Boundedness and asymptotics of a reaction-diffusion system with density-dependent motility

Hai Yang Jin, Shijie Shi, Zhi An Wang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

1 Citation (Scopus)

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⊂R2 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)∈C3([0,∞)),0<γ1≤γ(v)≤γ2,|γ(v)|≤η for all v≥0; • F(w)∈C1([0,∞)),F(0)=0,F(w)>0in(0,∞)andF(w)>0on[0,∞) for some positive constants γ12 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 languageEnglish
Pages (from-to)6758-6793
Number of pages36
JournalJournal of Differential Equations
Volume269
Issue number9
DOIs
Publication statusPublished - 15 Oct 2020

Keywords

  • Asymptotic stability
  • Density-dependent Motility
  • Global existence

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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