A new result for global existence and boundedness of solutions to a parabolic--parabolic Keller--Segel system with logistic source

2017/12/04 by Jiashan Zheng, Yanyan Li, Zheng, Jiashan +1 · 2 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · #Analysis of PDEs (math.AP) #Cellular Mechanics and Interactions #FOS: Mathematics #Gene Regulatory Network Analysis #Mathematical Biology Tumor Growth

paper · pdf · doi:10.48550/arxiv.1712.00906

openalex publication_date 2017/12/04 · openalex created_date 2017/12/22 · openalex updated_date 2026/07/28

Abstract

We consider the following fully parabolic Keller--Segel system with logistic source \u_t=Δu-χ∇⋅(u∇ v)+ au-μu², x∈ Ω, t · gt;0, \dispv_t=Δv- v +u, x∈ Ω, t · gt;0,.\eqno(KS) over a bounded domain Ω⊂ℝ^N(N≥1), with smooth boundary ∂Ω, the parameters a∈ ℝ, μ>0, χ>0. It is proved that if μ>0, then (KS) admits a global weak solution, while if μ>\frac(N-2)₊NχC^{(1)/((N)/(2)+1)}_(N)/(2)+1, then (KS) possesses a global classical solution which is bounded, where C^\frac1\fracN 2+1_(N)/(2)+1 is a positive constant which is corresponding to the maximal Sobolev regularity. Apart from this, we also show that if a = 0 and μ>\frac(N-2)₊NχC^{(1)/((N)/(2)+1)}_(N)/(2)+1, then both u(⋅, t) and v(⋅, t) decay to zero with respect to the norm in L^∞(Ω) as t→∞.

Citations

• Model for chemotaxis

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