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

Abstract

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 usage of inappropriate boundary conditions, in this article we study solutions to the stationary chemotaxis system 0 = ∆n − ∇ · (n∇c) 0 = ∆c − nc in bounded domains Ω ⊂ R N , N ≥ 1, under no-flux boundary conditions for n and the physically meaningful condition ∂ ν c = (γ − c)g on c, with given parameter γ > 0 and g ∈ C 1+β (Ω) satisfying g ≥ 0, g ≡ 0 on ∂Ω.We prove existence and uniqueness of solutions for any given mass Ω n > 0. These solutions are non-constant.

Prescribed signal concentration on the boundary: Weak solvability in a chemotaxis-Stokes system with proliferation

“…The author proves the existence of global classical solutions in 2D and global weak solutions in 3D. Additional results featuring a Robin boundary condition in fluidfree (i.e., u ≡ 0) variants of (1.1) have been investigated in [3] and [9]. The former considers a stationary (and hence doubly elliptic) system and establishes existence and uniqueness of a classical solution for any prescribed mass M := Ω n > 0.…”

“…Clearly, we can draw on previously established space-time bounds to extract additional space-time information on ∇ √ n ε from the previous Lemma, which in a second interpolation step can also be refined to a bound on ∇n ε in L 4 3 (Ω × (0, ∞)).…”

Radial solutions to a chemotaxis-consumption model involving prescribed signal concentrations on the boundary

“…Not least because of this, it has been suggested to use different, more realistic, inhomogeneous boundary conditions for the chemical signal (see [5] and [6], but also [29]), namely either Robin type boundary conditions where the rate of oxygen influx is controlled by the local oxygen concentration at the boundary or nonzero Dirichlet conditions directly prescribing the latter. (It has been confirmed [6,Prop. 5.1] that the latter kind of conditions arises as a limit case of the former.…”

“…Accordingly, only few results for chemotaxis-consumption models with boundary conditions different from homogeneous Neumann conditions are available: Those concerning the related system with slightly different chemotaxis and energy consumption studied in [16,17] with inhomogeneous Neumann and Dirichlet conditions are restricted to spatially one-dimensional domains. For Robin-type conditions of the form introduced in [5], also in higher dimensions, the stationary problem of (1.1) has been shown to be uniquely solvable (for any prescribed total mass Ω u of the first component, see [6]), and (under a moderate smallness condition) features as the limit of a parabolic-elliptic simplification of (1.1) (cf. [12]).…”

“…We have to leave open here the question how far information on the large time behaviour of the above solutions that goes beyond the boundedness features in (1.5) and in (1.7)-(1.9) can be derived, especially in the presence of large initial data. After all, a steady state analysis guided by the approach developed in [6] provides the following result which may be viewed as an indication for nontrivial dynamics involving structured states in (1.3):…”