### Electronic Research Archive

2020, Issue 3: 1143-1160. doi: 10.3934/era.2020063

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

• Received: 01 May 2020 Revised: 01 May 2020
• 35K57, 45G15, 35R35, 35B40

• 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[J]. Electronic Research Archive, 2020, 28(3): 1143-1160. doi: 10.3934/era.2020063

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• 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.

 [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 [2] Spreading speed revisited: Analysis of a free boundary model. Netw. Heterog. Media (2012) 7: 583-603. [3] The dynamics of a Fisher-KPP nonlocal diffusion model with free boundary. J. Funct. Anal. (2019) 277: 2772-2814. [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. [5] Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary. SIAM J. Math. Anal. (2013) 45: 1995-1996. [6] Y. Du, M. Wang and M. Zhao, Two species nonlocal diffusion systems with free boundaries, preprint, arXiv: 1907.04542. [7] The free boundary problem describing information diffusion in online social networks. J. Differ. Equ. (2013) 254: 1326-1341. [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 [9] The dynamics of nonlocal diffusion systems with different free boundaries. Comm. Pure Appl. Anal. (2020) 19: 3651-3672. [10] Entire solutions in the Fisher-KPP equation with nonlocal dispersal. Nonlinear Anal. Real World Appl. (2010) 11: 2302-2313. [11] Entire solutions of nonlocal dispersal equations with monostable nonlinearity in space periodic habitats. J. Differ. Equ. (2016) 261: 2472-2501. [12] Traveling waves and entire solutions for an epidemic model with asymmetric dispersal. Discrete Contin. Dyn. Syst. (2017) 37: 2483-2512. [13] Invasion entire solutions in a competition system with nonlocal dispersal. Discrete Contin. Dyn. Syst. (2015) 35: 1531-1560. [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 [15] Spreading-vanishing dichotomy in information diffusion in online social networks with intervention. Discrete Contin. Dyn. Syst. Ser. B (2019) 24: 1843-1865. [16] Entire solutions in nonlocal monostable equations: Asymmetric case. Commun. Pure Appl. Anal. (2019) 18: 1049-1072. [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 [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 [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 [20] Invasion by an inferior or superior competitor: A diffusive competition model with a free boundary in a heterogeneous environment. J. Math. Anal. Appl. (2015) 423: 377-398. [21] On some free boundary problems of the prey-predator model. J. Differ. Equ. (2014) 256: 3365-3394. [22] Free boundary problems for a Lotka-Volterra competition system. J. Dyn. Differ. Equ. (2014) 26: 655-672. [23] A free boundary problem for the predator-prey model with double free boundaries. J. Dyn. Differ. Equ. (2017) 29: 957-979. [24] Entire solution in an ignition nonlocal dispersal equation: Asymmetric kernel. Sci. China Math. (2017) 60: 1791-1804. [25] Entire solutions for nonlocal dispersal equations with bistable nonlinearity: Asymmetric case. Acta Math. Sin. (English Ser.) (2019) 35: 1771-1794. [26] Complex dynamic behavior of a rumor propagation model with spatial-temporal diffusion terms. Inform. Sci. (2016) 349–350: 119-136. [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
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