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2000 | 52 | 1 | 133-146
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Long-time asymptotics of solutions to some nonlinear wave equations

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Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
Rocznik
Tom
52
Numer
1
Strony
133-146
Opis fizyczny
Daty
wydano
2000
Twórcy
  • Instytut Matematyczny, Uniwersytet Wrocławski, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
Bibliografia
  • [AB] L. Abdelouhab, Nonlocal dispersive equations in weighted Sobolev spaces, Differential Integral Equations 5 (1992), 307-338.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [BL1] J. L. Bona and L. Luo, Decay of solutions to nonlinear, dispersive wave equations, Differential Integral Equations 6 (1993), 961-980.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [C] A. Carpio, Asymptotic behavior for the vorticity equations in dimensions two and three, Comm. Partial Differential Equations 19 (1994), 827-872.
  • [C2] A. Carpio, Large-time behavior in incompressible Navier-Stokes equations, SIAM J. Math. Anal. 27 (1996), 449-475.
  • [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.
  • [D2] D.B. Dix, Temporal asymptotic behavior of solutions to the Benjamin-Ono-Burgers equation, J. Differential Equations 90 (1991), 238-287.
  • [D] D. B. Dix, The dissipation of nonlinear dispersive waves: The case of asymptotically weak nonlinearity, Comm. Partial Differential Equations 17 (1992), 1665-1693.
  • [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.
  • [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.
  • [EZ] M. Escobedo and E. Zuazua, Large time behavior for convection-diffusion equations in $ℝ^N$, J. Funct. Anal. 100 (1991), 119-161.
  • [GKO] J. A. Goldstein, R. Kajikiya, and S. Oharu, On some nonlinear dispersive equations in several variables, Differential Integral Equations 3 (1990) 617-632.
  • [H] E. Hopf, The partial differential equation $u_t+uu_x=μu_xx$, Comm. Pure Appl. Math. 3 (1950), 201-230.
  • [K1] G. Karch, $L^p$-decay of solutions to dissipative-dispersive perturbations of conservation laws, Ann. Polon. Math. 67 (1997), 65-86.
  • [K2] G. Karch, Asymptotic behaviour of solutions to some pseudoparabolic equations, Math. Methods Appl. Sci. 20 (1997), 271-289.
  • [K3] G. Karch, Selfsimilar large time behavior of solutions to Korteweg-de Vries-Burgers equation, Nonlinear Analysis 35 (1999), 199-219.
  • [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
  • [LP] T. P. Liu and M. Pierre, Source solutions and asymptotic behavior in conservation laws, J. Diff. Eqns. 51 (1984), 419-441.
  • [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).
  • [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.
  • [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.
  • [Saut]HUKJ.-C. Saut, Sur quelques généralisations de l'équation de Korteweg-de Vries, J. Math. pures appl. 58 (1979), 21-61.
  • [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.
  • [W] C. E. Wayne, Invariant manifolds for parabolic partial differential equations on unbounded domains, Arch. Rational Mech. Anal. 138 (1997), 279-306.
  • [Z1] L. Zhang, Decay of solutions of generalized Benjamin-Bona-Mahony equations, Acta Math. Sinica, New Series 10 (1994), 428-438.
  • [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.
  • [Z] E. Zuazua, Weakly nonlinear large time behavior in scalar convection-diffusion equations, Differential Integral Equations 6 (1993), 1481-1491.
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Bibliografia
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