Existence of solutions to chemotaxis dynamics with logistic source

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

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, Japan

Received July 2014 Revised November 2014 Published November 2015

Abstract
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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: weak solutions, nonlinear m-accretive operators., Chemotaxis.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

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

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

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