A note on the Cauchy problem for the periodic two-component Novikov system

ABSTRACT

Considered herein is the initial value problem for the periodic two-component Novikov system. It is shown that the solution map of this problem is not uniformly continuous in Besov spaces $B_{2,r}^s\left(\mathbb{T}\right)\times B_{2,r}^{s-1}\left(\mathbb{T}\right)$ with $s>3/2,1\le p<\mathrm{\infty }$. Furthermore, the non-uniform continuous dependence on initial data is established in Besov space $B_{2,1}^{3/2}\left(\mathbb{T}\right)\times B_{2,1}^{1/2}\left(\mathbb{T}\right)$. Based on the well-posedness result and the lifespan for this problem, the method of approximate solutions is utilized.

COMMUNICATED BY:

• Adrian Constantin

KEYWORDS:

• The periodic two-component Novikov system
• approximate solutions
• Besov spaces

AMS SUBJECT CLASSIFICATION (2000):

• 35B30
• 35G25

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work of Y. Fu is supported by the National Natural Science Foundation of China [grant nos. 11471259, 11631007].