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  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 $
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