ON A PARTIALLY DEGENERATE REACTION-ADVECTION-DIFFUSION SYSTEM WITH FREE BOUNDARY CONDITIONS

Dandan Zhu; Ying Xu; Xueping Li

doi:10.11948/20230391

2025 Volume 15 Issue 1

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2025 > 15(1): 21-38

Next Article Previous Article

Dandan Zhu, Ying Xu, Xueping Li. ON A PARTIALLY DEGENERATE REACTION-ADVECTION-DIFFUSION SYSTEM WITH FREE BOUNDARY CONDITIONS[J]. Journal of Applied Analysis & Computation, 2025, 15(1): 21-38. doi: 10.11948/20230391

Citation:

Dandan Zhu, Ying Xu, Xueping Li. ON A PARTIALLY DEGENERATE REACTION-ADVECTION-DIFFUSION SYSTEM WITH FREE BOUNDARY CONDITIONS[J]. Journal of Applied Analysis & Computation, 2025, 15(1): 21-38. doi: 10.11948/20230391

ON A PARTIALLY DEGENERATE REACTION-ADVECTION-DIFFUSION SYSTEM WITH FREE BOUNDARY CONDITIONS

1.
School of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, China

Author Bio: Email: zzxuying@zzuli.edu.cn(Y. Xu); Email: lixueping@zzuli.edu.cn(X. Li)
Corresponding author: Email: zhudan@zzuli.edu.cn(D. Zhu)
Fund Project: The authors were supported by National Natural Science Foundation of China (11901541, 12101569) and the Doctor Foundation of Zhengzhou University of Light Industry (2020BSJJ039)

Abstract

Abstract

A partially degenerate reaction-diffusion system with advection term and free boundary conditions is investigated in this paper. Firstly, we present a spreading-vanishing dichotomy for the asymptotic behavior of solutions of the system. Then, we obtain criteria for spreading and vanishing. Moreover, numerical simulation is given to illustrate the theoretical results.
- Partially degenerate diffusion /
- free boundary problem /
- spreading and vanishing
MSC: 35K65, 35R35, 35B40

FullText(HTML)

References

[1]	I. Ahn, S. Beak and Z. G. Lin, The spreading fronts of an infective environment in a man-environment-man epidemic model, Appl. Math. Model, 2016, 40, 7082–7101. doi: 10.1016/j.apm.2016.02.038 CrossRef Google Scholar
[2]	J. F. Cao, Y. H. Du, F. Li and W. T. Li, The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries, J. Funct. Anal., 2019, 277, 2772–2814. doi: 10.1016/j.jfa.2019.02.013 CrossRef Google Scholar
[3]	V. Capasso and L. Maddalena, Convergence to equilibrium states for a reaction-diffusion system modeling the spatial spread of a class of bacterial and viral diseases, J. Math. Biol., 1981, 13, 173–184. doi: 10.1007/BF00275212 CrossRef Google Scholar
[4]	Q. L. Chen, F. Q. Li, Z. D. Teng and F. Wang, Global dynamics and asymptotic spreading speeds for a partially degenerate epidemic model with time delay and free boundaries, J. Dyn. Differential Equations, 2022, 34, 1209–1236. doi: 10.1007/s10884-020-09934-4 CrossRef Google Scholar
[5]	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., 2017, 82, 445–470. Google Scholar
[6]	W. Choi and I. Ahn, Non-uniform dispersal of logistic population models with free boundaries in a spatially heterogeneous environment, J. Math. Anal. Appl., 2019, 479, 283–314. doi: 10.1016/j.jmaa.2019.06.027 CrossRef Google Scholar
[7]	Y. H. Du and Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 2010, 42, 377–405. doi: 10.1137/090771089 CrossRef Google Scholar
[8]	J. Fang and X. Q. Zhao, Monotone wavefronts for partially degenerate reaction-diffusion systems, J. Diff. Eqs., 2009, 21, 663–680. doi: 10.1007/s10884-009-9152-7 CrossRef Google Scholar
[9]	H. Gu, Z. G. Lin and B. D. Lou, Long time behavior of solutions of a diffusion-advection logistic model with free boundaries, Appl. Math. Lett., 2014, 37, 49–53. doi: 10.1016/j.aml.2014.05.015 CrossRef Google Scholar
[10]	H. Gu, Z. G. Lin and B. D. Lou, Different asymptotic spreading speeds induced by advection in a diffusion problem with free boundaries, Proc. Amer. Math. Soc., 2015, 143, 1109–1117. Google Scholar
[11]	H. Gu, B. D. Lou and M. L. Zhou, Long time behavior of solutions of Fisher-KPP equation with advection and free boundaries, J. Funct. Anal., 2015, 269, 1714–1768. doi: 10.1016/j.jfa.2015.07.002 CrossRef Google Scholar
[12]	J. S. Guo and C. H. Wu, On a free boundary for a two-species weak competition system, J. Dyn. Differential Equations, 2012, 24, 873–895. doi: 10.1007/s10884-012-9267-0 CrossRef Google Scholar
[13]	J. S. Guo and C. H. Wu, Dynamics for a two-species competition-diffusion model with two free boundaries, Nonlinearity, 2015, 28, 1–27. doi: 10.1088/0951-7715/28/1/1 CrossRef Google Scholar
[14]	K. P. Hadeler and M. A. Lewis, Spatial dynamics of the diffusive logistic equation with a sedentary compartment, Canad. Appl. Math. Quart., 2002, 10, 473–499. Google Scholar
[15]	Y. Kaneko and H. Matsuzawa, Spreading speed and sharp asymptotic profiles of solutions in free boundary problems for nonlinear advection-diffusion equations, J. Math. Anal. Appl., 2015, 428, 43–76. doi: 10.1016/j.jmaa.2015.02.051 CrossRef Google Scholar
[16]	Y. Kaneko, H. Matsuzawa and Y. Yamada, A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions Ⅰ: Classification of asymptotic behavior, Discrete Contin. Dyn. Syst., 2020, 42, 2719–2745. Google Scholar
[17]	Y. Kaneko, H. Matsuzawa and Y. Yamada, A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions Ⅲ: General case, Discrete Contin. Dyn. Syst. S, 2024, 17, 742–761. doi: 10.3934/dcdss.2023089 CrossRef Google Scholar
[18]	F. Li, X. Liang and W. X. Shen, Diffusive KPP equations with free boundaries in time almost periodic environments: Ⅰ. Spreading and vanishing dichotomy, Discrete Contin. Dyn. Syst., 2016, 36, 3317–3338. Google Scholar
[19]	F. Li, X. Liang and W. X. Shen, Diffusive KPP equations with free boundaries in time almost periodic environments: Ⅱ. Spreading speeds and semi-wave solutions, J. Differential Equations, 2016, 261, 2403–2445. doi: 10.1016/j.jde.2016.04.035 CrossRef Google Scholar
[20]	L. Li, S. Y. Liu and M. X. Wang, A viral propagation model with a nonlinear infection rate and free boundaries, Sci. China Math., 2021, 64, 1971–1992. doi: 10.1007/s11425-020-1680-0 CrossRef Google Scholar
[21]	S. Y. Liu, H. M. Huang and M. X. Wang, A free boundary problem for a prey-predator model with degenerate diffusion and predator-stage structure, Discrete Contin. Dyn. Syst. Ser. B, 2020, 25, 1649–1670. Google Scholar
[22]	S. Y. Liu and M. X. Wang, Existence and uniqueness of solution of free boundary problem with partially degenerate diffusion, Nonlinear Anal. RWA, 2020, 54, 103097. doi: 10.1016/j.nonrwa.2020.103097 CrossRef Google Scholar
[23]	N. A. Maidana and H. Yang, Spatial spreading of West Nile virus described by traveling waves, J. Theoret. Biol., 2009, 258, 403–417. doi: 10.1016/j.jtbi.2008.12.032 CrossRef Google Scholar
[24]	H. Monobe and C. H. Wu, On a free boundary problem for a reaction-diffusion-advection logistic model in heterogeneous environment, J. Differential Equations, 2016, 261, 6144–6177. doi: 10.1016/j.jde.2016.08.033 CrossRef Google Scholar
[25]	S. Razvan and D. Gabriel, Numerical approximation of a free boundary problem for a predator-prey model, Numer. Anal. Appl., 2009, 5434, 548–555. Google Scholar
[26]	J. L. Ren, D. D. Zhu and H. Y. Wang, Spreading-vanishing dichotomy in information diffusion in online social networks with intervention, Discrete Contin. Dyn. Syst. Ser. B, 2019, 24, 1843–1865. Google Scholar
[27]	A. K. Tarboush, Z. G. Lin and M. Zhang, Spreading and vanishing in a West Nile virus model with expanding fronts, Sci. China Math., 2017, 60, 841–860. doi: 10.1007/s11425-016-0367-4 CrossRef Google Scholar
[28]	C. R. Tian and S. G. Ruan, A free boundary problem for Aedes aegypti mosquito invasion, Appl. Math. Model., 2017, 46, 203–217. doi: 10.1016/j.apm.2017.01.050 CrossRef Google Scholar
[29]	J. Wang and J. F. Cao, The spreading frontiers in partially degenerate reaction-diffusion systems, Nonlinear Analysis, 2015, 122, 215–238. doi: 10.1016/j.na.2015.04.003 CrossRef Google Scholar
[30]	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., 2015, 423, 377–398. doi: 10.1016/j.jmaa.2014.09.055 CrossRef Google Scholar
[31]	R. Wang and Y. H. Du, Long-time dynamics of a diffusive epidemic model with free boundaries, Discrete Contin. Dyn. Syst. Ser. B, 2021, 4, 2201–2238. Google Scholar
[32]	L. Wei, C. H. Zhang and M. L. Zhou, Long time behavior for solutions of the diffusive logistic equation with advection and free boundary, Calc. Var. Partial Differ. Eq., 2016, 55, 95–128. doi: 10.1007/s00526-016-1039-y CrossRef Google Scholar
[33]	S. L. Wu, Entire solutions in a bistable reaction-diffusion system modeling man-environment-man epidemics, Nonlinear Anal. RWA, 2012, 13, 1991–2005. doi: 10.1016/j.nonrwa.2011.12.020 CrossRef Google Scholar
[34]	D. Xu and X. Q. Zhao, Erratum to "Bistable waves in an epidemic model", J. Dyn. Differential Equations, 2005, 17, 219–247. doi: 10.1007/s10884-005-6294-0 CrossRef Google Scholar
[35]	D. W. Zhang and B. X. Dai, A free boundary problem for the diffusive intraguild predation model with intraspecific competition, J. Math. Anal. Appl., 2019, 474, 381–412. doi: 10.1016/j.jmaa.2019.01.050 CrossRef Google Scholar
[36]	H. T. Zhang, L. Li and M. X. Wang, Free boundary problems for the local-nonlocal diffusive model with different moving parameters, Discrete Contin. Dyn. Syst. Ser. B, 2023, 28, 474–498. doi: 10.3934/dcdsb.2022085 CrossRef Google Scholar
[37]	H. T. Zhang, L. Li and M. X. Wang, The dynamics of partially degenerate nonlocal diffusion systems with free boundaries, J. Math. Anal. Appl., 2022, 512, 126134. doi: 10.1016/j.jmaa.2022.126134 CrossRef Google Scholar
[38]	K. Zhang and X. Q. Zhao, Asymptotic behavior of a reaction-diffusion model with a quiescent stage, Proc. R. Soc. Lond. A, 2007, 463, 1029–1043. Google Scholar
[39]	P. A. Zhang and W. T. Li, Monotonicity and uniqueness of traveling waves for a reaction-diffusion model with a quiescent stage, Nonlinear Anal. TMA, 2010, 72, 2178–2189. doi: 10.1016/j.na.2009.10.016 CrossRef Google Scholar
[40]	W. Y. Zhang, Z. H. Liu and L. Zhou, A free boundary problem of a predator-prey model with a nonlocal reaction term, Z. Angew. Math. Phys., 2021, 72, 1–21. doi: 10.1007/s00033-020-01428-z CrossRef Google Scholar
[41]	Z. D. Zhang and A. K. Tarboush, The diffusive model for West Nile virus with advection and expanding fronts in a heterogeneous environment, International Journal of Biomathematic, 2020, 13, 2050057. doi: 10.1142/S1793524520500576 CrossRef Google Scholar
[42]	M. Zhao, W. T. Li and Y. Zhang, Dynamics of an epidemic model with advection and free boundaries, Math. Biosci. Eng., 2019, 16, 5991–6014. doi: 10.3934/mbe.2019300 CrossRef Google Scholar
[43]	X. Q. Zhao and W. D. Wang, Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. B, 2004, 4, 1117–1128. Google Scholar
[44]	Y. Zhao and M. X. Wang, A reaction-diffusion-advection equation with mixed and free boundary conditions, J. Dyn. Differential Equations, 2018, 30, 743–777. doi: 10.1007/s10884-017-9571-9 CrossRef Google Scholar
[45]	L. Zhou, S. Zhang and Z. H. Liu, A free boundary problem of a predator-prey model with advection in heterogeneous environment, Appl. Math. Comput., 2016, 289, 22–36. Google Scholar
[46]	P. Zhou and D. M. Xiao, The diffusive logistic model with a free boundary in heterogeneous environment, J. Differential Equations, 2014, 256, 1927–1954. Google Scholar
[47]	D. D. Zhu, J. L. Ren and H. P. Zhu, Spatial-temporal basic reproduction number and dynamics for a dengue disease diffusion model, Math. Method. Appl. Sci., 2018, 41, 5388–5403. Google Scholar