Approximation Analysis of a Parabolic-Parabolic Chemotaxis Model with Logarithmic Nonlinearity

TL;DR

This paper proves the global existence and boundedness of solutions for a chemotaxis model with logarithmic and polynomial nonlinearities, extending understanding of such systems in mathematical biology.

Contribution

It establishes the global boundedness of solutions for a Keller-Segel chemotaxis system with specific nonlinearities, under broad initial conditions.

Findings

Global existence of solutions is proven.

Solutions remain uniformly bounded over time.

The model accommodates natural cell growth and decay.

Abstract

We consider the Keller-Segel system with logical source \begin{align*} \begin{cases} u_t = \nabla \cdot (\phi(u)\nabla u) - \nabla \cdot (\psi(u)\nabla v)+f(u), & x \in \Omega, \; t > 0, v_t = \Delta v - v + u, & x \in \Omega, \; t > 0, \end{cases} \end{align*} in a smooth bounded domain \(\Omega \subset \mathbb{R}^n\) with \(n \geq 2\), the Neumann initial-boundary value problem admits a globally defined, uniformly bounded classic solution for all sufficiently regular non-negative initial data \(u_0\) and \(v_0\). In the first equation, assume that \(\phi\) and \(\psi\) are dominated by a logarithmic function and a polynomial respectively. The logical source \(f\) representing the natural growth and decay of cells satisfies \(f \in W^{1,\infty}_{\mathrm{loc}}(\Omega)\) and \(f(0) \geq 0\). Then we will see that the unique solution \(u \in C^{2,1}((\overline{\Omega}) \times [0,T] )\) and \(v \in W^{1,q}([0,T] ; C^{2,1}(\overline{\Omega}))\).