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

Qian Liu and Dan Li ()
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Qian Liu: School of Science, Shaoyang University, Shaoyang 422000, China
Dan Li: School of Mathematical Sciences, Chengdu University of Technology, Chengdu 610059, China

Mathematics, 2025, vol. 13, issue 16, 1-13

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 = ∫ 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 (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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