August  2018, 23(6): 2091-2119. doi: 10.3934/dcdsb.2018227

Traveling wave solutions in advection hyperbolic-parabolic system with nonlocal delay

1. 

School of Mathematics and Computational Science, Hunan First Normal University, Changsha, 410205, China

2. 

College of Science, National University of Defense Technology, Changsha, 410073, China

3. 

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, China

* Corresponding author: Jianhua Huang

Received  June 2016 Revised  May 2018 Published  August 2018 Early access  July 2018

This paper is concerned with a class of advection hyperbolic-parabolic systems with nonlocal delay. We prove that the wave profile is described by a hybrid system that consists of an integral transformation and an ordinary differential equation. By considering the same problem for a properly parameterized system and the continuous dependence of the wave speed on the parameter involved, we obtain the existence and uniqueness of traveling wave solutions in advection hyperbolic-parabolic system with nonlocal delay under bistable assumption. The influence of advection on the propagation speed is also considered.

Citation: Kun Li, Jianhua Huang, Xiong Li. Traveling wave solutions in advection hyperbolic-parabolic system with nonlocal delay. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2091-2119. doi: 10.3934/dcdsb.2018227
References:
[1]

N. D. AlikakosP. W. Bates and X. Chen, Periodic traveling waves and locating oscillating patterns in multidimensional domains, Trans. Amer. Math. Soc., 351 (1999), 2777-2805.  doi: 10.1090/S0002-9947-99-02134-0.

[2]

J. F. M. Al-Omari and S. A. Gourly, Monotone traveling fronts in age-structured reaction-diffusion model of a single species, J. Math. Biol., 45 (2002), 294-312.  doi: 10.1007/s002850200159.

[3]

J. F. M. Al-Omari and S. A. Gourley, Monotone wave-fronts in a structured population model with distributed maturation delay, IMA J. Appl. Math., 70 (2005), 858-879.  doi: 10.1093/imamat/hxh073.

[4]

P. B. AshwinM. V. BartuccelliT. J. Bridges and S. A. Gourly, Travelling fronts for the KPP equation with spatio-temporal delay, Z. Angew. Math. Phys., 53 (2002), 103-122.  doi: 10.1007/s00033-002-8145-8.

[5]

H. Berestycki, The influence of advection on the propagation of fronts in reaction-diffusion equations, in: H. Berestycki, Y. Pomeau (Eds. ), Nonlinear PDEs in Condensed Matter and Reactive Flows, in: NATO Sci. Ser. C, Kluwer, Dordrecht, 569 (2003), 11-48. doi: 10.1007/978-94-010-0307-0_2.

[6]

H. Berestycki and L. Nirenberg, Travelling fronts in cylinders, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 497-572.  doi: 10.1016/S0294-1449(16)30229-3.

[7]

J. Billingham, Dynamics of a strongly nonlocal reaction-diffusion population model, Nonlinearity, 17 (2004), 313-346.  doi: 10.1088/0951-7715/17/1/018.

[8]

N. F. Britton, Reaction-diffusion Equations and Their Applications to Biology, Academic Press, London, 1986. doi: 10.1002/bimj.4710310608.

[9]

N. F. Britton, Spatial structures and periodic traveling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688.  doi: 10.1137/0150099.

[10]

M. CenciniC. Lopez and D. Vergni, Reaction-Diffusion systems: Front propagation and spatial structures, Lecture Notes in Phys., 636 (2003), 187-210.  doi: 10.1007/978-3-540-39668-0_9.

[11]

F. Chen, Travelling waves for a neural network, Electron. J. Differential Equations, 2003 (2003), 1-4. 

[12]

X. Chen, Generation and propagation of interfaces in reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141.  doi: 10.1016/0022-0396(92)90146-E.

[13]

X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in non-local evolution equations, Adv. Differential Equations, 2 (1997), 125-160. 

[14]

X. Chen and J. S. Guo, Existence and asymptotic stability of travelling waves of discrete quasilinear monostable equations, J. Differential Equations, 184 (1992), 549-569.  doi: 10.1006/jdeq.2001.4153.

[15]

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

[16]

D. Daners and P. K. Medina, Abstract Evolution Equations, Periodic Problems and Applications, Pitman Res. Notes Math. Ser., vol. 279, Longman Scientific Technical, Harlow, 1992.

[17]

B. Ermentrout and J. B. McLeod, Existence and uniqueness of travelling waves for a neural network, Proc. Roy. Soc. Edinburgh Ser. A, 123 (1993), 461-478.  doi: 10.1017/S030821050002583X.

[18]

L. C. EvansH. M. Soner and P. E. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure. Appl. Math., 45 (1992), 1097-1123.  doi: 10.1002/cpa.3160450903.

[19]

P. C. Fife and J. B. McLeod, Phase transitions and generalized motion by mean curvature, Arch. Ration. Mech. Anal., 65 (1977), 355-361. 

[20]

B. H. Gilding and R. Kersner, Travelling Waves in Nonlinear Diffusion-Convection-Reaction, Progress in Nonlinear Differential Equations and their Applications, vol. 60. Birkh$ü$ser, Verlag, Basel, 2004. doi: 10.1007/978-3-0348-7964-4.

[21]

S. A. Gourley, Travelling fronts in the diffusive Nicholson's blowflies equation with distributed delays, Math. Comput. Model., 32 (2000), 843-853.  doi: 10.1016/S0895-7177(00)00175-8.

[22]

S. A. Gourley, Wave front solutions of a diffusive delay model for populations of Daphnia magna, Comput. Math. Appl., 42 (2001), 1421-1430.  doi: 10.1016/S0898-1221(01)00251-6.

[23]

S. A. Gourley and Y. Kuang, Wavefronts and global stability in a time-delayed population model with stage structure, Proc. R. Soc. Lond. Ser. A, 459 (2003), 1563-1579.  doi: 10.1098/rspa.2002.1094.

[24]

S. A. Gourley and S. Ruan, Convergence and travelling fronts in functional differential equations with nonlocal terms: A competition model, SIAM J. Math. Anal., 35 (2003), 806-822.  doi: 10.1137/S003614100139991.

[25]

X. Liang and J. Wu, Travelling waves and numerical approximations in a reaction advection diffusion equation with nonlocal delayed effects, J. Nonlinear Sci., 13 (2003), 289-310.  doi: 10.1007/s00332-003-0524-6.

[26]

S. Ma and J. Wu, Existence, uniqueness and asymptotic stability of traveling wavefronts in non-local delayed diffusion equation, J. Dynam. Differential Equations, 19 (2007), 391-436.  doi: 10.1007/s10884-006-9065-7.

[27]

S. Ma and X. Zou, Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay, J. Differential Equations, 217 (2005), 54-87.  doi: 10.1016/j.jde.2005.05.004.

[28]

S. Ma and X. Zou, Propagation and its failure in a lattice delayed differential equation with global interaction, J. Differential Equations, 212 (2005), 129-190.  doi: 10.1016/j.jde.2004.07.014.

[29]

L. Malaguti and C. Marcelli, Travelling wavefronts in reaction-diffusion equations with convection effects and non-regular terms, Math. Nachr., 242 (2002), 148-164.  doi: 10.1002/1522-2616(200207)242:1<148::AID-MANA148>3.0.CO;2-J.

[30]

L. Malaguti and C. Marcelli, The influence of convective effects on front propagation in certain diffusive models, in: V. Capasso (Ed. ), Mathematical Modelling and Computing in Biology and Medicine, 5th ESMTB Conference, 2002, Esculapio, Bologna, 1 (2003), 362-367.

[31]

L. MalagutiC. Marcelli and S. Matucci, Front propagation in bistable reaction-diffusion-advection equations, Adv. Differential Equations, 9 (2004), 1143-1166. 

[32]

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

[33]

C. Ou and J. Wu, Existence and uniqueness of a wavefront in a delayed hyperbolic-parabolic model, Nonlinear Anal., 63 (2005), 364-387.  doi: 10.1016/j.na.2005.05.025.

[34]

G. Raugel and J. Wu, Hyperbolic-parabolic equations with delayed non-local interaction: model derivation, wavefronts and global attractors, Preprint.

[35]

S. Ruan and D. Xiao, Stability of steady states and existence of traveling waves in a vector disease model, Proc. Roy. Soc. Edinburgh Ser. A, 134 (2004), 991-1011.  doi: 10.1017/S0308210500003590.

[36]

K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615.  doi: 10.1090/S0002-9947-1987-0891637-2.

[37]

W. Shen, Traveling waves in time almost periodic structures governed by bistable nonlinearities. I. Stability and uniqueness, J. Differential Equations, 159 (1999), 1-54.  doi: 10.1006/jdeq.1999.3651.

[38]

W. Shen, Traveling waves in time almost periodic structures governed by bistable nonlinearities. Ⅱ. Existence, J. Differential Equations, 159 (1999), 55-101.  doi: 10.1006/jdeq.1999.3652.

[39]

H. L. Smith and X. Q. Zhao, Global asymptotic stability of travelling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534.  doi: 10.1137/S0036141098346785.

[40]

H. L. Smith and H. R. Thieme, Strongly order preserving semiflows generated by functional differential equations, J. Differential Equations, 93 (1991), 332-363.  doi: 10.1016/0022-0396(91)90016-3.

[41]

J. W. H. SoJ. Wu and X. Zou, A reaction-diffusion model for a single species with age structure. I, Travelling wavefronts on unbounded domains, Proc. R. Soc. Lond. Ser. A, 457 (2001), 1841-1853.  doi: 10.1098/rspa.2001.0789.

[42]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Transl. Math. Monogr., vol. 140, Amer. Math. Soc., Providence, RI, 1994.

[43]

Z. C. WangW. T. Li and S. Ruan, Traveling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232.  doi: 10.1016/j.jde.2005.08.010.

[44]

Z. C. WangW. T. Li and S. Ruan, Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay, J. Differential Equations, 238 (2007), 153-200.  doi: 10.1016/j.jde.2007.03.025.

[45]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Applied Mathematical Sciences, vol. 119, Springer, NewYork, 1986. doi: 10.1007/978-1-4612-4050-1.

[46]

J. Wu and X. Zou, Traveling wave fronts of reaction diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687.  doi: 10.1023/A:1016690424892.

[47]

X. Zou, Delay induced traveling wave fronts in reaction diffusion equations of KPP-Fisher type, J. Comput. Appl. Math., 146 (2002), 309-321.  doi: 10.1016/S0377-0427(02)00363-1.

show all references

References:
[1]

N. D. AlikakosP. W. Bates and X. Chen, Periodic traveling waves and locating oscillating patterns in multidimensional domains, Trans. Amer. Math. Soc., 351 (1999), 2777-2805.  doi: 10.1090/S0002-9947-99-02134-0.

[2]

J. F. M. Al-Omari and S. A. Gourly, Monotone traveling fronts in age-structured reaction-diffusion model of a single species, J. Math. Biol., 45 (2002), 294-312.  doi: 10.1007/s002850200159.

[3]

J. F. M. Al-Omari and S. A. Gourley, Monotone wave-fronts in a structured population model with distributed maturation delay, IMA J. Appl. Math., 70 (2005), 858-879.  doi: 10.1093/imamat/hxh073.

[4]

P. B. AshwinM. V. BartuccelliT. J. Bridges and S. A. Gourly, Travelling fronts for the KPP equation with spatio-temporal delay, Z. Angew. Math. Phys., 53 (2002), 103-122.  doi: 10.1007/s00033-002-8145-8.

[5]

H. Berestycki, The influence of advection on the propagation of fronts in reaction-diffusion equations, in: H. Berestycki, Y. Pomeau (Eds. ), Nonlinear PDEs in Condensed Matter and Reactive Flows, in: NATO Sci. Ser. C, Kluwer, Dordrecht, 569 (2003), 11-48. doi: 10.1007/978-94-010-0307-0_2.

[6]

H. Berestycki and L. Nirenberg, Travelling fronts in cylinders, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 497-572.  doi: 10.1016/S0294-1449(16)30229-3.

[7]

J. Billingham, Dynamics of a strongly nonlocal reaction-diffusion population model, Nonlinearity, 17 (2004), 313-346.  doi: 10.1088/0951-7715/17/1/018.

[8]

N. F. Britton, Reaction-diffusion Equations and Their Applications to Biology, Academic Press, London, 1986. doi: 10.1002/bimj.4710310608.

[9]

N. F. Britton, Spatial structures and periodic traveling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688.  doi: 10.1137/0150099.

[10]

M. CenciniC. Lopez and D. Vergni, Reaction-Diffusion systems: Front propagation and spatial structures, Lecture Notes in Phys., 636 (2003), 187-210.  doi: 10.1007/978-3-540-39668-0_9.

[11]

F. Chen, Travelling waves for a neural network, Electron. J. Differential Equations, 2003 (2003), 1-4. 

[12]

X. Chen, Generation and propagation of interfaces in reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141.  doi: 10.1016/0022-0396(92)90146-E.

[13]

X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in non-local evolution equations, Adv. Differential Equations, 2 (1997), 125-160. 

[14]

X. Chen and J. S. Guo, Existence and asymptotic stability of travelling waves of discrete quasilinear monostable equations, J. Differential Equations, 184 (1992), 549-569.  doi: 10.1006/jdeq.2001.4153.

[15]

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

[16]

D. Daners and P. K. Medina, Abstract Evolution Equations, Periodic Problems and Applications, Pitman Res. Notes Math. Ser., vol. 279, Longman Scientific Technical, Harlow, 1992.

[17]

B. Ermentrout and J. B. McLeod, Existence and uniqueness of travelling waves for a neural network, Proc. Roy. Soc. Edinburgh Ser. A, 123 (1993), 461-478.  doi: 10.1017/S030821050002583X.

[18]

L. C. EvansH. M. Soner and P. E. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure. Appl. Math., 45 (1992), 1097-1123.  doi: 10.1002/cpa.3160450903.

[19]

P. C. Fife and J. B. McLeod, Phase transitions and generalized motion by mean curvature, Arch. Ration. Mech. Anal., 65 (1977), 355-361. 

[20]

B. H. Gilding and R. Kersner, Travelling Waves in Nonlinear Diffusion-Convection-Reaction, Progress in Nonlinear Differential Equations and their Applications, vol. 60. Birkh$ü$ser, Verlag, Basel, 2004. doi: 10.1007/978-3-0348-7964-4.

[21]

S. A. Gourley, Travelling fronts in the diffusive Nicholson's blowflies equation with distributed delays, Math. Comput. Model., 32 (2000), 843-853.  doi: 10.1016/S0895-7177(00)00175-8.

[22]

S. A. Gourley, Wave front solutions of a diffusive delay model for populations of Daphnia magna, Comput. Math. Appl., 42 (2001), 1421-1430.  doi: 10.1016/S0898-1221(01)00251-6.

[23]

S. A. Gourley and Y. Kuang, Wavefronts and global stability in a time-delayed population model with stage structure, Proc. R. Soc. Lond. Ser. A, 459 (2003), 1563-1579.  doi: 10.1098/rspa.2002.1094.

[24]

S. A. Gourley and S. Ruan, Convergence and travelling fronts in functional differential equations with nonlocal terms: A competition model, SIAM J. Math. Anal., 35 (2003), 806-822.  doi: 10.1137/S003614100139991.

[25]

X. Liang and J. Wu, Travelling waves and numerical approximations in a reaction advection diffusion equation with nonlocal delayed effects, J. Nonlinear Sci., 13 (2003), 289-310.  doi: 10.1007/s00332-003-0524-6.

[26]

S. Ma and J. Wu, Existence, uniqueness and asymptotic stability of traveling wavefronts in non-local delayed diffusion equation, J. Dynam. Differential Equations, 19 (2007), 391-436.  doi: 10.1007/s10884-006-9065-7.

[27]

S. Ma and X. Zou, Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay, J. Differential Equations, 217 (2005), 54-87.  doi: 10.1016/j.jde.2005.05.004.

[28]

S. Ma and X. Zou, Propagation and its failure in a lattice delayed differential equation with global interaction, J. Differential Equations, 212 (2005), 129-190.  doi: 10.1016/j.jde.2004.07.014.

[29]

L. Malaguti and C. Marcelli, Travelling wavefronts in reaction-diffusion equations with convection effects and non-regular terms, Math. Nachr., 242 (2002), 148-164.  doi: 10.1002/1522-2616(200207)242:1<148::AID-MANA148>3.0.CO;2-J.

[30]

L. Malaguti and C. Marcelli, The influence of convective effects on front propagation in certain diffusive models, in: V. Capasso (Ed. ), Mathematical Modelling and Computing in Biology and Medicine, 5th ESMTB Conference, 2002, Esculapio, Bologna, 1 (2003), 362-367.

[31]

L. MalagutiC. Marcelli and S. Matucci, Front propagation in bistable reaction-diffusion-advection equations, Adv. Differential Equations, 9 (2004), 1143-1166. 

[32]

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

[33]

C. Ou and J. Wu, Existence and uniqueness of a wavefront in a delayed hyperbolic-parabolic model, Nonlinear Anal., 63 (2005), 364-387.  doi: 10.1016/j.na.2005.05.025.

[34]

G. Raugel and J. Wu, Hyperbolic-parabolic equations with delayed non-local interaction: model derivation, wavefronts and global attractors, Preprint.

[35]

S. Ruan and D. Xiao, Stability of steady states and existence of traveling waves in a vector disease model, Proc. Roy. Soc. Edinburgh Ser. A, 134 (2004), 991-1011.  doi: 10.1017/S0308210500003590.

[36]

K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615.  doi: 10.1090/S0002-9947-1987-0891637-2.

[37]

W. Shen, Traveling waves in time almost periodic structures governed by bistable nonlinearities. I. Stability and uniqueness, J. Differential Equations, 159 (1999), 1-54.  doi: 10.1006/jdeq.1999.3651.

[38]

W. Shen, Traveling waves in time almost periodic structures governed by bistable nonlinearities. Ⅱ. Existence, J. Differential Equations, 159 (1999), 55-101.  doi: 10.1006/jdeq.1999.3652.

[39]

H. L. Smith and X. Q. Zhao, Global asymptotic stability of travelling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534.  doi: 10.1137/S0036141098346785.

[40]

H. L. Smith and H. R. Thieme, Strongly order preserving semiflows generated by functional differential equations, J. Differential Equations, 93 (1991), 332-363.  doi: 10.1016/0022-0396(91)90016-3.

[41]

J. W. H. SoJ. Wu and X. Zou, A reaction-diffusion model for a single species with age structure. I, Travelling wavefronts on unbounded domains, Proc. R. Soc. Lond. Ser. A, 457 (2001), 1841-1853.  doi: 10.1098/rspa.2001.0789.

[42]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Transl. Math. Monogr., vol. 140, Amer. Math. Soc., Providence, RI, 1994.

[43]

Z. C. WangW. T. Li and S. Ruan, Traveling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232.  doi: 10.1016/j.jde.2005.08.010.

[44]

Z. C. WangW. T. Li and S. Ruan, Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay, J. Differential Equations, 238 (2007), 153-200.  doi: 10.1016/j.jde.2007.03.025.

[45]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Applied Mathematical Sciences, vol. 119, Springer, NewYork, 1986. doi: 10.1007/978-1-4612-4050-1.

[46]

J. Wu and X. Zou, Traveling wave fronts of reaction diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687.  doi: 10.1023/A:1016690424892.

[47]

X. Zou, Delay induced traveling wave fronts in reaction diffusion equations of KPP-Fisher type, J. Comput. Appl. Math., 146 (2002), 309-321.  doi: 10.1016/S0377-0427(02)00363-1.

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