Large Time Asymptotic Analysis of Some Nonlinear Parabolic Equations – Some Constructive Approaches

  • Sachdev* P.L.
  • Srinivasa Rao Ch.
Part of the Springer Monographs in Mathematics book series (SMM)


This chapter describes some constructive approaches to the study of the asymptotic nature of solutions of some nonlinear partial differential equations of parabolic type: travelling waves, and self-similar or more general solutions obtained by the so-called balancing argument and its extensions. This is accomplished by first constructing these special solutions and hence showing their asymptotic nature analytically, numerically, or both. Sometimes it becomes possible to obtain asymptotic solutions in terms of power series in time with coefficient functions depending on the similarity variable. This approach is adopted to obtain solutions more general than self-similar or travelling waves. The analysis for such problems requires solutions of an infinite system of nonlinear ODEs. This class of solutions either ‘nonlinearise’ the linear solution of the given problem and/or embed in them limiting behaviour such as inviscid forms for convective–diffusive equations.


Large Time Asymptotic Solution Travel Wave Solution Burger Equation Nonlinear Diffusion 
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  1. 1.
    Abramowitz, M., Stegun, I. A. (1972) Handbook of Mathematical Functions, Dover, New York.Google Scholar
  2. 2.
    Bear, J. (1972) Dynamics of Fluids in Porous Media, Elsevier, New York.MATHGoogle Scholar
  3. 3.
    Bender, C. M., Orszag, S. A. (1978) Advanced Mathematical Methods for Scientists and Engineers, McGraw–Hill, New York.MATHGoogle Scholar
  4. 4.
    Berryman, J. G. (1977) Evolution of a stable profile for a class of nonlinear diffusion equations with fixed boundaries, J. Math. Phys. 18, 2108–2115.MATHCrossRefGoogle Scholar
  5. 5.
    Berryman, J. G., Holland, C. J. (1978a) Nonlinear diffusion problem arising in plasma physics, Phys. Rev. Lett. 40, 1720–1722.CrossRefMathSciNetGoogle Scholar
  6. 6.
    Berryman, J. G., Holland, C. J. (1978b) Evolution of a stable profile for a class of nonlinear diffusion equations II, J. Math. Phys. 19, 2476–2480.MATHCrossRefGoogle Scholar
  7. 7.
    Berryman, J. G., Holland, C. J. (1980) Stability of the separable solution for fast diffusion, Arch. Rational Mech. Anal. 74, 379–388.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Berryman, J. G., Holland, C. J. (1982) Asymptotic behavior of the nonlinear diffusion equation n t = (n −1 n x)x, J. Math. Phys. 23, 983–987.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Bowen, M., King, J. R. (2001) Asymptotic behaviour of the thin film equation in bounded domains, Euro. J. Appl. Math. 12, 135–157.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Bramson, M. D. (1978) Maximal displacement of branching Brownian motion, Comm. Pure Appl. Math. 31, 531–581.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Brezis, H., Peletier, L. A., Terman, D. (1986) A very singular solution of the heat equation with absorption, Arch. Rat. Mech. Anal. 95, 185–209.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Broadbridge, P., Knight, J. K., Rogers, C. (1988) Constant rate rain fall infiltration in a bounded profile: solution of a nonlinear model, Soil Sci. Soc. Am. J. 52, 1526–1533.CrossRefGoogle Scholar
  13. 13.
    Broadbridge, P., Rogers, C. (1990) Exact solution for vertical drainage and redistribution in soils, J. Engrg. Math. 24, 25–43.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Calogero, F., Lillo, S. De. (1991) The Burgers equation on the semiline with general boundary conditions at the origin, J. Math. Phys. 32, 99–105.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Carslaw, H. S., Jaeger, J. C. (1959) Conduction of Heat in Solids, Clarendon Press, Oxford.Google Scholar
  16. 16.
    Cazenave, T., Escobedo, M. (1994) A two-parameter shooting problem for a second-order differential equation, J. Differential Eq. 113, 418–451.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Clothier, B. E., Knight, J. H., White, I. (1981) Burgers’ equation: Application to field constant-flux infiltration, Soil Sci. 132, 255–261.CrossRefGoogle Scholar
  18. 18.
    Dawson, C. N., Van Duijn, C. J., Grundy, R. E. (1996) Large time asymptotics in contaminant transport in porous media, SIAM J. Appl. Math. 56, 965–993.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Dix, D. B. (2002) Large-time behaviour of solutions of Burgers’ equation, Proc. Roy. Soc. Edinburgh Sect. A 132, 843–878.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Drake, J. R. (1973) Plasma losses to an octupole hoop, Phys. Fluids 16, 1554–1555.CrossRefGoogle Scholar
  21. 21.
    Drake, J. R., Greenwood, J. R., Navratil, G. A., Post, R. S. (1977) Diffusion coefficient scaling in the Wisconsin levitated octupole, Phys. Fluids 20, 148–155.CrossRefGoogle Scholar
  22. 22.
    Escobedo, M., Zuazua, E. (1991) Large time behaviour for convection-diffusion equations in Rn, J. Funct. Anal. 100, 119–161.MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Escobedo, M., Vázquez, J. L., Zuazua, E. (1993) Asymptotic behaviour and source-type solutions for a diffusion-convection equation, Arch. Rational Mech. Anal. 124, 43–65.MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Esteban, J. R., Rodriguez, A., Vázquez, J. L. (1988) A nonlinear heat equation with singular diffusivity, Comm. Partial Differential Eq. 13, 985-1039.MATHCrossRefGoogle Scholar
  25. 25.
    Fife, P. C. (1979) Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics 28, Springer-Verlag, New York.MATHGoogle Scholar
  26. 26.
    Friedman, A., Kamin, S. (1980) The asymptotic behaviour of gas in an n-dimensional porous medium, Trans. Amer. Math. Soc. 262, 551-563.MATHMathSciNetGoogle Scholar
  27. 27.
    Galaktionov, V. A., Peletier, L. A., Vázquez J. L. (2000) Asymptotics of the fast diffusion equation with critical exponent, SIAM J. Math. Anal. 31, 1157–1174.MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Greenwood, J. R. (1975) Diffusion in the levitated toroidal octupole, Ph. D. Thesis, University of Wisconsin, Madison.Google Scholar
  29. 29.
    Grundy, R. E. (1988) Large time solution of the Cauchy problem for the generalized Burgers equation, Preprint, University of St. Andrews, UK.Google Scholar
  30. 30.
    Grundy, R. E., Sachdev, P. L., Dawson, C. N. (1994) Large time solution of an initial value problem for a generalized Burgers equation, in Nonlinear Diffusion Phenomenon, P. L. Sachdev and R. E. Grundy (Eds.), 68–83, Narosa, New Delhi.Google Scholar
  31. 31.
    Grundy, R. E., Van Duijn, C. J., Dawson, C. N. (1994a) Asymptotic profiles with finite mass in one-dimensional contaminant transport through porous media: The fast reaction case, Quart. J. Mech. Appl. Math. 47, 69–106.Google Scholar
  32. 32.
    Hadeler, K. P., Rothe, F. (1975) Travelling fronts in nonlinear diffusion equations, J. Math. Biol. 2, 251–263.MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Harris, S. E. (1996) Sonic shocks governed by the modified Burgers equation, Euro. J. Appl. Math. 7, 201–222.MATHCrossRefGoogle Scholar
  34. 34.
    Hopf, E. (1950) The partial differential equation u t + uu x = μu xx, Comm. Pure Appl. Math. 3, 201–230.MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Joseph, K. T., Sachdev, P. L. (1993) Exact analysis of Burgers equation on semiline with flux condition at the origin, Int. J. Non-Linear Mech., 28, 627–639.MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Khusnytdinova, N. V. (1967) The limiting moisture profile during infiltration in to a homogeneous soil, Prikl. Mat. Mekh. 31, 770–776.MathSciNetGoogle Scholar
  37. 37.
    King, J. R. (1993) Self-similar behaviour for the equation of fast nonlinear diffusion, Phil. Trans. Roy. Soc. London Ser. A 343, 337–375.MATHCrossRefGoogle Scholar
  38. 38.
    King, J. R., McCabe, P. M. (2003) On the Fisher–KPP equation with fast nonlinear diffusion, Proc. Roy. Soc Lond. A 459, 2529–2546.MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Kolmogorov, A. N., Petrovskii, I. G., Piskunov, N. S. (1937) Study of the diffusion equation with growth of the quantity of matter and its application to a biological problem, Mos. Uni. Bull. Math. 1–26; also in Applied Mathematics of Physical Phenomena F. Oliveira-Pinto and B. W. Conolly (Eds.), Horwood, Chichester (1982).Google Scholar
  40. 40.
    Kopell, N., Howard, L. N. (1973) Plane wave solutions to reaction-diffusion equations, Stud. Appl. Math. 52, 291–328.MATHMathSciNetGoogle Scholar
  41. 41.
    Krzyżański, M. (1959) Certaines inégalités relatives aux solutions de l’équation parabolique linéare normale, Bull. Acad. Polon. Sci., Ser. Math. Astr. Phys. 7, 131–135.MATHGoogle Scholar
  42. 42.
    Larson, D. A. (1978) Transient bounds and time-asymptotic behavior of solutions to nonlinear equations of Fisher type, SIAM J. Appl. Math. 34, 93–103.MATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Laurencot, Ph., Simondon, F. (1998) Long-time behaviour for porous medium equations with convection, Proc. Roy. Soc. Edin. 128A, 315–336.MathSciNetGoogle Scholar
  44. 44.
    Leach, J. A., Needham, D. J. (2001) The evolution of travelling waves in generalized Fisher equations via matched asymptotic expansions: Algebraic corrections, Quart. J. Mech. Appl. Math. 54, 157–175.MATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Lee-Bapty, I. P., Crighton, D. G. (1987) Nonlinear wave motion governed by the modified Burgers equation, Phil. Trans. Roy. Soc. Lond. A 323, 173–209.MATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Lees, M. (1966) A linear three-level difference scheme for quasilinear parabolic equations, Math. Comp. 20, 516–522.MATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    Mayil Vaganan, B. (1994) Exact analytic solutions for some classes of partial differential equations, Ph. D. Thesis, Indian Institute of Science, Bangalore.Google Scholar
  48. 48.
    McCabe, P. M., Leach, J. A., Needham, D. J. (2002) A note on the nonexistence of travelling waves in a class of singular reaction diffusion problems, Dynam. Sys. Int. J. 17, 131–135.MATHCrossRefMathSciNetGoogle Scholar
  49. 49.
    McKean, H. P. (1975) Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov, Comm. Pure Appl. Math. 28, 323–331.MATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    Merkin, J. H. Needham, D. J. (1989) Propagating reaction-diffusion waves in a simple isothermal quadratic chemical system, J. Engrg. Math. 23, 343-356.MATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    Merkin, J. H., Sadiq, M. A. (1996) The propagation of travelling waves in an open cubic autocatalytic chemical system, IMA J. Appl. Math. 57, 273–309.MATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    Needham, D. J. (1992) A formal theory concerning the generation and propagation of travelling wave-fronts in reaction-diffusion equations, Quart. J. Mech. Appl. Math. 45, 469–498.MATHCrossRefMathSciNetGoogle Scholar
  53. 53.
    Parker, D. F. (1981) An approximation for nonlinear acoustics of moderate amplitude, Acoustics Lett. 4, 239–244.Google Scholar
  54. 54.
    Peletier, L. A. (1970) Asymptotic behaviour of temperature profiles of a class of non-linear heat conduction problems, Quart. J. Mech. Appl. Math. 23, 441–447.MATHCrossRefMathSciNetGoogle Scholar
  55. 55.
    Philip, J. R. (1969) Theory of infiltration, Adv. Hydrosci. 5, 215–296.Google Scholar
  56. 56.
    Philip, J. R. (1970) Flow in porous media, Ann. Rev. Fluid Mech. 2, 177–204.CrossRefGoogle Scholar
  57. 57.
    Sachdev, P. L. (1987) Nonlinear Diffusive Waves, Cambridge University Press, Cambridge, UK.MATHCrossRefGoogle Scholar
  58. 58.
    Sachdev, P. L. (2000) Self-Similarity and Beyond. Exact Solutions of Nonlinear Problems, Chapman & Hall/CRC Press, New York.MATHCrossRefGoogle Scholar
  59. 59.
    Sachdev, P. L., Joseph, K. T. (1994) Exact representations of N-wave solutions of generalized Burgers equations, in Nonlinear Diffusion Phenomenon, P. L. Sachdev and R. E. Grundy (Eds.), 197–219, Narosa, New Delhi.Google Scholar
  60. 60.
    Sachdev, P. L., Nair, K. R. C. (1987) Generalized Burgers equations and Euler-Painlevé transcendents II, J. Math. Phys. 28, 997–1004.MATHCrossRefMathSciNetGoogle Scholar
  61. 61.
    Sachdev, P. L., Srinivasa Rao, Ch. (2000) N-wave solution of modified Burgers equation, Appl. Math. Lett. 13, 1–6.MATHCrossRefMathSciNetGoogle Scholar
  62. 62.
    Sachdev, P. L., Enflo, B. O., Srinivasa Rao, Ch., Mayil Vaganan, B., Poonam Goyal (2003) Large-time asymptotics for periodic solutions of some generalized Burgers equations, Stud. Appl. Math. 110, 181–204.MATHCrossRefMathSciNetGoogle Scholar
  63. 63.
    Sachdev, P. L., Joseph, K. T., Nair, K. R. C. (1994) Exact N-wave solutions for the nonplanar Burgers equation, Proc. Roy. Soc. London Ser. A 445, 501–517.MATHCrossRefMathSciNetGoogle Scholar
  64. 64.
    Sachdev, P. L., Joseph, K. T., Mayil Vaganan, B. (1996) Exact N-wave solutions of generalized Burgers equations, Stud. Appl. Math. 97, 349–367.MATHMathSciNetGoogle Scholar
  65. 65.
    Sachdev, P. L., Nair, K. R. C., Tikekar, V. G. (1986) Generalized Burgers equations and Euler-Painlevé transcendents I, J. Math. Phys. 27, 1506–1522.MATHCrossRefMathSciNetGoogle Scholar
  66. 66.
    Sachdev, P. L., Srinivasa Rao, Ch., Enflo, B. O. (2005) Large-time asymptotics for periodic solutions of the modified Burgers equation, Stud. Appl. Math. 114, 307–323.MATHCrossRefMathSciNetGoogle Scholar
  67. 67.
    Sachdev, P. L., Srinivasa Rao, Ch., Joseph, K. T. (1999) Analytic and numerical study of N-waves governed by the nonplanar Burgers equation, Stud. Appl. Math. 103, 89–120.MATHCrossRefMathSciNetGoogle Scholar
  68. 68.
    Shenker, Y., Roseman, J. J. (1995) On the exponential temporal decay of solutions and their derivatives for quasilinear parabolic equations, Z. angew. Math. Phys. 46, 198–223.MATHCrossRefMathSciNetGoogle Scholar
  69. 69.
    Sherratt, J. A. (2003) Periodic travelling wave selection by Dirichlet boundary conditions in oscillatory reaction–diffusion systems, SIAM J. Appl. Math. 63, 1520-1538.MATHCrossRefMathSciNetGoogle Scholar
  70. 70.
    Shih, Shagi-Di (1991) Shock layer structure of Burgers equation, in Conference on Nonlinear Analysis, Fon-Che Liu and Tai-Ping Liu (Eds.), 237–257, World Scientific, River Edge, NJ.Google Scholar
  71. 71.
    Smoller, J. (1989) Shock Waves and Reaction-Diffusion Equations, Springer Verlag, Berlin.Google Scholar
  72. 72.
    Sneyd, J., Dale, P. D., Duffy, A. (1998) Travelling waves in buffered systems: Applications to calcium waves, SIAM J. Appl. Math. 58, 1178–1192.MATHCrossRefMathSciNetGoogle Scholar
  73. 73.
    Srinivasa Rao, Ch., Satyanarayana, E. (2008a) Large time asymptotics for periodic solutions of a generalized Burgers equation, Int. J. Nonlinear Sci. 5, 237–245.MathSciNetGoogle Scholar
  74. 74.
    Srinivasa Rao, Ch., Satyanarayana, E. (2008b) Asymptotic N-wave solutions of the nonplanar Burgers equation, Stud. Appl. Math. 121, 191–221.MathSciNetGoogle Scholar
  75. 75.
    Srinivasa Rao, Ch., Sachdev, P. L., Mythily, R. (2002) Analysis of the self-similar solutions of the nonplanar Burgers equation, Nonlinear Anal. 51, 1447–1472.MATHCrossRefMathSciNetGoogle Scholar
  76. 76.
    Van Duijn, C. J., Grundy, R. E., Dawson, C. N. (1997) Large time profiles in reactive solute transport, Transp. Porous Media 27, 57–84.CrossRefGoogle Scholar
  77. 77.
    Van Dyke, M. (1975) Perturbation Methods in Fluid Mechanics, Parabolic Press, Stanford, CA.MATHGoogle Scholar
  78. 78.
    Vanaja, V., Sachdev, P. L. (1992) Asymptotic solutions of a generalized Burgers equation, Quart. Appl. Math. 50, 627–640.MATHMathSciNetGoogle Scholar
  79. 79.
    Weidman, P. D. (1976) On the spin-up and spin-down of a rotating fluid Part 2. Measurements and stability, J. Fluid Mech. 77, 709–735.CrossRefGoogle Scholar
  80. 80.
    Whitham, G. B. (1974) Linear and Nonlinear Waves, John Wiley & Sons, New York.MATHGoogle Scholar
  81. 81.
    Zel’dovich, Ya. B., Barenblatt, G. I. (1958) The asymptotic properties of self-modelling solutions of the non-stationary gas filtration equations, Sov. Phys. Dokl. 3, 44–47.Google Scholar

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Authors and Affiliations

  1. 1.Department of MathmeticsIndian Institute of Technology MadrasChennaiIndia

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