Global existence of solutions to the chemotaxis system with logistic source under nonlinear Neumann boundary conditions

We consider classical solutions to the chemotaxis system with logistic source, au−μu2, under nonlinear Neumann boundary conditions ∂u∂ν=|u|p with p>1 in a smooth convex bounded domain Ω⊂Rn, where n≥2. This paper aims to show that if p<32, and μ>0, n=2, or μ is sufficiently large when n≥3, then the parabolic-elliptic chemotaxis system admits a unique nonnegative global-in-time classical solution that is bounded in Ω×(0,∞). The similar result is also true if p<32, n=2, and μ>0 or p<75, n=3, and μ is sufficiently large for the parabolic-parabolic chemotaxis system.