Local well-posedness and blow-up of a two-component system in Besov spaces

The local well-posedness and blow-up phenomena for the Cauchy problem of a two-component integrable system are investigated. By employing the Littlewood-Paley decomposition and transport equation theory, together with logarithmic interpolation inequalities and the Osgood lemma, the local well-posedness of the system is established in the Besov space B_p, 1^1 / p \times B_p, 1^1 / p for p∈1,2. Moreover, a criterion determining the occurrence of finite-time blow-up for solutions of the system is derived. Finally, based on this blow-up criterion, necessary and sufficient conditions for solution blow-up are further obtained.