Eigenfunction expansion method and the long-time asymptotics for the damped boussinesq equation

Discrete and Continuous Dynamical Systems

Volumn 7, Issue 4, 2001, Pages 675-702

  Varlamov, Vladimir ^a  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

Construction of solutions; Damped Boussinesq equation; Higher order long time asymptotics

Indexed keywords

EID: 0035636684     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.2001.7.675     Document Type: Article

References (53)

1. M.J. Ablowitz and P.A. Clarkson, "Solitons, Nonlinear Evolution Equations, and the Inverse Scattering", Lond. Math. Soc., Cambridge Univ. Press, Cambridge, 1991.
2. M.J. Ablowitz and X.-P. Wang, Initial time layers and Kadomtsev-Petviashvili-type equations, Studies Appl. Maths, 98 (1997), 121-137.
3. M. Abramowitz and I.A. Stegun, Eds., "Handbook of Mathematical Functions", Dover Publ., New York, 1972.
4. A.V. Babine, M.I. Vishik, "Attractors of Evolutionary Equations", North-Holland, Amsterdam, 1992.
5. P. Biler, Time decay of solutions of semilinear strongly damped generalized wave equations, Math. Meth. Appl. Sci., 14 (1991), 427-443.
6. P. Biler, A singular perturbation problem for nonlinear damped hyperbolic equations, Proc. Roy. Soc. Edinburgh, 111A (1989), 1-31.
7. J. Bona and L. Luo, Initial-boundary value problem for model equations for the propagation of long waves, Lecture Notes in Pure and Applied Maths., 168 (1992), 65-94.
8. J. Bona and L. Luo, More results on the decay of solutions to nonlinear dispersive wave equations, Discrete & Continuous Dynamical Systems, 1 (1995), 151-193.
9. J. Bona and R. Sachs, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Comm. Math. Phys., 118 (1988), 15-29.
10. J. Boussinesq, Theorie des ondes et de remous qui se propagent le long d'un canal rectangulaire horizontal ... , J. Math. Pures Appl., Ser.2, 17 (1872), 55-108.
11. C. Cao, M.A. Rammaha, and E.S. Titi, Gevrey regularity for nonlinear analytic parabolic expansions on the sphere, J. Dynamics Diff. Eqns., to appear.
12. C. Cao, M.A. Rammaha, and E.S. Titi, The Navier-Stokes equations on the rotating sphere: Gevrey regularity and asymptotic degrees of freedom, ZAMP, 1999, to appear.
13. P. Deift, C. Tomei and E. Trubowitz, Inverse scattering and the Boussinesq equation, Comm. Pure Appl. Maths. 35 (1982), 567-628.
14. C. Foias and J.C. Saut, Linearization and normal form of the Navier-Stokes equations with potential forces, Ann. Inst. Henri Poicaré, Analyse Nonlineaire, 4, No. 1 (1987), 1-47.
15. C. Foias and J.C. Saut, Asymptotic integration of Navier-Stokes equations with potential forces, Indiana Univ. Math. Journal, 40, No. 1 (1991), 305-320.
16. V.M. Galkin, D.E. Pelinovsky, and Yu.A. Stepanyants, The structure of the rational solutions to the Boussinesq equation, Physica D, 80 (1995), 246-255.
17. R.B. Guenther and J.W. Lee, "Partial Differential Equations of Mathematical Physics and Integral Equations", Prentice Hall, NJ, 1988.
18. R. Hirota, Solutions of the classical Boussinesq equation and the spherical Boussinesq equation: the Wronskian technique, J. Phys. Soc. Japan, 85 (1986), 2137-2150.
19. E. Jahnke, F. Emde, and F. Lösch, "Tables of Higher Functions", 6 Ed., McGraw-Hill, New York, 1960.
20. V.K. Kalantarov and O.A. Ladyzhenskaya, The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types, J. Sov. Maths., 10 (1978), 53-70.
21. D.J. Korteweg and G. De Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Phil. Mag., 39 (1985), 422-443.
22. tt = -Au + F(u), Trans. Amer. Math. Soc., 192 (1974), 1-21.
23. F. Linares, Global existence of small solutions for a generalized Boussinesq equation, J. Diff. Eqns., 106 (1993), 257-293.
24. F. Linares and M. Scialom., Asymptotic behavior of solutions of a generalized Boussinesq-type equation, Nonlinear Analysis, 5 (1995), 1147-1158.
25. J.L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications", Springer-Verlag, New York, 1972.
26. F.-L. Liu and D. Russell, Solutions of the Boussinesq equation on a periodic domain, J. Math. Anal. Appl., 192 (1995), 194-219.
27. T.-P. Liu, "Nonlinear Hyperbolic-Dissipative Partial Differential Equations", Lecture Notes in Mathematics, Springer-Verlag, New York/Berlin, 1640 (1994), 103-146.
28. Y. Liu, Instability and blow up of solutions to a generalized Boussinesq equation, SIAM J. Math. Anal., 36 (1995), 1527-1546.
29. J.W. Miles, Solitary waves, Ann. Rev. Fluid Mech., 12 (1980), 11-43.
30. P.A. Milewski and J.B. Keller, Three-dimensional water waves, Studies Appl. Maths., 37 (1996), 149-166.
31. C. Müller, "Spherical Harmonics", Lecture Notes in Mathematics, Vol. 17, Springer-Verlag, New York/Berlin, 1966.
32. P.I. Naumkin and I.A. Shishmarev, "Nonlinear Nonlocal Equations in the Theory of Waves", Transl. Math. Monographs, Vol. 133, AMS, Providence, 1994.
33. F.W.J. Olver, "Introduction to Asymptotics and Special Functions", Acad. Press, New York, San Fransisco, London, 1974.
34. A.K. Pani, H. Saranga, Finite element Galerkin method for the "good " Boussinesq equation, Nonlinear Analysis 20 (1997), 937-956.
35. R.L. Pego and M.I. Weinstein, Eigenvalues and instabilities of solitary waves, Phil. Trans. Roy. Soc. London, A340 (1992), 47-94.
36. G. Sansone, "Orthogonal Functions", Interscience Publ., New York, 1959.
37. G. Tolstov, "Fourier Series", Dover Publ., New York, 1962.
38. M. Tsutsumi and T. Matahashi, On the Cauchy problem for the Boutsinesq-type equation, Math. Japonica, 36 (1991), 371-379.
39. V.V. Varlamov , On the Cauchy problem for the damped Boussinesq equation, Differential Integral Equations, 9, No.3 (1996), 619-634.
40. V.V. Varlamov, On spatially periodic solutions of the damped Boussinesq equation, Differential Integral Equations, 10, No. 6 (1997), 1197-1211.
41. V.V. Varlamov, On the initial-boundary value problem for the damped Boussinesq equation, Discrete & Continuous Dynamical Systems, 4, No.3 (1998), 431-444.
42. V.V. Varlamov, Long-time asymptotics of solutions of the second initial-boundary value problem for the damped Boussinesq equation, Abstract & Applied Analysis, 2, No. 3/4 (1998), 97-115.
43. V.V. Varlamov, Asymptotic behavior of solutions of the damped Boussinesq equation in two space dimensions, Int. J. Maths. & Math. Sci., 22, No.1 (1999), 131-145.
44. V.V. Varlamov, On the damped Boussinesq equation in a circle, Nonlinear Analysis, 38 (1999), 447-470.
45. V.V. Varlamov, On the third-order nonlinear evolution equation governing wave propagation in relaxing media, Studies Appl. Maths., 99 (1998), 25-48.
46. V.V. Varlamov, Nonlinear heat equation with a fractional Laplacian in a disk, Colloquium Mathematicum, 81 (1) 1999, 101-122.
47. V.V. Varlamov, Long-time asymptotics for the nonlinear heat equation with a fractional Laplacian in a ball, Studia Math., to appear.
48. S. Venakides, The Korteweg-de Vries equation with small dispersion: higher-order Lax-Levermore theory, Comm. Pure Appl. Maths., 43 (1990), 335-361.
49. S. Venakides, The zero dispersion limit of the Korteweg-de Vries equation for initial potentials with nontrivial reflection coefficients, Comm. Pure Appl. Maths., 38 (1985), 125-155.
50. G.N. Watson, "A Treatise on the Theory of Bessel Functions", Cambridge Univ. Press, London, 1966.
51. Webster, "Partial Differential Equations of Mathematical Physics", Hafner Publ., New York, 1950
52. Y.C. You, Inertial manifolds and stabilization in nonlinear elastic systems with structural damping, in "Differential Equations with Applications to Mathematical Physics", Acad. Press, Boston, 1993.
53. N.J. Zabusky, "Nonlinear Partial Differential Equations", Acad. Press, New York, 1967.