## Abstract

We consider the initial-boundary value problem of a system of reaction-diffusion equations with density-dependent motility [Formula presented] in a bounded domain Ω⊂R^{2} with smooth boundary, α and θ are non-negative constants and ν denotes the outward normal vector of ∂Ω. The random motility function γ(v) and functional response function F(w) satisfy the following assumptions: • γ(v)∈C^{3}([0,∞)),0<γ_{1}≤γ(v)≤γ_{2},|γ^{′}(v)|≤η for all v≥0; • F(w)∈C^{1}([0,∞)),F(0)=0,F(w)>0in(0,∞)andF^{′}(w)>0on[0,∞) for some positive constants γ_{1},γ_{2} and η. Based on the method of energy estimates and Moser iteration, we prove that the problem (⁎) has a unique classical global solution uniformly bounded in time. Furthermore we show that if θ>0, the solution (u,v,w) will converge to (0,0,w_{⁎}) in L^{∞} with some w_{⁎}>0 as time tends to infinity, while if θ=0, the solution (u,v,w) will asymptotically converge to (u_{⁎},u_{⁎},0) in L^{∞} with [Formula presented] if D>0 is suitably large.

Original language | English |
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Pages (from-to) | 6758-6793 |

Number of pages | 36 |

Journal | Journal of Differential Equations |

Volume | 269 |

Issue number | 9 |

DOIs | |

Publication status | Published - 15 Oct 2020 |

## Keywords

- Asymptotic stability
- Density-dependent Motility
- Global existence

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics