Well-posedness and persistence property for a four-component Novikov system with peakon solutions

Abstract

In this paper we mainly study the Cauchy problem of a four-component Novikov system. We first show the local well-posedness of the system in Besov spaces $$B^{s+1}_{p,r}\times B^{s+1}_{p,r}\times B^s_{p,r} \times B^s_{p,r}$$ with $$p,r\in [1,\infty ],~s>\max \{\frac{1}{p},\frac{1}{2}\}$$ by using the Littlewood–Paley theory and transport equations theory. Then, by virtue of logarithmic interpolation inequalities and the Osgood lemma, we prove the local well-posedness of the system in the critical Besov space $$B^{\frac{3}{2}}_{2,1}\times B^{\frac{3}{2}}_{2,1} \times B^{\frac{1}{2}}_{2,1}\times B^{\frac{1}{2}}_{2,1}$$ . Next, we establish two blow-up criteria for strong solutions to the system by using the structure of the system. Moreover, we investigate the persistence property for strong solutions to the system. Finally, we verify that the system possesses a special class of peakon solutions.