Some new asymptotic behaviors of a two-component b-family equations

In this article, we investigate some new asymptotic behaviors of the solution for a two-component b-family equations. We first prove the persistence properties of the solution for Eq. (1.1) when the initial data decay logarithmically, algebraically at infinity with the power \(\beta \in (0,\infty )\) $\beta \in (0,\infty )$ . Subsequently, we obtain the infinite propagation of the solution to Eq. (1.1). If the initial data satisfy certain compact condition, then the nontrivial solution u of Eq. (1.1) immediately loses compactly supported. Meanwhile, the solution u decays exponentially as \(|x|\rightarrow \infty \) $|x|\to \infty$ .