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Blow-Up and Boundedness in Quasilinear Parabolic-Elliptic Chemotaxis System with Nonlinear Signal Production

Blow-Up and Boundedness in Quasilinear Parabolic-Elliptic Chemotaxis System with Nonlinear Signal Production
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Year:    2023

Author:    Ruxi Cao, Zhongping Li

Journal of Partial Differential Equations, Vol. 36 (2023), Iss. 3 : pp. 262–285

Abstract

In this paper, we consider the quasilinear chemotaxis system of parabolic-elliptic type {ut=∇⋅(D(u)∇u)−∇⋅(f(u)∇v),x∈Ω, t>0,0=Δv−μ(t)+g(u),x∈Ω, t>0
under homogeneous Neumann boundary conditions in a smooth bounded domain  Ω⊂Rn, n≥1. The nonlinear diffusivity D(ξ) and chemosensitivity f(ξ) as well as nonlinear signal production g(ξ) are supposed to extend the prototypes D(ξ)=C0(1+ξ)−m,  f(ξ)=K(1+ξ)k,  g(ξ)=L(1+ξ)l,  C0>0,ξ≥0,K,k,L,l>0,m∈R. We proved that if m+k+l>1+2n, then there exists nonnegative radially symmetric initial data u0 such that the corresponding solutions blow up in finite time. However, the system admits a global bounded classical solution for arbitrary initial datum when m+k+l<1+2n.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jpde.v36.n3.2

Journal of Partial Differential Equations, Vol. 36 (2023), Iss. 3 : pp. 262–285

Published online:    2023-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    24

Keywords:    Chemotaxis nonlinear diffusion blow-up boundedness nonlinear signal production.

Author Details

Ruxi Cao

Zhongping Li