Global boundedness of solutions to a quasilinear parabolic–parabolic Keller–Segel system with logistic source. (February 2016)

• Record Type: Journal Article
• Authors: Zhang, Yinle; Zheng, Sining
• Is Part Of: Applied mathematics letters. Volume 52(2016:Feb.)
• Page Start: 15
• Page End: 20
• Publication Date: 2016-02
• Subjects: Keller–Segel system -- Chemotaxis -- Logistic source -- Global boundedness; Applied mathematics -- Periodicals; 519.05
• DOI: 10.1016/j.aml.2015.08.009
• Languages: English
• ISSNs: 0893-9659

Abstract

Abstract: We consider a quasilinear parabolic–parabolic Keller–Segel system with a logistic type source u t = ∇ ⋅ ( ϕ ( u ) ∇ u ) − ∇ ⋅ ( ψ ( u ) ∇ v ) + g ( u ), v t = Δ v − v + u in a smooth bounded domain Ω ⊂ R n, n ≥ 1, subject to nonnegative initial data and homogeneous Neumann boundary conditions, where ϕ, ψ and g are smooth positive functions satisfying c 1 s p ≤ ϕ ( s ) and c 1 s q ≤ ψ ( s ) ≤ c 2 s q for p, q ∈ R and s ≥ s 0 > 1, g ( s ) ≤ a s − μ s k for s > 0, with constants a ≥ 0, μ, c 1, c 2 > 0, and the extended logistic exponent k > 1 instead of the ordinary k = 2 . It is proved that if q < k − 1, or q = k − 1 with μ properly large that μ > μ 0 for some μ 0 > 0, then there exists a classical solution which is global in time and bounded. This shows the exact way of the logistic exponent k > 1 effecting the behavior of solutions.