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$ \{ {array} [c]{c} ∂ₜu+1/α ∂ₓ α -1∂ₓu = t-ν∂ₓu², t>0, x, \\ u(0, x) = u₀(x), x, {array} . $with the dispersion order $ α∈[ 3/2, 2) , $ we also assume that $ ν∈( 0, 1) $. Both the solution $ u $ and the initial data $ u₀ $ 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.