American Institute of Mathematical Sciences

Advanced
December  2017, 37(12): 6291-6318. doi: 10.3934/dcds.2017272

Asymptotic behavior of traveling waves for a three-component system with nonlocal dispersal and its application

Fang-Di Dong , Wan-Tong Li and  Jia-Bing Wang

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

Received  January 2017 Revised  June 2017 Published  August 2017

In this paper, we provide a general approach to study the asymptotic behavior of traveling wave solutions for a three-component system with nonlocal dispersal. Then as an important application, we establish a new type of entire solutions which behave as two traveling wave solutions coming from both sides of $x$-axis for a three-species Lotka-Volterra competition system.

Keywords: Asymptotic behavior, three-component system, traveling waves, entire solution, nonlocal dispersal.
Mathematics Subject Classification: Primary: 35K57, 37K65; Secondary: 92D30.
Citation: Fang-Di Dong, Wan-Tong Li, Jia-Bing Wang. Asymptotic behavior of traveling waves for a three-component system with nonlocal dispersal and its application. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6291-6318. doi: 10.3934/dcds.2017272
References:
[1]

P. W. Bates, P. C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Ration. Mech. Anal., 138 (1997), 105-136.  doi: 10.1007/s002050050037.  Google Scholar

[2]

J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439.  doi: 10.1090/S0002-9939-04-07432-5.  Google Scholar

[3]

X. Chen and J. S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), 123-146.  doi: 10.1007/s00208-003-0414-0.  Google Scholar

[4]

X. Chen and J. S. Guo, Existence and uniqueness of entire solutions for a reaction-diffusion equation, J. Differential Equations, 212 (2005), 62-84.  doi: 10.1016/j.jde.2004.10.028.  Google Scholar

[5]

X. Chen, J. S. Guo and H. Ninomiya, Entire solutions of reaction-diffusion equations with balanced bistable nonlinearities, Proc. R. Soc. Edinb. Sect. A, 136 (2006), 1207-1237.  doi: 10.1017/S0308210500004959.  Google Scholar

[6]

X. Chen, S. C. Fu and J. S. Guo, Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices, SIAM J. Math. Anal., 38 (2006), 233-258.  doi: 10.1137/050627824.  Google Scholar

[7]

A. De Masi, T. Gobron and E. Presutti, Travelling fronts in non-local evolution equations, Arch. Ration. Mech. Anal., 132 (1995), 143-205.  doi: 10.1007/BF00380506.  Google Scholar

[8]

A. De Masi, E. Orlandi, E. Presutti and L. Triolo, Stability of the interface in a model of phase separation, Proc. R.Soc. Edinb. A, 124 (1994), 1013-1022.  doi: 10.1017/S0308210500022472.  Google Scholar

[9]

O. Diekmann and H. Kaper, On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal., 2 (1978), 721-737.  doi: 10.1016/0362-546X(78)90015-9.  Google Scholar

[10]

B. Ermentrout and J. Mcleod, Existence and uniqueness of travelling waves for a neural network, Proc. R. Soc. Edinb. A, 123 (1993), 461-478.  doi: 10.1017/S030821050002583X.  Google Scholar

[11]

P. C. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in Nonlinear Analysis, (2003), 153-191.   Google Scholar

[12]

P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics 28, Springer, Berlin, 1979.  Google Scholar

[13]

Y. Fukao, Y. Morita and H. Ninomiya, Some entire solutions of the Allen-Cahn equation, Taiwanes J. Math., 8 (2004), 15-32.  doi: 10.11650/twjm/1500558454.  Google Scholar

[14]

J. S. Guo and C. H. Wu, Traveling wave front for a two-component lattice dynamical system arising in competition models, J. Differential Equations, 252 (2012), 4357-4391.  doi: 10.1016/j.jde.2012.01.009.  Google Scholar

[15]

J. S. Guo, Y. Wang, C. H. Wu and C. C. Wu, The minimal speed of traveling wave solutions for a diffusive three species competition system, Taiwan. J. Math., 19 (2015), 1805-1829.  doi: 10.11650/tjm.19.2015.5373.  Google Scholar

[16]

J. S. Guo and Y. C. Lin, Entire solutions for a discrete diffusive equation with bistable convolution type nonlinearity, Osaka J. Math., 50 (2013), 607-629.   Google Scholar

[17]

J. S. Guo and C. H. Wu, Entire solutions for a two-component competition system in a lattice, Tohoku. Math. J., 62 (2010), 17-28.  doi: 10.2748/tmj/1270041024.  Google Scholar

[18]

J. S. Guo and Y. Morita, Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations, Discrete Contin. Dyn. Syst., 12 (2005), 193-212.  doi: 10.3934/dcds.2005.12.193.  Google Scholar

[19]

F. Hamel and N. Nadirashvili, Entire solutions of the KPP equations, Comm. Pure Appl. Math., 52 (1999), 1255-1276.  doi: 10.1002/(SICI)1097-0312(199910)52:10<1255::AID-CPA4>3.0.CO;2-W.  Google Scholar

[20]

F. Hamel and N. Nadirashvili, Traveling fronts and entire solutions of the Fisher-KPP equation in $\mathbb{R}^{N}$, Arch. Rational Mech. Anal., 157 (2001), 91-163.  doi: 10.1007/PL00004238.  Google Scholar

[21]

W. T. Li, Y. J. Sun and Z. C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Anal. Real World Appl., 11 (2010), 2302-2313.  doi: 10.1016/j.nonrwa.2009.07.005.  Google Scholar

[22]

W. T. Li, J. B. Wang and L. Zhang, Entire solutions of nonlocal dispersal equations with monostable nonlinearity in space periodic habitats, J. Differential Equations, 261 (2016), 2472-2501.  doi: 10.1016/j.jde.2016.05.006.  Google Scholar

[23]

W. T. Li, Z. C. Wang and J. Wu, Entire solutions in monostable reaction-diffusion equations with delayed nonlinearity, J. Differential Equations, 245 (2008), 102-129.  doi: 10.1016/j.jde.2008.03.023.  Google Scholar

[24]

W. T. Li, L. Zhang and G. B. Zhang, Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Contin. Dyn. Syst., 35 (2015), 1531-1560.  doi: 10.3934/dcds.2015.35.1531.  Google Scholar

[25]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion equations, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590.  Google Scholar

[26]

Y. Morita and H. Ninomiya, Entire solution with merging fronts to reaction-diffusion equations, J. Dynam. Differential Equations, 18 (2006), 841-861.  doi: 10.1007/s10884-006-9046-x.  Google Scholar

[27]

Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217-2240.  doi: 10.1137/080723715.  Google Scholar

[28]

Y. J. Sun, W. T. Li and Z. C. Wang, Entire solutions in nonlocal dispersal equations with bistable nonlinearity, J. Differential Equations, 251 (2011), 551-581.  doi: 10.1016/j.jde.2011.04.020.  Google Scholar

[29]

M. Wang and G. Lv, Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delays, Nonlineariry, 23 (2010), 1609-1630.  doi: 10.1088/0951-7715/23/7/005.  Google Scholar

[30]

Z. C. Wang, W. T. Li and S. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations, 20 (2008), 563-607.  doi: 10.1007/s10884-008-9103-8.  Google Scholar

[31]

Z. C. Wang, W. T. Li and S. Ruan, Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity, Trans. Am. Math. Soc., 361 (2009), 2047-2084.  doi: 10.1090/S0002-9947-08-04694-1.  Google Scholar

[32]

Z. C. Wang, W. T. Li and J. Wu, Entire solutions in delayed lattice differential equations with monostable nonlinearity, SIAM J. Math. Anal., 40 (2009), 2392-2420.  doi: 10.1137/080727312.  Google Scholar

[33]

C. H. Wu, A general approach to the asymptotic behavior of traveling waves in a class of three-component lattice dynamical systems, J. Dynam. Differential Equations, 28 (2016), 317-338.  doi: 10.1007/s10884-016-9524-8.  Google Scholar

[34]

S. L. Wu and H. Wang, Front-like entire solutions for monostable reaction-diffusion systems, J. Dynam. Differential Equations, 25 (2013), 505-533.  doi: 10.1007/s10884-013-9293-6.  Google Scholar

[35]

H. Yagisita, Backward global solutions characterizing annihilation dynamics of travelling fronts, Publ. Res. Inst. Math. Sci., 39 (2003), 117-164.  doi: 10.2977/prims/1145476150.  Google Scholar

[36]

G. B. Zhang, W. T. Li and Z. C. Wang, Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity, J. Differential Equations, 252 (2012), 5096-5124.  doi: 10.1016/j.jde.2012.01.014.  Google Scholar

[37]

L. Zhang, W. T. Li and S. L. Wu, Multi-type entire solutions in a nonlocal dispersal epidemic model, J. Dynam. Differential Equations, 28 (2016), 189-224.  doi: 10.1007/s10884-014-9416-8.  Google Scholar

show all references

References:
[1]

P. W. Bates, P. C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Ration. Mech. Anal., 138 (1997), 105-136.  doi: 10.1007/s002050050037.  Google Scholar

[2]

J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439.  doi: 10.1090/S0002-9939-04-07432-5.  Google Scholar

[3]

X. Chen and J. S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), 123-146.  doi: 10.1007/s00208-003-0414-0.  Google Scholar

[4]

X. Chen and J. S. Guo, Existence and uniqueness of entire solutions for a reaction-diffusion equation, J. Differential Equations, 212 (2005), 62-84.  doi: 10.1016/j.jde.2004.10.028.  Google Scholar

[5]

X. Chen, J. S. Guo and H. Ninomiya, Entire solutions of reaction-diffusion equations with balanced bistable nonlinearities, Proc. R. Soc. Edinb. Sect. A, 136 (2006), 1207-1237.  doi: 10.1017/S0308210500004959.  Google Scholar

[6]

X. Chen, S. C. Fu and J. S. Guo, Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices, SIAM J. Math. Anal., 38 (2006), 233-258.  doi: 10.1137/050627824.  Google Scholar

[7]

A. De Masi, T. Gobron and E. Presutti, Travelling fronts in non-local evolution equations, Arch. Ration. Mech. Anal., 132 (1995), 143-205.  doi: 10.1007/BF00380506.  Google Scholar

[8]

A. De Masi, E. Orlandi, E. Presutti and L. Triolo, Stability of the interface in a model of phase separation, Proc. R.Soc. Edinb. A, 124 (1994), 1013-1022.  doi: 10.1017/S0308210500022472.  Google Scholar

[9]

O. Diekmann and H. Kaper, On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal., 2 (1978), 721-737.  doi: 10.1016/0362-546X(78)90015-9.  Google Scholar

[10]

B. Ermentrout and J. Mcleod, Existence and uniqueness of travelling waves for a neural network, Proc. R. Soc. Edinb. A, 123 (1993), 461-478.  doi: 10.1017/S030821050002583X.  Google Scholar

[11]

P. C. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in Nonlinear Analysis, (2003), 153-191.   Google Scholar

[12]

P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics 28, Springer, Berlin, 1979.  Google Scholar

[13]

Y. Fukao, Y. Morita and H. Ninomiya, Some entire solutions of the Allen-Cahn equation, Taiwanes J. Math., 8 (2004), 15-32.  doi: 10.11650/twjm/1500558454.  Google Scholar

[14]

J. S. Guo and C. H. Wu, Traveling wave front for a two-component lattice dynamical system arising in competition models, J. Differential Equations, 252 (2012), 4357-4391.  doi: 10.1016/j.jde.2012.01.009.  Google Scholar

[15]

J. S. Guo, Y. Wang, C. H. Wu and C. C. Wu, The minimal speed of traveling wave solutions for a diffusive three species competition system, Taiwan. J. Math., 19 (2015), 1805-1829.  doi: 10.11650/tjm.19.2015.5373.  Google Scholar

[16]

J. S. Guo and Y. C. Lin, Entire solutions for a discrete diffusive equation with bistable convolution type nonlinearity, Osaka J. Math., 50 (2013), 607-629.   Google Scholar

[17]

J. S. Guo and C. H. Wu, Entire solutions for a two-component competition system in a lattice, Tohoku. Math. J., 62 (2010), 17-28.  doi: 10.2748/tmj/1270041024.  Google Scholar

[18]

J. S. Guo and Y. Morita, Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations, Discrete Contin. Dyn. Syst., 12 (2005), 193-212.  doi: 10.3934/dcds.2005.12.193.  Google Scholar

[19]

F. Hamel and N. Nadirashvili, Entire solutions of the KPP equations, Comm. Pure Appl. Math., 52 (1999), 1255-1276.  doi: 10.1002/(SICI)1097-0312(199910)52:10<1255::AID-CPA4>3.0.CO;2-W.  Google Scholar

[20]

F. Hamel and N. Nadirashvili, Traveling fronts and entire solutions of the Fisher-KPP equation in $\mathbb{R}^{N}$, Arch. Rational Mech. Anal., 157 (2001), 91-163.  doi: 10.1007/PL00004238.  Google Scholar

[21]

W. T. Li, Y. J. Sun and Z. C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Anal. Real World Appl., 11 (2010), 2302-2313.  doi: 10.1016/j.nonrwa.2009.07.005.  Google Scholar

[22]

W. T. Li, J. B. Wang and L. Zhang, Entire solutions of nonlocal dispersal equations with monostable nonlinearity in space periodic habitats, J. Differential Equations, 261 (2016), 2472-2501.  doi: 10.1016/j.jde.2016.05.006.  Google Scholar

[23]

W. T. Li, Z. C. Wang and J. Wu, Entire solutions in monostable reaction-diffusion equations with delayed nonlinearity, J. Differential Equations, 245 (2008), 102-129.  doi: 10.1016/j.jde.2008.03.023.  Google Scholar

[24]

W. T. Li, L. Zhang and G. B. Zhang, Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Contin. Dyn. Syst., 35 (2015), 1531-1560.  doi: 10.3934/dcds.2015.35.1531.  Google Scholar

[25]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion equations, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590.  Google Scholar

[26]

Y. Morita and H. Ninomiya, Entire solution with merging fronts to reaction-diffusion equations, J. Dynam. Differential Equations, 18 (2006), 841-861.  doi: 10.1007/s10884-006-9046-x.  Google Scholar

[27]

Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217-2240.  doi: 10.1137/080723715.  Google Scholar

[28]

Y. J. Sun, W. T. Li and Z. C. Wang, Entire solutions in nonlocal dispersal equations with bistable nonlinearity, J. Differential Equations, 251 (2011), 551-581.  doi: 10.1016/j.jde.2011.04.020.  Google Scholar

[29]

M. Wang and G. Lv, Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delays, Nonlineariry, 23 (2010), 1609-1630.  doi: 10.1088/0951-7715/23/7/005.  Google Scholar

[30]

Z. C. Wang, W. T. Li and S. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations, 20 (2008), 563-607.  doi: 10.1007/s10884-008-9103-8.  Google Scholar

[31]

Z. C. Wang, W. T. Li and S. Ruan, Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity, Trans. Am. Math. Soc., 361 (2009), 2047-2084.  doi: 10.1090/S0002-9947-08-04694-1.  Google Scholar

[32]

Z. C. Wang, W. T. Li and J. Wu, Entire solutions in delayed lattice differential equations with monostable nonlinearity, SIAM J. Math. Anal., 40 (2009), 2392-2420.  doi: 10.1137/080727312.  Google Scholar

[33]

C. H. Wu, A general approach to the asymptotic behavior of traveling waves in a class of three-component lattice dynamical systems, J. Dynam. Differential Equations, 28 (2016), 317-338.  doi: 10.1007/s10884-016-9524-8.  Google Scholar

[34]

S. L. Wu and H. Wang, Front-like entire solutions for monostable reaction-diffusion systems, J. Dynam. Differential Equations, 25 (2013), 505-533.  doi: 10.1007/s10884-013-9293-6.  Google Scholar

[35]

H. Yagisita, Backward global solutions characterizing annihilation dynamics of travelling fronts, Publ. Res. Inst. Math. Sci., 39 (2003), 117-164.  doi: 10.2977/prims/1145476150.  Google Scholar

[36]

G. B. Zhang, W. T. Li and Z. C. Wang, Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity, J. Differential Equations, 252 (2012), 5096-5124.  doi: 10.1016/j.jde.2012.01.014.  Google Scholar

[37]

L. Zhang, W. T. Li and S. L. Wu, Multi-type entire solutions in a nonlocal dispersal epidemic model, J. Dynam. Differential Equations, 28 (2016), 189-224.  doi: 10.1007/s10884-014-9416-8.  Google Scholar

[1]

Guo-Bao Zhang, Fang-Di Dong, Wan-Tong Li. Uniqueness and stability of traveling waves for a three-species competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1511-1541. doi: 10.3934/dcdsb.2018218
[2]

Kun Li, Jianhua Huang, Xiong Li. Asymptotic behavior and uniqueness of traveling wave fronts in a delayed nonlocal dispersal competitive system. Communications on Pure & Applied Analysis, 2017, 16 (1) : 131-150. doi: 10.3934/cpaa.2017006
[3]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035
[4]

Wan-Tong Li, Li Zhang, Guo-Bao Zhang. Invasion entire solutions in a competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1531-1560. doi: 10.3934/dcds.2015.35.1531
[5]

Wan-Tong Li, Wen-Bing Xu, Li Zhang. Traveling waves and entire solutions for an epidemic model with asymmetric dispersal. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2483-2512. doi: 10.3934/dcds.2017107
[6]

Zhaohai Ma, Rong Yuan, Yang Wang, Xin Wu. Multidimensional stability of planar traveling waves for the delayed nonlocal dispersal competitive Lotka-Volterra system. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2069-2092. doi: 10.3934/cpaa.2019093
[7]

Yasumasa Nishiura, Takashi Teramoto, Xiaohui Yuan. Heterogeneity-induced spot dynamics for a three-component reaction-diffusion system. Communications on Pure & Applied Analysis, 2012, 11 (1) : 307-338. doi: 10.3934/cpaa.2012.11.307
[8]

Fei-Ying Yang, Yan Li, Wan-Tong Li, Zhi-Cheng Wang. Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1969-1993. doi: 10.3934/dcdsb.2013.18.1969
[9]

Jinling Zhou, Yu Yang, Cheng-Hsiung Hsu. Traveling waves for a nonlocal dispersal vaccination model with general incidence. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1469-1495. doi: 10.3934/dcdsb.2019236
[10]

Wei Luo, Zhaoyang Yin. Local well-posedness in the critical Besov space and persistence properties for a three-component Camassa-Holm system with N-peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5047-5066. doi: 10.3934/dcds.2016019
[11]

Xinglong Wu. On the Cauchy problem of a three-component Camassa--Holm equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2827-2854. doi: 10.3934/dcds.2016.36.2827
[12]

Yongsheng Mi, Chunlai Mu. On a three-Component Camassa-Holm equation with peakons. Kinetic & Related Models, 2014, 7 (2) : 305-339. doi: 10.3934/krm.2014.7.305
[13]

Yuncheng You. Dynamics of three-component reversible Gray-Scott model. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1671-1688. doi: 10.3934/dcdsb.2010.14.1671
[14]

Dashun Xu, Xiao-Qiang Zhao. Asymptotic speed of spread and traveling waves for a nonlocal epidemic model. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 1043-1056. doi: 10.3934/dcdsb.2005.5.1043
[15]

Jinling Zhou, Yu Yang. Traveling waves for a nonlocal dispersal SIR model with general nonlinear incidence rate and spatio-temporal delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1719-1741. doi: 10.3934/dcdsb.2017082
[16]

Xiongxiong Bao, Wenxian Shen, Zhongwei Shen. Spreading speeds and traveling waves for space-time periodic nonlocal dispersal cooperative systems. Communications on Pure & Applied Analysis, 2019, 18 (1) : 361-396. doi: 10.3934/cpaa.2019019
[17]

Bhargav Kumar Kakumani, Suman Kumar Tumuluri. Asymptotic behavior of the solution of a diffusion equation with nonlocal boundary conditions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 407-419. doi: 10.3934/dcdsb.2017019
[18]

Xiaojie Hou, Yi Li. Traveling waves in a three species competition-cooperation system. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1103-1120. doi: 10.3934/cpaa.2017053
[19]

Fengxin Chen. Stability and uniqueness of traveling waves for system of nonlocal evolution equations with bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 659-673. doi: 10.3934/dcds.2009.24.659
[20]

Belkacem Said-Houari, Radouane Rahali. Asymptotic behavior of the solution to the Cauchy problem for the Timoshenko system in thermoelasticity of type III. Evolution Equations & Control Theory, 2013, 2 (2) : 423-440. doi: 10.3934/eect.2013.2.423

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (54)
  • HTML views (33)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences