On the Cauchy problem of a weakly dissipative $\hat{I}¼$HS equation

Abstract

In this paper, we study the Cauchy problem of a weakly dissipative $\hat{I}¼$HS equation. We first establish the local well-posedness for the weakly dissipative $\hat{I}¼$HS equation by Kato's semigroup theory. Then, we derive the precise blow-up scenario for strong solutions to the equation. Moreover, we present some blow-up results for strong solutions to the equation. Finally, we give two global existence results to the equation.