Existence and uniqueness of solution to a hyperbolic-parabolic free boundary problem for biofilm growth

Biofilms are defined as a community of microorganisms embedded in a matrix of extracellular polymeric substances [1]. The experimental study of biofilms is relatively complex and expensive. Yet, a good understanding of the metabolic and chemical mechanisms taking place in and around the biofilm is important in several fields. In the medical field, for example, this understanding can facilitate the development of high-powered antibiotics since biofilms are known to be resistant to antibiotics. In the environmental field, biofilms are very important in wastewater treatment as they are responsible for various metabolic processes participating in waste degradation.
For several decades mathematical models have been proposed to contribute to the understanding of biofilms, evolving from 1D to 2D to 3D. Early models treated the biofilm as a continuum in one dimension and are exemplified by the founding stone biofilm model introduced by Wanner and Gujer [2]. Such model is based on a free boundary problem comprising a system of nonlinear hyperbolic systems which accounts for the biofilm growth, a system of diffusion-reaction equations for the substrates diffusion and conversion within the biofilm, and an ordinary differential equation that accounts for the biofilm thickness evolution. Although more sophisticated 2D and 3D models have been designed starting from the late 1990s with the aim to catch the highly complex heterogeneous 3D biofilm structure, these models have not found applications in engineering practice due to their complexity, and instead, there has been a pull-back towards 1D biofilm models for engineering applications [3].
In line with this trend, 1D Wanner-Gujer based models have been formulated and advocated for modelling biofilms in engineering systems. The phenomena modelled include, but are not limited to, biosorption [4], precipitation [5], invasion and attachment of new species [6], the role and interactions between the suspended microbial populations and the biofilm [7, 8]. In most of the models in literature, the forcing terms for the hyperbolic equations are modeled according to Monod kinetics which account for the microbial growth as a nonlinear function of the available nutrients. In the diffusion equations, these terms express the uptake or production of substrates by microbial activity. Due to the complexity of biofilm models, both with respect to mathematical structure and number of parameters, many works are limited to numerical simulations without ever considering the mathematical justification of the models or a rigorous analysis of its dynamics [9, 10, 7]. To cover this lack, recent works have been dedicated to the qualitative analysis of biofilm models of this type. Some of them focus on existence results for simplified models [11, 12]; others consider the pseudo stationarity of the diffusion equations due to time-scale arguments [13, 14].
In this work we study the existence and uniqueness of the complete 1D biofilm model as originally formulated in [2]. This study can be adapted to any kind of 1D biofilm from sorption to precipitation models. To achieve this, we use the method of characteristics and transpose the partial differential equations into integral systems. Then we state and prove an existence and uniqueness theorem of the solution to the problem in small and large time. Next, we briefly consider the case of the parabolic equation with a variable diffusion coefficient using the single layer potential.
The rest of this manuscript is organized as follows: in section 2, we present the model subject to this study, section 3 is devoted to the reduction of the PDEs to integral equations and to the existence theorem in small times. The extension of this result in large time is presented in section 4 while the case of parabolic equations with variable diffusivity case is discussed in section 6. We consider the Neumann-Robin boundary conditions in section 5 and conclude the paper and propose future recommendations in section 7.

The biofilm model considered in this work is formulated as a free boundary problem made of a system of nonlinear hyperbolic PDEs, a system of quasi-linear parabolic PDEs, a PDE for the velocity growth of the microbial mass, and the free boundary is given by an ODE modelling the evolution of the biofilm thickness over time.
In the sequel, we will denote by

• 1.

  $X_{i}(z,t)=\rho_{i}f_{i}(z,t)$ the concentration of biomass $i,\ i\in\{1,..,n\},\ \boldsymbol{X}=(X_{1},...,X_{n}),$ where $f_{i}$ are the volume fractions occupied by the biomasses in the biofilm and $\rho_{i}$ the density of the biomass, assumed constant for simplicity. In addition, the volume fractions $f_{i}$ are supposed to satisfy the incompressibility constraint

  $$\sum_{i=1}^{n}f_{i}=1;$$

  (1)

• 2.

  $C_{j}(z,t),\ j\in\{1,..,m\},\ \boldsymbol{C}=(C_{1},...,C_{m}),$ the concentration of nutrients (dissolved components) in the biofilm;

• 3.

  $u(z,t)$ the convective growth velocity of the biofilm;

• 4.

  $r_{M,i}(\boldsymbol{X},\boldsymbol{C})$ the reaction term in the hyperbolic equations which accounts for the biomass growth and decay rate;

• 5.

  $r_{C,j}(\boldsymbol{X},\boldsymbol{C})$ the reaction term of the parabolic equations which accounts for the impact of the consumption/production of the substrate (nutrient) $j$ by the biomass;

• 6.

  $L(t)$ the thickness of the biofilm, which is the unknown boundary (free boundary) of the mathematical problem, and $L_{0}$ the initial biofilm thickness.

The complete model considered can be stated as follows (details on the derivation and some applications can be found in [2]):

with the following initial condition:

By writing equations (2) in terms of volume fractions $f_{i}(z,t)$, summing over $i$ and using the constraint (1), we get the velocity equation below:

with boundary condition reflecting the zero biomass flux at the substratum ($z=0$)

The equation for the thickness is given by:

where $\sigma(t)$ is the attachment and detachment function which can assume positive or negative values. In last position we have the system of diffusion equations for the substrates, with initial and boundary conditions:

The diffusion system in a planar biofilm is now given by 7. We make a change of variable by setting $S_{j}=D_{j}C_{j}$, and after substituting in (7) we get the following equation:

where $F_{j}(\boldsymbol{X},\boldsymbol{S})=\hat{F}_{j}(\boldsymbol{X},\boldsymbol{C},D),~{}~{}\varphi_{j}(z)=D_{j}\varphi_{j}^{*}(z)$ and $\psi_{j}(t)=D_{j}\psi_{j}^{*}(t).$ In the sequel, we will also use $\tilde{H}_{i}(\boldsymbol{X},\boldsymbol{S})=\hat{H}_{i}(\boldsymbol{X},\boldsymbol{C},D),$ and $R(\boldsymbol{X},\boldsymbol{S})=\hat{R}(\boldsymbol{X},\boldsymbol{C},D)$ and the notation $S_{jx}$ will stand for $\frac{\partial S_{j}}{\partial x},$ with $x$ any variable when no confusion is possible.
Consider the following kernel of the heat equation [15, 16, 17] :

Define the following Green and Neumann functions in the half plane $z>0$ respectively given by:

To derive the solution of (8), we integrate the following Green’s identity on $0<\xi<L(t)$, $0<\tau<t$:

After introducing the boundary and initial conditions, we get:

where

and in equation (12), we have to determine $\Phi_{j}$ and $\theta_{j}$. By letting $z\rightarrow 0$ in equation (12) we get:

Next, we use the following property of the function K which is given in Theorem 1 of Chapter 5, page 137 of [15] and in Lemma 4.2.1, page 50 of [16] :

where $g$ is a continuous function. Taking the derivative of $S_{j}$ over $z$ in the domain $0<\xi<L(\tau)$, $0<\tau<t-\epsilon$ and letting $\epsilon\rightarrow 0$, and using relation 16 (or equivalently if we let $z\rightarrow L(t)$), we get:

where

In this way we have just proved that for any solution $S_{j}$ of (8) for all $t<\lambda$, $\theta_{j}$ (defined by (13)) has to satisfy the integral equation (17), where $L(t)$ is given by (19) (for $\sigma=0$) below:

Conversely, we consider that for some $\lambda>0,$ $\theta_{j}(t)$ is a continuous solution of the integral equation (17) for $0\leq t<\lambda$ and then we prove that $S_{j}(z,t)$ form a solution of equations (8) for $L(t)>0$ for all $t<\lambda$.
We can easily check that $S_{j}(z,t)$ satisfies (8a), (8b), and (8d). To verify that it satisfies the boundary condition (8c), we integrate Green’s identity 12 as before and we get:

where $\rho_{j}(t)=S_{jz}(0,t).$ Equation (20) is the same as (12) if the third integral of the right hand side is null. Since $G(z,t,0,\tau)$ vanishes, the third integral is zero. So, the condition is fulfilled. However, we still need to prove that $\rho_{j}(t)\equiv 0$ so that the boundary condition is satisfied. We thus have

for $0<z<L(t)$, $0<t<\lambda.$
We recall that

Hence, for any continuous function $h$

Since $\rho_{j}$ is continuous and bounded, we can write

and in particular that

Using the continuity of $\rho_{j}$, we get

with $C=C(t).$
Using Grönwall’s lemma, we conclude that $\rho(t)=0,~{}\forall t$ and thus, the boundary condition (8c) is satisfied.
In summary, we have proved the following theorem:

Next, we will derive integral equations for the hyperbolic equations rewritten here in compact form:

with the following initial condition:

Let us denote the characteristics of this system by $\eta(z_{0},t)$ so that they satisfy the following initial value problem:

Considering equation (6) for $\sigma(t)=0$, it follows immediately that $L(t)=\eta(L_{0},t).$ Thus, to get the integral equation for (22)–(23), we will use the characteristics coordinates $z=\eta(z_{0},t)$ that will be noted by $\eta$ for simplicity:

with $H_{i}(\boldsymbol{X},\boldsymbol{S})=\tilde{H}_{i}(\boldsymbol{X},\boldsymbol{S})-X_{i}R(\boldsymbol{X},\boldsymbol{S}).$ The integral equation for the free boundary is given by:

Finally, we convert equations (12) and (17) into characteristics coordinates. When $z=\eta(z_{0},t),$ $L(t)=\eta(L_{0},t)$, $L(\tau)=\eta(L_{0},\tau)$ and $\xi=\eta(\xi_{0},t)$ (to simplify the writing, we will drop the variable in brackets and write $\eta$ and $\tilde{\eta}$ for $\eta(z_{0},t)$ and $\eta(\xi_{0},t)$ respectively when no confusion is possible), we have:

and the equation for $\theta_{j}(t)$ is

where $\boldsymbol{X}^{0}$ and $\boldsymbol{S}^{0}$ denote respectively $\boldsymbol{X}(0,t)$ and $\boldsymbol{S}^{0}=(\Phi_{1}(t),...,\Phi_{m}(t))$. Moreover, $U_{j}(t)$ in characteristic coordinates assumes the following form:

Before we prove the existence and uniqueness theorem, we need the following lemmas to use the Banach-Caccioppoli fixed point theorem.
We consider the space of continuous functions

where $\|.\|_{\Omega}$ is given by

We choose the vector $\mathcal{U}$ such that $\mathcal{U}=\begin{pmatrix}u_{1}(\eta,t)\\ u_{2}(\eta,t)\\ u_{3}(\eta,t)\\ u_{4}(\eta,t)\end{pmatrix}=\begin{pmatrix}u(\eta,t)\\ X_{i}(\eta,t)\\ \eta(z_{0},t)\\ \theta_{j}(\eta,t)\end{pmatrix}$ and we define the map $\mathcal{T}:C_{\lambda,M}\rightarrow C_{\lambda,M}$ such that $\mathcal{T}(\mathcal{U}(\eta,t))=\begin{pmatrix}\mathcal{T}_{1}\left(\mathcal{U}(\eta,t)\right)\\ \mathcal{T}_{2}\left(\mathcal{U}(\eta,t)\right)\\ \mathcal{T}_{3}\left(\mathcal{U}(\eta,t)\right)\\ \mathcal{T}_{4}\left(\mathcal{U}(\eta,t)\right)\\ \end{pmatrix}$

where $\mathcal{T}_{i}\left(\mathcal{U}(\eta,t)\right)$ ($i=1,\cdots,4$) are respectively given by

We shall prove the properties of $\mathcal{T}$. We first need the following lemma whose proof is immediate.

P1. $\mathcal{T}$ maps $C_{\lambda,M}$ into itself. In what follows, we will mainly focus on the fourth component of $\mathcal{T}(\mathcal{U}(\eta,t)).$ The proof for the other components is similar to the one proposed in [14], so we skip it.

Proof: To prove (35), we choose $a$ such that $0<a<x<\lambda.$
We have

By using this last inequality and the definition of $U_{j},$ the inequality (35) follows easily. The inequality 36 is immediate and can be found in [18], same for 38.
To prove 37, we use the definition 11 and the inequality holds. The inequality 39 can also be proven in the same way. Finally, inequality 40 holds because $F_{j}$ is Lipschitz and $N(x,t,\xi,\tau)\leq\frac{1}{\sqrt{\pi(t-\tau)}}.$
To prove inequality 41 we use the following exponential inequality [18, 19]:

We have