Existence of  solutions to chemotaxis dynamics
           with logistic source

Tomomi  Yokota; Noriaki  Yoshino

doi:10.3934/proc.2015.1125

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

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
Related Papers
Under a Creative Commons license

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), 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.

show all references

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.

[1]	Dieter Bothe, Petra Wittbold. Abstract reaction-diffusion systems with $m$-completely accretive diffusion operators and measurable reaction rates. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2239-2260. doi: 10.3934/cpaa.2012.11.2239
[2]	Xiaoyu Chen, Jijie Zhao, Qian Zhang. Global existence of weak solutions for the 3D axisymmetric chemotaxis-Navier-Stokes equations with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2022 doi: 10.3934/dcds.2022062
[3]	Xiuting Li. The energy conservation for weak solutions to the relativistic Nordström-Vlasov system. Evolution Equations and Control Theory, 2016, 5 (1) : 135-145. doi: 10.3934/eect.2016.5.135
[4]	Yaiza Canzani, A. Rod Gover, Dmitry Jakobson, Raphaël Ponge. Nullspaces of conformally invariant operators. Applications to $\boldsymbol{Q_k}$-curvature. Electronic Research Announcements, 2013, 20: 43-50. doi: 10.3934/era.2013.20.43
[5]	Hafedh Bousbih. Global weak solutions for a coupled chemotaxis non-Newtonian fluid. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 907-929. doi: 10.3934/dcdsb.2018212
[6]	Georges Chamoun, Moustafa Ibrahim, Mazen Saad, Raafat Talhouk. Asymptotic behavior of solutions of a nonlinear degenerate chemotaxis model. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4165-4188. doi: 10.3934/dcdsb.2020092
[7]	Xia Huang. Stable weak solutions of weighted nonlinear elliptic equations. Communications on Pure and Applied Analysis, 2014, 13 (1) : 293-305. doi: 10.3934/cpaa.2014.13.293
[8]	Feng Li, Yuxiang Li. Global existence of weak solution in a chemotaxis-fluid system with nonlinear diffusion and rotational flux. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5409-5436. doi: 10.3934/dcdsb.2019064
[9]	Pan Zheng, Chunlai Mu, Xiaojun Song. On the boundedness and decay of solutions for a chemotaxis-haptotaxis system with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1737-1757. doi: 10.3934/dcds.2016.36.1737
[10]	Ludovic Dan Lemle. $L^1(R^d,dx)$-uniqueness of weak solutions for the Fokker-Planck equation associated with a class of Dirichlet operators. Electronic Research Announcements, 2008, 15: 65-70. doi: 10.3934/era.2008.15.65
[11]	Peiying Chen. Existence and uniqueness of weak solutions for a class of nonlinear parabolic equations. Electronic Research Announcements, 2017, 24: 38-52. doi: 10.3934/era.2017.24.005
[12]	Ismail Kombe. On the nonexistence of positive solutions to doubly nonlinear equations for Baouendi-Grushin operators. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5167-5176. doi: 10.3934/dcds.2013.33.5167
[13]	Jeffrey R. L. Webb. Positive solutions of nonlinear equations via comparison with linear operators. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5507-5519. doi: 10.3934/dcds.2013.33.5507
[14]	Martin Brokate, Pavel Krejčí. Weak differentiability of scalar hysteresis operators. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2405-2421. doi: 10.3934/dcds.2015.35.2405
[15]	Jiashan Zheng. Boundedness of solutions to a quasilinear higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 627-643. doi: 10.3934/dcds.2017026
[16]	Hong Yi, Chunlai Mu, Shuyan Qiu, Lu Xu. Global boundedness of radial solutions to a parabolic-elliptic chemotaxis system with flux limitation and nonlinear signal production. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3825-3849. doi: 10.3934/cpaa.2021133
[17]	Chao Ji. Ground state solutions of fractional Schrödinger equations with potentials and weak monotonicity condition on the nonlinear term. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6071-6089. doi: 10.3934/dcdsb.2019131
[18]	Fang Li, Bo You, Yao Xu. Dynamics of weak solutions for the three dimensional Navier-Stokes equations with nonlinear damping. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4267-4284. doi: 10.3934/dcdsb.2018137
[19]	Igor Chueshov, Irena Lasiecka. Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 777-809. doi: 10.3934/dcds.2006.15.777
[20]	Abdelmalek Aboussoror, Samir Adly, Vincent Jalby. Weak nonlinear bilevel problems: Existence of solutions via reverse convex and convex maximization problems. Journal of Industrial and Management Optimization, 2011, 7 (3) : 559-571. doi: 10.3934/jimo.2011.7.559

Impact Factor:

American Institute of Mathematical Sciences

Existence of solutions to chemotaxis dynamics with logistic source

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Existence of solutions to chemotaxis dynamics with logistic source

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors