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

Shijun Li, Yashuang Zhao, Shaopeng Xu, Shengjun Li

Abstract

We consider the Keller-Segel system with logical source \begin{align*} \begin{cases} u_t = \nabla \cdot (φ(u)\nabla u) - \nabla \cdot (ψ(u)\nabla v)+f(u), & x \in Ω, \; t > 0, v_t = Δv - v + u, & x \in Ω, \; t > 0, \end{cases} \end{align*} in a smooth bounded domain \(Ω\subset \mathbb{R}^n\) 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 \(φ\) and \(ψ\) are dominated by a logarithmic function and a polynomial respectively. The logical source \(f\) representing the natural growth and decay of cells satisfies \(f \in W^{1,\infty}_{\mathrm{loc}}(Ω)\) and \(f(0) \geq 0\). Then we will see that the unique solution \(u \in C^{2,1}((\overlineΩ) \times [0,T] )\) and \(v \in W^{1,q}([0,T] ; C^{2,1}(\overlineΩ))\).

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

Abstract

We consider the Keller-Segel system with logical source \begin{align*} \begin{cases} u_t = \nabla \cdot (φ(u)\nabla u) - \nabla \cdot (ψ(u)\nabla v)+f(u), & x \in Ω, \; t > 0, v_t = Δv - v + u, & x \in Ω, \; t > 0, \end{cases} \end{align*} in a smooth bounded domain with , the Neumann initial-boundary value problem admits a globally defined, uniformly bounded classic solution for all sufficiently regular non-negative initial data and . In the first equation, assume that and are dominated by a logarithmic function and a polynomial respectively. The logical source representing the natural growth and decay of cells satisfies \(f \in W^{1,\infty}_{\mathrm{loc}}(Ω)\) and \(f(0) \geq 0\). Then we will see that the unique solution \(u \in C^{2,1}((\overlineΩ) \times [0,T] )\) and \(v \in W^{1,q}([0,T] ; C^{2,1}(\overlineΩ))\).
Paper Structure (5 sections, 8 theorems, 129 equations)

This paper contains 5 sections, 8 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Approximate Equation
  3. Blow-up
  4. Main Results
  5. Conclusion

Key Result

Theorem 1

Let the following assumptions hold: Then there exists a maximal existence time $T_{\max} \in (0,+\infty]$ and a unique pair of nonnegative functions $(u_{\varepsilon}, v_{\varepsilon})$ such that Furthermore, if $T_{\max} < \infty$, then

Theorems & Definitions (16)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 6 more