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)
Well-posedness and persistence properties for the Novikov equation. (English) Zbl 1215.37051
The authors prove that the Cauchy problem for the Novikov equation $$\partial_tu-\partial_t\partial_x^2u+4u^2\partial_xu=3u\partial_xu\partial_x^2u+u^2\partial_x^3u,\quad u(0)=u_0,$$ is locally well-posed in the Besov spaces $B_{2,r}^{3/2}$ and in the Sobolev spaces $H^s$ with $s>3/2$. Some persistence properties for the strong solution of the above problem are also established.

37L05General theory, nonlinear semigroups, evolution equations
35Q53KdV-like (Korteweg-de Vries) equations
26A12Rate of growth of functions of one real variable, orders of infinity, slowly varying functions
Full Text: DOI
[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.; Escher, J.: Well-posedness, global existence and blow-up phenomena for a periodic quasi-linear hyperbolic equation, Comm. pure appl. Math. 51, 475-504 (1998) · Zbl 0934.35153 · doi:10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5
[3] Danchin, R.: A few remarks on the Camassa-Holm equation, Differential integral equations 14, No. 8, 953-988 (2001) · Zbl 1161.35329
[4] Danchin, R.: A note on well-posedness for the Camassa-Holm equation, J. differential equations 192, No. 2, 429-444 (2003) · Zbl 1048.35076 · doi:10.1016/S0022-0396(03)00096-2
[5] Degasperis, A.; Procesi, M.: Asymptotic integrability, , 23-37 (1999) · Zbl 0963.35167
[6] Himonas, A.; Misiolek, G.; Ponce, G.; Zhou, Y.: Persistence properties and unique continuation of solutions of the Camassa-Holm equation, Comm. math. Phys. 271, 511-522 (2007) · Zbl 1142.35078 · doi:10.1007/s00220-006-0172-4
[7] Hone, A. N. W.; Lundmark, H.; Szmigielski, J.: Explicit multipeakon solutions of Novikov’s cubically nonlinear integrable Camassa-Holm type equation, Dyn. partial differ. Equ. 6, No. 3, 253-289 (2009) · Zbl 1179.37092 · http://intlpress.com/DPDE/journal/DPDE-v06.php
[8] Hone, A. N. W.; Wang, J. P.: Integrable peakon equations with cubic nonlinearity, J. phys. Appl. math. Theor. 41, 372002 (2008) · Zbl 1153.35075 · doi:10.1088/1751-8113/41/37/372002
[9] Kato, T.: Quasi-linear equations of evolution with application to partial differential equations, , 25-70 (1975) · Zbl 0315.35077
[10] L. Ni, Y. Zhou, A new asymptotic behavior of solutions to the Camassa-Holm equation, Proc. Amer. Math. Soc. (2011), in press. · Zbl 1259.37046
[11] Novikov, V. S.: Generalisations of the Camassa-home equation, J. phys. A 42, No. 34, 342002 (2009) · Zbl 1181.37100 · doi:10.1088/1751-8113/42/34/342002
[12] Pazy, A.: Semigroups of linear operators and applications to partial differential equations, Appl. math. Sci. 44 (1983) · Zbl 0516.47023
[13] Triebel, H.: Theory of function spaces, Monogr. math. 78 (1983) · Zbl 0546.46027
[14] Zhou, Y.; Guo, Z.: Blow up and propagation speed of solutions to the DGH equation, Discrete contin. Dyn. syst. Ser. B 12, 657-670 (2009) · Zbl 1180.35473 · doi:10.3934/dcdsb.2009.12.657