Entire solutions of reaction-advection-diffusion equations with bistable nonlinearity in cylinders

Journal of Differential Equations

Volumn 246, Issue 11, 2009, Pages 4249-4267

  Liu, Nai Wei ^a   Li, Wan Tong ^a   Wang, Zhi Cheng ^a  

^a LANZHOU UNIVERSITY   (China)

Author keywords

Entire solution; Infinite cylinder; Reaction advection diffusion equation; Sub super solution; Traveling wave front

Indexed keywords

EID: 64149111652     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2008.12.005     Document Type: Article

References (43)

1. Aronson D.G., and Weinberger H.F. Multidimensional nonlinear diffusions arising in population genetics. Adv. Math. 30 (1978) 33-76
2. Bebernes J.W., Li C.M., and Li Y. Travelling fronts in cylinders and their stability. Rocky Mountain J. Math. 27 (1997) 123-150
3. Berestycki H., and Hamel F. Front propagation in periodic excitable media. Comm. Pure Appl. Math. 60 (2002) 949-1032
4. Berestycki H., and Hamel F. Generalized travelling waves for reaction-diffusion equations. Perspectives in Nonlinear Partial Differential Equations: In Honor of Haïm Brezis. Contemp. Math. vol. 446 (2007), Amer. Math. Soc. 101-123
5. Berestycki H., Larrouturou B., and Lions P.-L. Multidimensional travelling wave solutions of a flame propagation model. Arch. Ration. Mech. Anal. 111 (1990) 33-49
6. Berestycki H., Larrouturou B., and Roquejoffre J.M. Stability of travelling fronts in a model for flame propagation. Part I: Linear analysis. Arch. Ration. Mech. Anal. 117 (1992) 97-117
7. Berestycki H., and Nirenberg L. Travelling fronts in cylinders. Ann. Inst. H. Poincaré Anal. Non Linéaire 9 (1992) 497-572
8. Berestycki H. The influence of advection on the propagation of fronts in reaction-diffusion equations. In: Berestycki H., and Pomeau Y. (Eds). Nonlinear PDEs in Condensed Matter and Reactive Flows. NATO Sci. Ser. C vol. 569 (2003), Kluwer, Dordrecht
9. Chen X. Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations. Adv. Differential Equations 2 (1997) 125-160
10. Chen X., and Guo J.S. Existence and uniqueness of entire solutions for a reaction-diffusion equation. J. Differential Equations 212 (2005) 62-84
11. Chen X., Guo J.S., and Ninomiya H. Entire solutions of reaction-diffusion equations with balanced bistable nonlinearities. Proc. Roy. Soc. Edinburgh Sect. A 136 (2006) 1207-1237
12. Constantin P., Kiselev A., Oberman A., and Ryzhik L. Bulk burning rate in passive-reactive diffusion. Arch. Ration. Mech. Anal. 154 (2000) 53-91
13. Ei S.I. The motion of weakly interacting pulses in reaction-diffusion systems. J. Dynam. Differential Equations 14 (2002) 85-136
14. Ei S.I., Mimura M., and Nagayama M. Pulse-pulse interaction in reaction-diffusion systems. Phys. D 165 (2002) 176-198
15. Fife P.C., and McLeod J.B. The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch. Ration. Mech. Anal. 65 (1977) 335-361
16. Fisher R.A. The advance of advantageous genes. Ann. Eugenics 7 (1937) 355-369
17. Fukao Y., Morita Y., and Ninomiya H. Some entire solutions of the Allen-Cahn equation. Taiwanese J. Math. 8 (2004) 15-32
18. Guo J.S., and Morita Y. Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations. Discrete Contin. Dyn. Syst. 12 (2005) 193-212
19. Guo Y.J.L. Entire solutions for a discrete diffusive equation. J. Math. Anal. Appl. 347 (2008) 450-458
20. Hadeler K.P., and Rothe F. Travelling fronts in nonlinear diffusion equations. J. Math. Biol. 2 (1975) 251-263
21. Hamel F., and Nadirashvili N. Entire solutions of the KPP equation. Comm. Pure Appl. Math. 52 (1999) 1255-1276
22. N. Arch. Ration. Mech. Anal. 157 (2001) 91-163
23. Heinze S., Papanicolaou G., and Stevens A. Variational principles for propagation speeds in inhomogeneous media. SIAM J. Appl. Math. 62 (2001) 129-148
24. Henry D. Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math. (1981), Springer-Verlag, New York
25. Kawahara T., and Tanaka M. Interactions of traveling fronts: An exact solution of a nonlinear diffusion equation. Phys. Lett. A 97 8 (1983) 311-314
26. Kiselev A., and Ryzhik L. Enhancement of the traveling front speeds in reaction-diffusion equations with advection. Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2001) 309-358
27. Li W.T., Liu N.W., and Wang Z.C. Entire solutions in reaction-advection-diffusion equations in cylinders. J. Math. Pures Appl. 90 (2008) 492-504
28. Li W.T., Wang Z.C., and Wu J. Entire solutions in monostable reaction-diffusion equations with delayed nonlinearity. J. Differential Equations 245 (2008) 102-129
29. Morita Y., and Mimoto Y. Collision and collapse of layers in a 1D scalar reaction-diffusion equation. Phys. D 140 (2000) 151-170
30. Morita Y., and Ninomiya H. Entire solutions with merging fronts to reaction-diffusion equations. J. Dynam. Differential Equations 18 (2006) 841-861
31. Murray J.D. Mathematical Biology (1989), Springer-Verlag, Berlin
32. Roquejoffre J.M. Stability of travelling fronts in a model for flame propagation. Part II: Nonlinear stability. Arch. Ration. Mech. Anal. 117 (1992) 119-153
33. Roquejoffre J.M. Convergence to travelling waves for solutions of a class of semilinear parabolic equations. J. Differential Equations 108 (1994) 262-295
34. Roquejoffre J.M., Terman D., and Volpert V.A. Global stability of traveling fronts and convergence towards stacked families of waves in monotone parabolic systems. SIAM J. Math. Anal. 27 (1996) 1261-1269
35. Roquejoffre J.-M. Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders. Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997) 499-552
36. Vega J.M. The asymptotic behavior of the solutions of some semilinear elliptic equations in cylindrical domains. J. Differential Equations 102 (1993) 119-152
37. Volpert A.I., Volpert V.A., and Volpert V.A. Traveling Wave Solutions of Parabolic Systems. Transl. Math. Monogr. vol. 140 (1994), Amer. Math. Soc., Providence
38. Wang Z.C., Li W.T., and Ruan S. Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay. J. Differential Equations 238 (2007) 153-200
39. Wang Z.C., Li W.T., and Ruan S. Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity. Trans. Amer. Math. Soc. 361 (2009) 2047-2084
40. Z.C. Wang, W.T. Li, S. Ruan, Entire solutions in lattice delayed differential equations with global interaction: Bistable case, submitted for publication
41. Z.C. Wang, W.T. Li, J. Wu, Entire solutions in delayed lattice differential equations with monostable nonlinearity, SIAM J. Math. Anal., in press
42. Wu J. Theory and Applications of Partial Functional Differential Equations (1996), Springer-Verlag, New York
43. Yagisita H. Backward global solutions characterizing annihilation dynamics of travelling fronts. Publ. Res. Inst. Math. Sci. 39 (2003) 117-164