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Global solutions and blow‐up phenomena for the periodic b ‐equation
Author(s) -
Zhang S.,
Yin Z.
Publication year - 2010
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdq044
Subject(s) - sign (mathematics) , mathematics , initial value problem , mathematical analysis , periodic wave , wave equation , traveling wave
In the paper, we mainly study the Cauchy problem of a family of asymptotically equivalent shallow water wave equations, the so called periodic b ‐equation. We first establish the local well‐posedness for the periodic b ‐equation. Then we derive the precise blow‐up scenario and present two blow‐up results. Moreover, we show that the periodic b ‐equation has global strong solutions. Finally, we prove the existence of global weak solutions to the periodic b ‐equation, provided that initial data satisfy certain sign conditions.