Blow-up for a stochastic model of chemotaxis driven by conservative noise on R2

Avi Mayorcas; Milica Tomašević

doi:10.1007/s00028-023-00900-3

Blow-up for a stochastic model of chemotaxis driven by conservative noise on R²

Avi Mayorcas, Milica Tomašević

Department of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

Abstract

We establish criteria on the chemotactic sensitivity χ for the non-existence of global weak solutions (i.e., blow-up in finite time) to a stochastic Keller–Segel model with spatially inhomogeneous, conservative noise on R² . We show that if χ is sufficiently large then blow-up occurs with probability 1. In this regime, our criterion agrees with that of a deterministic Keller–Segel model with increased viscosity. However, for χ in an intermediate regime, determined by the variance of the initial data and the spatial correlation of the noise, we show that blow-up occurs with positive probability.

Original language	English
Article number	57
Journal	Journal of Evolution Equations
Volume	23
Issue number	3
Early online date	5 Aug 2023
DOIs	https://doi.org/10.1007/s00028-023-00900-3
Publication status	Published - 5 Aug 2023

Bibliographical note

Funding Information:
The authors warmly thank J. Norris, A. de Bouard, and L. Galeati for helpful discussions and B. Hambly for useful comments on the manuscript. The authors would like to express their gratitude to the French Centre National de Recherche Scientifique (CNRS) for the grant (PEPS JCJC) that supported this project. M.T. was partly supported by Fondation Mathématique Jacques Hadamard. Work on this paper was undertaken during A.M.’s tenure as INI-Simons Post Doctoral Research Fellow hosted by the Isaac Newton Institute for Mathematical Sciences (INI) participating in programme Frontiers in Kinetic Theory, and by the Department of Pure Mathematics and Mathematical Statistics (DPMMS) at the University of Cambridge. This author would like to thank INI and DPMMS for support and hospitality during this fellowship, which was supported by Simons Foundation (award ID 316017) and by Engineering and Physical Sciences Research Council (EPSRC) Grant Number EP/R014604/1.

Data Availability Statement

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Keywords

Blow-up criteria for SPDE
Keller–Segel equations of chemotaxis
SPDE with conservative noise

ASJC Scopus subject areas

Mathematics (miscellaneous)

Access to Document

10.1007/s00028-023-00900-3Licence: CC BY

Cite this

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note = "Funding Information: The authors warmly thank J. Norris, A. de Bouard, and L. Galeati for helpful discussions and B. Hambly for useful comments on the manuscript. The authors would like to express their gratitude to the French Centre National de Recherche Scientifique (CNRS) for the grant (PEPS JCJC) that supported this project. M.T. was partly supported by Fondation Math{\'e}matique Jacques Hadamard. Work on this paper was undertaken during A.M.{\textquoteright}s tenure as INI-Simons Post Doctoral Research Fellow hosted by the Isaac Newton Institute for Mathematical Sciences (INI) participating in programme Frontiers in Kinetic Theory, and by the Department of Pure Mathematics and Mathematical Statistics (DPMMS) at the University of Cambridge. This author would like to thank INI and DPMMS for support and hospitality during this fellowship, which was supported by Simons Foundation (award ID 316017) and by Engineering and Physical Sciences Research Council (EPSRC) Grant Number EP/R014604/1. ",

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