zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Global existence and asymptotic behaviour of solutions for the generalized Boussinesq equation. (English) Zbl 1161.35470
Summary: We prove the existence and the uniqueness of global solution for the Cauchy problem for the generalized Boussinesq equation. Under some assumptions, we also show that the $L_\infty $ norm of small solution of the Cauchy problem for the generalized Boussinesq equation decays to zero as $t$ tends to infinity.

MSC:
35Q35PDEs in connection with fluid mechanics
35B45A priori estimates for solutions of PDE
35B40Asymptotic behavior of solutions of PDE
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
WorldCat.org
Full Text: DOI
References:
[1] Akmel, D. G.: Global existence and decay for solution to the bad Boussinesq equation in two space dimensions. Applicable analysis 83, No. 1, 17-36 (2004) · Zbl 1049.35046
[2] Boussinesq, M. J.: Essai\dot{} sur la théorie des eaux courantes, mémoires présentés par divers savants á I. Académie des sciences inst. France, séries 2, No. 3, 1-680 (1877)
[3] Clarkson, P.: New exact solution of the Boussinesq equation. European journal of applied mathematics 1, 279-300 (1990) · Zbl 0721.35074
[4] Hrusa, W. J.: Global existence and asymptotic stability for a semilinear hyperbolic Volterra equation with large initial data. SIAM journal on mathematical analysis 16, 110-134 (1985) · Zbl 0571.45007
[5] Kalantarov, V. K.; Ladyzhenskaya, O. A.: The occurrence of collapse for quasilinear equation of parabolic and hyperbolic types. Journal of soviet mathematics 10, 53-70 (1978) · Zbl 0388.35039
[6] Levine, H. A.; Sleeman, B. O.: A note on the non-existence of global solutions of initial boundary value problems for the Boussinesq equation utt=uxx+3uxxxx-$12(u2)$xx. Journal of mathematical analysis and applications 107, 206-210 (1985) · Zbl 0591.35010
[7] Linares, F.; Scialom, M.: Asymptotic behavior of solutions of a generalized Boussinesq type equation. Nonlinear analysis 25, 1147-1158 (1995) · Zbl 0847.35109
[8] Linares, F.: Global existence of small solutions for a generalized Boussinesq equation. Journal of differential equations 106, 257-293 (1993) · Zbl 0801.35111
[9] Liu, Y.: Existence and blow up of a nonlinear pochhammercchree equation. Indiana university mathematics journal 45, No. 3, 797-816 (1996) · Zbl 0883.35116
[10] Liu, Y.: Decay and scattering of small solutions of a generalized Boussinesq equation. Journal of functional analysis 147, 51-68 (1997) · Zbl 0884.35129
[11] Li, T. T.; Chen, Y. M.: Global classical solution for nonlinear evolution equations. Pitman monographs and surveys in pure and applied mathematics 45 (1992)
[12] Schneider, G.; Eugene, C. W.: Kawahara dynamics in dispersive media. Physica D 152--153, 108-110 (2001)
[13] E.M. Stein, Singular integrals and differentiability properties of function, Princeton University, Princeton, NJ, 1970 · Zbl 0207.13501
[14] Tayler, M. E.: Partial differential equations III. Nonlinear equations (1996)
[15] Wang, S. B.; Chen, G. W.: Small amplitude solutions of the generalized imbq equation. Journal of mathematical analysis and applications 264, 846-866 (2002) · Zbl 1136.35425
[16] Wang, S. B.; Chen, G. W.: Cauchy problem of the generalized double dispersion equation. Nonlinear analysis theory, methods and applications 64, 159-173 (2006) · Zbl 1092.35056
[17] Wang, Y.; Mu, C. L.: Global existence and blow-up of the solutions for the multidimensional generalized Boussinesq equation. Mathematical methods in the applied sciences 30, 1403-1417 (2007) · Zbl 1127.35052
[18] Wang, Y.; Mu, C. L.: Blow-up and scattering of solution for a generalized Boussinesq equation. Applied mathematics and computation 188, 1131-1141 (2007) · Zbl 1118.35039