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Immediate regularization of measure-type population densities in a two-dimensional chemotaxis system with signal consumption

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Abstract

This paper deals with the Neumann initial-boundary value problem for a classical chemotaxis system with signal consumption in a disk. In contrast to previous studies which have established a comprehensive theory of global classical solutions for suitably regular nonnegative initial data, the focus in the present work is on the question to which extent initially prescribed singularities can be regularized despite the presence of the nonlinear cross-diffusive interaction. The main result in this paper asserts that at least in the framework of radial solutions immediate regularization occurs under an essentially optimal condition on the initial distribution of the population density. More precisely, it will turn out that for any radially symmetric initial data belonging to the space of regular signed Borel measures for the population density and to L2 for the signal density, there exists a classical solution to the Neumann initial-boundary value problem, which is smooth and approaches the given initial data in an appropriate trace sense.

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Acknowledgements

The first author was supported by the Applied Fundamental Research Program of Sichuan Province (Grant No. 2018JY0503) and Xihua University Scholars Training Program. The second author was supported by the Deutsche Forschungsgemeinschaft in the context of the project Emergence of Structures and Advantages in Cross-Diffusion Systems (Grant No. 411007140, GZ: WI 3707/5-1). The third author was supported by National Natural Science Foundation of China (Grant Nos. 11971093 and 11771045), the Applied Fundamental Research Program of Sichuan Province (Grant No. 20YYJC4388), and the Fundamental Research Funds for the Central Universities (Grant No. ZYGX2019J096). The authors are very grateful to the referees for their helpful suggestions. The third author expresses his hearty gratitude to Professor Tong Yang at the City University of Hong Kong for introducing the problem with measure valued initial data.

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Authors and Affiliations

  1. School of Science, Xihua University, Chengdu, 610039, China

    Yulan Wang

  2. Institut für Mathematik, Universität Paderborn, Paderborn, 33098, Germany

    Michael Winkler

  3. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China

    Zhaoyin Xiang

Authors
  1. Yulan Wang
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  2. Michael Winkler
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  3. Zhaoyin Xiang
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Correspondence to Zhaoyin Xiang.

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Wang, Y., Winkler, M. & Xiang, Z. Immediate regularization of measure-type population densities in a two-dimensional chemotaxis system with signal consumption. Sci. China Math. 64, 725–746 (2021). https://doi.org/10.1007/s11425-020-1708-0

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  • DOI: https://doi.org/10.1007/s11425-020-1708-0

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