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Planar quasi-homogeneous polynomial systems with a given weight degree
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 |
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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