# American Institute of Mathematical Sciences

December  2018, 23(10): 4117-4139. doi: 10.3934/dcdsb.2018128

## On a free boundary problem for a nonlocal reaction-diffusion model

 School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

Received  July 2017 Revised  November 2017 Published  April 2018

This paper is concerned with the spreading or vanishing dichotomy of a species which is characterized by a reaction-diffusion Volterra model with nonlocal spatial convolution and double free boundaries. Compared with classical reaction-diffusion equations, the main difficulty here is the lack of a comparison principle in nonlocal reaction-diffusion equations. By establishing some suitable comparison principles over some different parabolic regions, we get the sufficient conditions that ensure the species spreading or vanishing, as well as the estimates of the spreading speed if species spreading happens. Particularly, we establish the global attractivity of the unique positive equilibrium by a method of successive improvement of lower and upper solutions.

Citation: Jia-Feng Cao, Wan-Tong Li, Meng Zhao. On a free boundary problem for a nonlocal reaction-diffusion model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4117-4139. doi: 10.3934/dcdsb.2018128
##### References:
 [1] G. Bunting, Y. 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 [2] R. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, 2003. doi: 10.1002/0470871296. Google Scholar [3] X. Chen and A. Friedman, A free boundary problem arising in a model of wound healing, SIAM J. Math. Anal., 32 (2000), 778-800. doi: 10.1137/S0036141099351693. Google Scholar [4] X. Chen and A. Friedman, A free boundary problem for an elliptic-hyperbolic system: An application to tumor growth, SIAM J. Math. Anal., 35 (2003), 974-986. doi: 10.1137/S0036141002418388. Google Scholar [5] C. Corduneanu, Integral Equations and Stability of Feedback Systems, Academic Press, New York, London, 1973. Google Scholar [6] K. Deng and Y. Wu, Global stabilityfor a nonlocal reaction-diffusion population model, Nonlinear Anal. Real World Appl., 25 (2015), 127-136. doi: 10.1016/j.nonrwa.2015.03.006. Google Scholar [7] Y. Du and Z. Guo, Spreading-Vanishing dichotomy in a diffusive logistic model with a free boundary Ⅱ, J. Differential Equations, 250 (2011), 4336-4366. doi: 10.1016/j.jde.2011.02.011. Google Scholar [8] Y. Du and X. Liang, Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model, Ann.Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 279-305. doi: 10.1016/j.anihpc.2013.11.004. Google Scholar [9] Y. Du and Z. Lin, Spreading-Vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405. doi: 10.1137/090771089. Google Scholar [10] Y. Du and Z. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discrete Contin. Dyn. Syst.Ser. B, 19 (2014), 3105-3132. doi: 10.3934/dcdsb.2014.19.3105. Google Scholar [11] Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724. doi: 10.4171/JEMS/568. Google Scholar [12] Y. Du and L. Ma, Logistic type equations on $\mathbb{R}^N$ by a squeezing method involving boundary blow-up solutions, J. London Math. Soc., 64 (2001), 107-124. doi: 10.1017/S0024610701002289. Google Scholar [13] Y. Du, H. Matsuzawa and M. Zhou, Sharp estimate of the spreading speed determined by nonlinear free boundary problems, SIAM J. Math. Anal., 46 (2014), 375-396. doi: 10.1137/130908063. Google Scholar [14] R. A. Fisher, The wave of advance of advantageous, Ann. Eugenic., 7 (1937), 355-369. Google Scholar [15] J. Ge, K. Kim, Z. Lin and H. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differential Equations, 259 (2015), 5486-5509. doi: 10.1016/j.jde.2015.06.035. Google Scholar [16] J. S. Guo and C. H. Wu, On a free boundary problem for a two-species weak competition system, J. Dynam. Differential Equations, 24 (2012), 873-895. doi: 10.1007/s10884-012-9267-0. Google Scholar [17] H. Huang and M. Wang, The reaction-diffusion system for an SIR epidemic model with a free boundary, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2039-2050. doi: 10.3934/dcdsb.2015.20.2039. Google Scholar [18] A. N. Kolmogorov, I. G. Petrovski and N. S. Piskunov, Ètude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Univ. État. Moscou Sér. Intern. A 1 (1937), 1-26; English transl. in: P. Pelcé (Ed. ), Dynamics of Curved Fronts, Academic Press, 1988,105-130.Google Scholar [19] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Academic Press, New York, London, 1968. Google Scholar [20] Z. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892. doi: 10.1088/0951-7715/20/8/004. Google Scholar [21] R. Miller, On Volterra's population equation, SIAM J. Appl. Math., 14 (1966), 446-452. doi: 10.1137/0114039. Google Scholar [22] R. Peng and X. Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession, Discrete Contin. Dyn. Syst., 33 (2013), 2007-2031. doi: 10.3934/dcds.2013.33.2007. Google Scholar [23] R. Redlinger, On Volterra's population equation with diffusion, SIAM J. Math. Anal., 16 (1985), 135-142. doi: 10.1137/0516008. Google Scholar [24] L. I. Rubinstein, The Stefan Problem, American Mathematical Society, Providence, RI, 1971. Google Scholar [25] A. Schiaffino, On a diffusion Volterra equation, Nonlinear Anal., 3 (1979), 595-600. doi: 10.1016/0362-546X(79)90088-9. Google Scholar [26] A. Schiaffino and A. Tesei, Monotone methods and attractivity results for Volterra integro-partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 89 (1981), 135-142. doi: 10.1017/S0308210500032418. Google Scholar [27] A. Tesei, Stability properties for partial Volterra integro-differential equations, Ann. Mat. Pura Appl., 126 (1980), 103-115. doi: 10.1007/BF01762503. Google Scholar [28] V. Volterra, Lecons sur la Théorie Mathématique de la Lutte Pour la vie, Reprint of the 1931 original. Les Grands Classiques Gauthier-Villars. Éditions Jacques Gabay, Sceaux, 1990. Google Scholar [29] 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 [30] M. Wang, The diffusive logistic equation with a free boundary and sign-changing coefficient, J. Differential Equations, 258 (2015), 1252-1266. doi: 10.1016/j.jde.2014.10.022. Google Scholar [31] M. Wang and J. Zhao, Free boundary problem for a Lotka-Volterra competition system, J. Dynam. Differential Equations, 26 (2014), 655-672. doi: 10.1007/s10884-014-9363-4. Google Scholar [32] M. Wang, A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment, J. Funct. Anal., 270 (2016), 483-508. doi: 10.1016/j.jfa.2015.10.014. Google Scholar [33] Y. Yamada, On a certain class of semilinear Volterra diffusion equations, J. Math. Anal. Appl., 88 (1982), 433-451. doi: 10.1016/0022-247X(82)90205-0. Google Scholar [34] P. Zhou and Z. Lin, Global existence and blowup of a nonlocal problem in space with free boundary, J. Funct. Anal., 262 (2012), 3409-3429. doi: 10.1016/j.jfa.2012.01.018. Google Scholar

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##### References:
 [1] G. Bunting, Y. 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 [2] R. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, 2003. doi: 10.1002/0470871296. Google Scholar [3] X. Chen and A. Friedman, A free boundary problem arising in a model of wound healing, SIAM J. Math. Anal., 32 (2000), 778-800. doi: 10.1137/S0036141099351693. Google Scholar [4] X. Chen and A. Friedman, A free boundary problem for an elliptic-hyperbolic system: An application to tumor growth, SIAM J. Math. Anal., 35 (2003), 974-986. doi: 10.1137/S0036141002418388. Google Scholar [5] C. Corduneanu, Integral Equations and Stability of Feedback Systems, Academic Press, New York, London, 1973. Google Scholar [6] K. Deng and Y. Wu, Global stabilityfor a nonlocal reaction-diffusion population model, Nonlinear Anal. Real World Appl., 25 (2015), 127-136. doi: 10.1016/j.nonrwa.2015.03.006. Google Scholar [7] Y. Du and Z. Guo, Spreading-Vanishing dichotomy in a diffusive logistic model with a free boundary Ⅱ, J. Differential Equations, 250 (2011), 4336-4366. doi: 10.1016/j.jde.2011.02.011. Google Scholar [8] Y. Du and X. Liang, Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model, Ann.Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 279-305. doi: 10.1016/j.anihpc.2013.11.004. Google Scholar [9] Y. Du and Z. Lin, Spreading-Vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405. doi: 10.1137/090771089. Google Scholar [10] Y. Du and Z. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discrete Contin. Dyn. Syst.Ser. B, 19 (2014), 3105-3132. doi: 10.3934/dcdsb.2014.19.3105. Google Scholar [11] Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724. doi: 10.4171/JEMS/568. Google Scholar [12] Y. Du and L. Ma, Logistic type equations on $\mathbb{R}^N$ by a squeezing method involving boundary blow-up solutions, J. London Math. Soc., 64 (2001), 107-124. doi: 10.1017/S0024610701002289. Google Scholar [13] Y. Du, H. Matsuzawa and M. Zhou, Sharp estimate of the spreading speed determined by nonlinear free boundary problems, SIAM J. Math. Anal., 46 (2014), 375-396. doi: 10.1137/130908063. Google Scholar [14] R. A. Fisher, The wave of advance of advantageous, Ann. Eugenic., 7 (1937), 355-369. Google Scholar [15] J. Ge, K. Kim, Z. Lin and H. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differential Equations, 259 (2015), 5486-5509. doi: 10.1016/j.jde.2015.06.035. Google Scholar [16] J. S. Guo and C. H. Wu, On a free boundary problem for a two-species weak competition system, J. Dynam. Differential Equations, 24 (2012), 873-895. doi: 10.1007/s10884-012-9267-0. Google Scholar [17] H. Huang and M. Wang, The reaction-diffusion system for an SIR epidemic model with a free boundary, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2039-2050. doi: 10.3934/dcdsb.2015.20.2039. Google Scholar [18] A. N. Kolmogorov, I. G. Petrovski and N. S. Piskunov, Ètude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Univ. État. Moscou Sér. Intern. A 1 (1937), 1-26; English transl. in: P. Pelcé (Ed. ), Dynamics of Curved Fronts, Academic Press, 1988,105-130.Google Scholar [19] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Academic Press, New York, London, 1968. Google Scholar [20] Z. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892. doi: 10.1088/0951-7715/20/8/004. Google Scholar [21] R. Miller, On Volterra's population equation, SIAM J. Appl. Math., 14 (1966), 446-452. doi: 10.1137/0114039. Google Scholar [22] R. Peng and X. Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession, Discrete Contin. Dyn. Syst., 33 (2013), 2007-2031. doi: 10.3934/dcds.2013.33.2007. Google Scholar [23] R. Redlinger, On Volterra's population equation with diffusion, SIAM J. Math. Anal., 16 (1985), 135-142. doi: 10.1137/0516008. Google Scholar [24] L. I. Rubinstein, The Stefan Problem, American Mathematical Society, Providence, RI, 1971. Google Scholar [25] A. Schiaffino, On a diffusion Volterra equation, Nonlinear Anal., 3 (1979), 595-600. doi: 10.1016/0362-546X(79)90088-9. Google Scholar [26] A. Schiaffino and A. Tesei, Monotone methods and attractivity results for Volterra integro-partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 89 (1981), 135-142. doi: 10.1017/S0308210500032418. Google Scholar [27] A. Tesei, Stability properties for partial Volterra integro-differential equations, Ann. Mat. Pura Appl., 126 (1980), 103-115. doi: 10.1007/BF01762503. Google Scholar [28] V. Volterra, Lecons sur la Théorie Mathématique de la Lutte Pour la vie, Reprint of the 1931 original. Les Grands Classiques Gauthier-Villars. Éditions Jacques Gabay, Sceaux, 1990. Google Scholar [29] 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 [30] M. Wang, The diffusive logistic equation with a free boundary and sign-changing coefficient, J. Differential Equations, 258 (2015), 1252-1266. doi: 10.1016/j.jde.2014.10.022. Google Scholar [31] M. Wang and J. Zhao, Free boundary problem for a Lotka-Volterra competition system, J. Dynam. Differential Equations, 26 (2014), 655-672. doi: 10.1007/s10884-014-9363-4. Google Scholar [32] M. Wang, A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment, J. Funct. Anal., 270 (2016), 483-508. doi: 10.1016/j.jfa.2015.10.014. Google Scholar [33] Y. Yamada, On a certain class of semilinear Volterra diffusion equations, J. Math. Anal. Appl., 88 (1982), 433-451. doi: 10.1016/0022-247X(82)90205-0. Google Scholar [34] P. Zhou and Z. Lin, Global existence and blowup of a nonlocal problem in space with free boundary, J. Funct. Anal., 262 (2012), 3409-3429. doi: 10.1016/j.jfa.2012.01.018. Google Scholar
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