The dependences on initial data for the b-family equation in critical Besov space

Abstract

This paper is concerned with the periodic boundary problem for the b-family equation. At first, we use a different method to prove the local well-posedness in the critical Besov space B 3/2 2,1 . Then we show that if a weaker B q p,r -topology is used, the solution map becomes Hölder continuous. Moreover, we show that the dependence on initial data is optimal in B 3/2 2,1 in the sense that the solution map is continuous but not uniformly continuous. Finally, we obtain the periodic peaked solutions to the b-family equation and apply them to obtain the ill-posedness in B 3/2 2,∞ .

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