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Global boundedness and asymptotic stability of the Keller-Segel system with logistic-type source in the whole space

Qingchun Li, Haomeng Chen

Abstract

In this paper, we investigate the Cauchy problem of the parabolic-parabolic Keller-Segel system with the logistic-type term $au-bu^γ$ on $\mathbb{R}^N, N\geq2$. We discuss the global boundedness of classical solutions with nonnegative bounded and uniformly continuous initial functions when $γ>1$. Moreover, based on the persistence of classical solution we show the large time behavior of the positive constant equilibria with strictly positive initial function in the case of $γ\in(1,2)$.

Global boundedness and asymptotic stability of the Keller-Segel system with logistic-type source in the whole space

Abstract

In this paper, we investigate the Cauchy problem of the parabolic-parabolic Keller-Segel system with the logistic-type term on . We discuss the global boundedness of classical solutions with nonnegative bounded and uniformly continuous initial functions when . Moreover, based on the persistence of classical solution we show the large time behavior of the positive constant equilibria with strictly positive initial function in the case of .

Paper Structure

This paper contains 5 sections, 7 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Statements of the main results
  4. Global boundedness of classical solutions
  5. Asymptotic stability of constant equilibria

Key Result

Lemma 2.1

(DH0AP) For every $t>0$, holds for all $u\in L^\infty(\mathbb{R}^N)$, where $C_N>0$ is a constant depending only on $N$.

Theorems & Definitions (9)

  • Lemma 2.1
  • Lemma 2.2
  • Proposition 3.1
  • Theorem 3.1
  • Proposition 3.2
  • Remark 3.1
  • Theorem 3.2
  • Definition 4.1
  • Lemma 5.1