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In the paper, we consider the initial value problem to the Camassa-Holm andNovikov equations on the real-line case. We show that both the solution maps ofCamassa-Holm and Novikov equations are not uniformly continuous on the initialdata in Besov spaces $B_{p, r}^s(\mathbb{R})$ with $s>1$ and $1\leq p, r<\infty$. Our result improves the previous work given by Li et al. [J. Differ.Equ. 269 (2020); J. Math. Fluid Mech. 22:50 (2020); J. Math. Fluid Mech., 23:36(2021)].

Non-uniform dependence for the Camassa-Holm and Novikov equations in low regularity Besov spaces