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Blow-up, quenching, aggregation and collapse in a chemotaxis model with reproduction term

Blow-up, quenching, aggregation and collapse in a chemotaxis model with reproduction term In this paper, we consider the following chemotaxis model with ratio-dependent logistic reaction term $$\left\{ \begin{gathered} \tfrac{{\partial u}} {{\partial t}} = D\nabla (\nabla u - u\tfrac{{\nabla w}} {w}) + u(a - b\tfrac{u} {w}), (x,t) \in Q_T , \hfill \\ \tfrac{{\partial u}} {{\partial t}} = \beta u - \delta w, (x,t) \in Q_T , \hfill \\ u\nabla \ln (\tfrac{u} {w}) \cdot \vec n = 0, x \in \partial \Omega 0 < t < T, \hfill \\ u(x,0) = u_0 (x) > 0, x \in \bar \Omega , \hfill \\ w(x,0) = w_0 (x) > 0, x \in \bar \Omega , \hfill \\ \end{gathered} \right.$$ It is shown that the solution to the problem exists globally if b + β ≥ 0 and will blow up or quench if b + β < 0 by means of function transformation and comparison method. Various asymptotic behavior related to different coefficients and initial data is also discussed. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

Blow-up, quenching, aggregation and collapse in a chemotaxis model with reproduction term

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Publisher
Springer Journals
Copyright
Copyright © 2014 by Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/s10255-014-0406-8
Publisher site
See Article on Publisher Site

Abstract

In this paper, we consider the following chemotaxis model with ratio-dependent logistic reaction term $$\left\{ \begin{gathered} \tfrac{{\partial u}} {{\partial t}} = D\nabla (\nabla u - u\tfrac{{\nabla w}} {w}) + u(a - b\tfrac{u} {w}), (x,t) \in Q_T , \hfill \\ \tfrac{{\partial u}} {{\partial t}} = \beta u - \delta w, (x,t) \in Q_T , \hfill \\ u\nabla \ln (\tfrac{u} {w}) \cdot \vec n = 0, x \in \partial \Omega 0 < t < T, \hfill \\ u(x,0) = u_0 (x) > 0, x \in \bar \Omega , \hfill \\ w(x,0) = w_0 (x) > 0, x \in \bar \Omega , \hfill \\ \end{gathered} \right.$$ It is shown that the solution to the problem exists globally if b + β ≥ 0 and will blow up or quench if b + β < 0 by means of function transformation and comparison method. Various asymptotic behavior related to different coefficients and initial data is also discussed.

Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Nov 6, 2014

References