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A reducing mechanism on wave speed for chemotaxis systems with degenerate diffusion

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Abstract

This paper is concerned with traveling wave solutions for a chemotaxis model with degenerate diffusion of porous medium type. We establish the existence of semi-finite traveling waves, including the sharp type and \(C^1\) type semi-finite waves. Our results indicate that chemotaxis slows down the wave speed of semi-finite traveling wave, that is, the traveling wave speed for chemotaxis with porous medium (degenerate) diffusion is smaller than that for the porous medium equation without chemotaxis. As we know, this is a new result not shown in the existing literature. The result appears to be a little surprising since chemotaxis is a convective force. We prove our results by the Schauder’s fixed point theorem and estimate the wave speed by a variational approach.

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Acknowledgements

The research of S. Ji is supported by Guangdong Basic and Applied Basic Research Foundation Grant No. 2021A1515010367. The research of Z.A. Wang is supported by Hong Kong RGC GRF grant PolyU 153055/18P (Project ID P0005472). The research of T. Xu is supported by China Postdoctoral Science Foundation No. 2021M691070. The research of J. Yin was supported by NSFC Grant Nos. 11771156 & 12026220, Guangdong Basic and Applied Basic Research Foundation Grant No. 2020B1515310013, Science and Technology Program of Guangzhou No. 2019050001, and NSF of Guangzhou Grant No. 201804010391.

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Authors and Affiliations

  1. School of Mathematics, South China University of Technology, Guangzhou, 510641, Guangdong, China

    Shanming Ji & Tianyuan Xu

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

    Zhi-An Wang

  3. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, Guangdong, China

    Jingxue Yin

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  1. Shanming Ji
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  2. Zhi-An Wang
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  4. Jingxue Yin
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Correspondence to Tianyuan Xu.

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Communicated by M. Del Pino.

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Ji, S., Wang, ZA., Xu, T. et al. A reducing mechanism on wave speed for chemotaxis systems with degenerate diffusion. Calc. Var. 60, 178 (2021). https://doi.org/10.1007/s00526-021-01990-y

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