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On eigenvalue problems arising from nonlocal diffusion models
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 |
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.
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. Google Scholar |
[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. Google Scholar |
[12] |
L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, Providence, 1998. Google Scholar |
[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
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. Google Scholar |
[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. Google Scholar |
[12] |
L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, Providence, 1998. Google Scholar |
[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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