Long-time asymptotics of solutions to some nonlinear wave equations

Karch, Grzegorz

Banach Center Publications, Tome 51 (2000), p. 133-146 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2000-01-01

Zbl 0954.35140

EUDML-ID : urn:eudml:doc:209051

Bibliographie

[000] [AB] L. Abdelouhab, Nonlocal dispersive equations in weighted Sobolev spaces, Differential Integral Equations 5 (1992), 307-338. | Zbl 0786.35110

[001] [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. | Zbl 0790.35062

[002] [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. | Zbl 0634.35079

[003] [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. | Zbl 0689.35081

[004] [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. | Zbl 0229.35013

[005] [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. | Zbl 0561.35064

[006] [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.

[007] [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. | Zbl 0814.35115

[008] [BL1] J. L. Bona and L. Luo, Decay of solutions to nonlinear, dispersive wave equations, Differential Integral Equations 6 (1993), 961-980. | Zbl 0780.35098

[009] [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. | Zbl 0870.35088

[010] [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. | Zbl 0823.35018

[011] [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. | Zbl 0837.35130

[012] [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. | Zbl 0306.35027

[013] [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. | Zbl 0806.35067

[014] [C] A. Carpio, Asymptotic behavior for the vorticity equations in dimensions two and three, Comm. Partial Differential Equations 19 (1994), 827-872. | Zbl 0816.35108

[015] [C2] A. Carpio, Large-time behavior in incompressible Navier-Stokes equations, SIAM J. Math. Anal. 27 (1996), 449-475. | Zbl 0845.76019

[016] [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.

[017] [D2] D.B. Dix, Temporal asymptotic behavior of solutions to the Benjamin-Ono-Burgers equation, J. Differential Equations 90 (1991), 238-287. | Zbl 0784.35098

[018] [D] D. B. Dix, The dissipation of nonlinear dispersive waves: The case of asymptotically weak nonlinearity, Comm. Partial Differential Equations 17 (1992), 1665-1693. | Zbl 0762.35110

[019] [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. | Zbl 0807.35059

[020] [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. | Zbl 0791.35059

[021] [EZ] M. Escobedo and E. Zuazua, Large time behavior for convection-diffusion equations in $ℝ^N$, J. Funct. Anal. 100 (1991), 119-161. | Zbl 0762.35011

[022] [GKO] J. A. Goldstein, R. Kajikiya, and S. Oharu, On some nonlinear dispersive equations in several variables, Differential Integral Equations 3 (1990) 617-632. | Zbl 0735.35103

[023] [H] E. Hopf, The partial differential equation $u_t+uu_x=μu_xx$, Comm. Pure Appl. Math. 3 (1950), 201-230. | Zbl 0039.10403

[024] [K1] G. Karch, $L^p$-decay of solutions to dissipative-dispersive perturbations of conservation laws, Ann. Polon. Math. 67 (1997), 65-86. | Zbl 0882.35017

[025] [K2] G. Karch, Asymptotic behaviour of solutions to some pseudoparabolic equations, Math. Methods Appl. Sci. 20 (1997), 271-289. | Zbl 0869.35057

[026] [K3] G. Karch, Selfsimilar large time behavior of solutions to Korteweg-de Vries-Burgers equation, Nonlinear Analysis 35 (1999), 199-219. | Zbl 0923.35158

[027] [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

[028] [LP] T. P. Liu and M. Pierre, Source solutions and asymptotic behavior in conservation laws, J. Diff. Eqns. 51 (1984), 419-441. | Zbl 0545.35057

[029] [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).

[030] [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.

[031] [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. | Zbl 0890.35016

[032] [Saut]HUKJ.-C. Saut, Sur quelques généralisations de l'équation de Korteweg-de Vries, J. Math. pures appl. 58 (1979), 21-61. | Zbl 0449.35083

[033] [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. | Zbl 0836.35144

[034] [W] C. E. Wayne, Invariant manifolds for parabolic partial differential equations on unbounded domains, Arch. Rational Mech. Anal. 138 (1997), 279-306. | Zbl 0882.35061

[035] [Z1] L. Zhang, Decay of solutions of generalized Benjamin-Bona-Mahony equations, Acta Math. Sinica, New Series 10 (1994), 428-438. | Zbl 0813.35111

[036] [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. | Zbl 0838.35118

[037] [Z] E. Zuazua, Weakly nonlinear large time behavior in scalar convection-diffusion equations, Differential Integral Equations 6 (1993), 1481-1491. | Zbl 0805.35054