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February  2017, 37(2): 879-903. doi: 10.3934/dcds.2017036

On eigenvalue problems arising from nonlocal diffusion models

Fang Li 1, , Jerome Coville 2, and  Xuefeng Wang 1,3,,

1. 

Center for PDE, East China Normal University, 500 Dongchuan Road, Minhang 200241, Shanghai, China
2. 

Biostatistique et Processus Spatiaux, INRA, 84000, Avignon, France
3. 

Department of Mathematics, Southern University of Science and Technology, 1088 Xueyuan Road, Nanshan 518055, Shenzhen, China

FL is supported by NSF of China (No. 11431005), NSF of Shanghai (No. 16ZR1409600).JC is supported by the French ANR through the ANR JCJC project MODEVOL: ANR-13-JS01-0009 and the ANR project NONLOCAL: ANR-13-JS01-0009.XFW is supported by NSF of China (No. 11671190).

Received  June 2015 Revised  September 2015 Published  November 2016

We aim at saying as much as possible about the spectra of three classes of linear diffusion operators involving nonlocal terms. In all but one cases, we characterize the minimum $λ_p$ of the real part of the spectrum in two max-min fashions, and prove that in most cases $λ_p$ is an eigenvalue with a corresponding positive eigenfunction, and is algebraically simple and isolated; we also prove that the maximum principle holds if and only if $λ_p>0$ (in most cases) or $≥ 0$ (in one case). We prove these results by an elementary method based on the strong maximum principle, rather than resorting to Krein-Rutman theory as did in the previous papers. In one case when it is impossible to characterize $λ_p$ in the max-min fashion, we supply a complete description of the whole spectrum.

Keywords: Principal eigenvalue, nonlocal operators, max-min characterization, maximum principle.
Mathematics Subject Classification: Primary:45C05, 45K05;Secondary:35P1.
Citation: 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
References:
[1]

X. Bai and F. Li, Optimization of species survival for logistic models with non-local dispersal, Nonlinear Anal. Real World Appl., 21 (2015), 53-62.  doi: 10.1016/j.nonrwa.2014.06.006.

[2]

X. Bai and F. Li, Global dynamics of a competition model with nonlocal dispersal Ⅱ: The full system, J. Differential Equations, 258 (2015), 2655-2685.  doi: 10.1016/j.jde.2014.12.014.

[3]

P. Bates and G. Zhao, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl., 332 (2007), 428-440.  doi: 10.1016/j.jmaa.2006.09.007.

[4]

P. Bates and G. Zhao, Spectral convergence and turing patterns for nonlocal diffusion systems, preprints.

[5]

H. Berestycki, J. Coville and H.-H. Vo, Persistence criteria for populations with non-local dispersion, J. Math. Biol., 72 (2016), 1693-1745.  doi: 10.1007/s00285-015-0911-2.

[6]

H. Berestycki, L. Nirenberg and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92.  doi: 10.1002/cpa.3160470105.

[7]

J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differential Equations, 249 (2010), 2921-2953.  doi: 10.1016/j.jde.2010.07.003.

[8]

J. Coville, Singular measure as principal eigenfunction of some nonlocal operators, Appl. Math. Lett., 26 (2013), 831-835.  doi: 10.1016/j.aml.2013.03.005.

[9]

J. Coville, Nonlocal refuge model with a partial control, Discrete Contin. Dyn. Syst., 35 (2015), 1421-1446.  doi: 10.3934/dcds.2015.35.1421.

[10]

J. Coville, J. Davila and S. Martinez, Pulsating fronts for nonlocal dispersion and KPP nonlinearity, Ann. I. H. Poincare -AN, 30 (2013), 179-223.  doi: 10.1016/j.anihpc.2012.07.005.

[11]

J. F. Crow and M. Kimura, An Introduction to Population Genetics Theory, Burgess Pub. Co. , 1970.

[12]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, Providence, 1998.

[13]

V. Huston, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1007/s00285-003-0210-1.

[14] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995. 
[15]

F. Li, K. Nakashima and W.-M. Ni, Stability from the point of view of diffusion, relaxation and spatial inhomogeneity, Discrete Contin. Dyn. Syst., 20 (2008), 259-274. 

[16]

F. Li, Y. Lou and Y. Wang, Global dynamics of a competition model with non-local dispersal Ⅰ: the shadow system, J. Math. Anal. Appl., 412 (2014), 485-497.  doi: 10.1016/j.jmaa.2013.10.071.

[17]

H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.  doi: 10.1007/BF00277392.

[18]

W. Shen and X. Xie, On principal spectrum points/principal eigenvalues of nonlocal dispersal operators and applications, Discrete Contin. Dyn. Syst., 35 (2015), 1665-1696.  doi: 10.3934/dcds.2015.35.1665.

[19]

D. B. Smith, A sufficient condition for the existence of a principal eigenvalue for nonlocal diffusion equations with applications, J. Math. Anal. Appl., 418 (2014), 766-774.  doi: 10.1016/j.jmaa.2014.04.004.

[20]

L. Sun, J. Shi and Y. Wang, Existence and uniqueness of steady state solutions of a nonlocal diffusive logistic equation, Z. Angew. Math. Phys., 64 (2013), 1267-1278.  doi: 10.1007/s00033-012-0286-9.

[21]

J.-W. Sun, W.-T. Li and Z.-C. Wang, A nonlocal dispersal logistic equation with spatial degeneracy, Discrete Contin. Dyn. Syst., 35 (2015), 3217-3238.  doi: 10.3934/dcds.2015.35.3217.

[22]

J.-W. Sun, F.-Y. Yang and W.-T. Li, A nonlocal dispersal equation arising from a selection-migration model in genetics, J. Differential Equations, 257 (2014), 1372-1402.  doi: 10.1016/j.jde.2014.05.005.

[23]

Y. Yamada, On logistic diffusion equations with nonlocal interaction terms, Nonlinear Anal., 118 (2015), 51-62.  doi: 10.1016/j.na.2015.01.016.

show all references

FL is supported by NSF of China (No. 11431005), NSF of Shanghai (No. 16ZR1409600).
JC is supported by the French ANR through the ANR JCJC project MODEVOL: ANR-13-JS01-0009 and the ANR project NONLOCAL: ANR-13-JS01-0009.
XFW is supported by NSF of China (No. 11671190).

References:
[1]

X. Bai and F. Li, Optimization of species survival for logistic models with non-local dispersal, Nonlinear Anal. Real World Appl., 21 (2015), 53-62.  doi: 10.1016/j.nonrwa.2014.06.006.

[2]

X. Bai and F. Li, Global dynamics of a competition model with nonlocal dispersal Ⅱ: The full system, J. Differential Equations, 258 (2015), 2655-2685.  doi: 10.1016/j.jde.2014.12.014.

[3]

P. Bates and G. Zhao, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl., 332 (2007), 428-440.  doi: 10.1016/j.jmaa.2006.09.007.

[4]

P. Bates and G. Zhao, Spectral convergence and turing patterns for nonlocal diffusion systems, preprints.

[5]

H. Berestycki, J. Coville and H.-H. Vo, Persistence criteria for populations with non-local dispersion, J. Math. Biol., 72 (2016), 1693-1745.  doi: 10.1007/s00285-015-0911-2.

[6]

H. Berestycki, L. Nirenberg and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92.  doi: 10.1002/cpa.3160470105.

[7]

J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differential Equations, 249 (2010), 2921-2953.  doi: 10.1016/j.jde.2010.07.003.

[8]

J. Coville, Singular measure as principal eigenfunction of some nonlocal operators, Appl. Math. Lett., 26 (2013), 831-835.  doi: 10.1016/j.aml.2013.03.005.

[9]

J. Coville, Nonlocal refuge model with a partial control, Discrete Contin. Dyn. Syst., 35 (2015), 1421-1446.  doi: 10.3934/dcds.2015.35.1421.

[10]

J. Coville, J. Davila and S. Martinez, Pulsating fronts for nonlocal dispersion and KPP nonlinearity, Ann. I. H. Poincare -AN, 30 (2013), 179-223.  doi: 10.1016/j.anihpc.2012.07.005.

[11]

J. F. Crow and M. Kimura, An Introduction to Population Genetics Theory, Burgess Pub. Co. , 1970.

[12]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, Providence, 1998.

[13]

V. Huston, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1007/s00285-003-0210-1.

[14] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995. 
[15]

F. Li, K. Nakashima and W.-M. Ni, Stability from the point of view of diffusion, relaxation and spatial inhomogeneity, Discrete Contin. Dyn. Syst., 20 (2008), 259-274. 

[16]

F. Li, Y. Lou and Y. Wang, Global dynamics of a competition model with non-local dispersal Ⅰ: the shadow system, J. Math. Anal. Appl., 412 (2014), 485-497.  doi: 10.1016/j.jmaa.2013.10.071.

[17]

H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.  doi: 10.1007/BF00277392.

[18]

W. Shen and X. Xie, On principal spectrum points/principal eigenvalues of nonlocal dispersal operators and applications, Discrete Contin. Dyn. Syst., 35 (2015), 1665-1696.  doi: 10.3934/dcds.2015.35.1665.

[19]

D. B. Smith, A sufficient condition for the existence of a principal eigenvalue for nonlocal diffusion equations with applications, J. Math. Anal. Appl., 418 (2014), 766-774.  doi: 10.1016/j.jmaa.2014.04.004.

[20]

L. Sun, J. Shi and Y. Wang, Existence and uniqueness of steady state solutions of a nonlocal diffusive logistic equation, Z. Angew. Math. Phys., 64 (2013), 1267-1278.  doi: 10.1007/s00033-012-0286-9.

[21]

J.-W. Sun, W.-T. Li and Z.-C. Wang, A nonlocal dispersal logistic equation with spatial degeneracy, Discrete Contin. Dyn. Syst., 35 (2015), 3217-3238.  doi: 10.3934/dcds.2015.35.3217.

[22]

J.-W. Sun, F.-Y. Yang and W.-T. Li, A nonlocal dispersal equation arising from a selection-migration model in genetics, J. Differential Equations, 257 (2014), 1372-1402.  doi: 10.1016/j.jde.2014.05.005.

[23]

Y. Yamada, On logistic diffusion equations with nonlocal interaction terms, Nonlinear Anal., 118 (2015), 51-62.  doi: 10.1016/j.na.2015.01.016.

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