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February  2019, 24(2): 415-421. doi: 10.3934/dcdsb.2018179

Existence and uniqueness of solutions of free boundary problems in heterogeneous environments

Mingxin Wang ,

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

* Corresponding author: Mingxin Wang

Received  November 2017 Revised  January 2018 Published  February 2019 Early access  June 2018

Fund Project: This work was supported by NSFC Grants 11771110 and 11371113.

In this short paper we study the existence and uniqueness of solutions of free boundary problems coming from ecology in heterogeneous environments.

Keywords: Existence and uniqueness, free boundary problems, heterogeneous environments.
Mathematics Subject Classification: Primary: 35K51, 35R35, 92B05; Secondary: 35B40.
Citation: Mingxin Wang. Existence and uniqueness of solutions of free boundary problems in heterogeneous environments. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 415-421. doi: 10.3934/dcdsb.2018179
References:
[1]

J. F. Cao, W. T. Li and M. Zhao, A nonlocal diffusion model with free boundaries in spatial heterogeneous environment, J. Math. Anal. Appl., 449 (2017), 1015-1035.  doi: 10.1016/j.jmaa.2016.12.044.  Google Scholar

[2]

Q. L. Chen, F. Q. Li and F. Wang, The diffusive competition problem with a free boundary in heterogeneous time-periodic environment, J. Math. Anal. Appl., 433 (2016), 1594-1613.  doi: 10.1016/j.jmaa.2015.08.062.  Google Scholar

[3]

Q. L. Chen, F. Q. Li and F. Wang, A reaction-diffusion-advection competition model with two free boundaries in heterogeneous time-periodic environment, IMA J. Appl. Math., 82 (2017), 445-470.  doi: 10.1093/imamat/hxw059.  Google Scholar

[4]

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.  Google Scholar

[5]

C. X. Lei and Y. H. Du, Asymptotic profile of the solution to a free boundary problem arising in a shifting climate model, Discrete Contin. Dyn. Syst. B, 22 (2017), 895-911.  doi: 10.3934/dcdsb.2017045.  Google Scholar

[6]

H. Monobe and C-H. Wu, On a free boundary problem for a reaction-diffusion-advection logistic model in heterogeneous environment, J. Differential Equations, 261 (2016), 6144-6177.  doi: 10.1016/j.jde.2016.08.033.  Google Scholar

[7]

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

[8]

M. X. Wang, Sobolev Spaces, (in Chinese), Higher Education Press, Bejing, 2013. Google Scholar

[9]

M. X. 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

[10]

M. X. 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

[11]

M. X. Wang, Nonlinear Second Order Parabolic Equations, in: Lecture Notes. Google Scholar

[12]

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.  Google Scholar

[13]

M. X. Wang and J. F. Zhao, A free boundary problem for a predator-prey model with double free boundaries, J. Dyn. Diff. Equat., 29 (2017), 957-979.  doi: 10.1007/s10884-015-9503-5.  Google Scholar

[14]

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.  Google Scholar

[15]

M. Zhao, W. T. Li and J. F. Cao, A prey-predator model with a free boundary and sign-changing coefficient in time-periodic environment, Discrete Cont. Dyn. Syst. B, 22 (2017), 3295-3316.  doi: 10.3934/dcdsb.2017138.  Google Scholar

[16]

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.  Google Scholar

[17]

L. Zhou, S. Zhang and Z. H. Liu, An evolutional free-boundary problem of a reaction-diffusion-advection system, Proc. Royal Soc. Edinburgh A, 147 (2017), 615-648.  doi: 10.1017/S0308210516000226.  Google Scholar

show all references

References:
[1]

J. F. Cao, W. T. Li and M. Zhao, A nonlocal diffusion model with free boundaries in spatial heterogeneous environment, J. Math. Anal. Appl., 449 (2017), 1015-1035.  doi: 10.1016/j.jmaa.2016.12.044.  Google Scholar

[2]

Q. L. Chen, F. Q. Li and F. Wang, The diffusive competition problem with a free boundary in heterogeneous time-periodic environment, J. Math. Anal. Appl., 433 (2016), 1594-1613.  doi: 10.1016/j.jmaa.2015.08.062.  Google Scholar

[3]

Q. L. Chen, F. Q. Li and F. Wang, A reaction-diffusion-advection competition model with two free boundaries in heterogeneous time-periodic environment, IMA J. Appl. Math., 82 (2017), 445-470.  doi: 10.1093/imamat/hxw059.  Google Scholar

[4]

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.  Google Scholar

[5]

C. X. Lei and Y. H. Du, Asymptotic profile of the solution to a free boundary problem arising in a shifting climate model, Discrete Contin. Dyn. Syst. B, 22 (2017), 895-911.  doi: 10.3934/dcdsb.2017045.  Google Scholar

[6]

H. Monobe and C-H. Wu, On a free boundary problem for a reaction-diffusion-advection logistic model in heterogeneous environment, J. Differential Equations, 261 (2016), 6144-6177.  doi: 10.1016/j.jde.2016.08.033.  Google Scholar

[7]

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

[8]

M. X. Wang, Sobolev Spaces, (in Chinese), Higher Education Press, Bejing, 2013. Google Scholar

[9]

M. X. 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

[10]

M. X. 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

[11]

M. X. Wang, Nonlinear Second Order Parabolic Equations, in: Lecture Notes. Google Scholar

[12]

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.  Google Scholar

[13]

M. X. Wang and J. F. Zhao, A free boundary problem for a predator-prey model with double free boundaries, J. Dyn. Diff. Equat., 29 (2017), 957-979.  doi: 10.1007/s10884-015-9503-5.  Google Scholar

[14]

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.  Google Scholar

[15]

M. Zhao, W. T. Li and J. F. Cao, A prey-predator model with a free boundary and sign-changing coefficient in time-periodic environment, Discrete Cont. Dyn. Syst. B, 22 (2017), 3295-3316.  doi: 10.3934/dcdsb.2017138.  Google Scholar

[16]

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.  Google Scholar

[17]

L. Zhou, S. Zhang and Z. H. Liu, An evolutional free-boundary problem of a reaction-diffusion-advection system, Proc. Royal Soc. Edinburgh A, 147 (2017), 615-648.  doi: 10.1017/S0308210516000226.  Google Scholar

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