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.
The Bulletin of Society for Mathematical Services and Standards (Volume 12)
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