Stabilization in a chemotaxis-consumption model involving Robin-type boundary conditions

The chemotaxis-consumption model {u(t) = del . ((u+ 1)(alpha)del u) - del . (u(u + 1)(beta-1)del v), x is an element of Omega, t > 0 0 = Delta v - uv, x is an element of Omega, t > 0 is considered in a smooth bounded domain Omega subset of R-n under the boundary conditions (u + 1)(alpha)partial derivative(nu)u = u(u + 1)(beta-1)partial derivative V-nu and partial derivative V-nu = (gamma - v)g on partial derivative Omega, with parameters alpha, beta is an element of R, gamma > 0 and the nonnegative function g is an element of C1+ omega((Omega) over bar) for some omega is an element of(0, 1). If beta - alpha <= 1, it is proved that there exists a global bounded classical solution (u, v) for suitably regular initial data u(0). Furthermore, for the given mass m = integral(Omega)u(infinity) > 0, the corresponding stationary system admits a unique classical solution (u(infinity), v(infinity)), appearing as a large-time limit in the sense that if gamma > 0 is small enough, then |u(., t) - u(infinity)||L infinity(Omega) + ||v (., t) - v(infinity)||L-infinity ((Omega)) -> 0 as t -> infinity , with integral(Omega)u(infinity) = integral(Omega)u(0)= m > 0. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.