Long-time asymptotics of solutions to some nonlinear wave equations

Grzegorz Karch

Banach Center Publications (2000)

  • Volume: 52, Issue: 1, page 133-146
  • ISSN: 0137-6934

Abstract

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In this paper, we survey some recent results on the asymptotic behavior, as time tends to infinity, of solutions to the Cauchy problems for the generalized Korteweg-de Vries-Burgers equation and the generalized Benjamin-Bona-Mahony-Burgers equation. The main results give higher-order terms of the asymptotic expansion of solutions.

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Karch, Grzegorz. "Long-time asymptotics of solutions to some nonlinear wave equations." Banach Center Publications 52.1 (2000): 133-146. <http://eudml.org/doc/209051>.

@article{Karch2000,
abstract = {In this paper, we survey some recent results on the asymptotic behavior, as time tends to infinity, of solutions to the Cauchy problems for the generalized Korteweg-de Vries-Burgers equation and the generalized Benjamin-Bona-Mahony-Burgers equation. The main results give higher-order terms of the asymptotic expansion of solutions.},
author = {Karch, Grzegorz},
journal = {Banach Center Publications},
keywords = {Cauchy problems; generalized Korteweg-de Vries-Burgers equation; generalized Benjamin-Bona-Mahony-Burgers equation; higher-order terms of the asymptotic expansion},
language = {eng},
number = {1},
pages = {133-146},
title = {Long-time asymptotics of solutions to some nonlinear wave equations},
url = {http://eudml.org/doc/209051},
volume = {52},
year = {2000},
}

TY - JOUR
AU - Karch, Grzegorz
TI - Long-time asymptotics of solutions to some nonlinear wave equations
JO - Banach Center Publications
PY - 2000
VL - 52
IS - 1
SP - 133
EP - 146
AB - In this paper, we survey some recent results on the asymptotic behavior, as time tends to infinity, of solutions to the Cauchy problems for the generalized Korteweg-de Vries-Burgers equation and the generalized Benjamin-Bona-Mahony-Burgers equation. The main results give higher-order terms of the asymptotic expansion of solutions.
LA - eng
KW - Cauchy problems; generalized Korteweg-de Vries-Burgers equation; generalized Benjamin-Bona-Mahony-Burgers equation; higher-order terms of the asymptotic expansion
UR - http://eudml.org/doc/209051
ER -

References

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  1. [AB] L. Abdelouhab, Nonlocal dispersive equations in weighted Sobolev spaces, Differential Integral Equations 5 (1992), 307-338. Zbl0786.35110
  2. [Al] E. A. Alarcón, Existence and finite dimensionality of the global attractor for a class of nonlinear dissipative equations, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), 893-916. Zbl0790.35062
  3. [ABH] J. P. Albert, J. L. Bona, and D. Henry, Sufficient conditions for stability of solitary-wave solutions of model equation for long waves, Physica D 24 (1987), 343-366. Zbl0634.35079
  4. [ABS] Ch. J. Amick, J. L. Bona, and M. E. Schonbek, Decay of solutions of some nonlinear wave equations, J. Differential Equations 81 (1989), 1-49. Zbl0689.35081
  5. [BBM] T. B. Benjamin, J. L. Bona, and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Phil. Trans. Roy. Soc. London Ser. A 272 (1972), 47-78. Zbl0229.35013
  6. [B] P. Biler, Asymptotic behavior in time of solutions to some equations generalizing Korteweg-de Vries-Burgers equation, Bull. Polish Acad. Sci. ser. Math. 32 (1984), 275-282. Zbl0561.35064
  7. [BKW] P. Biler, G. Karch, and W. A. Woyczynski, Asymptotics for multifractal conservation laws, Report of the Mathematical Institute, University of Wrocław, no. 103 (1998) 1-25. 
  8. [BPM] V. Bisognin and G. Perla Menzala, Decay rates of the solutions of nonlinear dispersive equations, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), 1231-1246. Zbl0814.35115
  9. [BL1] J. L. Bona and L. Luo, Decay of solutions to nonlinear, dispersive wave equations, Differential Integral Equations 6 (1993), 961-980. Zbl0780.35098
  10. [BL2] J. L. Bona and L. Luo, More results on the decay of solutions to nonlinear, dispersive wave equations , Discrete Contin. Dynam. Systems 1 (1995), 151-193. Zbl0870.35088
  11. [BPW2] J. L. Bona, K. S. Promislow, and C. E. Wayne, On the asymptotic behaviour of solutions to nonlinear, dispersive, dissipative wave equations, J. Math. and Computers in Simulation 37 (1994), 264-277. Zbl0823.35018
  12. [BPW] J. L. Bona, K. S. Promislow, and C. E. Wayne, Higher-order asymptotics of decaying solutions of some nonlinear, dispersive, dissipative wave equations, Nonlinearity 8 (1995), 1179-1206. Zbl0837.35130
  13. [BS]HUKJ. L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation, Phil. Trans. Roy. Soc. London Ser. A 278 (1975), 555-601. Zbl0306.35027
  14. [BKL] J. Bricmont, A. Kupiainen, and G. Lin, Renormalization group and asymptotics of solutions of nonlinear parabolic equations, Comm. Pure Appl. Math. 47 (1994), 893-922. Zbl0806.35067
  15. [C] A. Carpio, Asymptotic behavior for the vorticity equations in dimensions two and three, Comm. Partial Differential Equations 19 (1994), 827-872. Zbl0816.35108
  16. [C2] A. Carpio, Large-time behavior in incompressible Navier-Stokes equations, SIAM J. Math. Anal. 27 (1996), 449-475. Zbl0845.76019
  17. [CL] I. L. Chern and T. P. Liu, Convergence to diffusion waves of solutions for viscous conservation laws, Comm. Math. Phys. 110 (1987), 1103-1133. 
  18. [D2] D.B. Dix, Temporal asymptotic behavior of solutions to the Benjamin-Ono-Burgers equation, J. Differential Equations 90 (1991), 238-287. Zbl0784.35098
  19. [D] D. B. Dix, The dissipation of nonlinear dispersive waves: The case of asymptotically weak nonlinearity, Comm. Partial Differential Equations 17 (1992), 1665-1693. Zbl0762.35110
  20. [EVZ1] M. Escobedo, J. L. Vázquez, and E. Zuazua, Asymptotic behaviour and source-type solutions for a diffusion-convection equation, Arch. Rat. Mech. Anal. 124 (1993), 43-65. Zbl0807.35059
  21. [EVZ2] M. Escobedo, J. L. Vázquez, and E. Zuazua, A diffusion-convection equation in several space dimensions, Indiana Univ. Math. J. 42 (1993), 1413-1440. Zbl0791.35059
  22. [EZ] M. Escobedo and E. Zuazua, Large time behavior for convection-diffusion equations in , J. Funct. Anal. 100 (1991), 119-161. Zbl0762.35011
  23. [GKO] J. A. Goldstein, R. Kajikiya, and S. Oharu, On some nonlinear dispersive equations in several variables, Differential Integral Equations 3 (1990) 617-632. Zbl0735.35103
  24. [H] E. Hopf, The partial differential equation , Comm. Pure Appl. Math. 3 (1950), 201-230. Zbl0039.10403
  25. [K1] G. Karch, -decay of solutions to dissipative-dispersive perturbations of conservation laws, Ann. Polon. Math. 67 (1997), 65-86. Zbl0882.35017
  26. [K2] G. Karch, Asymptotic behaviour of solutions to some pseudoparabolic equations, Math. Methods Appl. Sci. 20 (1997), 271-289. Zbl0869.35057
  27. [K3] G. Karch, Selfsimilar large time behavior of solutions to Korteweg-de Vries-Burgers equation, Nonlinear Analysis 35 (1999), 199-219. Zbl0923.35158
  28. [K4] G. Karch Large-time behavior of solutions to nonlinear wave equations: higher-order asymptotics, (1998) 1-24, Report of the Mathematical Institute, University of Wrocław, no. 96, sumitted. http://www.math.uni.wroc.pl/~karch 
  29. [LP] T. P. Liu and M. Pierre, Source solutions and asymptotic behavior in conservation laws, J. Diff. Eqns. 51 (1984), 419-441. Zbl0545.35057
  30. [NS]HUKP. I. Naumkin and I. A. Shishmarev, Nonlinear Nonlocal Equations in the Theory of Waves, Amer. Math. Soc. Series, Transl. of Math. Monographs, vol. 133 (1994). 
  31. [NS2]HUKP. I. Naumkin and I. A. Shishmarev, An asymptotic relationship between solutions of different nonlinear equations for large time values. I. Differentsialnye Uravneniya 30 (1994), no. 5, 873-881. Transl.: Differential Equations 30 (1994), no. 5, 806-814. 
  32. [PW] C.-A. Pillet and C. E. Wayne, Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations, J. Differential Equations 141 (1997), 310-326. Zbl0890.35016
  33. [Saut]HUKJ.-C. Saut, Sur quelques généralisations de l'équation de Korteweg-de Vries, J. Math. pures appl. 58 (1979), 21-61. Zbl0449.35083
  34. [SR]HUKM. E. Schonbek and S. V. Rajopadhye, Asymptotic behaviour of solutions to the Korteweg-de Vries-Burgers system, Ann. Inst. H. Poincaré, Analyse non linéaire 12 (1995), 425-457. Zbl0836.35144
  35. [W] C. E. Wayne, Invariant manifolds for parabolic partial differential equations on unbounded domains, Arch. Rational Mech. Anal. 138 (1997), 279-306. Zbl0882.35061
  36. [Z1] L. Zhang, Decay of solutions of generalized Benjamin-Bona-Mahony equations, Acta Math. Sinica, New Series 10 (1994), 428-438. Zbl0813.35111
  37. [Z2] L. Zhang, Decay of solutions of generalized Benjamin-Bona-Mahony-Burgers equations in n-space dimensions, Nonlinear Analysis T.M.A. 25 (1995), 1343-1396. Zbl0838.35118
  38. [Z] E. Zuazua, Weakly nonlinear large time behavior in scalar convection-diffusion equations, Differential Integral Equations 6 (1993), 1481-1491. Zbl0805.35054

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