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 - mu u(2), under nonlinear Neumann boundary conditions partial derivative u/partial derivative v = |u|(p) with p > 1 in a smooth convex bounded domain Omega subset of R-n, where n >= 2. This paper aims to show that if p < 3/2, and mu > 0, n = 2, or mu 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 Omega x (0, infinity). The similar result is also true if p < 3/2, n = 2, and mu > 0 or p < 7/5, n = 3, and mu is sufficiently large for the parabolic-parabolic chemotaxis system.(c) 2023 Elsevier Inc. All rights reserved.