2020/05/23 by Hai‐Yang Jin, Jin, Hai-Yang, Zhi‐An Wang +1
Mathematics · Biochemistry, Genetics and Molecular Biology · #Mathematical Biology Tumor Growth #Gene Regulatory Network Analysis #Microtubule and mitosis dynamics
paper · pdf · doi:10.48550/arxiv.2005.11462
The paper is concerned with the following chemotaxis system with nonlinear motility functions \begincases ut=∇ ⋅ (γ(v)∇ u- uχ(v)∇ v)+μu(1-u), amp;x∈ Ω, ~~tgt;0, 0=Δv+ u-v,amp; x∈ Ω, ~~tgt;0,
u(x,0)=u0(x), amp; x∈ Ω, \endcases with homogeneous Neumann boundary conditions in a bounded domain Ω⊂ \R2 with smooth boundary, where the motility functions γ(v) and χ(v) satisfy the following conditions \beginitemize \item \colorblack(γ,χ)∈ [C2[0,∞)]2 with γ(v)>0 and \colorblack (|χ(v)|2)/(γ(v)) is bounded for all v≥ 0. %for all v≥ 0 and limv→∞(|χ(v)|2)/(γ(v)) exists. \enditemize By employing the method of energy estimates , we establish the existence of globally bounded solutions of \eqref0-1 with μ>0 for any u0 ∈ W1, ∞(Ω). Then based on a Lyapunov function, we show that all solutions (u,v) of \eqref0-1 will exponentially converge to the unique constant steady state (1,1) provided μ>(K0)/(16) with K0=max0≤ v ≤ ∞(|χ(v)|2)/(γ(v)).