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Fractional diffusion with Neumann boundary conditions: The logistic equation
1. | Dipartimento di Matematica, Sapienza Università di Roma, Piazzale A. Moro 5, 00185 Roma |
2. | Dipartimento di Scienze e Tecnologie, Università degli Studi di Napoli Parthenope, Centro Direzionale Isola C4, 80143 Napoli, Italy |
3. | Dipartimento di Matematica "Francesco Brioschi", Politecnico di Milano, p.za Leonardo da Vinci 32, 20133 Milano, Italy |
References:
[1] |
Antonio Ambrosetti and Giovanni Prodi, "A Primer of Nonlinear Analysis,'' Cambridge Studies in Advanced Mathematics, 34, Cambridge University Press, Cambridge, 1995. |
[2] |
Fuensanta Andreu, José M. Mazón, Julio D. Rossi and Julián Toledo, The Neumann problem for nonlocal nonlinear diffusion equations, J. Evol. Equ., 8 (2008), 189-215.
doi: 10.1007/s00028-007-0377-9. |
[3] |
Henri Berestycki, Jean-Michel Roquejoffre and Luca Rossi, The periodic patch model for population dynamics with fractional diffusion, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 1-13.
doi: 10.3934/dcdss.2011.4.1. |
[4] |
Kenneth J. Brown, Local and global bifurcation results for a semilinear boundary value problem, J. Differential Equations, 239 (2007), 296-310.
doi: 10.1016/j.jde.2007.05.013. |
[5] |
Xavier Cabré and Jean-Michel Roquejoffre, Propagation de fronts dans les équations de Fisher-KPP avec diffusion fractionnaire, C. R. Math. Acad. Sci. Paris, 347 (2009), 1361-1366.
doi: 10.1016/j.crma.2009.10.012. |
[6] |
Xavier Cabré and Jinggang Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.
doi: 10.1016/j.aim.2010.01.025. |
[7] |
Luis A. Caffarelli and Luis Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[8] |
Luis A. Caffarelli, Sandro Salsa and Luis Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461.
doi: 10.1007/s00222-007-0086-6. |
[9] |
Robert S. Cantrell and Chris Cosner, The effects of spatial heterogeneity in population dynamics, J. Math. Biol., 29 (1991), 315-338.
doi: 10.1007/BF00167155. |
[10] |
Robert S. Cantrell and Chris Cosner, Conditional persistence in logistic models via nonlinear diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 267-281.
doi: 10.1017/S0308210500001621. |
[11] |
Antonio Capella, Juan Dávila, Louis Dupaigne and Yannick Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Comm. Partial Differential Equations, 36 (2011), 1353-1384.
doi: 10.1080/03605302.2011.562954. |
[12] |
Eleonora Di Nezza, Giampiero Palatucci and Enrico Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[13] |
Patricio Felmer and Alexander Quaas, Boundary blow up solutions for fractional elliptic equations, Asymptot. Anal., 78 (2012), 123-144. |
[14] |
Stathis Filippas, Luisa Moschini and Achilles Tertikas, Sharp trace Hardy-Sobolev-Maz'ya inequalities and the fractional Laplacian, Arch. Ration. Mech. Anal., 208 (2013), 109-161.
doi: 10.1007/s00205-012-0594-4. |
[15] |
Qing-Yang Guan and Zhi-Ming Ma, Reflected symmetric $\alpha$-stable processes and regional fractional Laplacian, Probab. Theory Related Fields, 134 (2006), 649-694.
doi: 10.1007/s00440-005-0438-3. |
[16] |
Peter Hess, "Periodic-Parabolic Boundary Value Problems and Positivity,'' Pitman Research Notes in Mathematics Series, 247, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991. |
[17] |
Nicolas E. Humphries, et al., Environmental context explains Levy and Brownian movement patterns of marine predators, Nature, 465 (2010), 1066-1069. |
[18] |
Gustavo Ferron Madeira and Arnaldo Simal do Nascimento, Bifurcation of stable equilibria and nonlinear flux boundary condition with indefinite weight, J. Differential Equations, 251 (2011), 3228-3247.
doi: 10.1016/j.jde.2011.07.020. |
[19] |
Adele Manes and Anna Maria Micheletti, Un'estensione della teoria variazionale classica degli autovalori per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital. (4), 7 (1973), 285-301. |
[20] |
Paul H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513.
doi: 10.1016/0022-1236(71)90030-9. |
[21] |
Andy M. Reynolds and Christopher J. Rhodes, The Lévy flight paradigm: Random search patterns and mechanisms, Ecology, 90 (2009), 877-887.
doi: 10.1890/08-0153.1. |
[22] |
J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218. |
[23] |
Pablo Raúl Stinga and José Luis Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122.
doi: 10.1080/03605301003735680. |
[24] |
Giovanni Maria Troianiello, "Elliptic Differential Equations and Obstacle Problems,'' The University Series in Mathematics, Plenum Press, New York, 1987. |
[25] |
Kenichiro Umezu, Behavior and stability of positive solutions of nonlinear elliptic boundary value problems arising in population dynamics, Nonlinear Anal., Ser. A: Theory Methods, 49 (2002), 817-840.
doi: 10.1016/S0362-546X(01)00142-0. |
[26] |
Enrico Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. S$\vece$MA, 49 (2009), 33-44. |
[27] |
Gandhimohan M. Viswanathan, et al., Levy flight search patterns of wandering albatrosses, Nature, 381 (1996), 413-415. |
show all references
References:
[1] |
Antonio Ambrosetti and Giovanni Prodi, "A Primer of Nonlinear Analysis,'' Cambridge Studies in Advanced Mathematics, 34, Cambridge University Press, Cambridge, 1995. |
[2] |
Fuensanta Andreu, José M. Mazón, Julio D. Rossi and Julián Toledo, The Neumann problem for nonlocal nonlinear diffusion equations, J. Evol. Equ., 8 (2008), 189-215.
doi: 10.1007/s00028-007-0377-9. |
[3] |
Henri Berestycki, Jean-Michel Roquejoffre and Luca Rossi, The periodic patch model for population dynamics with fractional diffusion, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 1-13.
doi: 10.3934/dcdss.2011.4.1. |
[4] |
Kenneth J. Brown, Local and global bifurcation results for a semilinear boundary value problem, J. Differential Equations, 239 (2007), 296-310.
doi: 10.1016/j.jde.2007.05.013. |
[5] |
Xavier Cabré and Jean-Michel Roquejoffre, Propagation de fronts dans les équations de Fisher-KPP avec diffusion fractionnaire, C. R. Math. Acad. Sci. Paris, 347 (2009), 1361-1366.
doi: 10.1016/j.crma.2009.10.012. |
[6] |
Xavier Cabré and Jinggang Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.
doi: 10.1016/j.aim.2010.01.025. |
[7] |
Luis A. Caffarelli and Luis Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[8] |
Luis A. Caffarelli, Sandro Salsa and Luis Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461.
doi: 10.1007/s00222-007-0086-6. |
[9] |
Robert S. Cantrell and Chris Cosner, The effects of spatial heterogeneity in population dynamics, J. Math. Biol., 29 (1991), 315-338.
doi: 10.1007/BF00167155. |
[10] |
Robert S. Cantrell and Chris Cosner, Conditional persistence in logistic models via nonlinear diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 267-281.
doi: 10.1017/S0308210500001621. |
[11] |
Antonio Capella, Juan Dávila, Louis Dupaigne and Yannick Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Comm. Partial Differential Equations, 36 (2011), 1353-1384.
doi: 10.1080/03605302.2011.562954. |
[12] |
Eleonora Di Nezza, Giampiero Palatucci and Enrico Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[13] |
Patricio Felmer and Alexander Quaas, Boundary blow up solutions for fractional elliptic equations, Asymptot. Anal., 78 (2012), 123-144. |
[14] |
Stathis Filippas, Luisa Moschini and Achilles Tertikas, Sharp trace Hardy-Sobolev-Maz'ya inequalities and the fractional Laplacian, Arch. Ration. Mech. Anal., 208 (2013), 109-161.
doi: 10.1007/s00205-012-0594-4. |
[15] |
Qing-Yang Guan and Zhi-Ming Ma, Reflected symmetric $\alpha$-stable processes and regional fractional Laplacian, Probab. Theory Related Fields, 134 (2006), 649-694.
doi: 10.1007/s00440-005-0438-3. |
[16] |
Peter Hess, "Periodic-Parabolic Boundary Value Problems and Positivity,'' Pitman Research Notes in Mathematics Series, 247, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991. |
[17] |
Nicolas E. Humphries, et al., Environmental context explains Levy and Brownian movement patterns of marine predators, Nature, 465 (2010), 1066-1069. |
[18] |
Gustavo Ferron Madeira and Arnaldo Simal do Nascimento, Bifurcation of stable equilibria and nonlinear flux boundary condition with indefinite weight, J. Differential Equations, 251 (2011), 3228-3247.
doi: 10.1016/j.jde.2011.07.020. |
[19] |
Adele Manes and Anna Maria Micheletti, Un'estensione della teoria variazionale classica degli autovalori per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital. (4), 7 (1973), 285-301. |
[20] |
Paul H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513.
doi: 10.1016/0022-1236(71)90030-9. |
[21] |
Andy M. Reynolds and Christopher J. Rhodes, The Lévy flight paradigm: Random search patterns and mechanisms, Ecology, 90 (2009), 877-887.
doi: 10.1890/08-0153.1. |
[22] |
J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218. |
[23] |
Pablo Raúl Stinga and José Luis Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122.
doi: 10.1080/03605301003735680. |
[24] |
Giovanni Maria Troianiello, "Elliptic Differential Equations and Obstacle Problems,'' The University Series in Mathematics, Plenum Press, New York, 1987. |
[25] |
Kenichiro Umezu, Behavior and stability of positive solutions of nonlinear elliptic boundary value problems arising in population dynamics, Nonlinear Anal., Ser. A: Theory Methods, 49 (2002), 817-840.
doi: 10.1016/S0362-546X(01)00142-0. |
[26] |
Enrico Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. S$\vece$MA, 49 (2009), 33-44. |
[27] |
Gandhimohan M. Viswanathan, et al., Levy flight search patterns of wandering albatrosses, Nature, 381 (1996), 413-415. |
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