# American Institute of Mathematical Sciences

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  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 & Continuous Dynamical Systems - B, 2018, 23 (6) : 2091-2119. doi: 10.3934/dcdsb.2018227
##### References:
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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. Google Scholar [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. Google Scholar [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. Google Scholar [8] N. F. Britton, Reaction-diffusion Equations and Their Applications to Biology, Academic Press, London, 1986. doi: 10.1002/bimj.4710310608. Google Scholar [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. Google Scholar [10] M. Cencini, C. 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. Google Scholar [11] F. Chen, Travelling waves for a neural network, Electron. J. Differential Equations, 2003 (2003), 1-4. Google Scholar [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. Google Scholar [13] X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in non-local evolution equations, Adv. Differential Equations, 2 (1997), 125-160. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [18] L. C. Evans, H. 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. Google Scholar [19] P. C. Fife and J. B. McLeod, Phase transitions and generalized motion by mean curvature, Arch. Ration. Mech. Anal., 65 (1977), 355-361. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [31] L. Malaguti, C. Marcelli and S. Matucci, Front propagation in bistable reaction-diffusion-advection equations, Adv. Differential Equations, 9 (2004), 1143-1166. Google Scholar [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. Google Scholar [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. Google Scholar [34] G. Raugel and J. Wu, Hyperbolic-parabolic equations with delayed non-local interaction: model derivation, wavefronts and global attractors, Preprint.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [41] J. W. H. So, J. 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. Google Scholar [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. Google Scholar [43] Z. C. Wang, W. 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. Google Scholar [44] Z. C. Wang, W. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

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##### References:
 [1] N. D. Alikakos, P. 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. Google Scholar [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. Google Scholar [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. Google Scholar [4] P. B. Ashwin, M. V. Bartuccelli, T. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [8] N. F. Britton, Reaction-diffusion Equations and Their Applications to Biology, Academic Press, London, 1986. doi: 10.1002/bimj.4710310608. Google Scholar [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. Google Scholar [10] M. Cencini, C. 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. Google Scholar [11] F. Chen, Travelling waves for a neural network, Electron. J. Differential Equations, 2003 (2003), 1-4. Google Scholar [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. Google Scholar [13] X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in non-local evolution equations, Adv. Differential Equations, 2 (1997), 125-160. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [18] L. C. Evans, H. 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. Google Scholar [19] P. C. Fife and J. B. McLeod, Phase transitions and generalized motion by mean curvature, Arch. Ration. Mech. Anal., 65 (1977), 355-361. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [31] L. Malaguti, C. Marcelli and S. Matucci, Front propagation in bistable reaction-diffusion-advection equations, Adv. Differential Equations, 9 (2004), 1143-1166. Google Scholar [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. Google Scholar [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. Google Scholar [34] G. Raugel and J. Wu, Hyperbolic-parabolic equations with delayed non-local interaction: model derivation, wavefronts and global attractors, Preprint.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [41] J. W. H. So, J. 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. Google Scholar [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. Google Scholar [43] Z. C. Wang, W. 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. Google Scholar [44] Z. C. Wang, W. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
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