# American Institute of Mathematical Sciences

2015, 2015(special): 1125-1133. doi: 10.3934/proc.2015.1125

## Existence of solutions to chemotaxis dynamics with logistic source

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

Received  July 2014 Revised  November 2014 Published  November 2015

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].
Citation: Tomomi Yokota, Noriaki Yoshino. Existence of solutions to chemotaxis dynamics with logistic source. Conference Publications, 2015, 2015 (special) : 1125-1133. doi: 10.3934/proc.2015.1125
##### 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.

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##### 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.
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