We study the global in time existence of small solutions for subcritical fractional modified Korteweg–de Vries equation \(\begin{aligned} \left\{ \begin{array}{c} \partial _{t}u+\frac{1}{\alpha }\left| \partial _{x}\right| ^{\alpha -1}\partial _{x}u=t^{\nu }\partial _{x}\left( u^{3}\right) ,\text t>0\textbf{,}x\in \mathbb {R},\\ u\left( 0,x\right) =u_{0}\left( x\right) , x\in \mathbb {R}\textbf{,} \end{array} \right. \end{aligned}\) where \(\alpha \in \left( \frac{3}{2},3\right) \) and \(\nu \in \left( 0,\nu _{\alpha }\right) ,\) \(\nu _{\alpha }=\frac{1}{24}\) for \(\frac{3}{2} <\alpha \le \frac{32}{11},\) \(\nu _{\alpha }=\frac{1}{3}\left( \frac{4}{\alpha }-\frac{5}{4}\right) \) for \(\frac{32}{11}\le \alpha <3\) , solutions u and the initial data \(u_{0}\) are the real-valued functions. We remark that \(\nu >0\) means that equation is subcritical in the sense of the large time asymptotic behavior of solutions. We assume that the initial data have an analytic extension on the sector and are small. Then we find the large time asymptotics of the solutions.
