On the Subcritical Fractional Modified Korteweg–de Vries Equation of Order α∈(0,1)∪(1,32]

We study the global in time existence of small solutions to the Cauchy problem for the subcritical fractional modified Korteweg–de Vries equation { ∂ t u + 1 α | ∂ x | α − 1 ∂ x u = t ν ∂ x ( u 3 ) , t > 0 , x ∈ R , u ( 0 , x ) = u 0 ( x ) , x ∈ R , where α ∈ ( 0 , 1 ) ∪ ( 1 , 3 2 ] , 0 < ν < 1 24 . We remark that the condition ν > 0 implies that equation is subcritical in the sense of the large-time asymptotic behavior of solutions. We assume that the initial data are small and admit an analytic extension on the sector and a strip. Then we find the large time asymptotics of solutions.