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

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 mc=8πδ, where δ represents the decay rate of the static individuals. When the total initial mass m=∫R2u0dx<mc, all solutions exist globally and remain bounded. In the critical case of m=mc, 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.