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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.

35Q53KdV-like (Korteweg-de Vries) equations
35B60Continuation of solutions of PDE
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.: 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 · 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