2015/03/09 by Cibing Yang, Xinru Cao, Yang, Cibing +5
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · #Analysis of PDEs (math.AP) #Cancer Cells and Metastasis #FOS: Mathematics #Gene Regulatory Network Analysis #Mathematical Biology Tumor Growth #math.AP
paper · pdf · doi:10.48550/arxiv.1503.02387
arxiv created 2015/03/09 · openalex publication_date 2015/03/09 · arxiv updated 2015/03/10 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
This paper deals with the higher dimension quasilinear parabolic-parabolic Keller-Segel system involving a source term of logistic type ut=∇⋅(ϕ(u)∇ u)-χ∇⋅(u∇ v)+g(u), τvt=Δv-v+u in Ω× (0,T), subject to nonnegative initial data and homogeneous Neumann boundary condition, where Ω is smooth and bounded domain in ℝn, n≥ 2, ϕ and g are smooth and positive functions satisfying ksp≤ϕ when s≥ s0>1, g(s) ≤ as - μs2 for s>0 with g(0)≥0 and constants a≥ 0, τ,χ,μ>0. It was known that the model without the logistic source admits both bounded and unbounded solutions, identified via the critical exponent (2)/(n). On the other hand, the model is just a critical case with the balance of logistic damping and aggregation effects, for which the property of solutions should be determined by the coefficients involved. In the present paper it is proved that there is θ0>0 such that the problem admits global bounded classical solutions, regardless of the size of initial data and diffusion whenever \fracχμ<θ0. This shows the substantial effect of the logistic source to the behavior of solutions.