Fractional Korteweg-de Vries equation and related ODE with multilinear operators

We investigate the global existence and large-time asymptotics of small solutions to the Cauchy problem for the fractional Korteweg-de Vries equation$ \left\{ \begin{array} [c]{c} \partial_{t}u+\frac{1}{\alpha}\left\vert \partial_{x}\right\vert ^{\alpha -1}\partial_{x}u = t^{-\nu}\partial_{x}u^{2}, t>0, x\in\mathbb{R}, \\ u(0, x) = u_{0}(x), x\in\mathbb{R}, \end{array} \right. $with the dispersion order $ \alpha\in\left[ \frac{3}{2}, 2\right) , $ we also assume that $ \nu\in\left( 0, 1\right) $. Both the solution $ u $ and the initial data $ u_{0} $ are assumed to be real-valued functions. We note that quadratic nonlinearities behave subcritically in the sense of the large-time asymptotics of solutions. However, we will show that the present model exhibits a markedly different asymptotic behavior due to the derivative structure of the nonlinear interaction along with the assumption that the initial data has zero mean value.