Fractional diffusion with Neumann boundary conditions: The logistic equation

Previous Article
Dead-core rates for the heat equation with a spatially dependent strong absorption
DCDS-B Home
This Issue
Next Article
Qualitative analysis of an age- and sex-structured vaccination model for human papillomavirus

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

Abstract
Related Papers

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 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.

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.

[1]	Juan-Luis Vázquez. Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 857-885. doi: 10.3934/dcdss.2014.7.857
[2]	Siwei Duo, Hong Wang, Yanzhi Zhang. A comparative study on nonlocal diffusion operators related to the fractional Laplacian. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 231-256. doi: 10.3934/dcdsb.2018110
[3]	Ravi P. Agarwal, Kanishka Perera, Zhitao Zhang. On some nonlocal eigenvalue problems. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 707-714. doi: 10.3934/dcdss.2012.5.707
[4]	Maria Fărcăşeanu, Mihai Mihăilescu, Denisa Stancu-Dumitru. Perturbed fractional eigenvalue problems. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6243-6255. doi: 10.3934/dcds.2017270
[5]	Sarah Constantin, Robert S. Strichartz, Miles Wheeler. Analysis of the Laplacian and spectral operators on the Vicsek set. Communications on Pure and Applied Analysis, 2011, 10 (1) : 1-44. doi: 10.3934/cpaa.2011.10.1
[6]	Kung-Ching Chang, Xuefeng Wang, Xie Wu. On the spectral theory of positive operators and PDE applications. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3171-3200. doi: 10.3934/dcds.2020054
[7]	Yutong Chen, Jiabao Su. Resonant problems for fractional Laplacian. Communications on Pure and Applied Analysis, 2017, 16 (1) : 163-188. doi: 10.3934/cpaa.2017008
[8]	J. R. L. Webb. Uniqueness of the principal eigenvalue in nonlocal boundary value problems. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 177-186. doi: 10.3934/dcdss.2008.1.177
[9]	Fei-Ying Yang, Wan-Tong Li, Jian-Wen Sun. Principal eigenvalues for some nonlocal eigenvalue problems and applications. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 4027-4049. doi: 10.3934/dcds.2016.36.4027
[10]	Fang Li, Jerome Coville, Xuefeng Wang. On eigenvalue problems arising from nonlocal diffusion models. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 879-903. doi: 10.3934/dcds.2017036
[11]	Wanbin Tong, Hongjin He, Chen Ling, Liqun Qi. A nonmonotone spectral projected gradient method for tensor eigenvalue complementarity problems. Numerical Algebra, Control and Optimization, 2020, 10 (4) : 425-437. doi: 10.3934/naco.2020042
[12]	Alexei Rybkin. On the boundary control approach to inverse spectral and scattering theory for Schrödinger operators. Inverse Problems and Imaging, 2009, 3 (1) : 139-149. doi: 10.3934/ipi.2009.3.139
[13]	Marta García-Huidobro, Raul Manásevich, J. R. Ward. Vector p-Laplacian like operators, pseudo-eigenvalues, and bifurcation. Discrete and Continuous Dynamical Systems, 2007, 19 (2) : 299-321. doi: 10.3934/dcds.2007.19.299
[14]	Kanishka Perera, Andrzej Szulkin. p-Laplacian problems where the nonlinearity crosses an eigenvalue. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 743-753. doi: 10.3934/dcds.2005.13.743
[15]	Tetsutaro Shibata. Global behavior of bifurcation curves for the nonlinear eigenvalue problems with periodic nonlinear terms. Communications on Pure and Applied Analysis, 2018, 17 (5) : 2139-2147. doi: 10.3934/cpaa.2018102
[16]	Marco Squassina. Preface: Recent progresses in the theory of nonlinear nonlocal problems. Discrete and Continuous Dynamical Systems - S, 2018, 11 (3) : i-i. doi: 10.3934/dcdss.201803i
[17]	Fang Liu. The eigenvalue problem for a class of degenerate operators related to the normalized $ p $-Laplacian. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2701-2720. doi: 10.3934/dcdsb.2021155
[18]	Patrizia Pucci, Mingqi Xiang, Binlin Zhang. A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 4035-4051. doi: 10.3934/dcds.2017171
[19]	Irene Benedetti, Valeri Obukhovskii, Valentina Taddei. Evolution fractional differential problems with impulses and nonlocal conditions. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1899-1919. doi: 10.3934/dcdss.2020149
[20]	John R. Graef, Lingju Kong, Qingkai Kong, Min Wang. Positive solutions of nonlocal fractional boundary value problems. Conference Publications, 2013, 2013 (special) : 283-290. doi: 10.3934/proc.2013.2013.283

2020 Impact Factor: 1.327

Tools

Metrics

Other articles
by authors

on AIMS
Eugenio Montefusco Benedetta Pellacci Gianmaria Verzini
on Google Scholar
Eugenio Montefusco Benedetta Pellacci Gianmaria Verzini

[Back to Top]