Abstract
Considered herein is the initial value problem for the two-component Novikov system with peakons. Based on the local well-posedness results for this problem, it is shown that the solution map $$z_{0}\mapsto z(t)$$ of this problem in the periodic case is not uniformly continuous in Besov spaces $$B^{s}_{p,r}({\mathbb {T}})\times B^{s}_{p,r}({\mathbb {T}}) $$ with $$s>\max \{5/2,2+1/p\}, 1\le p,r\le \infty $$ and $$B^{5/2}_{2,1}({\mathbb {T}})\times B^{5/2}_{2,1}({\mathbb {T}})$$ through the method of approximate solutions. Furthermore, it is in the non-periodic case that the non-uniform continuity of this solution map in Besov spaces $$B^{s}_{p,r}({\mathbb {R}})\times B^{s}_{p,r}({\mathbb {R}})$$ with $$s>\max \{5/2,2+1/p\}, 1\le p,r\le \infty $$ is discussed by constructing new subtle initial data. Finally, the Holder continuity of the solution map in Besov spaces is proved.
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