Principal eigenvalues for some nonlocal eigenvalue problems and applications

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

Principal eigenvalues for some nonlocal eigenvalue problems and applications

Fei-Ying Yang ^1, , Wan-Tong Li ^1, and Jian-Wen Sun ^1,

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

Abstract
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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.

Keywords: asymptotic behavior., principal eigenvalues, existence and uniqueness, positive solutions, Nonlocal operator.

Mathematics Subject Classification: Primary: 35B40, 45A05; Secondary: 47G2.

Citation: 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

References:

[1]	F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, AMS, Providence, Rhode Island, 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), 105-136. 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), 1119-1139. 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), 428-440. 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), 1-29. 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), 47-92. 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), 75-113. 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), 271-291. 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), 137-156. 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), 53-60. 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), 1693-1709. 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), 2921-2953. 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), 179-223. doi: 10.1016/j.anihpc.2012.07.005.
[14]	J. Coville, Nonlocal refuge model with a partial control, Discrete Contin. Dyn. Syst., 35 (2015), 1421-1446. doi: 10.3934/dcds.2015.35.1421.
[15]	R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Ser. Math. Comput. Biol., John Wiley and Sons, Chichester, UK, 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), 1-18. 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), 295-319. doi: 10.1006/jdeq.1996.0071.
[18]	P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153-191.
[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), 261-289. 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), 2037-2053. 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), 21-38. 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), 1299-1320.
[23]	V. Hutson, 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.
[24]	L. I. Ignat and J. D. Rossi, Decay estimates for nonlocal problems via energy methods, J. Math. Pures Appl., 92 (2009), 163-187. 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), 1241-1262. 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), 403-439. 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), 551-596. 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), 2302-2313. 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), 263-294. 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), 47-64. 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), 503-527. 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), 377-392. 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), 3501-3509. 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), 1372-1402. 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), 3217-3238. 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), 551-581. 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), 814-826. 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), 747-795. 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), 1681-1696. doi: 10.1090/S0002-9939-2011-11011-6.
[40]	J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218. 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), 4466-4474. 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, Providence, Rhode Island, 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), 105-136. 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), 1119-1139. 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), 428-440. 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), 1-29. 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), 47-92. 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), 75-113. 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), 271-291. 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), 137-156. 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), 53-60. 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), 1693-1709. 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), 2921-2953. 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), 179-223. doi: 10.1016/j.anihpc.2012.07.005.
[14]	J. Coville, Nonlocal refuge model with a partial control, Discrete Contin. Dyn. Syst., 35 (2015), 1421-1446. doi: 10.3934/dcds.2015.35.1421.
[15]	R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Ser. Math. Comput. Biol., John Wiley and Sons, Chichester, UK, 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), 1-18. 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), 295-319. doi: 10.1006/jdeq.1996.0071.
[18]	P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153-191.
[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), 261-289. 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), 2037-2053. 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), 21-38. 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), 1299-1320.
[23]	V. Hutson, 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.
[24]	L. I. Ignat and J. D. Rossi, Decay estimates for nonlocal problems via energy methods, J. Math. Pures Appl., 92 (2009), 163-187. 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), 1241-1262. 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), 403-439. 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), 551-596. 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), 2302-2313. 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), 263-294. 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), 47-64. 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), 503-527. 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), 377-392. 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), 3501-3509. 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), 1372-1402. 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), 3217-3238. 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), 551-581. 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), 814-826. 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), 747-795. 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), 1681-1696. doi: 10.1090/S0002-9939-2011-11011-6.
[40]	J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218. 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), 4466-4474. doi: 10.1016/j.na.2010.02.021.

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