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Wu, Xinglong; Yin, Zhaoyang

A note on the Cauchy problem of the Novikov equation. (English) Zbl 1273.35097

Appl. Anal. 92, No. 6, 1116-1137 (2013).
Summary: We mainly study the Cauchy problem of the Novikov equation. We first establish local well-posedness of the equation in Besov space \(B^s_{p,r},p,r\in [1,\infty ],s>\max \{\frac {3}{2},1+\frac {1}{p}\}\). Then we derive a lower bound for the maximal existence time and lower semicontinuity of the existence time. Finally, we prove that the equation is ill-posed in \(B^{3/2}_{2,\infty}\) by peakon solutions which exist globally in time to the equation.

Cited in 56 Documents

MSC:

35G25 Initial value problems for nonlinear higher-order PDEs
35B44 Blow-up in context of PDEs

Keywords:

local well-posedness; ill-posedness; Besov spaces; lower semicontinuity; peakon solutions; maximal existence time
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References:

[1] DOI: 10.1088/1751-8113/41/37/372002 · Zbl 1153.35075 · doi:10.1088/1751-8113/41/37/372002
[2] DOI: 10.1088/1751-8113/42/34/342002 · Zbl 1181.37100 · doi:10.1088/1751-8113/42/34/342002
[3] DOI: 10.1088/0305-4470/35/22/309 · Zbl 1039.35008 · doi:10.1088/0305-4470/35/22/309
[4] DOI: 10.1103/PhysRevLett.71.1661 · Zbl 0972.35521 · doi:10.1103/PhysRevLett.71.1661
[5] DOI: 10.1017/S0022112076002425 · Zbl 0351.76014 · doi:10.1017/S0022112076002425
[6] DOI: 10.1016/0167-2789(81)90004-X · Zbl 1194.37114 · doi:10.1016/0167-2789(81)90004-X
[7] DOI: 10.1098/rspa.2000.0701 · Zbl 0999.35065 · doi:10.1098/rspa.2000.0701
[8] DOI: 10.1016/S0065-2156(08)70254-0 · doi:10.1016/S0065-2156(08)70254-0
[9] DOI: 10.1007/s00205-006-0010-z · Zbl 1105.76013 · doi:10.1007/s00205-006-0010-z
[10] DOI: 10.1142/S0219530507000857 · Zbl 1139.35378 · doi:10.1142/S0219530507000857
[11] DOI: 10.5802/aif.1757 · doi:10.5802/aif.1757
[12] DOI: 10.1007/PL00004793 · Zbl 0954.35136 · doi:10.1007/PL00004793
[13] DOI: 10.1007/s00205-008-0128-2 · Zbl 1169.76010 · doi:10.1007/s00205-008-0128-2
[14] DOI: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D · Zbl 0940.35177 · doi:10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D
[15] DOI: 10.1007/s002200050801 · Zbl 1002.35101 · doi:10.1007/s002200050801
[16] 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
[17] DOI: 10.1023/A:1021186408422 · doi:10.1023/A:1021186408422
[18] Yin Z, Illinois J. Math. 47 pp 649– (2003)
[19] DOI: 10.1016/j.jfa.2003.07.010 · Zbl 1059.35149 · doi:10.1016/j.jfa.2003.07.010
[20] DOI: 10.1093/imanum/drm003 · Zbl 1246.76114 · doi:10.1093/imanum/drm003
[21] DOI: 10.1007/s00332-006-0803-3 · Zbl 1185.35194 · doi:10.1007/s00332-006-0803-3
[22] DOI: 10.1512/iumj.2007.56.3040 · Zbl 1124.35041 · doi:10.1512/iumj.2007.56.3040
[23] DOI: 10.1088/0266-5611/15/1/001 · Zbl 0923.35154 · doi:10.1088/0266-5611/15/1/001
[24] Holden H, Discr. Contin. Dyn. Syst. 14 pp 505– (2006)
[25] Hone ANW, Dyn. Partial Differ. Eqns. 6 pp 253– (2009) · Zbl 1179.37092 · doi:10.4310/DPDE.2009.v6.n3.a3
[26] Wu X, Annali Sc. Norm. Sup. Pisa
[27] Tiglay F, Int. Math. Res. Notices (2010)
[28] DOI: 10.1088/1751-8113/44/5/055202 · Zbl 1210.35217 · doi:10.1088/1751-8113/44/5/055202
[29] Danchin R, Fourier Analysis Methods for PDEs, Lecture Notes 14 November (2005)
[30] Danchin R, Differ. Integr. Eqns. 14 pp 953– (2001)
[31] Chemin JY, Phase Space Analysis of Partial Differential Equations pp 53– (2004)
[32] T. Kato,Quasi-linear equation of evolution, with applications to partial differential equations, inSpectral Theory and Differential Equation, Lecture Notes in Math., Vol. 488, Spring-Verlag, Berlin, 1975, pp. 25–70
[33] DOI: 10.1016/S0022-0396(03)00096-2 · Zbl 1048.35076 · doi:10.1016/S0022-0396(03)00096-2
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