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

Yanxia Wu; Yaping Wu

doi:10.3934/cpaa.2012.11.911

Article Contents

2012, Volume 11, Issue 3: 911-934. Doi: 10.3934/cpaa.2012.11.911

This issue Previous Article Heat equations and the Weighted $\bar\partial$-problem Next Article Nonexistence of solutions for nonlinear differential inequalities with gradient nonlinearities

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

Received: May 31, 2010

Revised: March 31, 2011

Published: April 2012

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 35K57, 35B25; Secondary: 35D15, 35K65.

Citation:

Related Papers

Cited by

References

[1]	J. Carr, "Applications of Centre Manifold Theory," Applied Mathematical Sciences, 35, Springer Verlag, New York, 1981.
[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.
[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.
[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.
[5]	Y. Hosono, Traveling waves for some biological systems with density dependent diffusion, Japan J. Appl. Math., 4 (1987), 297-359.
[6]	Y. Hosono, Traveling wave solutions for some density-dependent diffusion equations, Japan J. Appl. Math., 3 (1986), 163-196.
[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.
[8]	M. Ito, A remark on singular perturbation methods, Hiroshima Math. J., 14 (1985), 619-629.
[9]	Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion, Nonlinear Anal., 28 (1997), 145-164.
[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.
[11]	N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theret. Biol., 79 (1979), 83-99.
[12]	M. M. Tang and P. C. Fife, Propagation fronts for competiting species equations with diffusion, Arch. Rational. Mech. Anal., 73 (1980), 69-77.
[13]	Y. Wu, Traveling waves for a class of cross-diffusion systems with small parameters, J. Differential Equations, 123 (1995), 1-34.
[14]	Y. Wu, The existence of traveling waves for a cross-diffusion system with small parameter, Beijing Math., 3 (1997), 74-85.
[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.
[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.