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Free boundary problems with nonlocal and local diffusions Ⅱ: Spreading-vanishing and long-time behavior
School of Mathematics, Harbin Institute of Technology, Harbin 150001, China |
This is part Ⅱ of our study on the free boundary problems with nonlocal and local diffusions. In part Ⅰ, we obtained the existence, uniqueness, regularity and estimates of global solution. In part Ⅱ here, we show a spreading-vanishing dichotomy, and provide the criteria of spreading and vanishing, as well as the long time behavior of solution when spreading happens.
References:
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P. 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. |
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H. Berestycki, J. 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. |
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J.-F. Cao, Y. Du, F. 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. |
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J. F. Cao, W. T. Li and J. Wang, A Lotka-Volterra competition model with nonlocal diffusion and free boundaries, preprint, arXiv: 1905.09584, 2019. |
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J. Coville,
On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differential Equations, 249 (2010), 2921-2953.
doi: 10.1016/j.jde.2010.07.003. |
| [6] |
Y. Du, M. Wang and M. Zhao, Two species nonlocal diffusion systems with free boundaries, preprint, arXiv: 1907.04542v1, 2019. |
| [7] |
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. |
| [8] |
L. Li, W. Sheng and M. 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. |
| [9] |
J. Wang and M. Wang, Free boundary problems with nonlocal and local diffusions Ⅰ: Global solution, J. Math. Anal. Appl., (2020).
doi: 10.1016/j.jmaa.2020.123974. |
| [10] |
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. |
| [11] |
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. |
| [12] |
M. Wang and Q. 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. |
| [13] |
M. Wang and Y. Zhang,
Dynamics for a diffusive prey-predator model with different free boundaries, J. Differental Equations, 264 (2018), 3527-3558.
doi: 10.1016/j.jde.2017.11.027. |
| [14] |
M. 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. |
| [15] |
M. 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. |
| [16] |
M. Wang and J. 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. |
| [17] |
M. Wang and J. Zhao,
A free boundary problem for a predator-prey model with double free boundaries, J. Dynam. Differential Equations, 29 (2017), 957-979.
doi: 10.1007/s10884-015-9503-5. |
| [18] |
Y. Zhang and M. Wang,
A free boundary problem of the ratio-dependent prey-predator model, Appl. Anal., 94 (2015), 2147-2167.
doi: 10.1080/00036811.2014.979806. |
| [19] |
Y. Zhao and M. 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. |
show all references
References:
| [1] |
P. 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. |
| [2] |
H. Berestycki, J. 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. |
| [3] |
J.-F. Cao, Y. Du, F. 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. |
| [4] |
J. F. Cao, W. T. Li and J. Wang, A Lotka-Volterra competition model with nonlocal diffusion and free boundaries, preprint, arXiv: 1905.09584, 2019. |
| [5] |
J. Coville,
On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differential Equations, 249 (2010), 2921-2953.
doi: 10.1016/j.jde.2010.07.003. |
| [6] |
Y. Du, M. Wang and M. Zhao, Two species nonlocal diffusion systems with free boundaries, preprint, arXiv: 1907.04542v1, 2019. |
| [7] |
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. |
| [8] |
L. Li, W. Sheng and M. 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. |
| [9] |
J. Wang and M. Wang, Free boundary problems with nonlocal and local diffusions Ⅰ: Global solution, J. Math. Anal. Appl., (2020).
doi: 10.1016/j.jmaa.2020.123974. |
| [10] |
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. |
| [11] |
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. |
| [12] |
M. Wang and Q. 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. |
| [13] |
M. Wang and Y. Zhang,
Dynamics for a diffusive prey-predator model with different free boundaries, J. Differental Equations, 264 (2018), 3527-3558.
doi: 10.1016/j.jde.2017.11.027. |
| [14] |
M. 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. |
| [15] |
M. 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. |
| [16] |
M. Wang and J. 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. |
| [17] |
M. Wang and J. Zhao,
A free boundary problem for a predator-prey model with double free boundaries, J. Dynam. Differential Equations, 29 (2017), 957-979.
doi: 10.1007/s10884-015-9503-5. |
| [18] |
Y. Zhang and M. Wang,
A free boundary problem of the ratio-dependent prey-predator model, Appl. Anal., 94 (2015), 2147-2167.
doi: 10.1080/00036811.2014.979806. |
| [19] |
Y. Zhao and M. 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. |
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