On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators

Journal of Differential Equations

Volumn 249, Issue 11, 2010, Pages 2921-2953

  Coville, Jérôme ^a  

^a INRA   (France)

Author keywords

Asymptotic behaviour; Non trivial solution; Nonlocal diffusion operators; Principal eigenvalue

Indexed keywords

EID: 77956344620     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2010.07.003     Document Type: Article

References (40)

1. Alberti Giovanni, Bellettini Giovanni A nonlocal anisotropic model for phase transitions. I. The optimal profile problem. Math. Ann. 1998, 310(3):527-560.
2. Bates Peter W., Chmaj Adam An integrodifferential model for phase transitions: stationary solutions in higher space dimensions. J. Stat. Phys. 1999, 95(5-6):1119-1139.
3. Bates Peter W., Fife Paul C., Ren Xiaofeng, Wang Xuefeng Traveling waves in a convolution model for phase transitions. Arch. Ration. Mech. Anal. 1997, 138(2):105-136.
4. Bates Peter W., Zhao Guangyu Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal. J. Math. Anal. Appl. 2007, 332(1):428-440.
5. Berestycki H., Nirenberg L., Varadhan S.R.S. The principal eigenvalue and maximum principle for second-order elliptic operators in general domains. Comm. Pure Appl. Math. 1994, 47(1):47-92.
6. Berestycki Henri, Hamel François, Roques Lionel Analysis of the periodically fragmented environment model. I. Species persistence. J. Math. Biol. 2005, 51(1):75-113.
7. Berestycki Henri, Hamel François, Rossi Luca Liouville-type results for semilinear elliptic equations in unbounded domains. Ann. Mat. Pura Appl. (4) 2007, 186(3):469-507.
8. Browder Felix E. On the spectral theory of elliptic differential operators. I. Math. Ann. 1960/1961, 142:22-130.
9. Cain Michael L., Milligan Brook G., Strand Allan E. Long-distance seed dispersal in plant populations. Am. J. Bot. 2000, 87(9):1217-1227.
10. Chasseigne Emmanuel, Chaves Manuela, Rossi Julio D. Asymptotic behavior for nonlocal diffusion equations. J. Math. Pures Appl. (9) 2006, 86(3):271-291.
11. Chen Xinfu Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations. Adv. Differential Equations 1997, 2(1):125-160.
12. Chmaj Adam J.J., Ren Xiaofeng The nonlocal bistable equation: stationary solutions on a bounded interval. Electron. J. Differential Equations 2002, No. 02, 12 pp. (electronic).
13. Clark James S. Why trees migrate so fast: Confronting theory with dispersal biology and the paleorecord. Amer. Natural. 1998, 152(2):204-224.
14. Cortázar C., Coville J., Elgueta M., Martínez S. A nonlocal inhomogeneous dispersal process. J. Differential Equations 2007, 241(2):332-358.
15. Cortazar Carmen, Elgueta Manuel, Rossi Julio D. A nonlocal diffusion equation whose solutions develop a free boundary. Ann. Henri Poincaré 2005, 6(2):269-281.
16. Coville Jérôme On uniqueness and monotonicity of solutions of non-local reaction diffusion equation. Ann. Mat. Pura Appl. (4) 2006, 185(3):461-485.
17. Coville Jérôme Remarks on the strong maximum principle for nonlocal operators. Electron. J. Differential Equations 2008, No. 66, 10 pp.
18. Jérôme Coville, Harnack's inequality for some nonlocal equations and application, preprint du MPI, February 2008.
19. Jérôme Coville, Travelling fronts in asymmetric nonlocal reaction diffusion equation: The bistable and ignition case, preprint du CMM, July 2006.
20. Coville Jérôme, Dávila Juan, Martínez Salomé Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity. SIAM J. Math. Anal. 2008, 39(5):1693-1709.
21. Coville Jérôme, Dávila Juan, Martínez Salomé Nonlocal anisotropic dispersal with monostable nonlinearity. J. Differential Equations 2008, 244(12):3080-3118.
22. Coville Jérôme, Dupaigne Louis Propagation speed of travelling fronts in non local reaction-diffusion equations. Nonlinear Anal. 2005, 60(5):797-819.
23. Coville Jerome, Dupaigne Louis On a non-local equation arising in population dynamics. Proc. Roy. Soc. Edinburgh Sect. A 2007, 137(4):727-755.
24. De Masi A., Gobron T., Presutti E. Travelling fronts in non-local evolution equations. Arch. Ration. Mech. Anal. 1995, 132(2):143-205.
25. E. Deveaux, C. Klein, Estimation de la dispersion de pollen à longue distance à l'echelle d'un paysage agicole: une approche expérimentale, Publication du Laboratoire Ecologie, Systématique et Evolution, 2004.
26. Donsker Monroe D., Varadhan S.R.S. On a variational formula for the principal eigenvalue for operators with maximum principle. Proc. Natl. Acad. Sci. USA 1975, 72:780-783.
27. Edmunds D.E., Potter A.J.B., Stuart C.A. Non-compact positive operators. Proc. R. Soc. London Ser. A 1972, 328(1572):67-81.
28. Evans Lawrence C. Partial Differential Equations. Grad. Stud. Math. 1998, vol. 19. American Mathematical Society, Providence, RI.
29. Fife Paul C. Mathematical Aspects of Reacting and Diffusing Systems. Lecture Notes in Biomath. 1979, vol. 28. Springer-Verlag, Berlin.
30. Fife Paul C. An integrodifferential analog of semilinear parabolic PDEs. Lect. Notes Pure Appl. Math. 1996, vol. 177:137-145. Dekker, New York.
31. García-Melián Jorge, Rossi Julio D. On the principal eigenvalue of some nonlocal diffusion problems. J. Differential Equations 2009, 246(1):21-38.
32. Hutson V., Martinez S., Mischaikow K., Vickers G.T. The evolution of dispersal. J. Math. Biol. 2003, 47(6):483-517.
33. Medlock Jan, Kot Mark Spreading disease: integro-differential equations old and new. Math. Biosci. 2003, 184(2):201-222.
34. Murray J.D. Mathematical Biology. Biomathematics 1993, vol. 19. Springer-Verlag, Berlin. second ed.
35. Nussbaum Roger D. The radius of the essential spectrum. Duke Math. J. 1970, 37:473-478.
36. Nussbaum Roger D., Pinchover Yehuda On variational principles for the generalized principal eigenvalue of second order elliptic operators and some applications. J. Anal. Math. 1992, 59:161-177. Festschrift on the occasion of the 70th birthday of Shmuel Agmon.
37. Protter Murray H., Weinberger Hans F. Maximum Principles in Differential Equations 1967, Prentice Hall Inc., Englewood Cliffs, NJ.
38. Pucci Carlo Maximum and minimum first eigenvalues for a class of elliptic operators. Proc. Amer. Math. Soc. 1966, 17:788-795.
39. Schurr Frank M., Steinitz Ofer, Nathan Ran Plant fecundity and seed dispersal in spatially heterogeneous environments: models, mechanisms and estimation. J. Ecol. 2008, 96(4):628-641.
40. Zeidler Eberhard Nonlinear Functional Analysis and Its Applications. I: Fixed-Point Theorems 1986, Springer-Verlag, New York, translated from the German by Peter R. Wadsack.