doi: 10.3934/dcdsb.2022085
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Free boundary problems for the local-nonlocal diffusive model with different moving parameters

1. 

School of Mathematics, Harbin Institute of Technology, Harbin 150001, China

2. 

College of Science, Henan University of Technology, Zhengzhou 450001, China

3. 

School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China

* Corresponding author: Lei Li

Received  December 2021 Revised  March 2022 Early access May 2022

Fund Project: The work is supported by NSFC grants 12171120, 11971128

This paper concerns a class of local and nonlocal diffusion systems with double free boundaries possessing different moving parameters. We firstly obtain the existence, uniqueness and regularity of global solution and then prove that its dynamics are governed by a spreading-vanishing dichotomy. Then the sharp criteria for spreading and vanishing are established. Of particular importance is that long-time behaviors of solution in this problem are quite rich under the Lotka-Volterra type competition, prey-predator and mutualist growth conditions. Moreover, we also provide rough estimates of spreading speeds when spreading happens.

Citation: Heting Zhang, Lei Li, Mingxin Wang. Free boundary problems for the local-nonlocal diffusive model with different moving parameters. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022085
References:
[1]

S. Arora, M. T. Mohan and J. Dabas, Approximate controllability of the non-autonomous impulsive evolution equation with state-dependent delay in Banach space, Nonlinear Anal. Hybrid Syst., 39 (2021), Paper No. 100989, 23 pp. doi: 10.1016/j.nahs.2020.100989.

[2]

P. W. Bates and G. Y. 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. BerestyckiJ. Coville and H. H. Vo, On the definition and the properties of the principal eigenvalue of some nonlocal operators, J. Funct. Anal., 271 (2016), 2701-2751.  doi: 10.1016/j.jfa.2016.05.017.

[4]

G. BuntingY. H. 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.

[5]

J. F. Cao, W. T. Li and J. Wang, A free boundary problem of a diffusive SIRS model with nonlinear incidence, Z. Angew. Math. Phys., 68 (2017), Art. 39, 16 pp. doi: 10.1007/s00033-017-0786-8.

[6]

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

[7]

X. DongJ. P. Wang and M. X. Wang, Free boundary problems with local-nonlocal diffusions and different free boundaries Ⅱ: Spreading-vanishing and long-time behavior, Nonlinear Anal. Real World Appl., 64 (2022), 103445.  doi: 10.1016/j.nonrwa.2021.103445.

[8]

Y. H. DuF. Li and M. L. Zhou, Semi-wave and spreading speed of the nonlocal Fisher-KPP equation with free boundaries, J. Math. Pure Appl., 154 (2021), 30-66.  doi: 10.1016/j.matpur.2021.08.008.

[9]

Y. H. Du and Z. G. 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.

[10]

Y. H. Du and Z. G. 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.

[11]

Y. H. DuH. Matsuzawa and M. L. 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.

[12]

Y. H. Du and W. J. Ni, Analysis of a West Nile virus model with nonlocal diffusion and free boundaries, Nonlinearity, 33 (2020), 4407-4448.  doi: 10.1088/1361-6544/ab8bb2.

[13]

Y. H. DuM. X. Wang and M. Zhao, Two species nonlocal diffusion systems with free boundaries, Discrete Contin. Dyn. Syst., 42 (2022), 1127-1162.  doi: 10.3934/dcds.2021149.

[14]

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.

[15]

J. S. Guo and C. H. Wu, Dynamics for a two-species competition-diffusion model with two free boundaries, Nonlinearity, 28 (2015), 1-27.  doi: 10.1088/0951-7715/28/1/1.

[16]

H. M. Huang and M. X. 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.

[17]

Y. Kaneko and Y. Yamada, A free boundary problem for a reaction diffusion equation appearing in ecology, Adv. Math. Sci. Appl., 21 (2011), 467-492. 

[18]

C. Y. KaoY. Lou and W. X. Shen, Random dispersal vs. non-local dispersal, Discrete Contin. Dyn. Syst., 26 (2010), 551-596.  doi: 10.3934/dcds.2010.26.551.

[19]

L. LiS. Y. Liu and M. X. Wang, A viral propagation model with a nonlinear infection rate and free boundaries, Sci. China Math., 64 (2021), 1971-1992.  doi: 10.1007/s11425-020-1680-0.

[20]

L. Li, W. J. Sheng and M. X. Wang, Systems with nonlocal vs. local diffusions and free boundaries, J. Math. Anal. Appl., 483 (2020), 123646, 27 pp. doi: 10.1016/j.jmaa.2019.123646.

[21]

L. Li and M. X. Wang, Sharp estimates for a nonlocal diffusion problem with a free boundary, preprint, 2021, arXiv: 2108.09165.

[22]

L. Li and M. X. Wang, Dynamics for nonlocal diffusion problems with a free boundary and a fixed boundary, preprint, 2021, arXiv: 2105.13056.

[23]

L. Li and M. X. Wang, Free boundary problems of a mutualist model with nonlocal diffusions, J. Dynam. Differential Equations, 2022. https://doi.org/10.1007/s10884-022-10150-5.

[24]

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

[25]

W. X. Shen and X. 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.

[26]

J. P. Wang and M. X. Wang, Free boundary problems with nonlocal and local diffusions Ⅰ: Global solution, J. Math. Anal. Appl., 490 (2020), 123974, 24 pp. doi: 10.1016/j.jmaa.2020.123974.

[27]

J. P. Wang and M. X. Wang, Free boundary problems with nonlocal and local diffusion Ⅱ: Spreading-vanishing and long-time behavior, Discrete Contin. Dyn. Syst., Ser. B, 25 (2020), 4721-4736.  doi: 10.3934/dcdsb.2020121.

[28]

M. X. Wang, Erratum: Existence and uniqueness of solutions of free boundary problems in heterogeneous environments (vol 24, pg 415, 2019), Discrete Cont. Dyn. Syst. B, (2021). https://doi.org/10.3934/dcdsb.2021269.

[29] M. X. Wang, Nonlinear Second Order Parabolic Equations, CRC Press, Boca Raton, 2021.  doi: 10.1201/9781003150169.
[30]

M. X. Wang, Nonlinear Elliptic Equations (in Chinese), Science Press, Beijing, 2010.

[31]

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

[32]

M. X. Wang, Spreading and vanishing in the diffusive prey-predator model with a free boundary, Commun. Nonlinear Sci. Numer. Simul., 23 (2015), 311-327.  doi: 10.1016/j.cnsns.2014.11.016.

[33]

M. X. Wang and J. F. Zhao, Free boundary problems for a Lotka-Volterra competition system, J. Dynam. Differential Equations, 26 (2014), 655-672.  doi: 10.1007/s10884-014-9363-4.

[34]

M. X. Wang and J. F. Zhao, A free boundary problem for the predator-prey model with double free boundaries, J. Dynam. Differential Equations, 29 (2017), 957-979.  doi: 10.1007/s10884-015-9503-5.

[35]

M. X. Wang and Q. Y. Zhang, Dynamics for the diffusive Leslie-Gower model with double free boundaries, Discrete Contin. Dyn. Syst., 38 (2018), 2591-2607.  doi: 10.3934/dcds.2018109.

[36]

M. X. Wang and Y. Zhang, Two kinds of free boundary problems for the diffusive prey-predator model, Nonlinear Anal. Real World Appl., 24 (2015), 73-82.  doi: 10.1016/j.nonrwa.2015.01.004.

[37]

M. X. Wang and Y. Zhang, Dynamics for a diffusive prey-predator model with different free boundaries, J. Differential Equations, 264 (2018), 3527-3558.  doi: 10.1016/j.jde.2017.11.027.

[38]

M. X. Wang and Y. Zhang, The time-periodic diffusive competition models with a free boundary and sign-changing growth rates, Z. Angew. Math. Phys., 67 (2016), Art. 132, 24 pp. doi: 10.1007/s00033-016-0729-9.

[39]

R. Wang and Y. H. Du, Long-time dynamics of a diffusive epidemic model with free boundaries, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 2201-2238.  doi: 10.3934/dcdsb.2020360.

[40]

J. F. Zhao and M. X. Wang, A free boundary problem of a predator-prey model with higher dimension and heterogeneous environment, Nonlinear Anal. Real World Appl., 16 (2014), 250-263.  doi: 10.1016/j.nonrwa.2013.10.003.

[41]

M. ZhaoW. T. Li and Y. H. Du, The effect of nonlocal reaction in an epidemic model with nonlocal diffusion and free boundaries, Commun. Pure Appl. Anal., 19 (2020), 4599-4620.  doi: 10.3934/cpaa.2020208.

[42]

M. ZhaoY. ZhangW. T. Li and Y. H. Du, The dynamics of a degenerate epidemic model with nonlocal diffusion and free boundaries, J. Differential Equations, 269 (2020), 3347-3386.  doi: 10.1016/j.jde.2020.02.029.

[43]

Y. G. Zhao and M. X. Wang, Free boundary problems for the diffusive competition system in higher dimension with sign-changing coefficients, IMA J. Appl. Math., 81 (2016), 255-280.  doi: 10.1093/imamat/hxv035.

[44]

Y. G. Zhao and M. X. Wang, A reaction-diffusion-advection equation with mixed and free boundary conditions, J. Dynam. Differential Equations, 30 (2018), 743-777.  doi: 10.1007/s10884-017-9571-9.

[45]

Y. G. Zhao and M. X. Wang, Asymptotic behavior of solutions to a nonlinear Stefan problem with different moving parameters, Nonlinear Anal. Real World Appl., 31 (2016), 166-178.  doi: 10.1016/j.nonrwa.2016.02.001.

[46]

Y. Zhang and M. X. Wang, A free boundary problem of the ratio-dependent prey-predator model, Appl. Anal., 94 (2015), 2147-2167.  doi: 10.1080/00036811.2014.979806.

show all references

References:
[1]

S. Arora, M. T. Mohan and J. Dabas, Approximate controllability of the non-autonomous impulsive evolution equation with state-dependent delay in Banach space, Nonlinear Anal. Hybrid Syst., 39 (2021), Paper No. 100989, 23 pp. doi: 10.1016/j.nahs.2020.100989.

[2]

P. W. Bates and G. Y. 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. BerestyckiJ. Coville and H. H. Vo, On the definition and the properties of the principal eigenvalue of some nonlocal operators, J. Funct. Anal., 271 (2016), 2701-2751.  doi: 10.1016/j.jfa.2016.05.017.

[4]

G. BuntingY. H. 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.

[5]

J. F. Cao, W. T. Li and J. Wang, A free boundary problem of a diffusive SIRS model with nonlinear incidence, Z. Angew. Math. Phys., 68 (2017), Art. 39, 16 pp. doi: 10.1007/s00033-017-0786-8.

[6]

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

[7]

X. DongJ. P. Wang and M. X. Wang, Free boundary problems with local-nonlocal diffusions and different free boundaries Ⅱ: Spreading-vanishing and long-time behavior, Nonlinear Anal. Real World Appl., 64 (2022), 103445.  doi: 10.1016/j.nonrwa.2021.103445.

[8]

Y. H. DuF. Li and M. L. Zhou, Semi-wave and spreading speed of the nonlocal Fisher-KPP equation with free boundaries, J. Math. Pure Appl., 154 (2021), 30-66.  doi: 10.1016/j.matpur.2021.08.008.

[9]

Y. H. Du and Z. G. 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.

[10]

Y. H. Du and Z. G. 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.

[11]

Y. H. DuH. Matsuzawa and M. L. 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.

[12]

Y. H. Du and W. J. Ni, Analysis of a West Nile virus model with nonlocal diffusion and free boundaries, Nonlinearity, 33 (2020), 4407-4448.  doi: 10.1088/1361-6544/ab8bb2.

[13]

Y. H. DuM. X. Wang and M. Zhao, Two species nonlocal diffusion systems with free boundaries, Discrete Contin. Dyn. Syst., 42 (2022), 1127-1162.  doi: 10.3934/dcds.2021149.

[14]

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.

[15]

J. S. Guo and C. H. Wu, Dynamics for a two-species competition-diffusion model with two free boundaries, Nonlinearity, 28 (2015), 1-27.  doi: 10.1088/0951-7715/28/1/1.

[16]

H. M. Huang and M. X. 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.

[17]

Y. Kaneko and Y. Yamada, A free boundary problem for a reaction diffusion equation appearing in ecology, Adv. Math. Sci. Appl., 21 (2011), 467-492. 

[18]

C. Y. KaoY. Lou and W. X. Shen, Random dispersal vs. non-local dispersal, Discrete Contin. Dyn. Syst., 26 (2010), 551-596.  doi: 10.3934/dcds.2010.26.551.

[19]

L. LiS. Y. Liu and M. X. Wang, A viral propagation model with a nonlinear infection rate and free boundaries, Sci. China Math., 64 (2021), 1971-1992.  doi: 10.1007/s11425-020-1680-0.

[20]

L. Li, W. J. Sheng and M. X. Wang, Systems with nonlocal vs. local diffusions and free boundaries, J. Math. Anal. Appl., 483 (2020), 123646, 27 pp. doi: 10.1016/j.jmaa.2019.123646.

[21]

L. Li and M. X. Wang, Sharp estimates for a nonlocal diffusion problem with a free boundary, preprint, 2021, arXiv: 2108.09165.

[22]

L. Li and M. X. Wang, Dynamics for nonlocal diffusion problems with a free boundary and a fixed boundary, preprint, 2021, arXiv: 2105.13056.

[23]

L. Li and M. X. Wang, Free boundary problems of a mutualist model with nonlocal diffusions, J. Dynam. Differential Equations, 2022. https://doi.org/10.1007/s10884-022-10150-5.

[24]

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

[25]

W. X. Shen and X. 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.

[26]

J. P. Wang and M. X. Wang, Free boundary problems with nonlocal and local diffusions Ⅰ: Global solution, J. Math. Anal. Appl., 490 (2020), 123974, 24 pp. doi: 10.1016/j.jmaa.2020.123974.

[27]

J. P. Wang and M. X. Wang, Free boundary problems with nonlocal and local diffusion Ⅱ: Spreading-vanishing and long-time behavior, Discrete Contin. Dyn. Syst., Ser. B, 25 (2020), 4721-4736.  doi: 10.3934/dcdsb.2020121.

[28]

M. X. Wang, Erratum: Existence and uniqueness of solutions of free boundary problems in heterogeneous environments (vol 24, pg 415, 2019), Discrete Cont. Dyn. Syst. B, (2021). https://doi.org/10.3934/dcdsb.2021269.

[29] M. X. Wang, Nonlinear Second Order Parabolic Equations, CRC Press, Boca Raton, 2021.  doi: 10.1201/9781003150169.
[30]

M. X. Wang, Nonlinear Elliptic Equations (in Chinese), Science Press, Beijing, 2010.

[31]

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

[32]

M. X. Wang, Spreading and vanishing in the diffusive prey-predator model with a free boundary, Commun. Nonlinear Sci. Numer. Simul., 23 (2015), 311-327.  doi: 10.1016/j.cnsns.2014.11.016.

[33]

M. X. Wang and J. F. Zhao, Free boundary problems for a Lotka-Volterra competition system, J. Dynam. Differential Equations, 26 (2014), 655-672.  doi: 10.1007/s10884-014-9363-4.

[34]

M. X. Wang and J. F. Zhao, A free boundary problem for the predator-prey model with double free boundaries, J. Dynam. Differential Equations, 29 (2017), 957-979.  doi: 10.1007/s10884-015-9503-5.

[35]

M. X. Wang and Q. Y. Zhang, Dynamics for the diffusive Leslie-Gower model with double free boundaries, Discrete Contin. Dyn. Syst., 38 (2018), 2591-2607.  doi: 10.3934/dcds.2018109.

[36]

M. X. Wang and Y. Zhang, Two kinds of free boundary problems for the diffusive prey-predator model, Nonlinear Anal. Real World Appl., 24 (2015), 73-82.  doi: 10.1016/j.nonrwa.2015.01.004.

[37]

M. X. Wang and Y. Zhang, Dynamics for a diffusive prey-predator model with different free boundaries, J. Differential Equations, 264 (2018), 3527-3558.  doi: 10.1016/j.jde.2017.11.027.

[38]

M. X. Wang and Y. Zhang, The time-periodic diffusive competition models with a free boundary and sign-changing growth rates, Z. Angew. Math. Phys., 67 (2016), Art. 132, 24 pp. doi: 10.1007/s00033-016-0729-9.

[39]

R. Wang and Y. H. Du, Long-time dynamics of a diffusive epidemic model with free boundaries, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 2201-2238.  doi: 10.3934/dcdsb.2020360.

[40]

J. F. Zhao and M. X. Wang, A free boundary problem of a predator-prey model with higher dimension and heterogeneous environment, Nonlinear Anal. Real World Appl., 16 (2014), 250-263.  doi: 10.1016/j.nonrwa.2013.10.003.

[41]

M. ZhaoW. T. Li and Y. H. Du, The effect of nonlocal reaction in an epidemic model with nonlocal diffusion and free boundaries, Commun. Pure Appl. Anal., 19 (2020), 4599-4620.  doi: 10.3934/cpaa.2020208.

[42]

M. ZhaoY. ZhangW. T. Li and Y. H. Du, The dynamics of a degenerate epidemic model with nonlocal diffusion and free boundaries, J. Differential Equations, 269 (2020), 3347-3386.  doi: 10.1016/j.jde.2020.02.029.

[43]

Y. G. Zhao and M. X. Wang, Free boundary problems for the diffusive competition system in higher dimension with sign-changing coefficients, IMA J. Appl. Math., 81 (2016), 255-280.  doi: 10.1093/imamat/hxv035.

[44]

Y. G. Zhao and M. X. Wang, A reaction-diffusion-advection equation with mixed and free boundary conditions, J. Dynam. Differential Equations, 30 (2018), 743-777.  doi: 10.1007/s10884-017-9571-9.

[45]

Y. G. Zhao and M. X. Wang, Asymptotic behavior of solutions to a nonlinear Stefan problem with different moving parameters, Nonlinear Anal. Real World Appl., 31 (2016), 166-178.  doi: 10.1016/j.nonrwa.2016.02.001.

[46]

Y. Zhang and M. X. Wang, A free boundary problem of the ratio-dependent prey-predator model, Appl. Anal., 94 (2015), 2147-2167.  doi: 10.1080/00036811.2014.979806.

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