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, a u − μ u 2 , under nonlinear Neumann boundary conditions ∂ u ∂ ν = | u | p with p > 1 in a smooth convex bounded domain Ω ⊂ R n , where n ≥ 2. This paper aims to show that if p < 3 2 , 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 < 3 2 , n = 2 , and μ > 0 or p < 7 5 , n = 3 , and μ is sufficiently large for the parabolic-parabolic chemotaxis system. [ABSTRACT FROM AUTHOR]