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

Fractional diffusion with Neumann boundary conditions: The logistic equation

Eugenio Montefusco 1, , Benedetta Pellacci 2, and  Gianmaria Verzini 3,

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.
Keywords: Spectral fractional Laplacian, eigenvalue problems for nonlocal operators, bifurcation theory..
Mathematics Subject Classification: Primary: 35R11; Secondary: 35J65, 92D25, 35B3.
Citation: Eugenio Montefusco, Benedetta Pellacci, Gianmaria Verzini. Fractional diffusion with Neumann boundary conditions: The logistic equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2175-2202. doi: 10.3934/dcdsb.2013.18.2175
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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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Nicolas E. Humphries, et al., Environmental context explains Levy and Brownian movement patterns of marine predators, Nature, 465 (2010), 1066-1069. Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[22]

J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218.  Google Scholar

[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.  Google Scholar

[24]

Giovanni Maria Troianiello, "Elliptic Differential Equations and Obstacle Problems,'' The University Series in Mathematics, Plenum Press, New York, 1987.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

Gandhimohan M. Viswanathan, et al., Levy flight search patterns of wandering albatrosses, Nature, 381 (1996), 413-415. Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

Patricio Felmer and Alexander Quaas, Boundary blow up solutions for fractional elliptic equations, Asymptot. Anal., 78 (2012), 123-144.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

Nicolas E. Humphries, et al., Environmental context explains Levy and Brownian movement patterns of marine predators, Nature, 465 (2010), 1066-1069. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218.  Google Scholar

[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.  Google Scholar

[24]

Giovanni Maria Troianiello, "Elliptic Differential Equations and Obstacle Problems,'' The University Series in Mathematics, Plenum Press, New York, 1987.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

Gandhimohan M. Viswanathan, et al., Levy flight search patterns of wandering albatrosses, Nature, 381 (1996), 413-415. Google Scholar

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