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Addendum to the paper "Refined criteria toward boundedness in an attraction-repulsion chemotaxis system with nonlinear productions"

Silvia Frassu, Giuseppe Viglialoro

Abstract

These notes aim to provide a deeper insight on the specifics of the paper "Refined criteria toward boundedness in an attraction-repulsion chemotaxis system with nonlinear productions" by A. Columbu, S. Frassu and G. Viglialoro [\textit{Appl. Anal.} 2024, 103:2, 415--431].

Addendum to the paper "Refined criteria toward boundedness in an attraction-repulsion chemotaxis system with nonlinear productions"

Abstract

These notes aim to provide a deeper insight on the specifics of the paper "Refined criteria toward boundedness in an attraction-repulsion chemotaxis system with nonlinear productions" by A. Columbu, S. Frassu and G. Viglialoro [\textit{Appl. Anal.} 2024, 103:2, 415--431].
Paper Structure (9 sections, 8 theorems, 32 equations)

This paper contains 9 sections, 8 theorems, 32 equations.

Table of Contents

  1. Aim of the paper
  2. Presentation of the main theorem
  3. Existence of local-in-time solutions and a pricniple for boundedness
  4. Local existence statement
  5. Boundedness criterion
  6. A priori bounds; proof of the main result
  7. Some preparatory tools
  8. Achieving the boundedness criterion
  9. Proof of Theorem \ref{['TheoMain1']}

Key Result

Proposition 1

For $n\in\mathbb N$, let $\Omega\subset \mathbb R^n$ be a bounded domain with smooth boundary, $\rho>0$ and $q>\max\{1,\frac{1}{\rho}\}$. Then there is $C_\rho=C_\rho(\Omega,n,q)>0$ such that the following holds: Whenever $T\in(0,\infty]$, $I=[0,T)$, $h\in L^q(I;L^q(\Omega))$ and $\psi_0\in W^{2,q}_ satisfies

Theorems & Definitions (18)

  • Remark 1: On the origins and the meaning of model \ref{['problem']}
  • Proposition 1
  • proof
  • Remark 2: On the constant $C_\rho$ and the norm $\lVert \psi_0 \rVert_{q,1-\frac{1}{q}}$ in Proposition \ref{['PropositionConstantC']}
  • Theorem 2.1
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • ...and 8 more