divergence form), the equation has a global weak solution for small initial data. Furthermore, if the noise is not in a divergence form, we show that the solution has a finite time blowup (with nonzero probability) for any nonzero initial data. The results on the continuous dependence of solutions on the small random perturbations, alongside with the existence of local strong solutions, are also derived in this work.

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Acknowledgements

The research of Oleksandr Misiats was supported by Simons Collaboration Grant for Mathematicians No. 854856. The research of Ihsan Topaloglu was supported by Simons Collaboration Grant for Mathematicians No. 851065. The research of Oleksandr Stanzhytskyi was supported by Ukrainian Government Scientific Research Grant No. 210BF38-01.

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Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA, USA

    Oleksandr Misiats & Ihsan Topaloglu

  2. Department of Mathematics, Taras Shevchenko National University of Kiev, Kiev, Ukraine

    Oleksandr Stanzhytskyi

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  1. Oleksandr Misiats
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  2. Oleksandr Stanzhytskyi
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  3. Ihsan Topaloglu
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Correspondence to Ihsan Topaloglu.

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Misiats, O., Stanzhytskyi, O. & Topaloglu, I. On global existence and blowup of solutions of Stochastic Keller–Segel type equation. Nonlinear Differ. Equ. Appl. 29, 3 (2022). https://doi.org/10.1007/s00030-021-00735-2

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