April  2021, 26(4): 1797-1809. doi: 10.3934/dcdsb.2021028

Traveling wave solutions of a free boundary problem with latent heat effect

1. 

Department of Applied Mathematics, Tunghai University, Tunghai University, Taichung, 40704, Taiwan

2. 

Department of Mathematics, National Taiwan University, National Taiwan University, Taipei, 10617, Taiwan

* Corresponding author: Chueh-Hsin Chang

Received  July 2020 Revised  December 2020 Published  April 2021 Early access  January 2021

We study a free boundary problem of two competing species with latent heat effect. We establish the existence and uniqueness of the traveling wave solution and derive upper and lower bounds for the wave speed. Especially our results show that the latent heat retards propagation of the waves.

Citation: Chueh-Hsin Chang, Chiun-Chuan Chen, Chih-Chiang Huang. Traveling wave solutions of a free boundary problem with latent heat effect. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 1797-1809. doi: 10.3934/dcdsb.2021028
References:
[1]

C.-H. Chang and C.-C. Chen, Travelling wave solutions of a free boundary problem for a two-species competitive model, Communications on Pure and Applied Analysis, 12 (2013), 1065-1074.  doi: 10.3934/cpaa.2013.12.1065.  Google Scholar

[2]

C.-N. ChenC.-C. Chen and C.-C. Huang, Traveling waves for the FitzHugh-Nagumo system on an infinite channel, Journal of Differential Equations, 261 (2016), 3010-3041.  doi: 10.1016/j.jde.2016.05.014.  Google Scholar

[3]

E. N. DancerD. HilhorstM. Mimura and L. A. Peletier, Spatial segregation limit of a competition–diffusion system, European J. Appl. Math., 10 (1999), 97-115.  doi: 10.1017/S0956792598003660.  Google Scholar

[4]

Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 45 (2013), 1995-1996.  doi: 10.1137/090771089.  Google Scholar

[5]

Y. DuM. X. Wang and M. Zhou, Semi-wave and spreading speed for the diffusive competition model with a free boundary, J. Math.Pures Appl., 107 (2017), 253-287.  doi: 10.1016/j.matpur.2016.06.005.  Google Scholar

[6]

Y. Du and C. H. Wu, Spreading with two speeds and mass segregation in a diffusive competition system with free boundaries, Calc.Var. Partial Differential Equations, 57 (2018), Art. 52, 36pp. doi: 10.1007/s00526-018-1339-5.  Google Scholar

[7]

M. El-Hachem, S. W. McCue and M. J. Simpson, A sharp-front moving boundary model for malignant invasion, Physica D, 412 (2020), Article 132639, 11pp. doi: 10.1016/j.physd.2020.132639.  Google Scholar

[8]

S. Heinze, A Variational Approach to Traveling Waves, Technical Report 85, Max Planck Institute for Mathematical Sciences, 2001 Google Scholar

[9]

D. HilhorstM. IidaM. Mimura and H. Ninomiya, A competition–diffusion system approximation to the classical two-phase Stefan problem, Japan J. Indust. Appl. Math., 18 (2001), 161-180.  doi: 10.1007/BF03168569.  Google Scholar

[10]

D. HilhorstM. Mimura and R. Schätzle, Vanishing latent heat limit in a Stefan-like problem arising in biology, Nonlinear Anal. Real World Appl., 4 (2003), 261-285.  doi: 10.1016/S1468-1218(02)00009-3.  Google Scholar

[11]

A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Etude de l'equations de la diffusion avec croissance de la quantite de matiere et son application a un probleme biologique, Bull. Univ. Moscou S'er. Internat., A1 (1937), 1–26. English transl. in Dynamics of Curved Fronts, P. Pelc'e (ed.), Academic Press, 1988,105–130. Google Scholar

[12]

M. LuciaC. B. Muratov and M. Novaga, Linear vs. nonlinear selection for the propagation speed of the solutions of scalar reaction-diffusion equations invading an unstable equilibrium, Comm. Pure Appl. Math., 57 (2004), 616-636.  doi: 10.1002/cpa.20014.  Google Scholar

[13]

M. LuciaC. B. Muratov and M. Novaga, Existence of traveling waves of invasion for Ginzburg-Landau-type problems in infinite cylinders, Arch. Ration. Mech. Anal., 188 (2008), 475-508.  doi: 10.1007/s00205-007-0097-x.  Google Scholar

[14]

M. MimuraY. Yamada and S. Yotsutani, A free boundary problem in ecology, Japan J. Appl. Math., 2 (1985), 151-186.  doi: 10.1007/BF03167042.  Google Scholar

[15]

M. MimuraY. Yamada and S. Yotsutani, Stability analysis for free boundary problems in ecology, Hiroshima Math. J., 16 (1986), 477-498.  doi: 10.32917/hmj/1206130304.  Google Scholar

[16]

M. MimuraY. Yamada and S. Yotsutani, Free boundary problems for some reaction-diffusion equations, Hiroshima Math. J., 17 (1987), 241-280.  doi: 10.32917/hmj/1206130066.  Google Scholar

[17]

L. I. Rubinstein, The Stefan Problem, American Mathematical Society, Providence, RI, 1971.  Google Scholar

[18]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, 140, Amer. Math. Soc., Providence, 1994. doi: 10.1090/mmono/140.  Google Scholar

[19]

J. Yang, Asymptotic behavior of solutions for competitive models with a free boundary, Discrete Contin. Dyn. Syst., 35 (2015), 3253-3276.  doi: 10.3934/dcds.2015.35.3253.  Google Scholar

[20]

J. Yang, Traveling wave solutions of a time-periodic competitive system with a free boundary, J. Differential Equations, 265 (2018), 963-978.  doi: 10.1016/j.jde.2018.03.020.  Google Scholar

[21]

J. Yang and B. D. Lou, Traveling wave solutions of competitive models with free boundaries, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 817-826.  doi: 10.3934/dcdsb.2014.19.817.  Google Scholar

show all references

References:
[1]

C.-H. Chang and C.-C. Chen, Travelling wave solutions of a free boundary problem for a two-species competitive model, Communications on Pure and Applied Analysis, 12 (2013), 1065-1074.  doi: 10.3934/cpaa.2013.12.1065.  Google Scholar

[2]

C.-N. ChenC.-C. Chen and C.-C. Huang, Traveling waves for the FitzHugh-Nagumo system on an infinite channel, Journal of Differential Equations, 261 (2016), 3010-3041.  doi: 10.1016/j.jde.2016.05.014.  Google Scholar

[3]

E. N. DancerD. HilhorstM. Mimura and L. A. Peletier, Spatial segregation limit of a competition–diffusion system, European J. Appl. Math., 10 (1999), 97-115.  doi: 10.1017/S0956792598003660.  Google Scholar

[4]

Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 45 (2013), 1995-1996.  doi: 10.1137/090771089.  Google Scholar

[5]

Y. DuM. X. Wang and M. Zhou, Semi-wave and spreading speed for the diffusive competition model with a free boundary, J. Math.Pures Appl., 107 (2017), 253-287.  doi: 10.1016/j.matpur.2016.06.005.  Google Scholar

[6]

Y. Du and C. H. Wu, Spreading with two speeds and mass segregation in a diffusive competition system with free boundaries, Calc.Var. Partial Differential Equations, 57 (2018), Art. 52, 36pp. doi: 10.1007/s00526-018-1339-5.  Google Scholar

[7]

M. El-Hachem, S. W. McCue and M. J. Simpson, A sharp-front moving boundary model for malignant invasion, Physica D, 412 (2020), Article 132639, 11pp. doi: 10.1016/j.physd.2020.132639.  Google Scholar

[8]

S. Heinze, A Variational Approach to Traveling Waves, Technical Report 85, Max Planck Institute for Mathematical Sciences, 2001 Google Scholar

[9]

D. HilhorstM. IidaM. Mimura and H. Ninomiya, A competition–diffusion system approximation to the classical two-phase Stefan problem, Japan J. Indust. Appl. Math., 18 (2001), 161-180.  doi: 10.1007/BF03168569.  Google Scholar

[10]

D. HilhorstM. Mimura and R. Schätzle, Vanishing latent heat limit in a Stefan-like problem arising in biology, Nonlinear Anal. Real World Appl., 4 (2003), 261-285.  doi: 10.1016/S1468-1218(02)00009-3.  Google Scholar

[11]

A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Etude de l'equations de la diffusion avec croissance de la quantite de matiere et son application a un probleme biologique, Bull. Univ. Moscou S'er. Internat., A1 (1937), 1–26. English transl. in Dynamics of Curved Fronts, P. Pelc'e (ed.), Academic Press, 1988,105–130. Google Scholar

[12]

M. LuciaC. B. Muratov and M. Novaga, Linear vs. nonlinear selection for the propagation speed of the solutions of scalar reaction-diffusion equations invading an unstable equilibrium, Comm. Pure Appl. Math., 57 (2004), 616-636.  doi: 10.1002/cpa.20014.  Google Scholar

[13]

M. LuciaC. B. Muratov and M. Novaga, Existence of traveling waves of invasion for Ginzburg-Landau-type problems in infinite cylinders, Arch. Ration. Mech. Anal., 188 (2008), 475-508.  doi: 10.1007/s00205-007-0097-x.  Google Scholar

[14]

M. MimuraY. Yamada and S. Yotsutani, A free boundary problem in ecology, Japan J. Appl. Math., 2 (1985), 151-186.  doi: 10.1007/BF03167042.  Google Scholar

[15]

M. MimuraY. Yamada and S. Yotsutani, Stability analysis for free boundary problems in ecology, Hiroshima Math. J., 16 (1986), 477-498.  doi: 10.32917/hmj/1206130304.  Google Scholar

[16]

M. MimuraY. Yamada and S. Yotsutani, Free boundary problems for some reaction-diffusion equations, Hiroshima Math. J., 17 (1987), 241-280.  doi: 10.32917/hmj/1206130066.  Google Scholar

[17]

L. I. Rubinstein, The Stefan Problem, American Mathematical Society, Providence, RI, 1971.  Google Scholar

[18]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, 140, Amer. Math. Soc., Providence, 1994. doi: 10.1090/mmono/140.  Google Scholar

[19]

J. Yang, Asymptotic behavior of solutions for competitive models with a free boundary, Discrete Contin. Dyn. Syst., 35 (2015), 3253-3276.  doi: 10.3934/dcds.2015.35.3253.  Google Scholar

[20]

J. Yang, Traveling wave solutions of a time-periodic competitive system with a free boundary, J. Differential Equations, 265 (2018), 963-978.  doi: 10.1016/j.jde.2018.03.020.  Google Scholar

[21]

J. Yang and B. D. Lou, Traveling wave solutions of competitive models with free boundaries, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 817-826.  doi: 10.3934/dcdsb.2014.19.817.  Google Scholar

Figure 1.  The intersection between $ y = \lambda c $ and $ y = g(c) $, and the zeros of $ g(c) = 0 $ under three cases: Figure (a): $ g(0)>0 $, $ 0<c_{ \lambda }<c_{0}<c_{min,u} $. Figure (b): $ g(0) = 0 $, $ c_{\lambda } = c_{0} = 0 $, Figure (c): $ g(0)<0 $, $ c_{min,v}<c_{0}<c_{\lambda }<0 $
[1]

Chang-Hong Wu. Spreading speed and traveling waves for a two-species weak competition system with free boundary. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2441-2455. doi: 10.3934/dcdsb.2013.18.2441

[2]

Gary Bunting, Yihong Du, Krzysztof Krakowski. Spreading speed revisited: Analysis of a free boundary model. Networks & Heterogeneous Media, 2012, 7 (4) : 583-603. doi: 10.3934/nhm.2012.7.583

[3]

Hiroshi Matsuzawa. A free boundary problem for the Fisher-KPP equation with a given moving boundary. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1821-1852. doi: 10.3934/cpaa.2018087

[4]

Zhiguo Wang, Hua Nie, Yihong Du. Asymptotic spreading speed for the weak competition system with a free boundary. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5223-5262. doi: 10.3934/dcds.2019213

[5]

Harunori Monobe, Hirokazu Ninomiya. Traveling wave solutions with convex domains for a free boundary problem. Discrete & Continuous Dynamical Systems, 2017, 37 (2) : 905-914. doi: 10.3934/dcds.2017037

[6]

Gregoire Nadin. How does the spreading speed associated with the Fisher-KPP equation depend on random stationary diffusion and reaction terms?. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1785-1803. doi: 10.3934/dcdsb.2015.20.1785

[7]

Jiamin Cao, Peixuan Weng. Single spreading speed and traveling wave solutions of a diffusive pioneer-climax model without cooperative property. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1405-1426. doi: 10.3934/cpaa.2017067

[8]

Yoichi Enatsu, Emiko Ishiwata, Takeo Ushijima. Traveling wave solution for a diffusive simple epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 835-850. doi: 10.3934/dcdss.2020387

[9]

Meng Zhao, Wan-Tong Li, Wenjie Ni. Spreading speed of a degenerate and cooperative epidemic model with free boundaries. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 981-999. doi: 10.3934/dcdsb.2019199

[10]

Harunori Monobe, Hirokazu Ninomiya. Multiple existence of traveling waves of a free boundary problem describing cell motility. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 789-799. doi: 10.3934/dcdsb.2014.19.789

[11]

Jong-Shenq Guo, Ken-Ichi Nakamura, Toshiko Ogiwara, Chang-Hong Wu. The sign of traveling wave speed in bistable dynamics. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3451-3466. doi: 10.3934/dcds.2020047

[12]

Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003

[13]

Fang Li, Xing Liang, Wenxian Shen. Diffusive KPP equations with free boundaries in time almost periodic environments: I. Spreading and vanishing dichotomy. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3317-3338. doi: 10.3934/dcds.2016.36.3317

[14]

Tzong-Yow Lee and Fred Torcaso. Wave propagation in a lattice KPP equation in random media. Electronic Research Announcements, 1997, 3: 121-125.

[15]

Hua Chen, Wenbin Lv, Shaohua Wu. A free boundary problem for a class of parabolic type chemotaxis model. Kinetic & Related Models, 2015, 8 (4) : 667-684. doi: 10.3934/krm.2015.8.667

[16]

Hua Chen, Wenbin Lv, Shaohua Wu. A free boundary problem for a class of parabolic-elliptic type chemotaxis model. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2577-2592. doi: 10.3934/cpaa.2018122

[17]

Yoshiho Akagawa, Elliott Ginder, Syota Koide, Seiro Omata, Karel Svadlenka. A Crank-Nicolson type minimization scheme for a hyperbolic free boundary problem. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021153

[18]

Mohammed Mesk, Ali Moussaoui. On an upper bound for the spreading speed. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021210

[19]

Shitao Liu. Recovery of the sound speed and initial displacement for the wave equation by means of a single Dirichlet boundary measurement. Evolution Equations & Control Theory, 2013, 2 (2) : 355-364. doi: 10.3934/eect.2013.2.355

[20]

Zhenguo Bai, Tingting Zhao. Spreading speed and traveling waves for a non-local delayed reaction-diffusion system without quasi-monotonicity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4063-4085. doi: 10.3934/dcdsb.2018126

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (211)
  • HTML views (97)
  • Cited by (0)

[Back to Top]