2022/11/16 by Fei Gao, Gao, Fei, Hui Zhan +1
Computer Science · Mathematics · #26A33 #35M10 #35R11 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #G.1.8 #G.1.9 #Mathematical Biology Tumor Growth #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2211.08692
openalex publication_date 2022/11/16 · openalex created_date 2023/02/11 · openalex updated_date 2026/07/28
For the time-space fractional degenerate Keller-Segel equation \begincases ∂ tβu=-(-Δ)^\fracα2(ρ(v)u),amp; tgt;0
(-Δ)^\fracα2 v+v=u,amp; tgt;0 \endcases x∈Ω, Ω⊂ ℝn, β∈ (0,1),α∈ (1,2), we consider for n≥ 3 the problem of finding a time-independent upper bound of the classical solution such that as θ>0,C>0 ‖ u(⋅ ,t)-u0 ‖L∞ (Ω)+ ‖ v(⋅ ,t)-u0 ‖W1,∞ (Ω)≤ Ce^(-θ)1/βt, where u0=(1)/( | Ω |)∫ Ωu0dx. We find such solution in the special cases of time-independent upper bound of the concentration with Alikakos-Moser iteration and fractional differential inequality. In those cases the problem is reduced to a time-space fractional parabolic-elliptic equation which is treated with Lyapunov functional methods. A key element in our construction is a proof of the exponential stabilization toward the constant steady states by using fractional Duhamel type integral equation.