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  December 2018 Early access  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 and Continuous Dynamical Systems - B, 2018, 23 (10) : 4117-4139. doi: 10.3934/dcdsb.2018128
References:
[1]

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.

[2]

R. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, 2003. doi: 10.1002/0470871296.

[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.

[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.

[5]

C. Corduneanu, Integral Equations and Stability of Feedback Systems, Academic Press, New York, London, 1973.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

Y. DuH. 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.

[14]

R. A. Fisher, The wave of advance of advantageous, Ann. Eugenic., 7 (1937), 355-369. 

[15]

J. GeK. KimZ. 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.

[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.

[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.

[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.

[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.

[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.

[21]

R. Miller, On Volterra's population equation, SIAM J. Appl. Math., 14 (1966), 446-452.  doi: 10.1137/0114039.

[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.

[23]

R. Redlinger, On Volterra's population equation with diffusion, SIAM J. Math. Anal., 16 (1985), 135-142.  doi: 10.1137/0516008.

[24]

L. I. Rubinstein, The Stefan Problem, American Mathematical Society, Providence, RI, 1971.

[25]

A. Schiaffino, On a diffusion Volterra equation, Nonlinear Anal., 3 (1979), 595-600.  doi: 10.1016/0362-546X(79)90088-9.

[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.

[27]

A. Tesei, Stability properties for partial Volterra integro-differential equations, Ann. Mat. Pura Appl., 126 (1980), 103-115.  doi: 10.1007/BF01762503.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

show all references

References:
[1]

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.

[2]

R. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, 2003. doi: 10.1002/0470871296.

[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.

[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.

[5]

C. Corduneanu, Integral Equations and Stability of Feedback Systems, Academic Press, New York, London, 1973.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

Y. DuH. 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.

[14]

R. A. Fisher, The wave of advance of advantageous, Ann. Eugenic., 7 (1937), 355-369. 

[15]

J. GeK. KimZ. 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.

[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.

[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.

[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.

[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.

[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.

[21]

R. Miller, On Volterra's population equation, SIAM J. Appl. Math., 14 (1966), 446-452.  doi: 10.1137/0114039.

[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.

[23]

R. Redlinger, On Volterra's population equation with diffusion, SIAM J. Math. Anal., 16 (1985), 135-142.  doi: 10.1137/0516008.

[24]

L. I. Rubinstein, The Stefan Problem, American Mathematical Society, Providence, RI, 1971.

[25]

A. Schiaffino, On a diffusion Volterra equation, Nonlinear Anal., 3 (1979), 595-600.  doi: 10.1016/0362-546X(79)90088-9.

[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.

[27]

A. Tesei, Stability properties for partial Volterra integro-differential equations, Ann. Mat. Pura Appl., 126 (1980), 103-115.  doi: 10.1007/BF01762503.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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