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)
Some properties of solutions to the weakly dissipative Degasperis-Procesi equation. (English) Zbl 1170.35083
Summary: We consider the weakly dissipative Degasperis-Procesi equation. The present paper is concerned with some aspects of existence of global solutions, persistence properties and propagation speed. First we try to discuss the local well-posedness and blow-up scenario, then establish the sufficient conditions on global existence of the solution. Finally, persistence properties on strong solutions and the propagation speed for the weakly dissipative Degasperis-Procesi equation are also investigated.

MSC:
35Q53KdV-like (Korteweg-de Vries) equations
35B60Continuation of solutions of PDE
References:
[1]Camassa, R.; Holm, D.: An integrable shallow water equation with peaked solitons, Phys. rev. Lett. 71, 1661-1664 (1993) · Zbl 0972.35521 · doi:10.1103/PhysRevLett.71.1661
[2]Constantin, A.: The Cauchy problem for the periodic Camassa – Holm equation, J. differential equations 141, 218-235 (1997) · Zbl 0889.35022 · doi:10.1006/jdeq.1997.3333
[3]Dullin, H. R.; Gottwald, G. A.; Holm, D. D.: Korteweg – de Vries-5 and other asymptotically equivalent equations for shallow water waves, Fluid dynam. Res. 33, 73-79 (2003) · Zbl 1032.76518 · doi:10.1016/S0169-5983(03)00046-7
[4]Degasperis, A.; Holm, D. D.; Hone, A. N. W.: A new integrable equation with peakon solutions, Theoret. and math. Phys. 133, 1461-1472 (2002)
[5]Degasperis, A.; Procesi, M.: A.degasperisg.gaetasymmetry and perturbation theory, SPT 98, Symmetry and perturbation theory, SPT 98, 23 (1999)
[6]Guo, Z.: Blow up, global existence, and infinite propagation speed for the weakly dissipative Camassa – Holm equation, J. math. Phys. 49, 033516 (2008) · Zbl 1153.81368 · doi:10.1063/1.2885075
[7]Himonas, A.; Misiolek, G.; Ponce, G.; Zhou, Y.: Persistence properties and unique continuation of solutions of the Camassa – Holm equation, Comm. math. Phys. 271, 511-512 (2007) · Zbl 1142.35078 · doi:10.1007/s00220-006-0172-4
[8]Kato, T.: Spectral theory and differential equations, proc. Sympos., dedicated to konrad jorgens, Lecture notes in math. 48, 25 (1975)
[9]Liu, Y.; Yin, Z.: Global existence and blow-up phenomena for the Degasperis – Procesi equation, Comm. math. Phys. 267, 801-820 (2006) · Zbl 1131.35074 · doi:10.1007/s00220-006-0082-5
[10]Mckean, H. P.: Breakdown of a shallow water equation, Asian J. Math. 2, 767-774 (1998) · Zbl 0959.35140 · doi:http://www.intlpress.com/AJM/p/1998/2_4/AJM-2-4-867-874.pdf
[11]Misiolek, G.: Classical solutions of the periodic Camassa – Holm equation, Geom. funct. Anal. 12, No. 5, 1080-1104 (2002) · Zbl 1158.37311 · doi:10.1007/PL00012648
[12]Molinet, L.: On well-posedness results for Camassa – Holm equation on the line: A survey, J. nonlinear math. Phys. 11, No. 4, 521-533 (2004) · Zbl 1069.35076 · doi:10.2991/jnmp.2004.11.4.8
[13]Mustafa, O. G.: A note on the Degasperis – Procesi equation, J. nonlinear math. Phys. 12, 10-14 (2005) · Zbl 1067.35078 · doi:10.2991/jnmp.2005.12.1.2
[14]Whitham, G. B.: Linear and nonlinear waves, (1974) · Zbl 0373.76001
[15]Wu, S.; Yin, Z.: Blow-up and decay of the solution of the weakly dissipative Degasperis – Procesi equation, SIAM J. Math. anal. 40, No. 2, 475-490 (2008) · Zbl 1216.35126 · doi:10.1137/07070855X
[16]Wu, S.; Yin, Z.: Blow up, blow up rate and decay of the solution of the weakly dissipative Camassa – Holm equation, J. math. Phys. 47, 013504 (2006) · Zbl 1111.35067 · doi:10.1063/1.2158437
[17]Xin, Z.; Zhang, P.: On the uniqueness and large time behavior of the weak solution to a shallow water equation, Comm. partial differential equations 27, No. 9 – 10, 1815-1844 (2002) · Zbl 1034.35115 · doi:10.1081/PDE-120016129
[18]Zhou, Y.: Wave breaking for a shallow water equation, Nonlinear anal. 57, 137-152 (2004) · Zbl 1106.35070 · doi:10.1016/j.na.2004.02.004
[19]Zhou, Y.: Wave breaking for a periodic shallow water equation, J. math. Anal. appl. 290, 591-604 (2004) · Zbl 1042.35060 · doi:10.1016/j.jmaa.2003.10.017
[20]Zhou, Y.: Blow up phenomena for the integrable Degasperis – Procesi equation, Phys. lett. A 328, 157-162 (2004) · Zbl 1134.37361 · doi:10.1016/j.physleta.2004.06.027
[21]Y. Zhou, L. Zhu, On solutions to the Degasperis – Procesi equation, preprint, 2006