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doi: 10.3934/dcdsb.2022076
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Nonlocal dispersal equations in domains becoming unbounded

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

Received  February 2022 Revised  March 2022 Early access April 2022

This paper is concerned with a class of nonlocal dispersal equation in asymptotically cylindrical domain. Our aim is to investigate the asymptotic behavior of positive solutions in domain becoming infinite in some direction. We prove that the limiting behavior of positive solutions is only determined by the bounded part of whole domain.

Citation: Jian-Wen Sun. Nonlocal dispersal equations in domains becoming unbounded. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022076
References:
[1]

F. Andreu-Vaillo, J. M. Maz$\acute{o}$n, 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 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.

[3]

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.

[4]

H. Brézis and S. Kamin, Sublinear elliptic equations in ${\bf R}^n$, Manuscripta Math., 74 (1992), 87-106.  doi: 10.1007/BF02567660.

[5]

H. Brézis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal., 86 (2006), 271-291.  doi: 10.1016/0362-546X(86)90011-8.

[6]

E. ChasseigneM. 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.

[7]

M. ChipotJ. Dávila and M. del Pino, On the behavior of positive solutions of semilinear elliptic equations in asymptotically cylindrical domains, J. Fixed Point Theory Appl., 19 (2017), 205-213.  doi: 10.1007/s11784-016-0349-1.

[8]

C. CortázarM. ElguetaJ. 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.

[9]

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.

[10]

M. del Pino, Positive solutions of a semilinear elliptic equation on a compact manifold, Nonlinear Anal., 22 (1994), 1423-1430.  doi: 10.1016/0362-546X(94)90121-X.

[11]

Y. Du and S. Li, Positive solutions with prescribed patterns in some simple semilinear equations, Differential Integral Equations, 15 (2002), 805-822. 

[12]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in: Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153–191.

[13]

J. M. FraileP. Koch MedinaJ. 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.

[14]

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.

[15]

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.

[16]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes Math., vol. 840, Springer-Verlag, 1981.

[17]

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

[18]

C.-Y. KaoY. 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.

[19]

W. T. Li, J. López-Gómez and J. W. Sun, Sharp patterns of positive solutions for some weighted semilinear elliptic problems, Calc. Var. Partial Differential Equations, 60 (2021), Paper No. 85, 36 pp. doi: 10.1007/s00526-021-01993-9.

[20] J. López-Gómez, Metasolutions of Parabolic Equations in Population Dynamics, CRC Press, Boca Raton, 2016. 
[21]

J. López-Gómez and P. Rabinowitz, The effects of spatial heterogeneities on some multiplicity results, Discrete Contin. Dyn. Syst., 36 (2016), 941-952.  doi: 10.3934/dcds.2016.36.941.

[22]

Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations, 223 (2006), 400-426.  doi: 10.1016/j.jde.2005.05.010.

[23]

W. Shen and X. Xie, Approximations of random dispersal operators/equations by nonlocal dispersal operators/equations, J. Differential Equations, 259 (2015), 7375-7405.  doi: 10.1016/j.jde.2015.08.026.

[24]

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.

[25]

J.-W. Sun, Sharp profiles for periodic logistic equation with nonlocal dispersal, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 46, 19 pp. doi: 10.1007/s00526-020-1710-1.

[26]

J.-W. Sun, Effects of dispersal and spatial heterogeneity on nonlocal logistic equations, Nonlinearity, 34 (2021), 5434-5455.  doi: 10.1088/1361-6544/ac08e8.

[27]

J.-W. SunW.-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.

[28]

J.-W. SunW.-T. Li and F.-Y. Yang, 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.

[29]

J.-W. SunW.-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.

[30]

J.-B. Wang, W.-T. Li, F.-D. Dong and S.-X. Qiao, Recent developments on spatial propagation for diffusion equations in shifting environments, Discrete Contin. Dyn. Syst. Ser. B, (2022), in press. doi: 10.3934/dcdsb.2021266.

[31]

F.-Y. YangW.-T. Li and J.-W. Sun, Principal eigenvalues for some nonlocal eigenvalue problems and applications, Discrete Contin. Dyn. Syst., 36 (2016), 4027-4049.  doi: 10.3934/dcds.2016.36.4027.

[32]

G.-B. ZhangW.-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. Maz$\acute{o}$n, 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 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.

[3]

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.

[4]

H. Brézis and S. Kamin, Sublinear elliptic equations in ${\bf R}^n$, Manuscripta Math., 74 (1992), 87-106.  doi: 10.1007/BF02567660.

[5]

H. Brézis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal., 86 (2006), 271-291.  doi: 10.1016/0362-546X(86)90011-8.

[6]

E. ChasseigneM. 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.

[7]

M. ChipotJ. Dávila and M. del Pino, On the behavior of positive solutions of semilinear elliptic equations in asymptotically cylindrical domains, J. Fixed Point Theory Appl., 19 (2017), 205-213.  doi: 10.1007/s11784-016-0349-1.

[8]

C. CortázarM. ElguetaJ. 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.

[9]

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.

[10]

M. del Pino, Positive solutions of a semilinear elliptic equation on a compact manifold, Nonlinear Anal., 22 (1994), 1423-1430.  doi: 10.1016/0362-546X(94)90121-X.

[11]

Y. Du and S. Li, Positive solutions with prescribed patterns in some simple semilinear equations, Differential Integral Equations, 15 (2002), 805-822. 

[12]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in: Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153–191.

[13]

J. M. FraileP. Koch MedinaJ. 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.

[14]

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.

[15]

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.

[16]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes Math., vol. 840, Springer-Verlag, 1981.

[17]

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

[18]

C.-Y. KaoY. 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.

[19]

W. T. Li, J. López-Gómez and J. W. Sun, Sharp patterns of positive solutions for some weighted semilinear elliptic problems, Calc. Var. Partial Differential Equations, 60 (2021), Paper No. 85, 36 pp. doi: 10.1007/s00526-021-01993-9.

[20] J. López-Gómez, Metasolutions of Parabolic Equations in Population Dynamics, CRC Press, Boca Raton, 2016. 
[21]

J. López-Gómez and P. Rabinowitz, The effects of spatial heterogeneities on some multiplicity results, Discrete Contin. Dyn. Syst., 36 (2016), 941-952.  doi: 10.3934/dcds.2016.36.941.

[22]

Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations, 223 (2006), 400-426.  doi: 10.1016/j.jde.2005.05.010.

[23]

W. Shen and X. Xie, Approximations of random dispersal operators/equations by nonlocal dispersal operators/equations, J. Differential Equations, 259 (2015), 7375-7405.  doi: 10.1016/j.jde.2015.08.026.

[24]

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.

[25]

J.-W. Sun, Sharp profiles for periodic logistic equation with nonlocal dispersal, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 46, 19 pp. doi: 10.1007/s00526-020-1710-1.

[26]

J.-W. Sun, Effects of dispersal and spatial heterogeneity on nonlocal logistic equations, Nonlinearity, 34 (2021), 5434-5455.  doi: 10.1088/1361-6544/ac08e8.

[27]

J.-W. SunW.-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.

[28]

J.-W. SunW.-T. Li and F.-Y. Yang, 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.

[29]

J.-W. SunW.-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.

[30]

J.-B. Wang, W.-T. Li, F.-D. Dong and S.-X. Qiao, Recent developments on spatial propagation for diffusion equations in shifting environments, Discrete Contin. Dyn. Syst. Ser. B, (2022), in press. doi: 10.3934/dcdsb.2021266.

[31]

F.-Y. YangW.-T. Li and J.-W. Sun, Principal eigenvalues for some nonlocal eigenvalue problems and applications, Discrete Contin. Dyn. Syst., 36 (2016), 4027-4049.  doi: 10.3934/dcds.2016.36.4027.

[32]

G.-B. ZhangW.-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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