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. CaoW. 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. ChenF. 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. ChenF. 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. ZhaoW. 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. ZhouS. 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. CaoW. 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. ChenF. 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. ChenF. 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. ZhaoW. 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. ZhouS. 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

[1]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[2]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[3]

Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326

[4]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[5]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

[6]

Min Xi, Wenyu Sun, Jun Chen. Survey of derivative-free optimization. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 537-555. doi: 10.3934/naco.2020050

[7]

Vieri Benci, Marco Cococcioni. The algorithmic numbers in non-archimedean numerical computing environments. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020449

[8]

Yuxin Zhang. The spatially heterogeneous diffusive rabies model and its shadow system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020357

[9]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[10]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[11]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453

[12]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[13]

Monia Capanna, Jean C. Nakasato, Marcone C. Pereira, Julio D. Rossi. Homogenization for nonlocal problems with smooth kernels. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020385

[14]

Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 331-351. doi: 10.3934/dcdss.2020325

[15]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[16]

Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020052

[17]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119

[18]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267

[19]

Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260

[20]

Cheng Peng, Zhaohui Tang, Weihua Gui, Qing Chen, Jing He. A bidirectional weighted boundary distance algorithm for time series similarity computation based on optimized sliding window size. Journal of Industrial & Management Optimization, 2021, 17 (1) : 205-220. doi: 10.3934/jimo.2019107

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (353)
  • HTML views (470)
  • Cited by (6)

Other articles
by authors

[Back to Top]