Maximum principles, sliding techniques and applications to nonlocal equations

Electronic Journal of Differential Equations

Volumn 2007, Issue , 2007, Pages 1-23

  Coville, Jérôme ^a,b  

^a UNIVERSITÉ PARIS DAUPHINE   (France)

^b UNIVERSITY OF CHILE   (Chile)

Author keywords

Maximum principles; Nonlocal diffusion operators; Sliding methods

Indexed keywords

EID: 34249889875     PISSN: None     EISSN: 10726691     Source Type: Journal    
DOI: None     Document Type: Article

References (30)

1. Giovanni Alberti and Giovanni Bellettini. A nonlocal anisotropic model for phase transitions. I. The optimal profile problem. Math. Ann., 310(3): 527-560, 1998.
2. D. G. Aronson and H. F. Weinberger. Multidimensional nonlinear diffusion arising in population genetics. Adv. in Math., 30(1): 33-76, 1978.
3. Peter W. Bates, Paul C. Fife, Xiaofeng Ren, and Xuefeng Wang. Traveling waves in a convolution model for phase transitions. Arch. Rational Mech. Anal., 138(2): 105-136, 1997.
4. H. Berestycki and B. Larrouturou. Quelques aspects mathématiques de la propagation des flammes prémélangées. In Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. X (Paris, 1987-1988), volume 220 of Pitman Res. Notes Math. Ser., pages 65-129. Longman Sci. Tech., Harlow, 1991.
5. H. Berestycki, B. Larrouturou, and P.-L. Lions. Multi-dimensional travelling-wave solutions of a flame propagation model. Arch. Rational Mech. Anal., 111(1): 33-49, 1990.
6. H. Berestycki and L. Nirenberg. On the method of moving planes and the sliding method. Bol. Soc. Brasil. Mat. (N.S.), 22(1): 1-37, 1991.
7. Henri Berestycki and Louis Nirenberg. Travelling fronts in cylinders. Ann. Inst. H. Poincaré Anal. Non Linéaire, 9(5): 497-572, 1992.
8. Jack Carr and Adam Chmaj. Uniqueness of travelling waves for nonlocal monostable equations. Proc. Amer. Math. Soc., 132(8):2433-2439 (electronic), 2004.
9. Xinfu Chen. Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations. Adv. Differential Equations, 2(1): 125-160, 1997.
10. Xinfu Chen and Jong-Sheng Guo. Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics. Math. Ann., 326(1): 123-146, 2003.
11. Carmen Cortazar, Manuel Elgueta, and Julio D. Rossi. A nonlocal diffusion equation whose solutions develop a free boundary. Ann. Henri Poincaré, 6(2): 269-281, 2005.
12. Jérôme Coville and Louis Dupaigne. Propagation speed of travelling fronts in non local reaction-diffusion equations. Nonlinear Anal., 60(5): 797-819, 2005.
13. Jrme Coville. On the monotone behavior of solution of nonlocal reaction-diffusion equation. Ann. Mat. Pura Appl. (4), To appear, 2005.
14. Jrme Coville. équation de raction diffusion nonlocale. Thse de L'Universit Pierre et Marie Curie, Nov. 2003.
15. A. De Masi, T. Gobron, and E. Presutti. Travelling fronts in non-local evolution equations. Arch. Rational Mech. Anal., 132(2): 143-205, 1995.
16. A. De Masi, E. Orlandi, E. Presutti, and L. Triolo. Uniqueness and global stability of the instanton in nonlocal evolution equations. Rend. Mat. Appl. (7), 14(4): 693-723, 1994.
17. G. Bard Ermentrout and J. Bryce McLeod. Existence and uniqueness of travelling waves for a neural network. Proc. Roy. Soc. Edinburgh Sect. A, 123(3): 461-478, 1993.
18. Paul C. Fife. Mathematical aspects of reacting and diffusing systems, volume 28 of Lecture Notes in Biomathematics. Springer-Verlag, Berlin, 1979.
19. Paul C. Fife. An integrodifferential analog of semilinear parabolic PDEs. In Partial differential equations and applications, volume 177 of Lecture Notes in Pure and Appl. Math., pages 137-145. Dekker, New York, 1996.
20. R. A. Fisher. The genetical theory of natural selection. Oxford University Press, Oxford, variorum edition, 1999. Revised reprint of the 1930 original, Edited, with a foreword and notes, by J. H. Bennett.
21. David Gilbarg and Neil S. Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.
22. Brian H. Gilding and Robert Kersner. Travelling waves in nonlinear diffusion-convection reaction. Progress in Nonlinear Differential Equations and their Applications, 60. Birkhäuser Verlag, Basel, 2004.
23. A. N. Kolmogorov, I. G. Petrovsky, and N. S. Piskunov. étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Bulletin Université d'État à Moscow (Bjul. Moskowskogo Gos. Univ), Série Internationale(Section A): 1-26, 1937.
24. J. D. Murray. Mathematical biology, volume 19 of Biomathematics. Springer-Verlag, Berlin, second edition, 1993.
25. Murray H. Protter and Hans F. Weinberger. Maximum principles in differential equations. Prentice-Hall Inc., Englewood Cliffs, N.J., 1967.
26. Konrad Schumacher. Travelling-front solutions for integro-differential equations. I. J. Reine Angew. Math., 316: 54-70, 1980.
27. Panagiotis E. Souganidis. Interface dynamics in phase transitions. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), pages 1133-1144, Basel, 1995. Birkhäuser.
28. José M. Vega. On the uniqueness of multidimensional travelling fronts of some semilinear equations. J. Math. Anal. Appl., 177(2): 481-490, 1993.
29. H. F. Weinberger. Long-time behavior of a class of biological models. SIAM J. Math. Anal., 13(3): 353-396, 1982.
30. J. B. Zeldovich and D. A. Frank-Kamenetskii. A theory of thermal propagation of flame. Acta Physiochimica URSS, Série Internationale(9), 1938.