## Abstract and Applied Analysis

### Existence of global solution and nontrivial steady states for a system modeling chemotaxis

Zhenbu Zhang

#### Abstract

We consider a reaction-diffusion system modeling chemotaxis, which describes the situation of two species of bacteria competing for the same nutrient. We use Moser-Alikakos iteration to prove the global existence of the solution. We also study the existence of nontrivial steady state solutions and their stability.

#### Article information

Source
Abstr. Appl. Anal., Volume 2006 (2006), Article ID 81265, 23 pages.

Dates
First available in Project Euclid: 19 March 2007

https://projecteuclid.org/euclid.aaa/1174321965

Digital Object Identifier
doi:10.1155/AAA/2006/81265

Mathematical Reviews number (MathSciNet)
MR2270330

Zentralblatt MATH identifier
1145.35392

#### Citation

Zhang, Zhenbu. Existence of global solution and nontrivial steady states for a system modeling chemotaxis. Abstr. Appl. Anal. 2006 (2006), Article ID 81265, 23 pages. doi:10.1155/AAA/2006/81265. https://projecteuclid.org/euclid.aaa/1174321965

#### References

• N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Communications in Partial Differential Equations 4 (1979), no. 8, 827--868.
• H. Amann, Dynamic theory of quasilinear parabolic equations. I. Abstract evolution equations, Nonlinear Analysis. Theory, Methods & Applications 12 (1988), no. 9, 895--919.
• --------, Dynamic theory of quasilinear parabolic systems. III. Global existence, Mathematische Zeitschrift 202 (1989), no. 2, 219--250.
• --------, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential and Integral Equations 3 (1990), no. 1, 13--75.
• N. F. Britton, Reaction-Diffusion Equations and Their Applications to Biology, Academic Press, London, 1986.
• B. P. Conrad, Differential Equations with Boundary Value Problems, Prentice-Hall, New Jersey, 2003.
• J. Dockery, V. Hutson, K. Mischaikow, and M. Pernarowski, The evolution of slow dispersal rates: a reaction diffusion model, Journal of Mathematical Biology 37 (1998), no. 1, 61--83.
• L. Dung, Coexistence with chemotaxis, SIAM Journal on Mathematical Analysis 32 (2000), no. 3, 504--521.
• D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer, Berlin, 1981.
• E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology 26 (1970), no. 3, 399--415.
• K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcialaj Ekvacioj 44 (2001), no. 3, 441--469.
• J. Smoller, Shock Waves and Reaction-Diffusion Equations, Fundamental Principles of Mathematical Science, vol. 258, Springer, New York, 1983.
• X. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: effects of motility and chemotaxis and dynamics, SIAM Journal on Mathematical Analysis 31 (2000), no. 3, 535--560.
• X. Wang and Y. Wu, Qualitative analysis on a chemotactic diffusion model for two species competing for a limited resource, Quarterly of Applied Mathematics 60 (2002), no. 3, 505--531.
• Z. Zhang, Coexistence and stability of solutions for a class of reaction-diffusion systems, Electronic Journal Differential Equations 2005 (2005), no. 137, 1--16.