Global generalized solutions to a nonlinear Keller-Segel equation with singular sensitivity

2018/03/14 by Yan, Jianlu, Li, Yuxiang
#35A01 #35K55 #35Q92 #92C17 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1803.05213

Abstract

We consider the chemotaxis system \begincases \medskip u_t =Δu^m - ∇((u)/(v)∇ v), · amp; x∈Ω, t · gt;0, \medskip v_t =Δv -uv, · amp;x∈Ω, t · gt;0, \medskip (∂ u)/(∂ ν)=(∂ v)/(∂ν)=0, · amp;x∈∂Ω, t · gt;0, \medskip u(x,0)=u₀(x), v(x,0)=v₀(x), · amp;x∈Ω, \endcases in a smooth bounded domain Ω⊂ ℝⁿ, n≥2. In this work it is shown that for all reasonably regular initial data u₀≥0 and v₀>0, the corresponding Neumann initial-boundary value problem possesses a global generalized solution provided that m>1+(n-2)/(2n).

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