Asymptotic behavior of global solutions of aerotaxis equations

Steady‐state solutions of the aerotaxis problem

“…We investigate steady states of this model. Results concerning one‐dimensional version of aerotaxis problem are contained in earlier research 3,7,8 . Let us point out that the model considered here can be seen as a fluidless generalization of the one proposed in Tuval et al 6 Global solvability of chemotaxis‐fluid model in bounded domains, with homogeneous Neumann boundary condition imposed on $$$ p $$$, was studied by Winkler 9 and with inhomogeneous Robin boundary condition by Braukhoff and Tang 10 .…”

“…Results concerning one-dimensional version of aerotaxis problem are contained in earlier research. 3,7,8 Let us point out that the model considered here can be seen as a fluidless generalization of the one proposed in Tuval et al 6 Global solvability of chemotaxis-fluid model in bounded domains, with homogeneous Neumann boundary condition imposed on p, was studied by Winkler 9 and with inhomogeneous Robin boundary condition by Braukhoff and Tang. 10 Results concerning global well-posedness of chemotaxis-fluid model with the Dirichlet boundary condition imposed on p can be found in papers by Black and Winkler, 11 Black and Wu, 12,13 and Wang et al 14,15 Tao and Winkler 16 considered the following system { u t = ∇ • (∇u − u∇p), p t = Δp − pu, (8) in bounded domains under no-flux boundary condition on u and p. They proved that problem (8) in two-dimensional domains admits a global classical solution, and in three-dimensional domains, it has weak solutions that eventually become smooth.…”

“…Stationary solutions of problem (8) with nonzero Robin boundary condition imposed on p were considered by Braukhoff and Lenkeit,. 17 They proved that the problem has a unique stationary solution which is nonconstant.…”

Depleting the signal: Analysis of chemotaxis‐consumption models—A survey

“…For some further system variants closely related to (), results on global solvability were obtained in Refs. 107, 146–151, and corresponding steady‐state analysis can be found in Refs. 152, 153.…”