Global Existence for the Cauchy Problem of the Parabolic–Parabolic–ODE Chemotaxis Model with Indirect Signal Production on the Plane

Liu, Qian; Li, Dan

doi:10.3390/math13162624

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Global Existence for the Cauchy Problem of the Parabolic–Parabolic–ODE Chemotaxis Model with Indirect Signal Production on the Plane

by

Qian Liu

¹

and

Dan Li

^2,*

¹

School of Science, Shaoyang University, Shaoyang 422000, China

²

School of Mathematical Sciences, Chengdu University of Technology, Chengdu 610059, China

^*

Author to whom correspondence should be addressed.

Mathematics 2025, 13(16), 2624; https://doi.org/10.3390/math13162624

Submission received: 23 July 2025 / Revised: 11 August 2025 / Accepted: 14 August 2025 / Published: 15 August 2025

(This article belongs to the Section E: Applied Mathematics)

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Abstract

This paper establishes the global existence of solutions to a chemotaxis system with indirect signal production in the whole two-dimensional space. This system exhibits a mass threshold phenomenon governed by a critical mass

m_{c} = 8 π δ

, where

δ

represents the decay rate of the static individuals. When the total initial mass

m = \int_{R^{2}} u_{0} d x < m_{c}

, all solutions exist globally and remain bounded. In the critical case of

m = m_{c}

, the global existence or finite-time blow-up may occur depending on the initial conditions. The critical mass obtained in the whole space coincides with that previously derived in radially symmetric bounded domains. A key novelty lies in extending the analysis to the full plane, where the absence of compactness is overcome by constructing a suitable Lyapunov functional and employing refined Trudinger–Moser-type inequalities.

Keywords: global existence; Cauchy problem; indirect signal production; critical mass

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MDPI and ACS Style

Liu, Q.; Li, D. Global Existence for the Cauchy Problem of the Parabolic–Parabolic–ODE Chemotaxis Model with Indirect Signal Production on the Plane. Mathematics 2025, 13, 2624. https://doi.org/10.3390/math13162624

AMA Style

Liu Q, Li D. Global Existence for the Cauchy Problem of the Parabolic–Parabolic–ODE Chemotaxis Model with Indirect Signal Production on the Plane. Mathematics. 2025; 13(16):2624. https://doi.org/10.3390/math13162624

Chicago/Turabian Style

Liu, Qian, and Dan Li. 2025. "Global Existence for the Cauchy Problem of the Parabolic–Parabolic–ODE Chemotaxis Model with Indirect Signal Production on the Plane" Mathematics 13, no. 16: 2624. https://doi.org/10.3390/math13162624

APA Style

Liu, Q., & Li, D. (2025). Global Existence for the Cauchy Problem of the Parabolic–Parabolic–ODE Chemotaxis Model with Indirect Signal Production on the Plane. Mathematics, 13(16), 2624. https://doi.org/10.3390/math13162624

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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Global Existence for the Cauchy Problem of the Parabolic–Parabolic–ODE Chemotaxis Model with Indirect Signal Production on the Plane

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