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

TL;DR

This paper proves the existence and uniqueness of solutions for a complex free boundary problem modeling biofilm growth, involving hyperbolic and parabolic PDEs with evolving boundary conditions.

Contribution

It introduces a novel mathematical framework combining characteristics and fixed point methods to analyze biofilm models with free boundaries.

Findings

Existence and uniqueness of solutions established for the biofilm model.

Transformation of PDEs into integral equations using Green's functions.

Applicability of boundary conditions to variable boundary scenarios.

Abstract

This work presents the existence and uniqueness of solution to a free boundary value problem related to biofilm growth. The problem consists of a system of nonlinear hyperbolic partial differential equations governing the microbial species growth, and a system of parabolic partial differential equations describing the substrate dynamics. The free boundary evolution is governed by an ordinary differential equation that accounts for the thickness of the biofilm. We use the method of characteristics and fixed point strategies to prove the existence and uniqueness theorem in small and all times. All the equations are converted into integral equations, in particular this transformation is made for the parabolic equations by using the Green's functions. We consider Dirichlet-Neumann and Neumann-Robin boundary conditions for the substrates equations and their extension to the case with variable