September  2020, 28(3): 1143-1160. doi: 10.3934/era.2020063

The longtime behavior of the model with nonlocal diffusion and free boundaries in online social networks

College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China

* Corresponding author

Received  May 2020 Revised  May 2020 Published  July 2020

Fund Project: The research was partially supported by NSF of China (11671180)

In this paper we consider a free boundary problem with nonlocal diffusion describing information diffusion in online social networks. This model can be viewed as a nonlocal version of the free boundary problem studied by Ren et al. (Spreading-vanishing dichotomy in information diffusion in online social networks with intervention, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019) 1843–1865). We first show that this problem has a unique solution for all $ t>0 $, and then we show that its longtime behaviour is determined by a spreading-vanishing dichotomy. We also obtain sharp criteria for spreading and vanishing, and show that the spreading always happen if the diffusion rate of any one of the information is small, which is very different from the local diffusion model.

Citation: Meng Zhao. The longtime behavior of the model with nonlocal diffusion and free boundaries in online social networks. Electronic Research Archive, 2020, 28 (3) : 1143-1160. doi: 10.3934/era.2020063
References:
[1]

F. Andreu-Vaillo, J. M. Maz$\acute{o}$n, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, 165. American Mathematical Society, Providence, RI; Real Sociedad Matemática Española, Madrid, 2010. doi: 10.1090/surv/165.  Google Scholar

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G. BuntingY. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583-603.  doi: 10.3934/nhm.2012.7.583.  Google Scholar

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J.-F. CaoY. DuF. Li and W.-T. Li, The dynamics of a Fisher-KPP nonlocal diffusion model with free boundary, J. Funct. Anal., 277 (2019), 2772-2814.  doi: 10.1016/j.jfa.2019.02.013.  Google Scholar

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Y. Du, F. Li and M. Zhou, Semi-wave and spreading speed of the nonlocal Fisher-KPP equation with free boundaries, preprint, arXiv: 1909.03711. Google Scholar

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Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 45 (2013), 1995-1996.  doi: 10.1137/110822608.  Google Scholar

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Y. Du, M. Wang and M. Zhao, Two species nonlocal diffusion systems with free boundaries, preprint, arXiv: 1907.04542. Google Scholar

[7]

C. LeiZ. Lin and H. Wang, The free boundary problem describing information diffusion in online social networks, J. Differ. Equ., 254 (2013), 1326-1341.  doi: 10.1016/j.jde.2012.10.021.  Google Scholar

[8]

L. Li, W. Sheng and M. Wang, Systems with nonlocal vs. local diffusions and free boundaries, J. Math. Anal. Appl., 483 (2020), article no. 123646, 27 pp. doi: 10.1016/j.jmaa.2019.123646.  Google Scholar

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L. LiJ. Wang and M. Wang, The dynamics of nonlocal diffusion systems with different free boundaries, Comm. Pure Appl. Anal., 19 (2020), 3651-3672.  doi: 10.3934/cpaa.2020161.  Google Scholar

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W.-T. LiY.-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.  Google Scholar

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W.-T. LiJ.-B. Wang and L. Zhang, Entire solutions of nonlocal dispersal equations with monostable nonlinearity in space periodic habitats, J. Differ. Equ., 261 (2016), 2472-2501.  doi: 10.1016/j.jde.2016.05.006.  Google Scholar

[12]

W.-T. LiW.-B. Xu and L. Zhang, Traveling waves and entire solutions for an epidemic model with asymmetric dispersal, Discrete Contin. Dyn. Syst., 37 (2017), 2483-2512.  doi: 10.3934/dcds.2017107.  Google Scholar

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W.-T. LiL. Zhang and G.-B. Zhang, Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Contin. Dyn. Syst., 35 (2015), 1531-1560.  doi: 10.3934/dcds.2015.35.1531.  Google Scholar

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C. Peng, K. Xu, F. Wang and H. Wang, Predicting information diffusion initiated from multiple sources in online social networks, in 6th International Symposium on Computational Intelligence and Design(ISCID), (2013), 96–99. doi: 10.1109/ISCID.2013.138.  Google Scholar

[15]

J. RenD. Zhu and H. Wang, Spreading-vanishing dichotomy in information diffusion in online social networks with intervention, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1843-1865.  doi: 10.3934/dcdsb.2018240.  Google Scholar

[16]

Y.-J. SunL. ZhangW.-T. Li and Z.-C. Wang, Entire solutions in nonlocal monostable equations: Asymmetric case, Commun. Pure Appl. Anal., 18 (2019), 1049-1072.  doi: 10.3934/cpaa.2019051.  Google Scholar

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F. Wang, H. Wang and K. Xu, Diffusive logistic model towards predicting information diffusion in online social networks, in 32nd International Conference on Distributed Computing Systems Workshops (ICDCS), (2012), 133–139. doi: 10.1109/ICDCSW.2012.16.  Google Scholar

[18]

J. Wang and M. Wang, Free boundary problems with nonlocal and local diffusions Ⅰ: Global solution, J. Math. Anal. Appl., (2020), 123974. doi: 10.1016/j.jmaa.2020.123974.  Google Scholar

[19]

J. Wang and M. Wang, Free boundary problems with nonlocal and local diffusions Ⅱ: Spreading-vanishing and long-time behavior, Discrete Contin. Dyn. Syst. Ser. B, (2020). doi: 10.3934/dcdsb.2020121.  Google Scholar

[20]

J. Wang and L. Zhang, Invasion by an inferior or superior competitor: A diffusive competition model with a free boundary in a heterogeneous environment, J. Math. Anal. Appl., 423 (2015), 377-398.  doi: 10.1016/j.jmaa.2014.09.055.  Google Scholar

[21]

M. Wang, On some free boundary problems of the prey-predator model, J. Differ. Equ., 256 (2014), 3365-3394.  doi: 10.1016/j.jde.2014.02.013.  Google Scholar

[22]

M. Wang and J. Zhao, Free boundary problems for a Lotka-Volterra competition system, J. Dyn. Differ. Equ., 26 (2014), 655-672.  doi: 10.1007/s10884-014-9363-4.  Google Scholar

[23]

M. Wang and J. Zhao, A free boundary problem for the predator-prey model with double free boundaries, J. Dyn. Differ. Equ., 29 (2017), 957-979.  doi: 10.1007/s10884-015-9503-5.  Google Scholar

[24]

L. ZhangW.-T. Li and Z. C. Wang, Entire solution in an ignition nonlocal dispersal equation: Asymmetric kernel, Sci. China Math., 60 (2017), 1791-1804.  doi: 10.1007/s11425-016-9003-7.  Google Scholar

[25]

L. ZhangW. T. LiZ. C. Wang and Y. J. Sun, Entire solutions for nonlocal dispersal equations with bistable nonlinearity: Asymmetric case, Acta Math. Sin. (English Ser.), 35 (2019), 1771-1794.  doi: 10.1007/s10114-019-8294-8.  Google Scholar

[26]

L. ZhuH. Zhao and H. Wang, Complex dynamic behavior of a rumor propagation model with spatial-temporal diffusion terms, Inform. Sci., 349–350 (2016), 119-136.  doi: 10.1016/j.ins.2016.02.031.  Google Scholar

[27]

L. Zhu, H. Zhao and H. Wang, Partial differential equation modeling of rumor propagation in complex networks with higher order of organization, Chaos, 29 (2019), 053106, 23 pp. doi: 10.1063/1.5090268.  Google Scholar

show all references

References:
[1]

F. Andreu-Vaillo, J. M. Maz$\acute{o}$n, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, 165. American Mathematical Society, Providence, RI; Real Sociedad Matemática Española, Madrid, 2010. doi: 10.1090/surv/165.  Google Scholar

[2]

G. BuntingY. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583-603.  doi: 10.3934/nhm.2012.7.583.  Google Scholar

[3]

J.-F. CaoY. DuF. Li and W.-T. Li, The dynamics of a Fisher-KPP nonlocal diffusion model with free boundary, J. Funct. Anal., 277 (2019), 2772-2814.  doi: 10.1016/j.jfa.2019.02.013.  Google Scholar

[4]

Y. Du, F. Li and M. Zhou, Semi-wave and spreading speed of the nonlocal Fisher-KPP equation with free boundaries, preprint, arXiv: 1909.03711. Google Scholar

[5]

Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 45 (2013), 1995-1996.  doi: 10.1137/110822608.  Google Scholar

[6]

Y. Du, M. Wang and M. Zhao, Two species nonlocal diffusion systems with free boundaries, preprint, arXiv: 1907.04542. Google Scholar

[7]

C. LeiZ. Lin and H. Wang, The free boundary problem describing information diffusion in online social networks, J. Differ. Equ., 254 (2013), 1326-1341.  doi: 10.1016/j.jde.2012.10.021.  Google Scholar

[8]

L. Li, W. Sheng and M. Wang, Systems with nonlocal vs. local diffusions and free boundaries, J. Math. Anal. Appl., 483 (2020), article no. 123646, 27 pp. doi: 10.1016/j.jmaa.2019.123646.  Google Scholar

[9]

L. LiJ. Wang and M. Wang, The dynamics of nonlocal diffusion systems with different free boundaries, Comm. Pure Appl. Anal., 19 (2020), 3651-3672.  doi: 10.3934/cpaa.2020161.  Google Scholar

[10]

W.-T. LiY.-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.  Google Scholar

[11]

W.-T. LiJ.-B. Wang and L. Zhang, Entire solutions of nonlocal dispersal equations with monostable nonlinearity in space periodic habitats, J. Differ. Equ., 261 (2016), 2472-2501.  doi: 10.1016/j.jde.2016.05.006.  Google Scholar

[12]

W.-T. LiW.-B. Xu and L. Zhang, Traveling waves and entire solutions for an epidemic model with asymmetric dispersal, Discrete Contin. Dyn. Syst., 37 (2017), 2483-2512.  doi: 10.3934/dcds.2017107.  Google Scholar

[13]

W.-T. LiL. Zhang and G.-B. Zhang, Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Contin. Dyn. Syst., 35 (2015), 1531-1560.  doi: 10.3934/dcds.2015.35.1531.  Google Scholar

[14]

C. Peng, K. Xu, F. Wang and H. Wang, Predicting information diffusion initiated from multiple sources in online social networks, in 6th International Symposium on Computational Intelligence and Design(ISCID), (2013), 96–99. doi: 10.1109/ISCID.2013.138.  Google Scholar

[15]

J. RenD. Zhu and H. Wang, Spreading-vanishing dichotomy in information diffusion in online social networks with intervention, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1843-1865.  doi: 10.3934/dcdsb.2018240.  Google Scholar

[16]

Y.-J. SunL. ZhangW.-T. Li and Z.-C. Wang, Entire solutions in nonlocal monostable equations: Asymmetric case, Commun. Pure Appl. Anal., 18 (2019), 1049-1072.  doi: 10.3934/cpaa.2019051.  Google Scholar

[17]

F. Wang, H. Wang and K. Xu, Diffusive logistic model towards predicting information diffusion in online social networks, in 32nd International Conference on Distributed Computing Systems Workshops (ICDCS), (2012), 133–139. doi: 10.1109/ICDCSW.2012.16.  Google Scholar

[18]

J. Wang and M. Wang, Free boundary problems with nonlocal and local diffusions Ⅰ: Global solution, J. Math. Anal. Appl., (2020), 123974. doi: 10.1016/j.jmaa.2020.123974.  Google Scholar

[19]

J. Wang and M. Wang, Free boundary problems with nonlocal and local diffusions Ⅱ: Spreading-vanishing and long-time behavior, Discrete Contin. Dyn. Syst. Ser. B, (2020). doi: 10.3934/dcdsb.2020121.  Google Scholar

[20]

J. Wang and L. Zhang, Invasion by an inferior or superior competitor: A diffusive competition model with a free boundary in a heterogeneous environment, J. Math. Anal. Appl., 423 (2015), 377-398.  doi: 10.1016/j.jmaa.2014.09.055.  Google Scholar

[21]

M. Wang, On some free boundary problems of the prey-predator model, J. Differ. Equ., 256 (2014), 3365-3394.  doi: 10.1016/j.jde.2014.02.013.  Google Scholar

[22]

M. Wang and J. Zhao, Free boundary problems for a Lotka-Volterra competition system, J. Dyn. Differ. Equ., 26 (2014), 655-672.  doi: 10.1007/s10884-014-9363-4.  Google Scholar

[23]

M. Wang and J. Zhao, A free boundary problem for the predator-prey model with double free boundaries, J. Dyn. Differ. Equ., 29 (2017), 957-979.  doi: 10.1007/s10884-015-9503-5.  Google Scholar

[24]

L. ZhangW.-T. Li and Z. C. Wang, Entire solution in an ignition nonlocal dispersal equation: Asymmetric kernel, Sci. China Math., 60 (2017), 1791-1804.  doi: 10.1007/s11425-016-9003-7.  Google Scholar

[25]

L. ZhangW. T. LiZ. C. Wang and Y. J. Sun, Entire solutions for nonlocal dispersal equations with bistable nonlinearity: Asymmetric case, Acta Math. Sin. (English Ser.), 35 (2019), 1771-1794.  doi: 10.1007/s10114-019-8294-8.  Google Scholar

[26]

L. ZhuH. Zhao and H. Wang, Complex dynamic behavior of a rumor propagation model with spatial-temporal diffusion terms, Inform. Sci., 349–350 (2016), 119-136.  doi: 10.1016/j.ins.2016.02.031.  Google Scholar

[27]

L. Zhu, H. Zhao and H. Wang, Partial differential equation modeling of rumor propagation in complex networks with higher order of organization, Chaos, 29 (2019), 053106, 23 pp. doi: 10.1063/1.5090268.  Google Scholar

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