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Existence results for a biofilm free-boundary problem with dominant detachment

Dieudonné Zirhumananana

TL;DR

The paper studies existence and uniqueness for a Wanner–Gujer free‑boundary model of biofilms with dominant detachment. It develops a fixed-domain reformulation and uses maximal $L^p$ regularity, semigroup theory, and a contraction mapping to prove local well‑posedness and continuous dependence. It obtains global existence via invariance-region arguments and energy estimates, showing that detachment dissipation prevents blow‑up and keeps the free boundary bounded. The results bridge applied biofilm modeling with rigorous PDE analysis and provide reusable lemmas for related transport–diffusion systems.

Abstract

This work addresses the existence and uniqueness of a Wanner-Gujer free-boundary problem that models biofilms under conditions of prevailing detachment. This result significantly extends previous findings in both tumor growth modeling and the biofilm modeling field. Besides establishing local existence and uniqueness, we also prove the continuous dependence of the solution on initial and boundary data. Furthermore, global existence is deduced using a combination of invariance regions and energy estimates. The proof for local existence is obtained by utilizing fixed point arguments combined with semigroup theory.

Existence results for a biofilm free-boundary problem with dominant detachment

TL;DR

The paper studies existence and uniqueness for a Wanner–Gujer free‑boundary model of biofilms with dominant detachment. It develops a fixed-domain reformulation and uses maximal regularity, semigroup theory, and a contraction mapping to prove local well‑posedness and continuous dependence. It obtains global existence via invariance-region arguments and energy estimates, showing that detachment dissipation prevents blow‑up and keeps the free boundary bounded. The results bridge applied biofilm modeling with rigorous PDE analysis and provide reusable lemmas for related transport–diffusion systems.

Abstract

This work addresses the existence and uniqueness of a Wanner-Gujer free-boundary problem that models biofilms under conditions of prevailing detachment. This result significantly extends previous findings in both tumor growth modeling and the biofilm modeling field. Besides establishing local existence and uniqueness, we also prove the continuous dependence of the solution on initial and boundary data. Furthermore, global existence is deduced using a combination of invariance regions and energy estimates. The proof for local existence is obtained by utilizing fixed point arguments combined with semigroup theory.

Paper Structure

This paper contains 5 sections, 4 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Notations and reformualtion of the problem
  3. Primary lemmas and local existence
  4. Global existence via invariance regions and energy estimates
  5. Statements and Declarations

Key Result

Lemma 3.1

Under the assumptions (i)--(vi) there exists $T^* \in (0, T_1]$ and a unique solution $\boldsymbol{C} \in W^{2,1}_p(\Omega \times (0,T^*))^m$ to eq:main-problem-final. Moreover the following estimate holds where $M > 0$ depends on $p, T^*, \|\boldsymbol{B}\|_{C^\alpha([0,T^*])}$, and $L_H$.

Theorems & Definitions (4)

  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Theorem 4.1