Transition Layers to Chemotaxis-Consumption Models with Volume-Filling Effect (Journal of Mathematical Study, 2024)
We are interested in the dynamical behaviors of solutions to a parabolic-parabolic chemotaxis-consumption model with a volume-filling effect on a boundedinterval, where the physical no-flux boundary condition for the bacteria and mixedDirichlet-Neumann boundary condition for the oxygen are prescribed. By taking acontinuity argument, we first show that the model admits a unique nonconstant steadystate. Then we use Helly’s compactness theorem to show that the asymptotic profile ofsteady state is a transition layer as the chemotactic coefficient goes to infinity. Finally,based on the energy method along with a cancellation structure of the model, we showthat the steady state is nonlinearly stable under appropriate perturbations. Moreover,we do not need any assumption on the parameters in showing the stability of steadystate.
Keywords: Chemotaxis; volume-filling; transition layer; stability; physical boundary conditions.