# American Institute of Mathematical Sciences

## On the generalized Burgers-Huxley equation: Existence, uniqueness, regularity, global attractors and numerical studies

 Department of Mathematics, Indian Institute of Technology Roorkee-IIT Roorkee, Haridwar Highway, Roorkee, Uttarakhand 247667, India

*Corresponding author: Manil T. Mohan

Received  January 2020 Revised  July 2020 Published  September 2020

Fund Project: The first author is supported by DST-INSPIRE Faculty Award (IFA17-MA110)

In this work, we consider the forced generalized Burgers-Huxley equation and establish the existence and uniqueness of a global weak solution using a Faedo-Galerkin approximation method. Under smoothness assumptions on the initial data and external forcing, we also obtain further regularity results of weak solutions. Taking external forcing to be zero, a positivity result as well as a bound on the classical solution are also established. Furthermore, we examine the long-term behavior of solutions of the generalized Burgers-Huxley equations. We first establish the existence of absorbing balls in appropriate spaces for the semigroup associated with the solutions and then show the existence of a global attractor for the system. The inviscid limits of the Burgers-Huxley equations to the Burgers as well as Huxley equations are also discussed. Next, we consider the stationary Burgers-Huxley equation and establish the existence and uniqueness of weak solution by using a Faedo-Galerkin approximation technique and compactness arguments. Then, we discuss about the exponential stability of stationary solutions. Concerning numerical studies, we first derive error estimates for the semidiscrete Galerkin approximation. Finally, we present two computational examples to show the convergence numerically.

Citation: Manil T. Mohan, Arbaz Khan. On the generalized Burgers-Huxley equation: Existence, uniqueness, regularity, global attractors and numerical studies. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020270
##### References:
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Wayne, Using global invariant manifolds to understand metastability in the Burgers equation with small viscosity, SIAM J. Appl. Dyn. Syst., 8 (2009), 1043-1065.  doi: 10.1137/08073651X.  Google Scholar [6] Y. Benia and B.-K. Sadallah, Existence of solutions of Burgers equations in domains that can be transformed to rectangles, Electronic Journal of Differential Equations, (2016), No. 157, pp. 1–13.  Google Scholar [7] N. Bressan and A. Quarteroni, An implicit/explicit spectral method for Burgers equation, Calcolo, 23 (1987), 265-284.  doi: 10.1007/BF02576532.  Google Scholar [8] J. M. Burgers, A mathematical model illustrating the theory of turbulence, Advances in Applied Mechanics, (1948), 171–199.  Google Scholar [9] J. M. Burgers, The Nonlinear Diffusion Equation, Reidel, Dordrecht, 1974. doi: 10.1007/978-94-010-1745-9.  Google Scholar [10] H. R. Clark, M. A. Rincon and A. 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Appl., 465 (2018), 500-530.  doi: 10.1016/j.jmaa.2018.05.010.  Google Scholar [15] M. T. Darvishi, S. Kheybari and F. Khani, Spectral collocation method and Darvishi's preconditionings to solve the generalized Burgers-Huxley equation, Communications in Nonlinear Science and Numerical Simulation, 13 (2008), 2091-2103.  doi: 10.1016/j.cnsns.2007.05.023.  Google Scholar [16] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 5, Springer-Verlag, 1992. doi: 10.1007/978-3-642-58090-1.  Google Scholar [17] D. B. Dix, Nonuniqueness and uniqueness in the initial-value problem for Burgers' equation, SIAM J. Math. Anal, 27 (1996), 708-724.  doi: 10.1137/0527038.  Google Scholar [18] T. S. El-Danaf, Solitary wave solutions for the generalized Burgers-Huxley equation, International Journal of Nonlinear Sciences and Numerical Simulation, 8 (2007), 315-318.  doi: 10.1515/IJNSNS.2007.8.3.315.  Google Scholar [19] V. J. Ervin, J. E. 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Golbabai, A new domain decomposition algorithm for generalized Burgers-Huxley equation based on Chebyshev polynomials and preconditioning, Chaos, Solitons & Fractals, 39 (2009), 849-857.  doi: 10.1016/j.chaos.2007.01.099.  Google Scholar [28] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Lezioni Lincee, Cambridge University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511569418.  Google Scholar [29] J. E. Macás-Dáz, J. Ruiz-Ramírez and J. Villa, The numerical solution of a generalized Burgers-Huxley equation through a conditionally bounded and symmetry-preserving method, Computers and Mathematics with Applications, 61 (2011), 3330-3342.  doi: 10.1016/j.camwa.2011.04.022.  Google Scholar [30] M. T. Mohan, Deterministic and stochastic equations of motion arising in Oldroyd fluids of order one: Existence, uniqueness, exponential stability and invariant measures, Stochastic Analysis and Applications, 38 (2020), 1-61. doi: 10.1080/07362994.2019.1646138.  Google Scholar [31] A. Molabahramia and F. Khani, The homotopy analysis method to solve the Burgers-Huxley equation, Nonlinear Analysis: Real World Applications, 10 (2009), 589-600.  doi: 10.1016/j.nonrwa.2007.10.014.  Google Scholar [32] M. Morandi Cecchi, R. Nociforo and P. Patuzzo Grego, Burgers problems - Theoretical results, Ital. J. Pure Appl. Math., (1997), 159-174.  Google Scholar [33] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 3 (1959), 115-162.   Google Scholar [34] J. C. Robinson, Attractors and finite-dimensional behavior in the 2D Navier-Stokes equations, ISRN Mathematical Analysis, Volume 2013, Article ID 291823, 29 pages. doi: 10.1155/2013/291823.  Google Scholar [35] M. Sari and G. Gürarslan, Numerical solutions of the generalized Burgers-Huxley equation by a differential quadrature method, Math. Probl. Eng., 2009, Art. ID 370765, 11 pp. doi: 10.1155/2009/370765.  Google Scholar [36] J. Satsuma, Topics in Soliton Theory and Exactly Solvable Nonlinear Equations, World Scientific, Singapore, 1987. Google Scholar [37] W. E. Schiesser, Burgers-Huxley equation, Chapter 14 in Spline Collocation Methods for Partial Differential Equations: With Applications in $\mathbb{R}$, Wiley, 2017. doi: 10.1002/9781119301066.ch14.  Google Scholar [38] J. Simon, Compact sets in the space $\mathrm{L}^p(0, T;\mathrm{B})$, Annali di Matematica Pura ed Applicata, 146 (1986), 65-96.  doi: 10.1007/BF01762360.  Google Scholar [39] N. Smaoui, Analyzing the dynamics of the forces Burgers equation, Journal of Applied Mathematics and Stochastic Analysis, 13 (2000), 269-285.  doi: 10.1155/S1048953300000241.  Google Scholar [40] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, 1984.  Google Scholar [41] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, Second Edition, CBMS-NSF Regional Conference Series in Applied Mathematics, 1995. doi: 10.1137/1.9781611970050.  Google Scholar [42] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar [43] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Second Edition, Springer, 1997. doi: 10.1007/978-3-662-03359-3.  Google Scholar [44] S. Tomasiello, Numerical solutions of the Burgers-Huxley equation by the IDQ method, International Journal of Computer Mathematics, 87 (2010), 129-140.  doi: 10.1080/00207160801968762.  Google Scholar [45] X.-Y. Wang, Nerve propagation and wall in liquid crystals, Physics Letters A, 112 (1985), 402-406.  doi: 10.1016/0375-9601(85)90411-6.  Google Scholar [46] X. Y. Wang, Z. S. Zhu and Y. K. Lu, Solitary wave solutions of the generalized Burgers-Huxley equation, Journal of Physics, A, 23 (1990), 271-274.  doi: 10.1088/0305-4470/23/3/011.  Google Scholar [47] A.-M. Wazwaz, Travelling wave solutions of generalized forms of Burgers, Burgers-KdV and Burgers-Huxley equations, Applied Mathematics and Computation, 169 (2005), 639-656.  doi: 10.1016/j.amc.2004.09.081.  Google Scholar [48] A.-M. Wazwaz, Analytic study on Burgers, Fisher, Huxley equations and combined forms of these equations, Applied Mathematics and Computation, 195 (2008), 754-761.  doi: 10.1016/j.amc.2007.05.020.  Google Scholar [49] A.-M. Wazwaz, Burgers, Fisher and related equations., In: Partial Differential Equations and Solitary Waves Theory. Nonlinear Physical Scienc, Springer, Berlin, Heidelberg, 2009. doi: 10.1007/978-3-642-00251-9.  Google Scholar [50] O. Yu. Efimova and N. A. Kudryashov, Exact solutions of the Burgers-Huxley equation, Journal of Applied Mathematics and Mechanics, 68 (2004), 413-420.  doi: 10.1016/S0021-8928(04)00055-3.  Google Scholar [51] S. Zibaei, M. Zeinadini and M. Namjoo, Numerical solutions of Burgers-Huxley equation by exact finite difference and NSFD schemes, Journal of Difference Equations and Applications, 22 (2016), 1098-1113.  doi: 10.1080/10236198.2016.1173687.  Google Scholar

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##### References:
 [1] N. Alibaud and B. Andreianov, Non-uniqueness of weak solutions for the fractal Burgers equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 997-1016.  doi: 10.1016/j.anihpc.2010.01.008.  Google Scholar [2] H. Bateman, Some recent researches on the motion of fluids, Mon. Weather Rev., 43 (1915), 163-170.  doi: 10.1175/1520-0493(1915)43<163:SRROTM>2.0.CO;2.  Google Scholar [3] B. Batiha, M. S. M. Noorani and I. Hashim, Application of variational iteration method to the generalized Burgers-Huxley equation, Chaos, Solitons & Fractals, 36 (2008), 660-663.  doi: 10.1016/j.chaos.2006.06.080.  Google Scholar [4] B. Batiha, M. S. M. Noorani and I. Hashim, Numerical simulation of the generalized Huxley equation by He's variational iteration method, Applied Mathematics and Computation, 186 (2007), 1322-1325.  doi: 10.1016/j.amc.2006.07.166.  Google Scholar [5] M. Beck and C. E. Wayne, Using global invariant manifolds to understand metastability in the Burgers equation with small viscosity, SIAM J. Appl. Dyn. Syst., 8 (2009), 1043-1065.  doi: 10.1137/08073651X.  Google Scholar [6] Y. Benia and B.-K. Sadallah, Existence of solutions of Burgers equations in domains that can be transformed to rectangles, Electronic Journal of Differential Equations, (2016), No. 157, pp. 1–13.  Google Scholar [7] N. Bressan and A. Quarteroni, An implicit/explicit spectral method for Burgers equation, Calcolo, 23 (1987), 265-284.  doi: 10.1007/BF02576532.  Google Scholar [8] J. M. Burgers, A mathematical model illustrating the theory of turbulence, Advances in Applied Mechanics, (1948), 171–199.  Google Scholar [9] J. M. Burgers, The Nonlinear Diffusion Equation, Reidel, Dordrecht, 1974. doi: 10.1007/978-94-010-1745-9.  Google Scholar [10] H. R. Clark, M. A. Rincon and A. Silva, Analysis and numerical simulation of viscous Burgers equation, Numerical Functional Analysis and Optimization, 32 (2011), 695-716.  doi: 10.1080/01630563.2011.580873.  Google Scholar [11] J. Cole, On a quasi-linear parabolic equation occurring in aerodynamics, Quarterly of Applied Mathematics, 9 (1951), 225-236.  doi: 10.1090/qam/42889.  Google Scholar [12] J. Cyranka, Existence of globally attracting fixed points of viscous Burgers equation with constant forcing. A computer assisted proof, Topol. Methods Nonlinear Anal., 45 (2015), 655-697.  doi: 10.12775/TMNA.2015.031.  Google Scholar [13] J. Cyranka and P. Zgliczyński, Existence of globally attracting fixed points of viscous Burgers equation with nonautonomous forcing. A computer assisted proof, SIAM J. Appl. Dyn. Syst., 14 (2015), 787-821.  doi: 10.1137/14096699X.  Google Scholar [14] J. Cyranka and P. B. Mucha, A construction of two different solutions to an elliptic system, J. Math. Anal. Appl., 465 (2018), 500-530.  doi: 10.1016/j.jmaa.2018.05.010.  Google Scholar [15] M. T. Darvishi, S. Kheybari and F. Khani, Spectral collocation method and Darvishi's preconditionings to solve the generalized Burgers-Huxley equation, Communications in Nonlinear Science and Numerical Simulation, 13 (2008), 2091-2103.  doi: 10.1016/j.cnsns.2007.05.023.  Google Scholar [16] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 5, Springer-Verlag, 1992. doi: 10.1007/978-3-642-58090-1.  Google Scholar [17] D. B. Dix, Nonuniqueness and uniqueness in the initial-value problem for Burgers' equation, SIAM J. Math. Anal, 27 (1996), 708-724.  doi: 10.1137/0527038.  Google Scholar [18] T. S. El-Danaf, Solitary wave solutions for the generalized Burgers-Huxley equation, International Journal of Nonlinear Sciences and Numerical Simulation, 8 (2007), 315-318.  doi: 10.1515/IJNSNS.2007.8.3.315.  Google Scholar [19] V. J. Ervin, J. E. Macías-Díaz and J. Ruiz-Ramíreza, A positive and bounded finite element approximation of the generalized Burgers-Huxley equation, Journal of Mathematical Analysis and Applications, 424 (2015), 1143-1160.  doi: 10.1016/j.jmaa.2014.11.047.  Google Scholar [20] L. C. Evans, Partial Differential Equations, Grad. Stud. Math., vol. 19, Amer. Math. Soc., Providence, RI, 1998.  Google Scholar [21] I. Hashim, M. S. M. Noorani and M. R. Said Al-Hadidi, Solving the generalized Burgers-Huxley equation using the Adomian decomposition method, Mathematical and Computer Modelling, 43 (2006), 1404-1411.  doi: 10.1016/j.mcm.2005.08.017.  Google Scholar [22] I. Hashim, M. S. M. Noorani and B. Batiha, A note on the Adomian decomposition method for the generalized Huxley equation, Applied Mathematics and Computation, 181 (2006), 1439-1445.  doi: 10.1016/j.amc.2006.03.011.  Google Scholar [23] E. Hopf, The partial differential equationy $u_t+uu_x = \mu u_xx$, Communications on Pure and Applied Mathematics, 3 (1950), 201-230.  doi: 10.1002/cpa.3160030302.  Google Scholar [24] C.-H. Hsia and X. Wang, On a Burgers' type equation, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1121-1139.  doi: 10.3934/dcdsb.2006.6.1121.  Google Scholar [25] H. N. A. Ismail, K. Raslan and A. A. Abd Rabboh, Adomian decomposition method for Burgers-Huxley and Burgers-Fisher equations, Applied Mathematics and Computation, 159 (2004), 291-301.  doi: 10.1016/j.amc.2003.10.050.  Google Scholar [26] M. Javidi, A numerical solution of the generalized Burgers-Huxley equation by spectral collocation method, Applied Mathematics and Computation, 178 (2006), 338-344.  doi: 10.1016/j.amc.2005.11.051.  Google Scholar [27] M. Javidi and A. Golbabai, A new domain decomposition algorithm for generalized Burgers-Huxley equation based on Chebyshev polynomials and preconditioning, Chaos, Solitons & Fractals, 39 (2009), 849-857.  doi: 10.1016/j.chaos.2007.01.099.  Google Scholar [28] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Lezioni Lincee, Cambridge University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511569418.  Google Scholar [29] J. E. Macás-Dáz, J. Ruiz-Ramírez and J. Villa, The numerical solution of a generalized Burgers-Huxley equation through a conditionally bounded and symmetry-preserving method, Computers and Mathematics with Applications, 61 (2011), 3330-3342.  doi: 10.1016/j.camwa.2011.04.022.  Google Scholar [30] M. T. Mohan, Deterministic and stochastic equations of motion arising in Oldroyd fluids of order one: Existence, uniqueness, exponential stability and invariant measures, Stochastic Analysis and Applications, 38 (2020), 1-61. doi: 10.1080/07362994.2019.1646138.  Google Scholar [31] A. Molabahramia and F. Khani, The homotopy analysis method to solve the Burgers-Huxley equation, Nonlinear Analysis: Real World Applications, 10 (2009), 589-600.  doi: 10.1016/j.nonrwa.2007.10.014.  Google Scholar [32] M. Morandi Cecchi, R. Nociforo and P. Patuzzo Grego, Burgers problems - Theoretical results, Ital. J. Pure Appl. Math., (1997), 159-174.  Google Scholar [33] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 3 (1959), 115-162.   Google Scholar [34] J. C. Robinson, Attractors and finite-dimensional behavior in the 2D Navier-Stokes equations, ISRN Mathematical Analysis, Volume 2013, Article ID 291823, 29 pages. doi: 10.1155/2013/291823.  Google Scholar [35] M. Sari and G. Gürarslan, Numerical solutions of the generalized Burgers-Huxley equation by a differential quadrature method, Math. Probl. Eng., 2009, Art. ID 370765, 11 pp. doi: 10.1155/2009/370765.  Google Scholar [36] J. Satsuma, Topics in Soliton Theory and Exactly Solvable Nonlinear Equations, World Scientific, Singapore, 1987. Google Scholar [37] W. E. Schiesser, Burgers-Huxley equation, Chapter 14 in Spline Collocation Methods for Partial Differential Equations: With Applications in $\mathbb{R}$, Wiley, 2017. doi: 10.1002/9781119301066.ch14.  Google Scholar [38] J. Simon, Compact sets in the space $\mathrm{L}^p(0, T;\mathrm{B})$, Annali di Matematica Pura ed Applicata, 146 (1986), 65-96.  doi: 10.1007/BF01762360.  Google Scholar [39] N. Smaoui, Analyzing the dynamics of the forces Burgers equation, Journal of Applied Mathematics and Stochastic Analysis, 13 (2000), 269-285.  doi: 10.1155/S1048953300000241.  Google Scholar [40] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, 1984.  Google Scholar [41] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, Second Edition, CBMS-NSF Regional Conference Series in Applied Mathematics, 1995. doi: 10.1137/1.9781611970050.  Google Scholar [42] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar [43] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Second Edition, Springer, 1997. doi: 10.1007/978-3-662-03359-3.  Google Scholar [44] S. Tomasiello, Numerical solutions of the Burgers-Huxley equation by the IDQ method, International Journal of Computer Mathematics, 87 (2010), 129-140.  doi: 10.1080/00207160801968762.  Google Scholar [45] X.-Y. Wang, Nerve propagation and wall in liquid crystals, Physics Letters A, 112 (1985), 402-406.  doi: 10.1016/0375-9601(85)90411-6.  Google Scholar [46] X. Y. Wang, Z. S. Zhu and Y. K. Lu, Solitary wave solutions of the generalized Burgers-Huxley equation, Journal of Physics, A, 23 (1990), 271-274.  doi: 10.1088/0305-4470/23/3/011.  Google Scholar [47] A.-M. Wazwaz, Travelling wave solutions of generalized forms of Burgers, Burgers-KdV and Burgers-Huxley equations, Applied Mathematics and Computation, 169 (2005), 639-656.  doi: 10.1016/j.amc.2004.09.081.  Google Scholar [48] A.-M. Wazwaz, Analytic study on Burgers, Fisher, Huxley equations and combined forms of these equations, Applied Mathematics and Computation, 195 (2008), 754-761.  doi: 10.1016/j.amc.2007.05.020.  Google Scholar [49] A.-M. Wazwaz, Burgers, Fisher and related equations., In: Partial Differential Equations and Solitary Waves Theory. Nonlinear Physical Scienc, Springer, Berlin, Heidelberg, 2009. doi: 10.1007/978-3-642-00251-9.  Google Scholar [50] O. Yu. Efimova and N. A. Kudryashov, Exact solutions of the Burgers-Huxley equation, Journal of Applied Mathematics and Mechanics, 68 (2004), 413-420.  doi: 10.1016/S0021-8928(04)00055-3.  Google Scholar [51] S. Zibaei, M. Zeinadini and M. Namjoo, Numerical solutions of Burgers-Huxley equation by exact finite difference and NSFD schemes, Journal of Difference Equations and Applications, 22 (2016), 1098-1113.  doi: 10.1080/10236198.2016.1173687.  Google Scholar
Plots of solution (Case I) for the fixed time step size $k = 1/10$ and the different values of spatial mesh size (a) $h = 1/2$; (b) $h = 1/4$; (c) $h = 1/8$; (d) $h = 1/16$ at $t = 1$
Plots of solution (Case I) for the fixed spatial mesh size $h = 1/32$ and the different values of time step size (a) $k = 1/2$; (b) $k = 1/6$; (c) $k = 1/10$ at $t = 1$
Plots of solution (Case II) for the fixed time step size $k = 1/100$ and the different values of the spatial mesh size (a)$h = 1/2$; (b) $h = 1/4$; (c) $h = 1/8$; (d) $h = 1/16$ at $t = 1$
Plots of solution (Case II) for the fixed spatial mesh size $h = 1/32$ and the different values of time step size (a) $k = 1/2$; (b) $k = 1/6$; (c) $k = 1/10$ at $t = 1$
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