Stationary solutions to a chemotaxis-consumption model with realistic boundary conditions for the oxygen

Previous studies of chemotaxis models with consumption of the chemoattractant (with or without fluid) have not been successful in explaining pattern formation even in the simplest form of concentration near the boundary, which had been experimentally observed. Following the suggestions that the main reason for that is the usage of inappropriate boundary conditions, in this paper we study the solutions to the stationary chemotaxis system {0 = Delta n - del . (n del c), 0 = Delta c - nc in bounded domains Omega subset of R-N, N >= 1, under the no-flux boundary conditions for n and the physically meaningful condition partial derivative(nu)c = (gamma - c)g on c, with the given parameter gamma > 0 and g is an element of C1+beta*(Omega), beta(*) is an element of (0, 1), satisfying g >= 0, g not equivalent to 0 on partial derivative Omega. We prove the existence and uniqueness of solutions for any given mass integral(Omega) n > 0. These solutions are nonconstant.