zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Existence and scattering of small solutions to a Boussinesq type equation of sixth order. (English) Zbl 1195.35229
Summary: We consider the existence and uniqueness of the global small solution as well as the small data scattering result to the Cauchy problem for a Boussinesq type equation of sixth order with the nonlinear term $f(u)$ behaving as $u^p$ $(p>1)$ as $u\to 0$ in $\Bbb R^n$, $n\ge 1$. The main method and techniques used in our paper are the Littlewood-Paley dyadic decomposition, the stationary phase estimate and some properties of Bessel functions.

35L76Semilinear higher-order hyperbolic equations
35L30Higher order hyperbolic equations, initial value problems
35Q35PDEs in connection with fluid mechanics
Full Text: DOI
[1] Boussinesq, M. J.: Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, J. math. Pures appl. 17, 55-108 (1872) · Zbl 04.0493.04 · http://gallica.bnf.fr/ark:/12148/bpt6k164163.f00000063
[2] Makhankov, V. G.: On stationary solutions of the Schrödinger equation with a self-consistent potential satisying Boussinesq’s equation, Phys. lett. A 50, 42-44 (1974)
[3] Makhankov, V. G.: Dynamics of classical solitons (in non-integrabla systems), physics reports, Phys. lett. C 35, 1-128 (1978)
[4] Clarkson, A.; Leveque, R. J.; Saxton, R.: Solitary wave intercation in elastic rods, Stud. appl. Math. 75, 95-122 (1986) · Zbl 0606.73028
[5] Mott, J.: Elastic waves propagation in an infinite isotropic solid cylinder, J. acoust. Soc. E 54, 1129-1135 (1973)
[6] Dodd, R. K.; Eilbeck, J. D.; Gibbon, J. D.; Morris, H. C.: Solitons and nonlinear waves, (1982) · Zbl 0496.35001
[7] M.J. Ablowitz, H. Segur, Solitons and the inverse scattering transform, SIAM Studies in Applied Mathematics, Philadelphia, 1981. · Zbl 0472.35002
[8] Bona, J. L.; Sachs, R. L.: Global existence of smooth solutions and stability of solitary wave for a generalized Boussinesq equation, Comm. math. Phys. 118, 15-29 (1988) · Zbl 0654.35018 · doi:10.1007/BF01218475
[9] Tsutsumi, M.; Matahashi, T.: On the Cauchy problem for the Boussinesq type equations, Math. japonica 36, 371-379 (1991) · Zbl 0734.35082
[10] Linares, F.: Global existence of small solutions for a generalized Boussinesq equation, J. differential equations 106, 257-293 (1993) · Zbl 0801.35111 · doi:10.1006/jdeq.1993.1108
[11] Chen, G.; Wang, S.: Existence and nonexistence of global solutions for the generalized imbq equations, Nonlinear anal. TMA 36, 961-980 (1999) · Zbl 0920.35005 · doi:10.1016/S0362-546X(97)00710-4
[12] Chen, G.; Xing, J.; Yang, Z.: Cauchy problem for generalized imbq equation with several variables, Nonlinear anal. TMA 26, 1255-1270 (1996) · Zbl 0866.35094 · doi:10.1016/0362-546X(94)00332-C
[13] Linares, F.; Scialom, M.: Asymptotic behaviour of solutions of a generalized Boussinesq type equation, Nonlinear anal. 25, 1147-1158 (1995) · Zbl 0847.35109 · doi:10.1016/0362-546X(94)00236-B
[14] Liu, Y.: Decay and scattering of small solutions of a generalized Boussinesq equation, J. funct. Anal. 147, 51-68 (1997) · Zbl 0884.35129 · doi:10.1006/jfan.1996.3052
[15] Wang, S.; Chen, G.: The Cauchy problem for the generalized imbq equation in ws,$p(Rn)$, J. math. Anal. appl. 266, 38-54 (2002) · Zbl 1043.35118 · doi:10.1006/jmaa.2001.7670
[16] Wang, S.; Chen, G.: Small amplitude solutions of the generalized imbq equation, J. math. Anal. appl. 274 (2002) · Zbl 1136.35425 · doi:10.1016/S0022-247X(02)00401-8
[17] Cho, Y.; Ozawa, T.: On small amplitude solutions to the generalized Boussinesq equations, Discrete contin. Dyn. syst. 17, 691-711 (2007) · Zbl 1157.35087 · doi:10.3934/dcds.2007.17.691
[18] Schneider, Guido; Wayne, C. Eugene: Kawahara dynamics in dispersive media, Physica D 152--153, 384-394 (2001) · Zbl 0977.35123 · doi:10.1016/S0167-2789(01)00181-6
[19] Bergh, J.; Löfström, J.: Interpolation spaces, (1976) · Zbl 0344.46071
[20] Miao, C.: Harmonic analysis method for PDE, Monographs on modern pure mathematics 107 (2008)
[21] Stein, E. M.: Harmonic analysis, (1993) · Zbl 0821.42001
[22] Ben-Artzi, M.; Koch, H.; Saut, J. C.: Dispersion estimates for fourth order Schrödinger equations, C. R. Acad. sci. Paris, série I 330, 87-92 (2000) · Zbl 0942.35160 · doi:10.1016/S0764-4442(00)00120-8
[23] Guo, Z.; Peng, L.; Wang, B.: Decay estimates for a class of wave equation, J. funct. Anal. 254, 1642-1660 (2008) · Zbl 1145.35032 · doi:10.1016/j.jfa.2007.12.010
[24] Grafakos, L.: Classical and modern Fourier analysis, (2004) · Zbl 1148.42001
[25] Miao, C.: Harmonic analysis and applications to pdes, Monographs on modern pure mathematics 89, 3 (2004)
[26] Stein, E. M.: An introduction to Fourier analysis on Euclidean spaces, (1971) · Zbl 0232.42007
[27] Jhon, F.: Plane waves and spherical means, applied to partial differential equations, (1955) · Zbl 0067.32101
[28] Christ, F. M.; Weinstein, M. I.: Dispersion of small amplitude solution of the generalized Korteweg-de Vries equation, J. funct. Anal. 100, 87-109 (1991) · Zbl 0743.35067 · doi:10.1016/0022-1236(91)90103-C
[29] Kato, T.: On nonlinear Schrödinger equations II. Hs-solutions and unconditional wellposedness, J. d’anal. Math. 67, 281-306 (1995) · Zbl 0848.35124 · doi:10.1007/BF02787794
[30] Ginibre, J.; Ozawa, T.; Velo, G.: On the existence of the wave operators for a class of nonlinear Schrödinger equations, Ann. inst. H. Poincaré phys. Théor. 60, 211-239 (1994) · Zbl 0808.35136 · numdam:AIHPA_1994__60_2_211_0