# American Institute of Mathematical Sciences

October  2013, 18(8): 2175-2202. doi: 10.3934/dcdsb.2013.18.2175

## 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

Received  February 2013 Revised  May 2013 Published  July 2013

Motivated by experimental studies on the anomalous diffusion of biological populations, we study the spectral square root of the Laplacian in bounded domains with Neumann homogeneous boundary conditions. Such operator arises in the continuous limit for long jumps random walks with reflecting barriers. Existence and uniqueness results for positive solutions are proved in the case of indefinite nonlinearities of logistic type by means of bifurcation theory.
Citation: Eugenio Montefusco, Benedetta Pellacci, Gianmaria Verzini. Fractional diffusion with Neumann boundary conditions: The logistic equation. Discrete and Continuous Dynamical Systems - B, 2013, 18 (8) : 2175-2202. doi: 10.3934/dcdsb.2013.18.2175
##### 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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##### 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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