Subscribe

Subscribe to our Newsletter and get informed about new publication regulary and special discounts for subscribers!

BSMaSS > Volume 12 > Blow-Up Phenomena and Global Existence to a Weakly...
< Back to Volume

Blow-Up Phenomena and Global Existence to a Weakly Dissipative Shallow Water Equation

Full Text PDF

Abstract:

We first establish the local well-posedness for a weakly dissipative shallow water equation which includes both the weakly dissipative Camassa-Holm equation and the weakly dissipative Degasperis-Procesi equation as its special cases. Then two blow-up results are derived for certain initial profiles. Finally, We study the long time behavior of the solutions.

Info:

Periodical:
The Bulletin of Society for Mathematical Services and Standards (Volume 12)
Pages:
10-20
Citation:
J. B. Zhou et al., "Blow-Up Phenomena and Global Existence to a Weakly Dissipative Shallow Water Equation", The Bulletin of Society for Mathematical Services and Standards, Vol. 12, pp. 10-20, 2014
Online since:
December 2014
Export:
Distribution:
References:

A. Degasperis, M. Procesi, Asymptotic integrability, in: A. Degasperis, G. Gaeta (Eds. ), Symmetry and Perturbation Theory, World Scientific, Singapore, 1999, pp.23-37.

H. P. Mckean, Integrable Systems and Algebraic Curves, Global Analysis, Springer Lecture Notes in Mathematics, vol. 755, 1979, pp.83-200.

C. E. Kenig, G. Ponce, L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Commun. Pur. Appl. Math. 46 (1993) 527-620.

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations, Part II: The KdV equation, Geom. Funct. Anal. 3 (1993) 209-262.

T. Tao, Low-regularity global solutions to nonlinear dispersive equations, Surveys in analysis and operator theory (Canberra, 2001), 19-48, Proc. Centre Math. Appl. Austral. Nat. Univ., 40, Austral. Nat. Univ., Canberra, (2002).

C. E. Kenig, G. Ponce, L. Vega, A bilinear estimate with applications to the KdV equation, J. Am. Math. Soc. 9 (1996) 573-603.

G. B. Whitham, Linear and Nonlinear Waves, J. Wiley \& Sons, New York, (1980).

R. Camassa, D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993) 1661-1664.

B. Fuchssteniner, A. S. Fokas, symplectic structures, their Backlund transformation and hereditary symmetries, Phys. D 4 (1981) 47-66.

B. Fuchssteniner, some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation, Phys. D 4 (1996) 229-243.

A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. R. Soc. Lond. A 457 (2008) 953-970.

J. Lenells, Traveling wave solutions of the Camassa-Holm equation, J. Differential Equations 217 (2005) 393-430.

A. Constantin, W. A. Strauss, Stability of the Camassa-Holm solitons, J. Nonlinear Sci. 12 (2002) 415-422.

A. Constantin, W. A. Strauss, Stability of peakons, Comm. Pur. Appl. Math. 53 (2000) 603610.

R. S. Johnson, On solutions of the Camassa-Holm equation, Proc. R. Soc. Lond. A 459 (2003) 1687-1708.

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. I. Fourier (Grenoble) 50 (2000) 321-362.

G. Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal. 46 (2001) 309-327.

A. Constantin, J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pur. Appl. Math. 51 (1998) 475-504.

H. Dai, K. Kwek, H. Gao, C. Qu, On the Cauchy problem of the Camassa-Holm equation, Front. Math. China 1 (2006) 144-159.

A. Constantin, J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm-Sci. 26 (1998) 303-328.

Y. Li, P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differ. Equations 162 (2000) 27-63.

A. Bressan, A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal. 183 (2007) 215-239.

Z. Xin, P. Zhang, On the weak solutions to a shallow water equation, Comm. Pur. Appl. Math. 53 (2000) 1411-1433.

A. Degasperis, D. Holm, A. Hone, A new integral equation with peakon solutions, Theoret. Math. Phys. 133 (2002) 1461-1472.

H. R. Dullin, G. A. Gottwald, D. D. Holm, Camassa-Holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves, Fluid Dynam. Res. 33 (2003) 73-95.

D. Holm, M. F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J. Appl. Dyn. Syst. 2 (2002) 323-380.

H. Lundmark, J. Szmigielski, Multi-peakon solutions of the Degasperis-Procesi equation, Inverse Probl. 19 (2003) 1241-1245.

Y. Matsuno, Multisoliton solutions of the Degasperis-Procesi equation and their peakon limit, Inverse Probl. 21 (2005) 1553- 1570.

Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math. 47 (2003) 649-666.

Z. Yin, Global existence for a new periodic integrable equation, J. Math. Anal. Appl. 283 (2003) 129-139.

Z. Yin, Global solutions to a new integrable equation with peakons, Indiana Univ. Math. J. 53 (2004) 1189-1210.

Z. Yin, Global weak solutions to a new periodic integrable equation with peakon solutions, J. Funct. Anal. 212 (2004) 182-194.

Y. Liu, Z. Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation, Comm. Math. Phys. 267 (2006) 801-820.

J. Escher, Y. Liu, Z. Yin, Global weak solutions and blow-up structure for the DegasperisProcesi equation, J. Funct. Anal. 241 (2006) 457-485.

J. Escher, Y. Liu, Z. Yin, Shock waves and blow-up phenomena for the periodic DegasperisProcesi equation, Indiana Univ. Math. J. 56 (2007) 87-117.

G. M. Coclite, K. H. Karlsen, On the well-posedness of the Degasperis-Procesi equation, J. Funct. Anal. 223 (2006) 60-91.

A. Constantin, D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal. 192 (2009) 165-186.

S. Y. Lai, Y. H. Wu, Global solutions and blow-up phenomena to a shallow water equation, J. Differ. Equations 249 (2010) 693-706.

Z. Yin, S. Y. Lai, Y. X. Guo, Global existence of weak solutions for a shallow water equation, Comput. Math. Appl. 60 (2010) 2645-2652.

E. Ott, R. N. Sudan, Damping of solitary waves, Phys. Fluids 13 (1970) 1432-1434.

J. M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time, J. Differ. Equations 74 (1988) 369-390.

S. Wu, Z. Yin, Global existence and blow-up phenomena for the weakly dissipative CamassaHolm equation, J. Differ. Equations 246 (2009) 4309-4321.

S. Wu, Z. Yin, Blow-up, blow-up rate and decay of the solution of the weakly dissipative Camassa-Holm equation, J. Math. Phys. 47 (2006) 1-12.

S. Wu, Z. Yin, Blow-up and decay of the solution of the weakly dissipative Degasperis-Procesi equation, SIAM J. Math. Anal. 40 (2008) 475-490.

S. Wu, J. Escher, Z. Yin, Global existence and blow-up phenomena for a weakly dissipative Degasperis-Procesi equation, Discrete Cont. Dyn-B 12 (2009) 633-645.

S. Wu, Z. Yin, Blow-up phenomena and decay for the periodic Degasperis-Procesi equation with weak dissipation, J. Nonlinear Math. Phys. 15 (2008) 28-49.

W. Yan, Y. Li, Y. Zhang, Global existence and blow-up phenomena for the weakly dissipative Novikov equation, Nonlinear Anal. 75 (2012) 2464-2473.

T. Kato, On the Korteweg-de Vries equation, Manuscripta Math. 28 (1979) 89-99.

W. S. Niu, S. H. Zhang, Blow-up phenomena and global existence for the nonuniform weakly dissipative b-equation, J. Math. Anal. Appl. 374 (2011) 166-177.

Show More Hide