# American Institute of Mathematical Sciences

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

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

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

* Corresponding author: Mingxin Wang

Received  November 2017 Revised  January 2018 Published  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.

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. [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. [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. [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. [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. [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. [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. [8] M. X. Wang, Sobolev Spaces, (in Chinese), Higher Education Press, Bejing, 2013. [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. [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. [11] M. X. Wang, Nonlinear Second Order Parabolic Equations, in: Lecture Notes. [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. [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. [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. [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. [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. [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.

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##### 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. [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. [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. [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. [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. [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. [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. [8] M. X. Wang, Sobolev Spaces, (in Chinese), Higher Education Press, Bejing, 2013. [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. [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. [11] M. X. Wang, Nonlinear Second Order Parabolic Equations, in: Lecture Notes. [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. [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. [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. [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. [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. [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.
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