×

Blow-up phenomenon for the integrable Degasperis-Procesi equation. (English) Zbl 1134.37361

Summary: We investigate a new integrable equation derived recently by Degasperis and Procesi. Analogous to the Camassa-Holm equation, this new equation possesses the blow-up phenomenon. Under the special structure of this equation, we establish sufficient conditions on the initial data to guarantee the formulation of a singularity in the sense that the derivative of the solution blows up in finite time. Moreover, a global existence result is found.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35Q58 Other completely integrable PDE (MSC2000)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bona, J. L.; Smith, R., Philos. Trans. R. Soc. London, Ser. A, 278, 555 (1975)
[2] Bourgain, J., Geom. Funct. Anal., 3, 209 (1993)
[3] Camassa, R.; Holm, D. D., Phys. Rev. Lett., 71, 1661 (1993)
[4] Camassa, R.; Holm, D. D.; Hyman, J., Adv. Appl. Mech., 31, 1 (1994)
[5] Constantin, A.; Escher, J., Acta Math., 181, 229 (1998)
[6] Constantin, A.; Escher, J., Commun. Pure Appl. Math., 51, 475 (1998)
[7] Degasperis, A.; Holm, D. D.; Hone, A. N.W., Theor. Math. Phys., 133, 1461 (2002)
[8] Degasperis, A.; Procesi, M., (Degasperis, A.; Gaeta, G., Symmetry and Perturbation Theory, SPT 98, Rome, 16-22 December 1998 (1999), World Scientific: World Scientific River Edge, NJ), 23
[9] Dullin, H. R.; Gottwald, G. A.; Holm, D. D., Fluid Dyn. Res., 33, 7395 (2003)
[10] Evans, L. C., Partial Differential Equations, Graduate Studies in Mathematics, vol. 19 (1998), American Mathematical Society: American Mathematical Society Providence, RI
[11] Kato, T., Manuscripta Math., 28, 89 (1979)
[12] Kato, T., in: Spectral Theory and Differential Equations (Proc. Sympos., Dundee, 1974, dedicated to Konrad Jorgens), Lecture Notes in Mathematics, vol. 48 (1975), Springer-Verlag: Springer-Verlag Berlin, p. 25
[13] Li, Y.; Olver, P., J. Differential Equations, 162, 27 (2000)
[14] Lundmark, H.; Szmigielski, J., Inverse Problems, 19, 1241 (2003)
[15] Shkoller, S., J. Funct. Anal., 160, 337 (1998)
[16] Whitham, G. B., Linear and Nonlinear Waves (1974), Wiley: Wiley New York · Zbl 0373.76001
[17] Xin, Z.; Zhang, P., Commun. Pure Appl. Math., 53, 1411 (2000)
[18] Xin, Z.; Zhang, P., Comm. Partial Differential Equations, 27, 1815 (2002)
[19] Zhou, Y., J. Math. Anal. Appl., 290, 591 (2004)
[20] Zhou, Y., Nonlinear Analysis, 57, 137 (2004)
[21] Y. Zhou, Math. Nachr. (2004), in press; Y. Zhou, Math. Nachr. (2004), in press
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.