Abstract
In this paper, we investigate the speed selection mechanism of traveling wave solutions for a reaction–diffusion–advection equation with high-order terms in a cylindrical domain. The study focuses the problem under two cases for Neumann boundary condition and Dirichlet boundary condition. By using the upper and lower solutions method, general conditions for both linear and nonlinear selections are obtained. When the equation is expanded to higher dimensions, literature examining this particular topic is scarce. In light of this, new results have been obtained for both linear and nonlinear speed selections of the equation with high-order terms. For different power exponents m and n, specific sufficient conditions for linear and nonlinear selections with the minimal wave speed are derived by selecting suitable upper and lower solutions. The impact of the power exponents m and n on speed selection is analyzed.
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Bradshaw-Hajek, B.H., Broadbridge, P.: A robust cubic reaction–diffusion system for gene propagation. Math. Comput. Model. 39, 1151–1163 (2004)
Gutiérrez, P., Escaff, D., Descalzi, O.: Transition from pulses to fronts in the cubic–quintic complex Ginzburg–Landau equation. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 367, 3227–3238 (2009)
Murray, J.D.: Mathematical Biology II: Spatial Models and Biomedical Applications, vol. 3. Springer, New York (2001)
Descalzi, O., Cisternas, J., Brand, H.R.: Collisions of pulses can lead to holes via front interaction in the cubic–quintic complex Ginzburg–Landau equation in an annular geometry. Phys. Rev. E 74(6), 065201 (2006)
Gutiérrez, P., Descalzi, O.: Existence range of pulses in the quintic complex Ginzburg–Landau equation. In: AIP Conference Proceedings, vol. 913, pp. 127–132. American Institute of Physics (2007)
Berestycki, H., Hamel, F.: Front propagation in periodic excitable media. Commun. Pure Appl. Math. A J. Issued Courant Inst. Math. Sci. 55(8), 949–1032 (2002)
Li, S., Akbar, S., Sohail, M., Nazir, U., Singh, A., Alanazi, M., Hassan, A.M.: Influence of buoyancy and viscous dissipation effects on 3d magneto hydrodynamic viscous hybrid nano fluid (MgO-TiO\(_{2}\)) under slip conditions. Case Stud. Therm. Eng. 49, 103281 (2023)
Akbar, S., Sohail, M.: Three dimensional MHD viscous flow under the influence of thermal radiation and viscous dissipation. Int. J. Emerg. Multidiscip. Math. 1(3), 106–117 (2022)
Nazir, U., Sohail, M., Mukdasai, K., Singh, A., Alahmadi, R.A., Galal, A.M., Eldin, S.M.: Applications of variable thermal properties in Carreau material with ion slip and Hall forces towards cone using a non-Fourier approach via FE-method and mesh-free study. Front. Mater. 9, 1054138 (2022)
Bradshaw, B., Broadbridge, P., Fulford, G.R., Aldis, G.K.: Huxley and fisher equations for gene propagation: an exact solution. ANZIAM J. 44, 11–20 (2002)
Bazykin, A.D.: Hypothetical mechanism of speciation. Evolution 23, 685–687 (1969)
Khater, M.M.: Novel computational simulation of the propagation of pulses in optical fibers regarding the dispersion effect. Int. J. Mod. Phys. B 37(09), 2350083 (2023)
Khater, M.M.A.: A hybrid analytical and numerical analysis of ultra-short pulse phase shifts. Chaos Solitons Fractals 169, 113232 (2023)
Huang, Z., Ou, C.: Speed selection for traveling waves of a reaction–diffusion–advection equation in a cylinder. Phys. D 402, 132225 (2020)
Li, W., Liu, N., Wang, Z.-C.: Entire solutions in reaction–advection–diffusion equations in cylinders. J. Math. Pures Appl. 90, 492–504 (2008)
Berestycki, H., Nirenberg, L.: Travelling fronts in cylinders. Ann. Inst. Henri Poincare Anal. Non Lineaire 9, 497–572 (1992)
Meng, H.: The existence and non-existence of traveling waves of scalar reaction–diffusion–advection equation in unbounded cylinder. Comput. Math. Math. Phys. 53, 1644–1652 (2013)
Berestycki, H., Larrouturou, B., Roquejoffre, J.-M.: Stability of travelling fronts in a model for flame propagation part I: linear analysis. Arch. Ration. Mech. Anal. 117, 97–117 (1992)
Roquejoffre, J.-M.: Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders. Ann. Inst. Henri Poincaré C Anal. Non Linéaire 14, 499–552 (1997)
Sheng, W.-J., Wang, J.-B.: Entire solutions of time periodic bistable reaction–advection–diffusion equations in infinite cylinders. J. Math. Phys. 56(8), 081501 (2015)
Ma, Z., Wang, Z.-C.: The trichotomy of solutions and the description of threshold solutions for periodic parabolic equations in cylinders. J. Dyn. Differ. Equ. 35, 3665–3689 (2023)
Wang, F., Khan, M.N., Ahmad, I., Ahmad, H., Abu-Zinadah, H., Chu, Y.-M.: Numerical solution of traveling waves in chemical kinetics: time-fractional fishers equations. Fractals 30(02), 2240051 (2022)
Wang, F., Zheng, K., Ahmad, I., Ahmad, H.: Gaussian radial basis functions method for linear and nonlinear convection–diffusion models in physical phenomena. Open Phys. 19(1), 69–76 (2021)
Zhang, J., Wang, F., Nadeem, S., Sun, M.: Simulation of linear and nonlinear advection–diffusion problems by the direct radial basis function collocation method. Int. Commun. Heat Mass Transf. 130, 105775 (2022)
Liang, X., Zhao, X.-Q.: Spreading speeds and traveling waves for abstract monostable evolution systems. J. Funct. Anal. 259(4), 857–903 (2010)
Ebert, U., Saarloos, W.: Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts. Phys. D 146(1–4), 1–99 (2000)
Sabelnikov, V., Lipatnikov, A.: Speed selection for traveling-wave solutions to the diffusion–reaction equation with cubic reaction term and burgers nonlinear convection. Phys. Rev. E 90(3), 033004 (2014)
Weinberger, H.: On sufficient conditions for a linearly determinate spreading speed. Discrete Contin. Dyn. Syst. Ser. B 17(6), 2267–2280 (2012)
Ma, M., Ou, C.: The minimal wave speed of a general reaction–diffusion equation with nonlinear advection. Z. Angew. Math. Phys. 72, 1–14 (2021)
Muratov, C., Novaga, M.: Front propagation in infinite cylinders. I. A variational approach. Commun. Math. Sci. 6(4), 799–826 (2008)
Trofimchuk, E., Pinto, M., Trofimchuk, S.: Pushed traveling fronts in monostable equations with monotone delayed reaction. Discrete Contin. Dyn. Syst. Ser. 33, 2169–2187 (2013)
Lucia, M., Muratov, C.B., Novaga, M.: Linear vs. nonlinear selection for the propagation speed of the solutions of scalar reaction–diffusion equations invading an unstable equilibrium. Commun. Pure Appl. Math. A J. Issued Courant Inst. Math. Sci. 57(5), 616–636 (2004)
Wu, S.-L., Niu, T.-C., Hsu, C.-H.: Global asymptotic stability of pushed traveling fronts for monostable delayed reaction–diffusion equations. Discrete Contin. Dyn. Syst. 37(6), 3467–3486 (2017)
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SH and CP analyzed the method and revised the manuscript text together, and SH wrote and prepared the original draft, HW supervised the writing. All authors have read and agreed to the published version of the manuscript.
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Pan, C., Hu, S. & Wang, H. Speed Selection of Traveling Waves of a Reaction–Diffusion–Advection Equation with High-Order Terms. Qual. Theory Dyn. Syst. 23, 75 (2024). https://doi.org/10.1007/s12346-023-00923-8
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DOI: https://doi.org/10.1007/s12346-023-00923-8