A logistic equation with refuge and nonlocal diffusion

Communications on Pure and Applied Analysis

Volumn 8, Issue 6, 2009, Pages 2037-2053

  García Melián, Jorge ^a   Rossi, Julio D ^b  

^a UNIVERSIDAD DE LA LAGUNA   (Spain)

^b UNIVERSIDAD DE BUENOS AIRES   (Argentina)

Author keywords

Logistic problems; Nonlocal diffusion

Indexed keywords

EID: 73349140331     PISSN: 15340392     EISSN: 15535258     Source Type: Journal    
DOI: 10.3934/cpaa.2009.8.2037     Document Type: Article

References (35)

1. F. Andreu, J. M. Mazon, J. D. Rossi and J. Toledo, The Neumann problem for nonlocal nonlinear diffusion equations, J. Evol. Eqns., 8 (2008), 189-215.
2. G. Bachman and L. Narici, "Functional Analysis," Dover, New York, 2000.
3. P. Bates, X. Chen and A. Chmaj, Heteroclinic solutions of a van der Waals model with indefinite nonlocal interactions, Calc. Var., 24 (2005), 261-281.
4. P. Bates and A. Chmaj, An integrodifferential model for phase transitions: stationary solutions in higher space dimensions, J. Stat. Phys., 95 (1999), 1119-1139.
5. P. Bates, P. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rat. Mech. Anal., 138 (1997), 105-136.
6. J. Carr and A. Chmaj, Uniqueness of traveling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439.
7. E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl., 86 (2006), 271-291.
8. X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Diff. Eqns., 2 (1997), 125-160.
9. A. Chmaj and X. Ren, Homoclinic solutions of an integral equation: existence and stability, J. Diff. Eqns., 155 (1999), 17-43.
10. A. Chmaj and X. Ren, The nonlocal bistable equation: stationary solutions on a bounded interval, Electronic J. Diff. Eqns., 2002 (2002), 1-12.
11. C. Cortázar, J. Coville, M. Elgueta and S. Martínez, A non local inhomogeneous dispersal process, J. Diff. Eqns., 241 (2007), 332-358.
12. C. Cortázar, M. Elgueta and J. D. Rossi, A nonlocal diffusion equation whose solutions develop a free boundary, Ann. Henri Poincaré, 6 (2005), 269-281.
13. C. Cortázar, M. Elgueta, J. D. Rossi and N. Wolanski, Boundary fluxes for nonlocal diffusion, J. Diff. Eqns., 234 (2007), 360-390.
14. C. Cortázar, M. Elgueta, J. D. Rossi and N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Arch. Rat. Mech. Anal., 187 (2008) 137-156.
15. J. Coville, On uniqueness and monotonicity of solutions on non-local reaction diffusion equations, Ann. Mat. Pura Appl., 185 (2006), 461-485.
16. J. Coville, Maximum principles, sliding techniques and applications to nonlocal equations, Electron. J. Diff. Eqns., 68 (2007), 1-23.
17. J. Coville, J. Dávila and S. Martínez,Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, SIAM J. Math. Anal., 39 (2008), 1693-1709.
18. J. Coville and L. Dupaigne, Propagation speed of travelling fronts in nonlocal reaction diffusion equations, Nonl. Anal., 60 (2005), 797-819.
19. J. Coville and L. Dupaigne, On a nonlocal equation arising in population dynamics, Proc. Roy. Soc. Edinburgh, 137 (2007), 1-29.
20. M. A. del Pino, Positive solutions of a semilinear elliptic equation on a compact manifold, Nonlinear Anal., 22 (1994), 1423-1430.
21. P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in "Trends in Nonlinear Analysis," pp. 153-191, Springer-Verlag, Berlin, 2003.
22. J. M. Fraile, P. Koch-Medina, J. Lopez-Gomez and S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear equation, J. Diff. Eqns., 127 (1996), 295-319.
23. J. García-Melián, R. Gomez-Reñasco, J. Lopez-Gomez and J. Sabina de Lis, Point-wise growth and uniqueness of positive solutions for a class of sublinear elliptic problems where bifurcation from infinity occurs, Arch. Rat. Mech. Anal., 145 (1998), 261-289.
24. J. García-Melián, R. Letelier-Albornoz and J. Sabina de Lis, Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up, Proc. Amer. Math. Soc., 129 (2001), 3593-3602.
25. J. García-Melián and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems, J. Diff. Eqns., 246 (2009), 21-38.
26. J. García-Melián and J. Sabina de Lis, On the perturbation of eigenvalues for the p-Laplacian, C. R. Acad. Sci. Paris Sér. I Math., 332 (2001), 893-898.
27. V. Hutson, S. Martínez, K. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.
28. L. I. Ignat and J. D. Rossi, Refined asymptotic expansions for nonlocal evolution equations, J. Evol. Eqns., 8 (2008), 617-629.
29. L. I. Ignat and J. D. Rossi, A nonlocal convection-diffusion equation, J. Funct. Anal., 251 (2007), 399-437.
30. M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Amer. Math. Soc. Transl., 10 (1962), 199-325.
31. J. Lopez-Gomez and J. Sabina de Lis, First variations of principal eigenvalues with respect to the domain and point-wise growth of positive solutions for problems where bifurcation from infinity occurs, J. Diff. Eqns., 148 (1998), 47-64.
32. p = 0 on the compact manifolds, Trans. Amer. Math. Soc., 331 (1992), 503-527.
33. A. F. Pazoto and J. D. Rossi, Asymptotic behavior for a semilinear nonlocal equation, Asympt. Anal., 52 (2007), 143-155.
34. K. Schumacher, Travelling-front solutions for integro-differential equations I, J. Reine Angew. Math., 316 (1980), 54-70.
35. L. Zhang, Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neural networks, J. Diff. Eqns., 197 (2004), 162-196.