The initial value problem for a multi-component Novikov equation

In this article, we investigate the initial value problem for a multi-component Novikov equation. At first, by using a Galerkin-type approximation method and commutator estimate, we prove the existence, uniqueness, and blow-up scenario of the solution to the equation in Sobolev space $ H^{s, p}(\mathbb{R}) $, $ s>\max\{\frac{5}{2}, 2+\frac{1}{p}\} $, $ p\in(1, \infty) $. Then, we investigate that the data-to-solution map of the equation is continuous. In fact, we generalize the well-posedness results of the two-component Novikov equation derived by Himonas and Mantzavinos (JMP 57 (2016) 071503, Nonlinearity 25 (2012)). Finally, by the conservation laws, the global solution for some two-component case is derived.