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Annales Polonici Mathematici

1997 | 67 | 1 | 65-86
Tytuł artykułu

$L^p$-decay of solutions to dissipative-dispersive perturbations of conservation laws

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We study the decay in time of the spatial $L^p$-norm (1 ≤ p ≤ ∞) of solutions to parabolic conservation laws with dispersive and dissipative terms added
uₜ - uₓₓₜ - νuₓₓ + buₓ = f(u)ₓ or uₜ + uₓₓₓ - νuₓₓ + buₓ = f(u)ₓ,
and we show that under general assumptions about the nonlinearity, solutions of the nonlinear equations have the same long time behavior as their linearizations at the zero solution.
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
65-86
Opis fizyczny
Daty
wydano
1997
otrzymano
1996-02-12
Twórcy
autor
• Institute of Mathematics, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
Bibliografia
• [1] L. Abdelouhab, Nonlocal dispersive equations in weighted Sobolev spaces, Differential Integral Equations 5 (1992), 307-338.
• [2] 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.
• [3] J. Albert, On the decay of solutions of the generalized Benjamin-Bona-Mahony equation, J. Math. Anal. Appl. 141 (1989), 527-537.
• [4] C. J. Amick, J. L. Bona, and M. E. Schonbek, Decay of solutions of some nonlinear wave equations, J. Differential Equations 81 (1989), 1-49.
• [5] T. B. Benjamin, J. L. Bona, and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. A 272 (1972), 47-78.
• [6] P. Biler, Asymptotic behaviour in time of solutions to some equations generalizing the Korteweg-de Vries-Burgers equation, Bull. Polish Acad. Sci. Math. 32 (1984), 275-282.
• [7] J. L. Bona and L. Luo, Decay of solutions to nonlinear, dispersive wave equations, Differential Integral Equations 6 (1993), 961-980.
• [8] F. M. Christ and M. I. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal. 100 (1991), 87-109.
• [9] M. G. Crandall and L. Tartar, Some relations between nonexpansive and order preserving mappings, Proc. Amer. Math. Soc. 78 (1980), 385-390.
• [10] E. DiBenedetto and M. Pierre, On the maximum principle for pseudoparabolic equations, Indiana Univ. Math. J. 30 (1981), 821-854.
• [11] D. B. Dix, The dissipation of nonlinear dispersive waves: The case of asymptotically weak nonlinearity, Comm. Partial Differential Equations 17 (1992), 1665-1693.
• [12] J. Dziubański and G. Karch, Nonlinear scattering for some dispersive equations generalizing Benjamin-Bona-Mahony equations, Monatsh. Math. 122 (1996), 35-43.
• [13] M. Escobedo and E. Zuazua, Large time behavior for convection-diffusion equations in $ℝ^N$, J. Funct. Anal. 100 (1991), 119-161.
• [14] T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, in: Studies in Applied Math., Adv. in Math. Suppl. Stud. 8, Academic Press, 1983, 93-128.
• [15] C. E. Kenig, G. Ponce, and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation, J. Amer. Math. Soc. 4 (1991), 323-347.
• [16] B. L. Lucier, On Sobolev regularizations of hyperbolic conservation laws, Comm. Partial Differential Equations 10 (1985), 1-28.
• [17] M. E. Schonbek, Decay of solutions to parabolic conservation laws, Comm. Partial Differential Equations 5 (1980), 449-473.
• [18] M. E. Schonbek, Uniform decay rates for parabolic conservation laws, Nonlinear Anal. 10 (1986), 943-953.
• [19] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, 1993.
• [20] L. Zhang, Decay of solutions of generalized Benjamin-Bona-Mahony equations, Acta Math. Sinica (N.S.) 10 (1994), 428-438.
• [21] L. Zhang, Decay of solutions of generalized Benjamin-Bona-Mahony-Burgers equations in n-space dimensions, Nonlinear Anal. 25 (1995), 1343-1369.
Typ dokumentu
Bibliografia
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