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)
Blow-up and global solutions to a new integrable model with two components. (English) Zbl 1205.35045

The author considers the two-component generalized Camassa-Holm system

u t -u xxt +3uu x -2u x u xx -uu xxx +σρρ x =0,t>0,x;
ρ t +(ρu) x =0,t>0,x,

which takes the equivalent form of a quasilinear evolution equation of hyperbolic type:

u t +uu x + x G*u 2 +1 2u x 2 +σ 2ρ 2 =0,t>0,x;
ρ t +(ρu) x =0,t>0,x,

where the sign * denotes the spatial convolution, G(s) is the associated Green function of the operator (1- x 2 ) -1 , σ can be chosen to 1 or -1. For this system, the global existence and blow-up phenomena questions are investigated. The blow-up criteria for the nonperiodic case are also obtained.

MSC:
35G25Initial value problems for nonlinear higher-order PDE
35B30Dependence of solutions of PDE on initial and boundary data, parameters
35B44Blow-up (PDE)
References:
[1]Bressan, A.; Constantin, A.: Global conservative solutions of the Camassa-Holm equation, Arch. ration. Mech. anal. 183, 215-239 (2007) · Zbl 1105.76013 · doi:10.1007/s00205-006-0010-z
[2]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
[3]Constantin, A.; Escher, J.: Well-posedness, global existence and blow-up phenomenon 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
[4]Constantin, A.; Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations, Acta math. 181, 229-243 (1998) · Zbl 0923.76025 · doi:10.1007/BF02392586
[5]Constantin, A.; Escher, J.: On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z. 233, 75-91 (2000) · Zbl 0954.35136 · doi:10.1007/PL00004793
[6]Constantin, A.; Escher, J.: Global weak solutions for a shallow water equation, Indiana univ. Math. J. 47, 1527-1545 (1998) · Zbl 0930.35133 · doi:10.1512/iumj.1998.47.1466
[7]Constantin, A.; Escher, J.: Global existence and blow-up for a shallow water equation, Ann. sc. Norm. super. Pisa cl. Sci. 26, 303-328 (1998) · Zbl 0918.35005 · doi:numdam:ASNSP_1998_4_26_2_303_0
[8]Constantin, A.; Ivanov, R.: On an integrable two-component Camassa-Holm shallow water system, Phys. lett. A 372, 7129-7132 (2008) · Zbl 1227.76016 · doi:10.1016/j.physleta.2008.10.050
[9]Chen, M.; Liu, S.; Zhang, Y.: A two-component generalization of the Camassa-Holm equation and its solutions, Lett. math. Phys. 75, 1-15 (2006) · Zbl 1105.35102 · doi:10.1007/s11005-005-0041-7
[10]Constantin, A.; Strauss, W.: Stability of peakons, Comm. pure appl. Math. 53, 603-610 (2000) · Zbl 1049.35149 · doi:10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L
[11]Escher, J.; Liu, Y.; Yin, Z.: Global weak solutions ans blow-up structure for the Degasperis-Procesi equation, J. funct. Anal. 241, 457-485 (2006) · Zbl 1126.35053 · doi:10.1016/j.jfa.2006.03.022
[12]Escher, J.; Lechtenfeld, O.; Yin, Z.: Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation, Discrete contin. Dyn. syst. 19, 493-513 (2007) · Zbl 1149.35307
[13]Falqui, G.: On a Camassa-Holm type equation with two dependent variables, J. phys. A 39, 327-342 (2006) · Zbl 1084.37053 · doi:10.1088/0305-4470/39/2/004
[14]Fuchssteiner, B.; Fokas, A. S.: Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D 4, No. 1, 47-66 (1981/1982) · Zbl 1194.37114 · doi:10.1016/0167-2789(81)90004-X
[15]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
[16]Guo, Z.: Some properties of solutions to the weakly dissipative Degasperis-Procesi equation, J. differential equations 246, 4332-4344 (2009) · Zbl 1170.35083 · doi:10.1016/j.jde.2009.01.032
[17]Galaktionov, V.; Vazquez, J.: The problem of blow-up in nonlinear parabolic equations, Discrete contin. Dyn. syst. 8, 399-433 (2002) · Zbl 1010.35057 · doi:10.3934/dcds.2002.8.399
[18]Guo, Z.; Zhou, Y.: On solutions to a two-component generalized Camassa-Holm system, Stud. appl. Math. 124, 307-322 (2010) · Zbl 1189.35255 · doi:10.1111/j.1467-9590.2009.00472.x
[19]Himonas, A.; Misiolek, G.: The Cauchy problem for an integrable shallow water equation, Differential integral equations 14, 821-831 (2001) · Zbl 1009.35075
[20]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
[21]Johnson, R. S.: Camassa-Holm, Korteweg-de Vries and related models for water waves, J. fluid mech. 455, 63-82 (2002) · Zbl 1037.76006 · doi:10.1017/S0022112001007224
[22]L. Jin, Z. Guo, On a two-component Degasperis-Procesi shallow water system, Nonlinear Anal. Real World Appl. (2010), doi:10.1016/j.nonrwa.2010.05.003
[23]Kato, T.: Spectral theory and differential equations, proc. Sympos., Lecture notes in math. 48, 25 (1975)
[24]Li, Y.; Olver, P.: Well-posedness and blow-up solutions for an integrable nonlinear dispersive model wave equation, J. differential equations 162, 27-63 (2000) · Zbl 0958.35119 · doi:10.1006/jdeq.1999.3683
[25]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
[26]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
[27]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
[28]Ni, L.: The Cauchy problem for a two-component generalized θ-equations, Nonlinear anal. 73, 1338-1349 (2010) · Zbl 1195.37052 · doi:10.1016/j.na.2010.04.064
[29]Mustafa, O. G.: On smooth traveling waves of an integrable two-component Camassa-Holm shallow water system, Wave motion 46, 397-402 (2009) · Zbl 1231.76063 · doi:10.1016/j.wavemoti.2009.06.011
[30]Olver, P.; Rosenau, P.: Tri-Hamiltonian duality between solitons and solitary wave solutions having compact support, Phys. rev. E 53, 1900-1906 (1996)
[31]Popowicz, Z.: A 2-component or N=2 supersymmetric Camassa-Holm equation, Phys. lett. A 354, 110-114 (2006)
[32]Wunsch, M.: The generalized Hunter-Saxton system, SIAM J. Math. anal. 42, 1286-1304 (2010) · Zbl 1223.35092 · doi:10.1137/090768576
[33]Xin, Z.; Zhang, P.: On the weak solution to a shallow water equation, Comm. pure appl. Math. 53, 1411-1433 (2000) · Zbl 1048.35092 · doi:10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5
[34]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
[35]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
[36]Zhou, Y.: Blow-up of solutions to the DGH equation, J. funct. Anal. 250, 227-248 (2007) · Zbl 1124.35079 · doi:10.1016/j.jfa.2007.04.019
[37]Zhou, Y.: Local well-posedness and blow-up criteria of solutions for a rod equation, Math. nachr. 278, No. 14, 172-1739 (2005) · Zbl 1125.35103 · doi:10.1002/mana.200310337
[38]Zhou, Y.: Blow-up of solutions to a nonlinear dispersive rod equation, Calc. var. Partial differential equations 25, 63-77 (2006) · Zbl 1172.35504 · doi:10.1007/s00526-005-0358-1