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Blow-up, blow-up rate and decay of the solution of the weakly dissipative Camassa-Holm equation. (English) Zbl 1111.35067

Summary: We mainly study several problems on the weakly dissipative periodic Camassa-Holm equation. At first, the local well-posedness of the equation is obtained by Kato’s theorem, a necessary and sufficient condition of the blow-up of the solution and some criteria guaranteeing the blow-up of the solution are established. Then, the blow-up rate of the solution is discussed. Moreover, we prove that the equation has global solutions and these global solutions decay to zero as time goes to infinite provided the potentials associated to their initial date are of one sign.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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[1] DOI: 10.1006/aima.1998.1768 · Zbl 0919.35118 · doi:10.1006/aima.1998.1768
[2] DOI: 10.1088/0266-5611/15/1/001 · Zbl 0923.35154 · doi:10.1088/0266-5611/15/1/001
[3] DOI: 10.1103/PhysRevLett.71.1661 · Zbl 0972.35521 · doi:10.1103/PhysRevLett.71.1661
[4] Camassa R., Adv. Appl. Math. 31 pp 1– (1994)
[5] DOI: 10.1006/jdeq.1997.3333 · Zbl 0889.35022 · doi:10.1006/jdeq.1997.3333
[6] DOI: 10.1098/rspa.2000.0701 · Zbl 0999.35065 · doi:10.1098/rspa.2000.0701
[7] Constantin A., Ann. Scuola Norm. Sup. Pisa, Cl. Sci. 26 pp 303– (1998)
[8] DOI: 10.1007/BF02392586 · Zbl 0923.76025 · doi:10.1007/BF02392586
[9] DOI: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5 · Zbl 0934.35153 · doi:10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5
[10] DOI: 10.1007/s002080050228 · Zbl 0923.76028 · doi:10.1007/s002080050228
[11] DOI: 10.1007/PL00004793 · Zbl 0954.35136 · doi:10.1007/PL00004793
[12] DOI: 10.1007/s002200050801 · Zbl 1002.35101 · doi:10.1007/s002200050801
[13] DOI: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L · Zbl 1049.35149 · doi:10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L
[14] DOI: 10.1007/s00332-002-0517-x · Zbl 1022.35053 · doi:10.1007/s00332-002-0517-x
[15] DOI: 10.1016/0167-2789(81)90004-X · Zbl 1194.37114 · doi:10.1016/0167-2789(81)90004-X
[16] DOI: 10.1016/0167-2789(96)00048-6 · Zbl 0900.35345 · doi:10.1016/0167-2789(96)00048-6
[17] DOI: 10.1016/0022-0396(88)90010-1 · Zbl 0668.35084 · doi:10.1016/0022-0396(88)90010-1
[18] Himonas A., Diff. Integral Eq. 14 pp 821– (2001)
[19] DOI: 10.1007/BFb0067080 · doi:10.1007/BFb0067080
[20] DOI: 10.1006/jdeq.1999.3683 · Zbl 0958.35119 · doi:10.1006/jdeq.1999.3683
[21] DOI: 10.1016/j.na.2004.10.006 · Zbl 1063.35137 · doi:10.1016/j.na.2004.10.006
[22] DOI: 10.1063/1.1693097 · doi:10.1063/1.1693097
[23] DOI: 10.1016/S0362-546X(01)00791-X · Zbl 0980.35150 · doi:10.1016/S0362-546X(01)00791-X
[24] DOI: 10.1007/s00013-004-1199-4 · Zbl 1126.35056 · doi:10.1007/s00013-004-1199-4
[25] DOI: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5 · Zbl 1048.35092 · doi:10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5
[26] DOI: 10.3934/dcds.2004.11.393 · Zbl 1061.35123 · doi:10.3934/dcds.2004.11.393
[27] DOI: 10.1007/s00028-004-0166-7 · Zbl 1073.35064 · doi:10.1007/s00028-004-0166-7
[28] DOI: 10.1016/j.jmaa.2003.10.017 · Zbl 1042.35060 · doi:10.1016/j.jmaa.2003.10.017
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