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TRAVELLING WAVE SOLUTIONS FOR DOUBLY DEGENERATE REACTION–DIFFUSION EQUATIONS

Part of: Mathematical biology in general Parabolic equations and systems

Published online by Cambridge University Press:  10 March 2011

M. B. A. MANSOUR
M. B. A. MANSOUR*
Affiliation:
Department of Mathematics, Faculty of Science at Qena, South Valley University, Qena, Egypt (email: m.mansour4@hotmail.com)
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Abstract

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This paper concerns a nonlinear doubly degenerate reaction–diffusion equation which appears in a bacterial growth model and is also of considerable mathematical interest. A travelling wave analysis for the equation is carried out. In particular, the qualitative behaviour of both sharp and smooth travelling wave solutions is analysed. This travelling wave behaviour is also verified by some numerical computations for a special case.

Keywords

density-dependent diffusionnonlinear degenerate parabolic equationstravelling waves

MSC classification

Secondary: 35K65: Degenerate parabolic equations 92B05: General biology and biomathematics
Type
Research Article
Information
The ANZIAM Journal , Volume 52 , Issue 1 , July 2010 , pp. 101 - 109
Copyright
Copyright © Australian Mathematical Society 2011

References

[1]Arrowsmith, D. K. and Place, C. M., An introduction to dynamical systems (Cambridge University Press, Cambridge, 1990).Google Scholar
[2]Atkinson, C., Reuter, G. E. H. and Ridler-Rowe, C. J., “Traveling wave solutions for some nonlinear diffusion equations”, SIAM J. Math. Anal. 12 (1981) 880–892.CrossRefGoogle Scholar
[3]Garduno, F. S. and Maini, P. K., “An approximation to a sharp front type solution of a density dependent reaction–diffusion equation”, Appl. Math. Lett. 7 (1994) 47–51.CrossRefGoogle Scholar
[4]Kawasaki, K., Mochizuki, A., Matsushita, M., Umeda, T. and Shigesada, N., “Modeling spatio-temporal patterns generated by Bacillus subtilis”, J. Theoret. Biol. 188 (1997) 177–185.CrossRefGoogle ScholarPubMed
[5]Malaguti, L. and Marcelli, C., “Sharp profiles in degenerate and doubly degenerate Fisher–KPP equations”, J. Differential Equations 195 (2003) 471–496.CrossRefGoogle Scholar
[6]Mansour, M. B. A., “Accurate computation of traveling wave solutions of some nonlinear diffusion equations”, Wave Motion 44 (2007) 222–230.CrossRefGoogle Scholar
[7]Mansour, M. B. A., “Traveling wave solutions of a nonlinear reaction–diffusion–chemotaxis model for bacterial pattern formation”, Appl. Math. Modelling 32 (2008) 240–247.CrossRefGoogle Scholar
[8]Ohgiwari, M., Matsushita, M. and Matsuyama, T., “Morphological changes in growth phenomena of bacterial colony patterns”, J. Phys. Soc. Japan 61 (1992) 816–822.CrossRefGoogle Scholar
[9]Press, H. W., Flannery, B. P., Teukolsky, S. A. and Vetterling, W. T., Numerical recipes: the art of scientific computing (Cambridge University Press, New York, 1986).Google Scholar
[10]Satnoianu, P. A., Maini, P. K., Garduno, F. S. and Armitage, J. P., “Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation”, Discrete Contin. Dyn. Syst. Ser. B 1 (2001) 339–362.Google Scholar
[11]Sherratt, J. A., “On the transition from initial data to traveling waves in the Fisher–KPP equation”, Dyn. Stab. Syst. 13 (1998) 167–174.CrossRefGoogle Scholar
[12]Sherratt, J. A., “On the form of smooth-front traveling waves in a reaction–diffusion equation with degenerate nonlinear diffusion”, Math. Model. Nat. Phenom. 5 (2010) 63–78.CrossRefGoogle Scholar
[13]Sherratt, J. A. and Marchant, B. P., “Nonsharp traveling wave fronts in the Fisher equation with degenerate nonlinear diffusion”, Appl. Math. Lett. 9 (1996) 33–38.CrossRefGoogle Scholar