July  2016, 36(7): 4027-4049. doi: 10.3934/dcds.2016.36.4027

Principal eigenvalues for some nonlocal eigenvalue problems and applications

1. 

School of Mathematics and Statistics, Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, Gansu 730000, China, China

Received  April 2015 Revised  November 2015 Published  March 2016

This paper is concerned with the principal eigenvalues of some nonlocal operators. We first derive a result on the limit of certain sequences of principal eigenvalues associated with some nonlocal eigenvalue problems. Then, such a result is used to study the existence, uniqueness and asymptotic behavior of positive solutions to a nonlocal stationary problem with a parameter. Finally, the long-time behavior of the solutions of the corresponding nonlocal evolution equation and the asymptotic behavior of positive stationary solutions on parameter are discussed.
Citation: Fei-Ying Yang, Wan-Tong Li, Jian-Wen Sun. Principal eigenvalues for some nonlocal eigenvalue problems and applications. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 4027-4049. doi: 10.3934/dcds.2016.36.4027
References:
[1]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs,, AMS, (2010). doi: 10.1090/surv/165.

[2]

P. Bates, P. Fife, X. Ren and X. Wang, Travelling waves in a convolution model for phase transitions,, Arch. Ration. Mech. Anal., 138 (1997), 105. doi: 10.1007/s002050050037.

[3]

P. Bates and A. Chmaj, An integrodifferential model for phase transitions: Stationary solutions in higher space dimensions,, J. Statist. Phys., 95 (1999), 1119. doi: 10.1023/A:1004514803625.

[4]

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

[5]

H. Berestycki, Le nombre de solutions de certains problèmes semi-linéaires elliptiques,, J. Funct. Anal., 40 (1981), 1. doi: 10.1016/0022-1236(81)90069-0.

[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. doi: 10.1002/cpa.3160470105.

[7]

H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model. I. Species persistence,, J. Math. Biol., 51 (2005), 75. doi: 10.1007/s00285-004-0313-3.

[8]

E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equation,, J. Math. Pures Appl., 86 (2006), 271. doi: 10.1016/j.matpur.2006.04.005.

[9]

C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems,, Arch. Ration. Mech. Anal., 187 (2008), 137. doi: 10.1007/s00205-007-0062-8.

[10]

C. Cortázar, M. Elgueta and J. D. Rossi, Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions,, Israel J. Math., 170 (2009), 53. doi: 10.1007/s11856-009-0019-8.

[11]

J. Coville, J. Dávila and S. Martínez, Existence and uniqueness of solutions to nonlocal equation with monostable nonlinearity,, SIAM J. Math. Anal., 39 (2008), 1693. doi: 10.1137/060676854.

[12]

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

[13]

J. Coville, J. Dávila and S. Martínez, Pulsating fronts for nonlocal dispersion and KPP nonlinearity,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 179. doi: 10.1016/j.anihpc.2012.07.005.

[14]

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

[15]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations,, Ser. Math. Comput. Biol., (2003). doi: 10.1002/0470871296.

[16]

Y. Du and Q. Huang, Blow-up solutions for a class of semilinear elliptic and parabolic equations,, SIAM J. Math. Anal., 31 (1999), 1. doi: 10.1137/S0036141099352844.

[17]

J. M. Fraile, P. Koch Medina, J. López-Gómez and S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation,, J. Differential Equations, 127 (1996), 295. doi: 10.1006/jdeq.1996.0071.

[18]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions,, in Trends in Nonlinear Analysis, (2003), 153.

[19]

J. García-Melián, R. Gómez-Reñasco, J. López-Gómez and J. C. Sabina de Lis, Pointwise growth and uniqueness of positive solutions for a class of sublinear elliptic problems where bifurcation from infinity occurs,, Arch. Ration. Mech. Anal., 145 (1998), 261. doi: 10.1007/s002050050130.

[20]

J. García-Melián and J. D. Rossi, A logistic equation with refuge and nonlocal diffusion,, Commun. Pure Appl. Anal., 8 (2009), 2037. doi: 10.3934/cpaa.2009.8.2037.

[21]

J. García-Melián and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems,, J. Differential Equations, 246 (2009), 21. doi: 10.1016/j.jde.2008.04.015.

[22]

M. Grinfeld, G. Hines, V. Hutson, K. Mischaikow, G. T. Vickers, Non-local dispersal,, Differential Integral Equations, 18 (2005), 1299.

[23]

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

[24]

L. I. Ignat and J. D. Rossi, Decay estimates for nonlocal problems via energy methods,, J. Math. Pures Appl., 92 (2009), 163. doi: 10.1016/j.matpur.2009.04.009.

[25]

Y. Jin and M. A. Lewis, Seasonal influences on population spread and persistence in streams: Critical domain size,, SIAM J. Appl. Math., 71 (2011), 1241. doi: 10.1137/100788033.

[26]

Y. Jin and M. A. Lewis, Seasonal influences on population spread and persistence in streams: Spreading speeds,, J. Math. Biol., 65 (2012), 403. doi: 10.1007/s00285-011-0465-x.

[27]

C. Y. Kao, Y. Lou and W. Shen, Random dispersal vs non-local dispersal,, Discrete Contin. Dyn. Syst., 26 (2010), 551. doi: 10.3934/dcds.2010.26.551.

[28]

W. T. Li, Y. J. Sun and Z. C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal,, Nonlinear Anal. Real World Appl., 11 (2010), 2302. doi: 10.1016/j.nonrwa.2009.07.005.

[29]

J. López-Gómez, The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems,, J. Differential Equations, 127 (1996), 263. doi: 10.1006/jdeq.1996.0070.

[30]

J. López-Gómez and J. C. Sabina de Lis, First variations of principal eigenvalues with respect to the domain and point-wise growth of positive solutions for problems where bifurcation from infinity occurs,, J. Differential Equations, 148 (1998), 47. doi: 10.1006/jdeq.1998.3456.

[31]

T. Ouyang, On the positive solutions of semilinear equations $\Delta u+\lambda u-hu^p=0$,, Trans. Amer. Math. Soc., 331 (1992), 503. doi: 10.2307/2154124.

[32]

S. Pan, W. T. Li and G. Lin, Travelling wave fronts in nonlocal reaction-diffusion systems and applications,, Z. Angew. Math. Phys., 60 (2009), 377. doi: 10.1007/s00033-007-7005-y.

[33]

J. W. Sun, W. T. Li and F. Y. Yang, Approximate the Fokker-Planck equation by a class of nonlocal dispersal problems,, Nonlinear Anal., 74 (2011), 3501. doi: 10.1016/j.na.2011.02.034.

[34]

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. doi: 10.1016/j.jde.2014.05.005.

[35]

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

[36]

Y. J. Sun, W. T. Li and Z. C. Wang, Entire solutions in nonlocal dispersal equations with bistable nonlinearity,, J. Differential Equations, 251 (2011), 551. doi: 10.1016/j.jde.2011.04.020.

[37]

Y. J. Sun, W. T. Li and Z. C. Wang, Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonlinearity,, Nonlinear Anal., 74 (2011), 814. doi: 10.1016/j.na.2010.09.032.

[38]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats,, J. Differential Equations, 249 (2010), 747. doi: 10.1016/j.jde.2010.04.012.

[39]

W. Shen and A. Zhang, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats,, Proc. Amer. Math. Soc., 140 (2012), 1681. doi: 10.1090/S0002-9939-2011-11011-6.

[40]

J. G. Skellam, Random dispersal in theoretical populations,, Biometrika, 38 (1951), 196. doi: 10.1093/biomet/38.1-2.196.

[41]

G. B. Zhang, W. T. Li and Y. J. Sun, Asymptotic behavior for nonlocal dispersal equations,, Nonlinear Anal., 72 (2010), 4466. doi: 10.1016/j.na.2010.02.021.

show all references

References:
[1]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs,, AMS, (2010). doi: 10.1090/surv/165.

[2]

P. Bates, P. Fife, X. Ren and X. Wang, Travelling waves in a convolution model for phase transitions,, Arch. Ration. Mech. Anal., 138 (1997), 105. doi: 10.1007/s002050050037.

[3]

P. Bates and A. Chmaj, An integrodifferential model for phase transitions: Stationary solutions in higher space dimensions,, J. Statist. Phys., 95 (1999), 1119. doi: 10.1023/A:1004514803625.

[4]

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

[5]

H. Berestycki, Le nombre de solutions de certains problèmes semi-linéaires elliptiques,, J. Funct. Anal., 40 (1981), 1. doi: 10.1016/0022-1236(81)90069-0.

[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. doi: 10.1002/cpa.3160470105.

[7]

H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model. I. Species persistence,, J. Math. Biol., 51 (2005), 75. doi: 10.1007/s00285-004-0313-3.

[8]

E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equation,, J. Math. Pures Appl., 86 (2006), 271. doi: 10.1016/j.matpur.2006.04.005.

[9]

C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems,, Arch. Ration. Mech. Anal., 187 (2008), 137. doi: 10.1007/s00205-007-0062-8.

[10]

C. Cortázar, M. Elgueta and J. D. Rossi, Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions,, Israel J. Math., 170 (2009), 53. doi: 10.1007/s11856-009-0019-8.

[11]

J. Coville, J. Dávila and S. Martínez, Existence and uniqueness of solutions to nonlocal equation with monostable nonlinearity,, SIAM J. Math. Anal., 39 (2008), 1693. doi: 10.1137/060676854.

[12]

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

[13]

J. Coville, J. Dávila and S. Martínez, Pulsating fronts for nonlocal dispersion and KPP nonlinearity,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 179. doi: 10.1016/j.anihpc.2012.07.005.

[14]

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

[15]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations,, Ser. Math. Comput. Biol., (2003). doi: 10.1002/0470871296.

[16]

Y. Du and Q. Huang, Blow-up solutions for a class of semilinear elliptic and parabolic equations,, SIAM J. Math. Anal., 31 (1999), 1. doi: 10.1137/S0036141099352844.

[17]

J. M. Fraile, P. Koch Medina, J. López-Gómez and S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation,, J. Differential Equations, 127 (1996), 295. doi: 10.1006/jdeq.1996.0071.

[18]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions,, in Trends in Nonlinear Analysis, (2003), 153.

[19]

J. García-Melián, R. Gómez-Reñasco, J. López-Gómez and J. C. Sabina de Lis, Pointwise growth and uniqueness of positive solutions for a class of sublinear elliptic problems where bifurcation from infinity occurs,, Arch. Ration. Mech. Anal., 145 (1998), 261. doi: 10.1007/s002050050130.

[20]

J. García-Melián and J. D. Rossi, A logistic equation with refuge and nonlocal diffusion,, Commun. Pure Appl. Anal., 8 (2009), 2037. doi: 10.3934/cpaa.2009.8.2037.

[21]

J. García-Melián and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems,, J. Differential Equations, 246 (2009), 21. doi: 10.1016/j.jde.2008.04.015.

[22]

M. Grinfeld, G. Hines, V. Hutson, K. Mischaikow, G. T. Vickers, Non-local dispersal,, Differential Integral Equations, 18 (2005), 1299.

[23]

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

[24]

L. I. Ignat and J. D. Rossi, Decay estimates for nonlocal problems via energy methods,, J. Math. Pures Appl., 92 (2009), 163. doi: 10.1016/j.matpur.2009.04.009.

[25]

Y. Jin and M. A. Lewis, Seasonal influences on population spread and persistence in streams: Critical domain size,, SIAM J. Appl. Math., 71 (2011), 1241. doi: 10.1137/100788033.

[26]

Y. Jin and M. A. Lewis, Seasonal influences on population spread and persistence in streams: Spreading speeds,, J. Math. Biol., 65 (2012), 403. doi: 10.1007/s00285-011-0465-x.

[27]

C. Y. Kao, Y. Lou and W. Shen, Random dispersal vs non-local dispersal,, Discrete Contin. Dyn. Syst., 26 (2010), 551. doi: 10.3934/dcds.2010.26.551.

[28]

W. T. Li, Y. J. Sun and Z. C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal,, Nonlinear Anal. Real World Appl., 11 (2010), 2302. doi: 10.1016/j.nonrwa.2009.07.005.

[29]

J. López-Gómez, The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems,, J. Differential Equations, 127 (1996), 263. doi: 10.1006/jdeq.1996.0070.

[30]

J. López-Gómez and J. C. Sabina de Lis, First variations of principal eigenvalues with respect to the domain and point-wise growth of positive solutions for problems where bifurcation from infinity occurs,, J. Differential Equations, 148 (1998), 47. doi: 10.1006/jdeq.1998.3456.

[31]

T. Ouyang, On the positive solutions of semilinear equations $\Delta u+\lambda u-hu^p=0$,, Trans. Amer. Math. Soc., 331 (1992), 503. doi: 10.2307/2154124.

[32]

S. Pan, W. T. Li and G. Lin, Travelling wave fronts in nonlocal reaction-diffusion systems and applications,, Z. Angew. Math. Phys., 60 (2009), 377. doi: 10.1007/s00033-007-7005-y.

[33]

J. W. Sun, W. T. Li and F. Y. Yang, Approximate the Fokker-Planck equation by a class of nonlocal dispersal problems,, Nonlinear Anal., 74 (2011), 3501. doi: 10.1016/j.na.2011.02.034.

[34]

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. doi: 10.1016/j.jde.2014.05.005.

[35]

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

[36]

Y. J. Sun, W. T. Li and Z. C. Wang, Entire solutions in nonlocal dispersal equations with bistable nonlinearity,, J. Differential Equations, 251 (2011), 551. doi: 10.1016/j.jde.2011.04.020.

[37]

Y. J. Sun, W. T. Li and Z. C. Wang, Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonlinearity,, Nonlinear Anal., 74 (2011), 814. doi: 10.1016/j.na.2010.09.032.

[38]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats,, J. Differential Equations, 249 (2010), 747. doi: 10.1016/j.jde.2010.04.012.

[39]

W. Shen and A. Zhang, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats,, Proc. Amer. Math. Soc., 140 (2012), 1681. doi: 10.1090/S0002-9939-2011-11011-6.

[40]

J. G. Skellam, Random dispersal in theoretical populations,, Biometrika, 38 (1951), 196. doi: 10.1093/biomet/38.1-2.196.

[41]

G. B. Zhang, W. T. Li and Y. J. Sun, Asymptotic behavior for nonlocal dispersal equations,, Nonlinear Anal., 72 (2010), 4466. doi: 10.1016/j.na.2010.02.021.

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