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- Publisher
- Springer Journals
- Copyright
- Copyright © Springer Nature B.V. 2019
- Subject
- Mathematics; Computational Mathematics and Numerical Analysis; Applications of Mathematics; Partial Differential Equations; Probability Theory and Stochastic Processes; Calculus of Variations and Optimal Control; Optimization
- ISSN
- 0167-8019
- eISSN
- 1572-9036
- DOI
- 10.1007/s10440-019-00275-z
- Publisher site
- See Article on Publisher Site
Abstract
This paper deals with the quasilinear degenerate chemotaxis system with flux limitation{ut=∇⋅(up∇uu2+|∇u|2)−χ∇⋅(uq∇v1+|∇v|2),x∈Ω,t>0,0=Δv−μ+u,x∈Ω,t>0,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document} $$\begin{aligned} \textstyle\begin{cases} u_{t} = \nabla \cdot \biggl(\frac{u^{p} \nabla u}{\sqrt{u^{2} + | \nabla u|^{2}}} \biggr) -\chi \nabla \cdot \biggl( \frac{u^{q} \nabla v}{\sqrt{1 + |\nabla v|^{2}}} \biggr), &x\in \varOmega ,\ t>0, \\ 0 = \Delta v - \mu + u, &x\in \varOmega ,\ t>0, \end{cases}\displaystyle \end{aligned}$$ \end{document} where Ω:=BR(0)⊂Rn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\varOmega := B_{R}(0) \subset \mathbb{R}^{n}$\end{document} (n∈N\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$n \in \mathbb{N}$\end{document}) is a ball with some R>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$R>0$\end{document}, and χ>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\chi >0$\end{document}, p,q≥1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$p,q\geq 1$\end{document}, μ:=1|Ω|∫Ωu0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\mu := \frac{1}{| \varOmega |} \int _{\varOmega }u_{0}$\end{document} and u0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$u_{0}$\end{document} is an initial data of an unknown function u\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$u$\end{document}. Bellomo–Winkler (Trans. Am. Math. Soc. Ser. B 4, 31–67, 2017) established existence of an initial data such that the corresponding solution blows up in finite time when p=q=1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$p=q=1$\end{document}. This paper gives existence of blow-up solutions under some condition for χ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\chi $\end{document} and u0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$u_{0}$\end{document} when 1≤p≤q\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$1\leq p\leq q$\end{document}.
Journal
Acta Applicandae Mathematicae – Springer Journals
Published: Jun 8, 2020