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May  2012, 11(3): 911-934. doi: 10.3934/cpaa.2012.11.911

Existence of traveling waves with transition layers for some degenerate cross-diffusion systems

Yanxia Wu 1, and  Yaping Wu 1,

1. 

Department of Mathematics, Capital Normal University, Beijing 100048, China

Received  June 2010 Revised  April 2011 Published  December 2011

This paper is concerned with the existence of traveling waves for a class of degenerate reaction-diffusion systems with cross-diffusion. By applying the analytic singular perturbation method and the theorem of center manifold approximation, we prove the existence of traveling waves with transition layers for the more general degenerate systems with cross-diffusion. Especially for the degenerate S-K-T competition system with cross-diffusion we prove that some new wave patterns exhibiting competition exclusion are induced by the cross-diffusion. We also extend some of the existence results in [5] for the non-cross diffusion systems to the more general degenerate biological systems with cross-diffusion, however the detailed fast-slow structure of the waves for the systems with cross-diffusion is a little different from those for the systems without cross-diffusion.
Keywords: degenerate reaction-diffusion system., traveling wave solutions, cross-diffusion, transition layers, Existence.
Mathematics Subject Classification: Primary: 35K57, 35B25; Secondary: 35D15, 35K6.
Citation: Yanxia Wu, Yaping Wu. Existence of traveling waves with transition layers for some degenerate cross-diffusion systems. Communications on Pure & Applied Analysis, 2012, 11 (3) : 911-934. doi: 10.3934/cpaa.2012.11.911
References:
[1]

J. Carr, "Applications of Centre Manifold Theory," Applied Mathematical Sciences, 35, Springer Verlag, New York, 1981.  Google Scholar

[2]

C. Conley and R. A. Gardner, An application of the generalized Morse index to travelling wave solutions of a competitive reaction-diffusion model, Indiana Univ. Math. J., 33 (1984), 319-343. Google Scholar

[3]

R. A. Gardner, Existence and stability of traveling wave solutions of competition models: A degree theoretical approach, J. Differential Equations, 44 (1982), 343-364. Google Scholar

[4]

Y. Hosono, Traveling waves for a diffusive Lotka-Volterra competition model. I. Singular perturbations, Discrete Contin. Dyn. Syst. Ser. B., 3 (2003), 79-95. Google Scholar

[5]

Y. Hosono, Traveling waves for some biological systems with density dependent diffusion, Japan J. Appl. Math., 4 (1987), 297-359. Google Scholar

[6]

Y. Hosono, Traveling wave solutions for some density-dependent diffusion equations, Japan J. Appl. Math., 3 (1986), 163-196. Google Scholar

[7]

Y. Hosono and M. Mimura, Singular perturbation approach to traveling waves in competing and diffusion species models, J. Math. Kyoto Univ., 22 (1982), 435-461. Google Scholar

[8]

M. Ito, A remark on singular perturbation methods, Hiroshima Math. J., 14 (1985), 619-629. Google Scholar

[9]

Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion, Nonlinear Anal., 28 (1997), 145-164. Google Scholar

[10]

Y. Kan-on and Q. Fang, Stability of monotone traveling waves for competition-diffusion equation, Japan, J. Indust. Appl. Math., 13 (1996), 343-349. Google Scholar

[11]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theret. Biol., 79 (1979), 83-99. Google Scholar

[12]

M. M. Tang and P. C. Fife, Propagation fronts for competiting species equations with diffusion, Arch. Rational. Mech. Anal., 73 (1980), 69-77. Google Scholar

[13]

Y. Wu, Traveling waves for a class of cross-diffusion systems with small parameters, J. Differential Equations, 123 (1995), 1-34. Google Scholar

[14]

Y. Wu, The existence of traveling waves for a cross-diffusion system with small parameter, Beijing Math., 3 (1997), 74-85. Google Scholar

[15]

Y. Wu and X. Zhao, The existence and stability of traveling waves with transition layers for some singular cross diffusion systems, Phys. D, 200 (2005), 325-358. Google Scholar

[16]

Y. P. Wu and Y. Zhao, The existence and stability of traveling waves with transition layers for the S-K-T competition model with cross-diffusion, Sci. China Math., 53 (2010), 1161-1184. Google Scholar

show all references

References:
[1]

J. Carr, "Applications of Centre Manifold Theory," Applied Mathematical Sciences, 35, Springer Verlag, New York, 1981.  Google Scholar

[2]

C. Conley and R. A. Gardner, An application of the generalized Morse index to travelling wave solutions of a competitive reaction-diffusion model, Indiana Univ. Math. J., 33 (1984), 319-343. Google Scholar

[3]

R. A. Gardner, Existence and stability of traveling wave solutions of competition models: A degree theoretical approach, J. Differential Equations, 44 (1982), 343-364. Google Scholar

[4]

Y. Hosono, Traveling waves for a diffusive Lotka-Volterra competition model. I. Singular perturbations, Discrete Contin. Dyn. Syst. Ser. B., 3 (2003), 79-95. Google Scholar

[5]

Y. Hosono, Traveling waves for some biological systems with density dependent diffusion, Japan J. Appl. Math., 4 (1987), 297-359. Google Scholar

[6]

Y. Hosono, Traveling wave solutions for some density-dependent diffusion equations, Japan J. Appl. Math., 3 (1986), 163-196. Google Scholar

[7]

Y. Hosono and M. Mimura, Singular perturbation approach to traveling waves in competing and diffusion species models, J. Math. Kyoto Univ., 22 (1982), 435-461. Google Scholar

[8]

M. Ito, A remark on singular perturbation methods, Hiroshima Math. J., 14 (1985), 619-629. Google Scholar

[9]

Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion, Nonlinear Anal., 28 (1997), 145-164. Google Scholar

[10]

Y. Kan-on and Q. Fang, Stability of monotone traveling waves for competition-diffusion equation, Japan, J. Indust. Appl. Math., 13 (1996), 343-349. Google Scholar

[11]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theret. Biol., 79 (1979), 83-99. Google Scholar

[12]

M. M. Tang and P. C. Fife, Propagation fronts for competiting species equations with diffusion, Arch. Rational. Mech. Anal., 73 (1980), 69-77. Google Scholar

[13]

Y. Wu, Traveling waves for a class of cross-diffusion systems with small parameters, J. Differential Equations, 123 (1995), 1-34. Google Scholar

[14]

Y. Wu, The existence of traveling waves for a cross-diffusion system with small parameter, Beijing Math., 3 (1997), 74-85. Google Scholar

[15]

Y. Wu and X. Zhao, The existence and stability of traveling waves with transition layers for some singular cross diffusion systems, Phys. D, 200 (2005), 325-358. Google Scholar

[16]

Y. P. Wu and Y. Zhao, The existence and stability of traveling waves with transition layers for the S-K-T competition model with cross-diffusion, Sci. China Math., 53 (2010), 1161-1184. Google Scholar

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