On the well‐posedness of mathematical models for multicomponent biofilms

Abstract

Abstract. Bacterial biofilms are microbial depositions on immersed surfaces. Their mathematical description leads to degenerate diffusion-reaction equations with two non-Fickian effects:(i) a porous medium equation like degeneracy where the biomass density vanishes, and (ii) a super-diffusion singularity if the biomass density reaches its threshold density. In the case of multi-species interactions, several such equations are coupled, both in the reaction terms and in the nonlinear diffusion operator. In this paper we generalize previous work on existence and uniqueness of solutions of this type of models and give a general, relatively easy to apply criterion for well-posedness. The use of the criterion is illustrated in several examples from the biofilm modeling literature.

Mathematical analysis of a quorum sensing induced biofilm dispersal model and numerical simulation of hollowing effects

“…We consider smooth non-degenerate approximations for system (4) and show that their solutions converge to the solution of the degenerate problem (4). The ideas are based on the proof developed for scalar degenerate reaction-diffusion equations of porous medium type in [2], the solution theory in [17] for the single species biofilm model and the ideas applied in [10,24,38] and [39] for multi-species biofilm models. For small ε > 0, we define…”

Well-posedness of singular-degenerate porous medium type equations and application to biofilm models

“…holds for almost every τ 1 , τ 2 ∈ (0, T ) and hence, the map τ → ´Ω Ψ ⋆ (u(τ ), ū) can be represented by an absolutely continuous function and its derivative is given by (21).…”

“…which is justified by Remark 3.13. Moreover, we can rewrite the term involving the time derivative using (21) with ū(x) = ũ(x, t 2 ), which implies that…”

“…Similarly, for fixed t 1 we apply (7) for the supersolution ũ and the non-positive test function η(x, t 2 ) = ψ − δ ( w(x, t 2 ) − w(x, t 1 )) . Moreover, using (21) with ū(x) = u(x, t 1 ), it follows that…”

“…The biofilm growth model [5] has been studied in a series of papers, mainly in simulation studies, and has been further extended to take additional biofilm processes into account, see e.g. [6], [9], [17], [21]. This led to more involved, strongly coupled systems involving several dissolved substrates and multiple types of biomass.…”