Identification of a chemotactic sensitivity in a coupled system

Abstract

Chemotaxis is the process by which cells behave in a way that follows the chemical gradient. Applications to bacteria growth, tissue inflammation, and vascular tumors provide a focus on optimization strategies. Experiments can characterize the form of possible chemotactic sensitivities. This paper addresses the recovery of the chemotactic sensitivity from these experiments while allowing for nonlinear dependence of the parameter on the state variables. The existence of solutions to the forward problem is analyzed. The identification of a chemotactic parameter is determined by inverse problem techniques. Tikhonov regularization is investigated and appropriate convergence results are obtained. Numerical results of concentration dependent chemotactic terms are explored.

Identification of Chemotaxis Models with Volume-Filling

“…Since V is open and ρ is continuous, there exists δ > 0 such that the ball B δ (x,t) ⊂ V and ρ(x, t) ∈ (ρ−ε,ρ+ε) for all (x, t) ∈ B δ (x,t). From this and (12) we conclude that d(t) = d(t) for all |t −t| < δ. Using a continuation argument, we obtain that d(t) = d(t) for all t ∈ (t 0 , t 1 ), which was to be shown.…”

“…with some continuous function d depending only on t. We will show below, that d is in fact constant on V , which by (12) and the fact that ρ attains all possible values on V yields the assertion of the theorem.…”

“…To the best of our knowledge, only few results on inverse problems in chemotaxis are available to date. In [12] the case f = f (ρ, c) = ρ f (c) is considered where the function f is to be identified. The special structure of the cross-diffusion term div(ρ f (c)∇c) is an important ingredient for the analysis in [12], and we need different techniques to prove uniqueness for the inverse problems (i) and (ii) here.…”

“…In [12] the case f = f (ρ, c) = ρ f (c) is considered where the function f is to be identified. The special structure of the cross-diffusion term div(ρ f (c)∇c) is an important ingredient for the analysis in [12], and we need different techniques to prove uniqueness for the inverse problems (i) and (ii) here. After establishing the identifiability, we also discuss the possibility to reconstruct the parameters by numerical methods.…”

On uniqueness and stable estimation of multiple parameters in the Cahn–Hilliard equation

“…The simultaneous identification of multiple parameters in nonlinear elliptic and parabolic problems has been investigated, for instance, in [7,10,16,23,39], and uniqueness questions as well as numerical methods for the stable solution have been studied. Related results have also been derived in the context of chemotaxis [17,24]. The recent work [32,33] addresses the identification of multiple scalar parameters in a phase-field model for tumour growth, which is an extended version of the Cahn-Hilliard system above.…”