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)
On the asymptotic behavior of solution for the generalized IBq equation with hydrodynamical damped term. (English) Zbl 1237.35021
The authors study the Cauchy problem for the generalized IBq equation with hydrodynamical damped term in $n$-dimensional space. They observe that the dissipative structure of the linearized equation is of the regularity-loss type. This means that they have the optimal decay estimates of solutions under the additional regularity assumption on the initial data. Under smallness condition on the initial data, the authors prove the global existence and decay of the small amplitude solution in the Sobolev space.

35B40Asymptotic behavior of solutions of PDE
35L30Higher order hyperbolic equations, initial value problems
35Q35PDEs in connection with fluid mechanics
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
Full Text: DOI
[1] Arévalo, E.; Gaididei, Yu.; Mertens, F. G.: Soliton dynamics in damped and forced Boussinesq equations, Eur. phys. J. B 27, 63-74 (2002)
[2] Bergh, J.; Löfström, J.: Interpolation spaces, (1976) · Zbl 0344.46071
[3] Boussinesq, J.: Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. math. Pures appl. 17, No. 2, 55-108 (1872) · Zbl 04.0493.04 · http://gallica.bnf.fr/ark:/12148/bpt6k164163.f00000063
[4] Boussinesq, J.: Essai sur la théorie des eaux courantes, Mém. acad. Sci. inst. Nat. France 23, No. 1, 1-680 (1877)
[5] Chen, G.; Wang, S.: Existence and nonexistence of global solutions for generalized imbq equation, Nonlinear anal. 36, 961-980 (1999) · Zbl 0920.35005 · doi:10.1016/S0362-546X(97)00710-4
[6] Cho, Y.; Ozawa, T.: Remarks on modified improved Boussinesq equations in one space dimension, Proc. R. Soc. lond. Ser. A math. Phys. eng. Sci. 462, 1949-1963 (2006) · Zbl 1149.76609 · doi:10.1098/rspa.2006.1675
[7] 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
[8] Clarkson, A.; Leveque, R. J.; Saxton, R.: Solitary-wave interactions in elastic rods, Stud. appl. Math. 75, 95-122 (1986) · Zbl 0606.73028
[9] Fabrizio, M.; Lazzari, B.: On the existence and the asymptotic stability of solutions for linear viscoelastic solids, Arch. ration. Mech. anal. 116, 139-152 (1991) · Zbl 0766.73013 · doi:10.1007/BF00375589
[10] Karch, G.: Selfsimilar profiles in large time asymptotics of solutions to damped wave equations, Studia math. 143, 175-197 (2000) · Zbl 0964.35022
[11] Kato, T.: On nonlinear Schrödinger equations II. Hs-solutions and unconditional wellposedness, J. anal. Math. 67, 281-306 (1995) · Zbl 0848.35124 · doi:10.1007/BF02787794
[12] Lin, Q.; Wu, Y.; Loxton, R.: On the Cauchy problem for a generalized Boussinesq equation, J. math. Anal. appl. 353, 186-195 (2009) · Zbl 1168.35336 · doi:10.1016/j.jmaa.2008.12.002
[13] Liu, Y.: Existence and blow up of solutions of a nonlinear Pochhammer-chree equation, Indiana univ. Math. J. 45, 797-816 (1996) · Zbl 0883.35116
[14] Liu, Z.; Zheng, S.: On the exponential stability of linear viscoelasticity and thermoviscoelasticity, Quart. appl. Math. 54, 21-31 (1996) · Zbl 0868.35011
[15] Farah, L. G.; Linares, F.: Global rough solutions to the cubic nonlinear Boussinesq equation, J. lond. Math. soc. 81, No. 1, 241-254 (2010) · Zbl 1187.35207 · doi:10.1112/jlms/jdp069
[16] Ferreira, Lucas C. F.: Existence and scattering theory for Boussinesq type equations with singular data, J. differential equations 250, 2372-2388 (2011) · Zbl 1210.35208 · doi:10.1016/j.jde.2010.11.013
[17] Machihara, S.; Ozawa, T.: Interpolation inequalities in Besov spaces, Proc. amer. Math. soc. 131, 1553-1556 (2002) · Zbl 1022.46018 · doi:10.1090/S0002-9939-02-06715-1
[18] Makhankov, V. G.: Dynamics of classical solitons (in non-integrable systems), Phys. lett. C 35, 1-128 (1978)
[19] Matsumura, A.: On the asymptotic behavior of solutions of semi-linear wave equations, Publ. res. Inst. math. Sci. 12, 169-189 (1976) · Zbl 0356.35008 · doi:10.2977/prims/1195190962
[20] Polat, N.: Existence and blow up of solution of Cauchy problem of the generalized damped multidimensional improved modified Boussinesq equation, Z. naturforsch. A 63a, 543-552 (2008)
[21] Nishihara, K.: Lp-lq estimates of solutions to the damped wave equation in 3-dimensional space and their applications, Math. Z. 244, 631-649 (2003) · Zbl 1023.35078 · doi:10.1007/s00209-003-0516-0
[22] Rivera, J. E. M.: Asymptotic behavior in linear viscoelasticity, Quart. appl. Math. 52, 628-648 (1994) · Zbl 0814.35009
[23] Rosenau, P.: Dynamics of nonlinear mass-spring chains near continuum limit, Phys. lett. A 118, 222-227 (1986)
[24] Rosenau, P.: Dynamics of dense lattices, Phys. rev. B 36, 5868-5876 (1987)
[25] Russell, J. Scott: On waves, , 311-390 (1845)
[26] Todorova, G.; Yordanov, B.: Critical exponent for a nonlinear wave equation with damping, J. differential equations 174, 464-489 (2000) · Zbl 0994.35028 · doi:10.1006/jdeq.2000.3933
[27] Varlamov, V. V.: Eigenfunction expansion method and the long-time asymptotics for the damped Boussinesq equation, Discrete contin. Dyn. syst. 7, No. 4, 675-702 (2001) · Zbl 1021.35089 · doi:10.3934/dcds.2001.7.675
[28] 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
[29] Wang, S.; Chen, G.: Small amplitude solutions of the generalized imbq equation, J. math. Anal. appl. 274, 846-866 (2002) · Zbl 1136.35425 · doi:10.1016/S0022-247X(02)00401-8