Existence of  solutions to chemotaxis dynamics           with logistic source

Tomomi  Yokota; Noriaki  Yoshino

doi:10.3934/proc.2015.1125

Article Contents

2015, Volume 2015: 1125-1133. Doi: 10.3934/proc.2015.1125 open access

This issue Previous Article Classification of periodic orbits in the planar equal-mass four-body problem Next Article Pullback uniform dissipativity of stochastic reversible Schnackenberg equations

Existence of solutions to chemotaxis dynamics with logistic source

Tomomi Yokota^1, and
Noriaki Yoshino^1,

1.
Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601

Received: July 2014

Revised: November 2014

Published: November 2015

Abstract / Introduction Related Papers Cited by

Abstract

This paper is concerned with a chemotaxis system with nonlinear diffusion and logistic growth term $f(b) = \kappa b-\mu |b|^{\alpha-1}b$ with $\kappa>0$, $\mu>0$ and $\alpha > 1$ under the no-flux boundary condition. It is shown that there exists a local solution to this system for any $L^2$-initial data and that under a stronger assumption on the chemotactic sensitivity there exists a global solution for any $L^2$-initial data. The proof is based on the method built by Marinoschi [8].

Keywords:

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation:

Related Papers

Cited by

References

[1]	M. Aida, T. Tsujikawa, M. Efendiev, A. Yagi and M. Mimura, Lower estimate of the attractor dimension for a chemotaxis growth system, J. London Math. Soc. 74 (2006), 453-474.
[2]	E. Ardeleanu, G. Marinoschi, An asymptotic solution to a nonlinear reaction-diffusion system with chemotaxis, Numer. Funct. Anal. Optim. 34 (2013), 117-148.
[3]	V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York, 2010.
[4]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.
[5]	S. Dragomir, Some Gronwall Type Inequalities and Applications, Nova Science Publishers, Inc., Hauppauge, NY, 2003.
[6]	E. F. Keller, L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399-415.
[7]	J. L. Lions, Quelques Méthodes de Résollution des Problemes aux Limites non Linéaires, Dunod, Paris, 1969.
[8]	G. Marinoschi, Well-posedness for chemotaxis dynamics with nonlinear cell diffusion, J. Math. Anal. Appl. 402 (2013), 415-439.
[9]	J. I. Tello, Mathematical analysis and stability of a chemotaxis model with logistic term, Math. Methods Appl. Sci. 27 (2004), 1865-1880.
[10]	J. I. Tello, M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations 32 (2007), 849-877.
[11]	T. Yokota, N. Yoshino, Existence of solutions to chemotaxis dynamics with Lipschitz diffusion, J. Math. Anal. Appl. 419 (2014), 756-774.